Estimating Foreign Exchange Exposure in the Department of National Defence P.E. Desmier Director, Materiel Group Operational Research

DRDC CORA TR 2006–23 January 2007

Defence R&D Canada Centre for Operational Research and Analysis Materiel Group Operational Research Assistant Deputy Minister (Materiel)

National Defence

Défense nationale

Estimating Foreign Exchange Exposure in the Department of National Defence P.E. Desmier Director, Materiel Group Operational Research

This publication contains sensitive information that shall be protected in accordance with prescribed regulations. Release of or access to this information is subject to the provisions of the Access to Information Act, the Privacy Act, and other statutes as appropriate.

DRDC – Centre for Operational Research and Analysis Technical Report DRDC CORA TR 2006–23 January 2007

Author

P.E. Desmier

Approved by

P.J. Comeau Section Head (Joint & Common)

Approved for release by

R.G. Dickinson Director, Operational Research (Joint & Common)

The information contained herein has been derived and determined through best practice and adherence to the highest levels of ethical, scientific and engineering investigative principles. The reported results, their interpretation, and any opinions expressed therein, remain those of the authors and do not represent, or otherwise reflect, any official opinion or position of DND or the Government of Canada.

© Her Majesty the Queen as represented by the Minister of National Defence, 2007 © Sa majesté la reine, représentée par le ministre de la Défense nationale, 2007

Abstract Quantifying foreign exchange risk is not a trivial process although it is generally accepted that the standard method for reporting financial risk today is the “Value-at-Risk” or VaR method. Simply put, VaR is defined as the predicted worst-case loss at a specific confidence level over a certain period of time. Thus VaR provides a quantitative measure of the downside risk of exposure in all foreign currency transactions. This report documents the theory and application of a model that is built on forecasting expenditures for the ADM(Mat) National Procurement and Capital (equipment) accounts and the time-varying volatilities of foreign currency returns. These diverse methodologies are then combined into an overall VaR model to determine the maximum expected loss from adverse exchange rate fluctuations over the budget year. This is recognized as the first step any organization must take before considering risk mitigation strategies to reduce and hopefully eliminate foreign transaction exposure. This study also illuminates certain policy implications for functional finance and performance/risk management specialists in the department. In particular, the VCDS Group through the Director Force Planning and Programme Coordination (DFPPC), and ADM(Fin CS) through Director Budget and Director Strategic Finance and Costing (DSFC), may want to examine the possibility of adjusting corporate budget allocations (quarterly) based on the results of the VaR model. The department should also examine opportunities to apply the VaR analytical approach to quantifying the financial risk in other budget expenditure areas subject to market/price risk such as bulk fuels, energy/hydro, and certain commodities (e.g., steel, ballistic materials, etc.) where expenditure amounts warrant. As the department embarks on massive multi-year capital acquisitions and continues to be engaged in sizeable, complex overseas deployments, the need to measure and accurately assess financial risk has never been greater.

Résumé Quantifier le risque de change n’est pas une opération simple, mais on s’accorde généralement pour utiliser aujourd’hui la méthode de la valeur à risque, ou méthode VAR, pour mesurer ce risque. En termes simples, la VAR correspond au montant de pertes qui ne devrait être dépassé qu’avec une probabilité donnée (niveau de confiance) sur un horizon temporel donné. La VAR fournit donc une mesure quantitative du risque de perte dans les opérations de change. Ce rapport expose les dimensions théorique et pratique d’un modèle qui repose sur la prévision des dépenses pour le compte de l’approvisionnement national et le compte de biens d’investissement du SMA(Mat) et la variabilité temporelle des rendements de change. Les différentes méthodes sont combinées en un modèle VAR afin de calculer le montant de

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pertes prévu maximal résultant des fluctuations défavorables du taux de change au cours de l’année budgétaire. C’est là la première étape que doit accomplir toute organisation avant de considérer des stratégies visant à réduire et, de préférence, à éliminer le risque lié aux opérations en devises. De plus, cette étude contient des enseignements sur le plan de l’action pour les spécialistes des finances et de la gestion de la performance/du risque au Ministère. En particulier, le groupe du VCEMD, par l’intermédiaire du Directeur - Planification des Forces et coordination du programme (DPFCP), et le SMA(Fin SM), par l’intermédiaire du Directeur Budget et du Directeur - Finances et établissement des coûts (Stratégie) (DFECS), pourraient vouloir étudier la possibilité de rajuster les affectations du budget ministériel (trimestriellement) sur la base du résultat du modèle VAR. Le Ministère pourrait aussi étudier la possibilité d’utiliser la méthode VAR pour calculer le risque financier pour d’autres postes de dépenses budgétaires exposés au risque de prix, comme les carburants en vrac, l’hydroénergie et certains produits tels que l’acier, le matériel ballistique, etc., lorsque le montant des dépenses le justifie. Au moment où le Ministère s’engage dans un vaste programme pluriannuel d’acquisition de biens et continue de mener outre mer des missions importantes et délicates, la nécessité d’évaluer avec précision le risque financier n’a jamais été aussi grande.

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Executive summary Like any institution engaged in international trade, the Department of National Defence (DND) is often required to make payment in foreign currency when acquiring new equipment and supplies to support military operations at home and abroad. Foreign currency exposure refers to the sensitivity of an organization’s cash flows to changes in the exchange rates. Given the high volatility of various currencies, it is generally accepted that exchange rate exposure, if not quantified and managed properly, may expose the organization to significant budgetary consequences due to poor responsiveness to foreign exchange volatility. Given the historical and continued importance of foreign exchange exposure to the department, this study focused primarily on the measurement and evaluation of this form of risk. Value-at-Risk, as applied to foreign exchange exposure, is an estimation of the probability of losses that could arise from changes in exchange rates. It has become very popular in financial risk management because it is easily understood and condenses a vast amount of information into a single, summary statistical measure of market risk under normal market conditions. It is an estimate of risk that is based on historical data that relies on the concept that the future will be like the past with extreme events predictable, albeit with an equally extreme likelihood of occurrence. And that might be its one weakness, it says nothing about the size of losses once the VaR limit (5% in this study) has been exceeded, and which reside in the tail of the distribution. While no risk measure is perfect, VaR is certainly better than the alternative, which is no risk measure at all. Nevertheless, the model developed and documented in this report accounts for extreme losses by not only reporting the 5th percentile VaR, but also the expected maximum loss. Measuring the risk of exposure is recognized as the first step any organization must take before considering risk mitigation strategies to reduce and hopefully eliminate foreign transaction exposure. Currently, DND has made no allowance for foreign currency hedging (although certain financial instruments have been proposed) for the reduction and possible elimination of foreign exchange losses. Without such instruments in place, there is great uncertainty as to what the expected foreign exchange transaction losses will be in the future. Budget rates, although based on viable time series analysis, provide only point estimates of what the future exchange rate may be, without any direction as to how this estimate will vary. The end result is that procurement/budget managers within capital equipment projects and in-service equipment management teams must provide a “best guess” as to how much funds they need to hold back (reserves) to account for unforecasted losses. VaR analysis helps by providing a quantitative, statistical aid to support them in their estimates and ultimately reduce the dependency of holding more money than is necessary for foreign currency losses that may or may not materialize. Therefore, quantifying and managing exchange rate exposure properly means managers can now exercise proper responsiveness to foreign exchange volatility. This study highlights certain policy implications for functional finance and performance/risk management specialists in the department. In particular, the VCDS Group through the Director Force Planning and Programme Coordination (DFPPC), and ADM(Fin CS) through

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Director Budget and Director Strategic Finance and Costing (DSFC), may want to examine the possibility of adjusting corporate budget allocations (quarterly) based on the results of the VaR model. In fact, these groups should consider adopting the VaR methodology as part of the department’s integrated risk management framework for managing budgetary risk attributed to exposure to foreign currency fluctuations. Under this scenario, the quarterly foreign exchange reports produced by DSFC would continue to provide a benchmark forecast exchange rate for all anticipated foreign currency denominated expenditures in the department from which the VaR model would measure the risk of loss on exchange. By extension, the department should also examine opportunities to apply the VaR analytical approach to quantifying financial risk in other budget expenditure areas subject to market/price risk such as bulk fuels, energy/hydro, and certain commodities (e.g., steel, ballistic materials, etc.) where expenditure amounts warrant. As the department embarks on massive multi-year capital acquisitions and continues to be engaged in sizeable, complex overseas deployments, the need to measure and accurately assess financial risk has never been greater. Moreover, should the department decide to seek central government agency concurrence to implement (or pilot) a financial hedging strategy to limit foreign exchange risk, the ability to measure and report exchange rate risk would be fundamental for successful hedging. Notwithstanding, this study does illustrate the practical application of the VaR method to arguably the largest department financial risk area, foreign currency exposure, and it is hoped that it will contribute to a better understanding of this risk parameter and how it can be more consistently and accurately measured, reported and ultimately controlled. P.E. Desmier; 2007; Estimating Foreign Exchange Exposure in the Department of National Defence; DRDC CORA TR 2006–23; DRDC – Centre for Operational Research and Analysis.

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Sommaire Comme toute institution qui fait du commerce international, le ministère de la Défense nationale (MDN) doit souvent effectuer des paiements en monnaie étrangère lorsqu’il achète le matériel et les fournitures nécessaires à la conduite des opérations militaires au pays et à l’étranger. L’exposition au risque de change évoque la sensibilité des flux de trésorerie d’une organisation à la variation des taux de change. Étant donné la forte variabilité des changes, on reconnaît généralement que l’exposition au risque de change peut avoir des conséquences budgétaires importantes pour une organisation si ce risque est mal évalué et mal géré et si l’organisation a une faible capacité de réaction. Compte tenu de l’importance constante de ce problème pour le Ministère, cette étude porte principalement sur l’évaluation de ce type de risque. Dans ce contexte, la méthode de la valeur à risque sert à estimer la perte que l’on pourrait subir par suite de la variation des taux de change, étant donné un certain niveau de probabilité. Cette méthode est maintenant couramment utilisée en gestion du risque financier parce qu’elle est facile à comprendre et qu’elle résume une grande quantité d’informations en une mesure statistique globale du risque du marché. Cette estimation du risque est établie à partir de données historiques et repose sur l’idée que le futur ressemblera au passé et qu’il est donc possible de prévoir les événements exceptionnels, même si leur probabilité d’occurrence est extrêmement faible. Voilà la faiblesse que peut présenter cette méthode : elle ne renseigne aucunement sur le montant des pertes une fois le seuil de probabilité franchi (5% dans la présente étude), ces pertes se trouvant dans la queue de la distribution. Si aucune mesure du risque n’est parfaite, la méthode VAR est sûrement mieux que la seconde branche de l’alternative, à savoir l’absence totale de mesure. Cela dit, le modèle élaboré dans ce rapport rend compte des pertes extraordinaires en indiquant non seulement le résultat correspondant au 5e percentile, mais aussi le montant de pertes prévu maximal sur la base de 10 000 itérations. Mesurer le degré d’exposition au risque de change est la première étape que doit accomplir toute organisation avant de considérer des stratégies visant à réduire et, de préférence, à éliminer le risque lié aux opérations en devises. À l’heure actuelle, le MDN ne dispose d’aucun instrument de couverture du risque de change (bien que certains instruments financiers lui aient été proposés) pour atténuer, voire prévenir, les pertes liées aux opérations en devises. Sans de tels instruments, il sera très difficile de prévoir quelles seront ces pertes dans l’avenir. Bien qu’ils soient fondés sur une analyse chronologique viable, les taux budgétaires ne sont que des estimations ponctuelles des taux de change futurs, qui ne disent pas comment évolueront ces taux. En conséquence, les gestionnaires d’approvisionnement ou de budget affectés aux projets d’acquisition de biens d’équipement ou aux équipes de gestion de l’équipement en service sont obligés d’" estimer au meilleur de leur connaissance " le montant qu’il doivent mettre en réserve pour couvrir les pertes non prévues. La méthode VAR leur fournit donc un outil statistique précieux qui leur permettra d’estimer avec plus de précision le montant à mettre en réserve et, par conséquent, d’éviter de mettre de côté plus qu’il ne faut d’argent pour couvrir des pertes plus ou moins hypothétiques. Ainsi donc, l’évaluation et la gestion judicieuses du risque de change impliquent que les gestionnaires peuvent désormais mieux réagir aux variations des taux de change.

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Cette étude contient des enseignements sur le plan de l’action pour les spécialistes des finances et de la gestion de la performance/du risque au Ministère. En particulier, le groupe du VCEMD, par l’intermédiaire du Directeur - Planification des Forces et coordination du programme (DPFCP), et le SMA(Fin SM), par l’intermédiaire du Directeur - Budget et du Directeur - Finances et établissement des coûts (Stratégie) (DFECS), pourraient vouloir étudier la possibilité de rajuster les affectations du budget ministériel (trimestriellement) sur la base du résultat du modèle VAR correspondant au 5e percentile. De fait, ces groupes devraient envisager d’inclure la méthode VAR dans le cadre de gestion intégrée du risque du Ministère, afin de gérer le risque budgétaire associé à l’exposition aux fluctuations monétaires. Si ce scénario se matérialisait, les rapports trimestriels du DFECS sur les opérations de change continueraient de fournir un taux de change de référence pour toutes les prévisions de dépenses libellées en monnaie étrangère au Ministère et on évaluerait le risque de perte dans les opérations de change à l’aide du modèle VAR. Par extension, le Ministère pourrait aussi étudier la possibilité d’utiliser la méthode VAR pour calculer le risque financier pour d’autres postes de dépenses budgétaires exposés au risque de prix, comme les carburants en vrac, l’hydro-énergie et certains produits tels que l’acier, le matériel ballistique, etc., lorsque le montant des dépenses le justifie. Au moment où le Ministère s’engage dans un vaste programme pluriannuel d’acquisition de biens et continue de mener outre mer des missions importantes et délicates, la nécessité d’évaluer avec précision le risque financier n’a jamais été aussi grande. En outre, si le Ministère devait décider de solliciter l’approbation d’un organisme du gouvernement central en vue de mettre en œuvre (ou d’expérimenter) une stratégie de couverture des risques de change, son aptitude à évaluer et à décrire le risque de change serait indispensable au succès de cette stratégie. Cela dit, cette étude illustre l’application de la méthode VAR au type de risque financier sans doute le plus important au Ministère : le risque de change, et il est à espérer que cette méthode permettra de mieux comprendre ce risque et de déterminer comment on peut le mesurer et le décrire avec plus de régularité et de précision et, en fin de compte, comment on peut le maîtriser. P.E. Desmier; 2007; Estimating Foreign Exchange Exposure in the Department of National Defence; DRDC CORA TR 2006–23; RDDC – Centre pour la recherche et l’analyse opérationnelles.

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Table of contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Figures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1

The Budgeting Process and Historical Rate Variances . . . . . . . .

4

2.2

Expenditure Data . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.1

The Rules for Data Filtering . . . . . . . . . . . . . . . . .

9

2.2.2

Data Integrity . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.3

Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.4

Final Expenditure Data . . . . . . . . . . . . . . . . . . . .

13

Foreign Exchange Data . . . . . . . . . . . . . . . . . . . . . . . .

15

The Expenditure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.1

An MA(4) Model for the USD National Procurement Account . . .

16

3.1.1

USD NP: Transforming the Series . . . . . . . . . . . . . .

17

3.1.2

USD NP: Fitting the ARMA Model . . . . . . . . . . . . .

19

3.1.3

USD NP: Testing for Invertibility and Stationarity . . . . . .

21

2.3 3

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3.1.4

4

USD NP: Testing the Residuals . . . . . . . . . . . . . . . .

21

3.2

An ARMA(2,3) Model for the USD Capital Account . . . . . . . .

22

3.3

An ARMA(4,3) Model for the GBP National Procurement Account .

22

3.4

An MA(1) Model for the GBP Capital Account . . . . . . . . . . .

23

3.5

An ARMA(2,6) Model for the EURO National Procurement Account 24

3.6

An MA(1) Model for the EURO Capital Account . . . . . . . . . .

24

Forecasting Foreign Exchange . . . . . . . . . . . . . . . . . . . . . . . . .

26

4.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

4.2

The Shape of the Returns Distribution . . . . . . . . . . . . . . . .

26

4.3

Univariate vs. Multivariate Approach . . . . . . . . . . . . . . . . .

27

4.4

The Univariate GARCH(1,1) Model . . . . . . . . . . . . . . . . .

30

4.4.1

Maximum Likelihood Estimation (MLE) with t˜(d) . . . . .

31

4.4.2

Validation of Non-Normality Assumption . . . . . . . . . .

32

Validating the GARCH(1,1) Variance Models . . . . . . . . . . . .

35

4.5.1

Backtesting the GARCH models - Type I Testing . . . . . .

35

4.5.2

Backtesting the GARCH models - Conditional Coverage Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

In-Sample Check on the Autocorrelations . . . . . . . . . .

39

The VaR Model for DND . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

5.1

Background: VaR Methodologies . . . . . . . . . . . . . . . . . . .

41

5.2

Filtered Historical Simulation For Returns . . . . . . . . . . . . . .

42

5.3

Filtered Historical Simulation For Expenditures . . . . . . . . . . .

45

5.4

Building the VaR Model . . . . . . . . . . . . . . . . . . . . . . . .

46

Simulation Results: Running the FOREX Model . . . . . . . . . . . . . . .

48

6.1

The Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

6.2

Forecasting Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . .

48

4.5

4.5.3 5

6

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6.3

Forecasting Expenditures . . . . . . . . . . . . . . . . . . . . . . .

50

6.3.1

Forecasted Expenditure Validation . . . . . . . . . . . . . .

50

6.4

Forecasting Performance for Currency Returns . . . . . . . . . . . .

55

6.5

Assessing Convergence . . . . . . . . . . . . . . . . . . . . . . . .

58

6.5.1

Convergence of Results . . . . . . . . . . . . . . . . . . . .

58

6.5.2

Test for VaR Convergence . . . . . . . . . . . . . . . . . .

60

7

Discussion on Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

8

Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . .

64

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Annexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

A

Fitting the Expenditure Models . . . . . . . . . . . . . . . . . . . . . . . .

71

A.1

USD Capital Account . . . . . . . . . . . . . . . . . . . . . . . . .

71

A.1.1

USD Capital: Transforming the Series . . . . . . . . . . . .

71

A.1.2

USD Capital: Fitting the ARMA Model . . . . . . . . . . .

73

A.1.3

USD Capital: Testing for Invertibility and Stationarity . . .

74

A.1.4

USD Capital: Testing the Residuals . . . . . . . . . . . . .

75

GBP NP Account . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

A.2.1

GBP NP: Transforming the Series . . . . . . . . . . . . . .

75

A.2.2

GBP NP: Fitting the ARMA Model . . . . . . . . . . . . .

76

A.2.3

GBP NP: Testing for Invertibility and Stationarity . . . . . .

79

A.2.4

GBP NP: Testing the Residuals . . . . . . . . . . . . . . . .

79

GBP Capital Account . . . . . . . . . . . . . . . . . . . . . . . . .

80

A.3.1

GBP Capital: Transforming the Series . . . . . . . . . . . .

80

A.3.2

GBP Capital: Fitting the ARMA Model . . . . . . . . . . .

80

A.3.3

GBP Capital: Testing for Invertibility and Stationarity . . .

82

A.2

A.3

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A.3.4

GBP Capital: Testing the Residuals . . . . . . . . . . . . .

83

EUR NP Account . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

A.4.1

EUR NP: Transforming the Series . . . . . . . . . . . . . .

83

A.4.2

EUR NP: Fitting the ARMA Model . . . . . . . . . . . . .

84

A.4.3

EUR NP: Testing for Invertibility and Stationarity . . . . . .

86

A.4.4

EUR NP: Testing the Residuals . . . . . . . . . . . . . . . .

87

EUR Capital Account . . . . . . . . . . . . . . . . . . . . . . . . .

88

A.5.1

EUR Capital: Transforming the Series . . . . . . . . . . . .

88

A.5.2

EUR Capital: Fitting the ARMA Model . . . . . . . . . . .

88

A.5.3

EUR Capital: Testing for Invertibility and Stationarity . . .

90

A.5.4

EUR Capital: Testing the Residuals . . . . . . . . . . . . .

91

The FOREX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

B.1

Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

B.2

The Risk Analysis Software . . . . . . . . . . . . . . . . . . . . . .

92

B.3

Starting the Model . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

B.3.1

The SplashScreen . . . . . . . . . . . . . . . . . . . . . . .

93

B.3.2

The SwitchBoard . . . . . . . . . . . . . . . . . . . . . . .

93

Input Data Options . . . . . . . . . . . . . . . . . . . . . . . . . .

94

B.4.1

The Enter Actuals Form . . . . . . . . . . . . . . . . . . .

95

Model Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

B.5.1

The Report Sheet . . . . . . . . . . . . . . . . . . . . . . .

95

FOREX Graphical Output . . . . . . . . . . . . . . . . . . . . . . .

98

A.4

A.5

B

B.4

B.5

B.6

List of symbols/abbreviations/acronyms/initialisms . . . . . . . . . . . . . . . . . . 100

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Figures 1

Rates and Canadian Dollar Variance on U.S. Dollar Liquidated Obligations (NP and Capital (equipment) Accounts). Left-hand scale shows exchange rate; Right-hand scale shows variance. . . . . . . . . . . . . . . . . . . . . .

6

Rates and Canadian Dollar Variance on U.K. Pound Sterling Liquidated Obligations (NP and Capital (equipment) Accounts). Left-hand scale shows exchange rate; Right-hand scale shows variance. . . . . . . . . . . . . . . .

7

Rates and Canadian Dollar Variance on EURO Liquidated Obligations (NP and Capital (equipment) Accounts). Left-hand scale shows exchange rate; Right-hand scale shows variance. . . . . . . . . . . . . . . . . . . . . . . .

8

USD, GBP and EUR Outliers for the National Procurement and Capital Accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

USD, GBP and EUR Liquidated Obligations for the National Procurement and Capital Accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

6

USD, GBP and EUR Exchange Rates in Canadian Dollars . . . . . . . . . .

15

7

(a)Time plot of monthly USD NP liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-12 and lag-1 differenced series. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from MA(4) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals. . . . . . . . . . . . .

18

8

Return distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

9

Correlogram of cross products of CAD/USD, GBP and EUR returns. . . . .

28

10

(a–c)Time plots of CAD/USD, GBP and EUR exchange rates. (d–f) Time plots of CAD/USD, GBP and EUR returns. . . . . . . . . . . . . . . . . . .

29

Quantile-Quantile plots of daily CAD/USD, CAD/GBP and CAD/EUR returns (a-b); (d-f) returns standardized by GARCH(1,1) against the normal distribution; (g-i) returns standardized by GARCH(1,1) against the student-t distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

12

BackTesting GARCH models at the 95% confidence level . . . . . . . . . .

37

13

Autocorrelation of CAD/USD, GBP and EUR standardized squared returns (dashed line) and squared returns (solid line). . . . . . . . . . . . . . . . . .

40

Extraction of monthly exchange rates . . . . . . . . . . . . . . . . . . . . .

43

2

3

4

5

11

14

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15

The FHS process for returns . . . . . . . . . . . . . . . . . . . . . . . . . .

44

16

The FHS process for expenditures . . . . . . . . . . . . . . . . . . . . . . .

45

17

Value-at-Risk distributions for CAD/USD national procurement (a) and capital accounts (b) for one month ahead from 31 March 2006. Red areas to left and right of average correspond to the lower and upper 5% of results respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Cumulative expenditure distribution for USD national procurement (a) and capital accounts (b) for one month ahead from 31 March 2006. . . . . . . . .

54

18

19

Return Distributions for (a) CAD/USD (b) CAD/GBP and (c) CAD/EUR exchanges for one month ahead from 31 March 2006. Red areas to left and right of average correspond to the lower and upper 5% of results respectively. 56

20

Convergence results for 5th percentile of CAD/USD national procurement account on April 30 2006. . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

(a)Time plot of monthly USD Capital liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-1 differenced series. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from ARMA(2,3) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals. . . . . . . . . . . . . . . . . . . . . . .

72

(a)Time plot of monthly GBP national procurement liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-1 differenced series. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from ARMA(4,3) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals. . . . . . . . . . . .

77

(a)Time plot of monthly GBP Capital liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-1 differenced series. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from MA(1) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals. . . . . . . . . . . . . . . . . . . . . . . . .

81

(a)Time plot of monthly EUR national procurement liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed, lag-12 and lag-1 differenced subset starting 01 June 2000. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from ARMA(2,6) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals. . .

85

A.1

A.2

A.3

A.4

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A.5

(a)Time plot of monthly EUR Capital liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-1 differenced subset starting 01 November 2001. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from MA(1) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals. . . . . . . . . . . .

89

B.1

The FOREX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

B.2

The FOREX Model Splash Screen . . . . . . . . . . . . . . . . . . . . . . .

93

B.3

The SwitchBoard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

B.4

@Risk Simulating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

B.5

Close Excel Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

B.6

Input form to enter actual expenditures, liquidated foreign exchange rates and budget rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

B.7

The FOREX Model Report Sheet (partial) . . . . . . . . . . . . . . . . . . .

97

B.8

FOREX Model VaR Graphical Output (USD NP) . . . . . . . . . . . . . . .

99

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Tables 1

Annual Realized Variances in $CAD for NP and Capital (equipment) Accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

USD, GBP and EUR Main Outlier Contributors for the NP and Capital Accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

3

USD NP model selection statistics . . . . . . . . . . . . . . . . . . . . . . .

20

4

Return and squared return statistics . . . . . . . . . . . . . . . . . . . . . .

28

5

Cross-product of returns statistics . . . . . . . . . . . . . . . . . . . . . . .

28

6

Coefficients for the GARCH(1,1) models . . . . . . . . . . . . . . . . . . .

33

7

Type I testing results of number of excessions found for each currency using the GARCH model, a 50 day observation period and a 95% confidence level

36

8

Conditional Coverage testing for each currency . . . . . . . . . . . . . . . .

39

9

Forecast budget rates as of 09 March 2006 . . . . . . . . . . . . . . . . . .

47

10

5th, 50th and 0th percentiles for national procurement and capital accounts .

49

11

Results of interpolation of actual expenditures to the forecasted cumulative distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

12

Expenditure percentiles for national procurement and capital accounts . . . .

53

13

Results of interpolation of actual returns to the forecasted cumulative distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Forecasting performance of CAD/USD, CAD/GBP and CAD/EUR monthly returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

15

Simulation results for CAD/USD 5th percentile one month ahead VaR . . . .

60

A.1

USD Capital model selection statistics . . . . . . . . . . . . . . . . . . . . .

73

A.2

GBP NP model selection statistics . . . . . . . . . . . . . . . . . . . . . . .

76

A.3

GBP Capital model selection statistics . . . . . . . . . . . . . . . . . . . . .

82

A.4

EUR NP model selection statistics . . . . . . . . . . . . . . . . . . . . . . .

84

A.5

EUR Capital model selection statistics . . . . . . . . . . . . . . . . . . . . .

90

2

14

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Acknowledgements Sincere appreciation is extended to: Maj. Richard Groves, DMG Compt 5-4, for his foresight in proposing the project, providing input to the report, and arranging briefings with DMG Compt and ADM(Fin CS) staff, and the Defence Comptrollership Council (DCC); Mr. Spencer C. Lee, DMG Compt 5-4-2, for providing all raw FMAS downloads for the national procurement and capital accounts and providing input to the report; Dr. Binyam Solomon, DG Strat G 2-3/DRDC CORA, for reviewing the report and providing useful methodological comments; and, Professor Peter F. Christoffersen, McGill University Finance Research Centre, for providing input to the modelling and results and for proofreading the manuscript and providing many valuable comments.

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1 1.1

Introduction Background

Like any institution engaged in international trade, the Department of National Defence (DND) is often required to make payment in foreign currency when acquiring new equipment and supplies to support military operations at home and abroad. Between fiscal years (FY) 99/00 and 05/06, DND expended $4.8 billion in foreign currencies from the National Procurement (NP) and Capital accounts [1]. Of this, 82.3% ($4.0B) was spent in U.S. Dollars (USD), 7.97% ($385.2M) in British Pound Sterling (GBP) and 4.95% ($239.2M) in the Euro (EUR)1 . The remaining 4.78% was divided between 12 countries ranging from the German Deutschmark ($91.3M)) to the New Zealand Dollar ($7.0K). Currently, foreign currency transactions account for between 7-10% of DND spending. Foreign exchange exposure refers to the sensitivity of an organization’s cash flows to changes in the exchange rates. Its impact and mitigation strategies have been studied for a number of years. In the 1990’s, for example, a number of empirical analyses of the foreign exchange rate exposure of both Canadian and foreign multinationals found low or even negligible levels of exposure [2, 3, 4]. However, it has been argued that the absence of evidence may be due more to restrictions imposed on the data sample and an aggregation of economic measures, than an industry level trend. Firms least likely to be exposed would have had access to both operational and financial hedging strategies to counter foreign currency risk [5, 6]. Now, given the high volatility of various currencies, it is generally accepted that exchange rate exposure, if not quantified and managed properly, may expose the organization to significant budgetary consequences due to poor responsiveness to foreign exchange volatility [7]. There are a number of techniques that could be used to risk manage foreign currency exposure. Forward contracts, futures contracts and money market hedges are all available for the organization that wants to eliminate foreign transaction exposure of less than one-year [8, 9]. Within DND, forward contracts hedging techniques have already been proposed [10, 11], but DND has adopted a “risk-neutral” approach most likely due to political and institutional constraints towards the generally perceived speculative nature of the foreign currency markets. As a consequence, and as a result of Canada’s small industrial base, DND finds itself exposed to substantial foreign exchange risk in the acquisition and maintenance of equipment and supplies [11]. Quantifying risk is not a trivial process although it is generally accepted that the standard method for reporting risk today is the “Value-at-Risk” or VaR method first used in the J.P. Morgan G-30 report in July 1993 and theorized in their RiskMetrics framework for quantifying market risk [12], but originally developed from the pioneering work on Modern Portfolio Theory by Markowitz and Sharpe2 , as well as analysis of the derivatives market 1 On 01 January 1999 the euro became the official currency of the eleven (now 12) participating countries, however, national currencies remained as denominations of the euro and continued to be used as a matter of convenience until 2002. On 01 January 2002, “E-Day”, the circulation of euro banknotes and coins started. 2 Harry M. Markowitz, William F. Sharpe and Merton Miller shared the 1990 Nobel Prize in Economics “. . .

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1

by U.S. banks in the 1980’s. Simply put, VaR is defined as the predicted worst-case loss at a specific confidence level over a certain period of time. While most financial institutions report the VaR at the one-day 95% probability, any parameter of the distribution (e.g., standard deviation of the portfolio return) could be used. Thus VaR can provide a quantitative measure of the downside risk of exposure in all foreign currency transactions. As requested by the Director Materiel Group Comptroller (DMG Compt) [13], the Director Materiel Group Operational Research was asked to develop and demonstrate the utility of using VaR analysis within ADM(Mat) for forecasting the potential impact of foreign currency fluctuations of the USD, GBP and EUR exchanges on the ADM(Mat) national procurement and capital (equipment) accounts, and apply VaR techniques to determine the maximum expected loss from adverse exchange rate fluctuations over the remaining periods of the budget year. The model should be able to provide a forecast of the potential additional expenditures that may need to be requested so that budgets can be more successfully managed.

1.2

Scope

In this report we apply VaR methodologies together with advanced linear and non-linear time series analysis to predict the amount of foreign exchange risk that DND faces in both its national procurement and capital accounts. While VaR is a very simple and intuitive concept, its measurement is a very challenging statistical problem. Consequently, to do full justice to this analysis, the theoretical details are fully developed in this paper and consolidated in sections 3 & 4 of which the casual reader may glance over without any (assumed) loss of continuity. This report is divided into eight sections and two annexes. Following the introduction, section 2 describes the DND budgeting process; presents a simplified form of the equation that defines VaR for the department; and, develops a complete data analysis for the two main variables that make up the VaR: Expenditures for the national procurement and capital accounts, and foreign exchange rates for the three main currencies that have accounted for 95.2% of all foreign transactions in the last seven years, i.e., USD, GBP and EUR. Section 3 describes the mathematical derivation of the six expenditure models (two accounts × three currencies). Each currency/account is modelled as either a pure autoregressive (AR) or pure moving average (MA) model or a combined ARMA model. To illustrate the details involved in deriving each model, Section 3 describes the complete derivation and testing of the USD national procurement expenditures model. The remaining five models are presented in final form, with complete derivation and testing, in Annex A. Section 4 discusses financial returns; the shape of the return distributions and develops models for the financial return series that accurately model the characteristics of each curfor their pioneering work in the theory of financial economics.”

2

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rency, such as time-varying volatilities (heteroskedasticity), volatility clustering and nonnormal error distributions. A univariate GARCH(1,1) model is developed for each currency with standardized t˜(d) error distributions, and parameters derived through maximized likelihood estimation. Section 4 also contains full validation of each model including the nonnormality assumption and complete backtesting where the GARCH models were validated to determine the accuracy of predicting both the frequency and size of expected losses. In Section 5 all the details in the previous sections are assembled to build the VaR model for DND that uses Filtered Historical Simulation for returns and expenditures and takes into account the changes in past and current volatilities of historical returns with the least number of assumptions about the statistical properties of future price changes. Section 6 focuses on the results of running the FOREX, or FOReign EXchange, risk model. Results are given for forecasted VaR, forecasted expenditures and currency returns as well as “out-of-sample” testing of the model with actual data post March 2006. Section 6 also includes convergence testing of the model with complete statistical validation of VaR convergence. Section 7 provides a discussion on the linear (expenditures) and non-linear (currency returns) modelling. Possible future analysis, including increasing the data sample for expenditures and moving towards a multivariate approach for the currency exchanges, is also discussed. The report concludes with a discussion on the impact of VaR analysis to the department and recommendations for the use and development of the FOREX model.

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2

Data Analysis

2.1

The Budgeting Process and Historical Rate Variances

In DND, the budget planning process starts in March-June of the preceding year where budgetary requirements are estimated in Canadian dollars by using the DND Director Budget (DB) forecasted rates for specific currencies [11]. In the overall process, the vast majority of foreign exchange exposure comes from the variance (difference) between exchange rates existing when obligations are budgeted and that existing when obligations are liquidated. These differences are generally absorbed within the local budgets that were used to procure the service or equipment. Therefore, being able to predict the rate variances with reasonable accuracy would ensure proper management of public funds by minimizing the effects of adverse currency movements. The methodology used by the DB in forecasting currencies was, until Economic Model (EM) 2004-05, mostly based on analysis where simple Autoregressive Integrated Moving Average (ARIMA) models were applied. Since the forecast was completed only once per year for the production of the EM (released in May - June), each EM forecast included April of the current FY as actuals and the remaining eleven months as forecasts [14]. Each ARIMA model was developed with historical quarterly data going back to 1990. Up to and including EM 2004, forecasts were developed for each quarter but only a FY average was reported for use in the EM; monthly or quarterly forecasts were not provided. Starting in September 2004, quarterly update reports with monthly forecasts were finally reported for use [15]. By comparing the budget rates to the rates at which obligations are liquidated (accounts paid), the potential for a funding gap and an impact on local budgets becomes evident. Using the Canadian dollar as the base currency, Figures 1 - 3 compare the budget rate against the liquidated rate for the three currencies: USD, GBP and EUR, and for the two accounts National Procurement and Capital (equipment). Consideration was limited to these three currencies as they represented approximately 95% of all foreign exchange transactions from the past seven years and demonstrated reasonable expenditure distributions for both accounts. The expenditure amount and rate at liquidation are proxied by the sum of expenditures at month end and the average monthly rate for each currency. The monthly realized budget variance (V) is simply the difference between the budget rate (b) and the liquidated rate (p) multiplied by the expenditure (E), i.e., V = E × (b − p) .

(1)

Equation 1, in its simplified form, is the basic relationship that defines all VaR calculations for this study3 . Therefore, if the liquidated exchange rate is greater than the budget rate, a negative variance (loss) is forecasted and a shortfall is presented to the local budget for which funds must be acquired from other sources. 3 In

forecasting the variance, uncertainty results from two sources, the forecasted expenditure, E, and the liquidated rate, p. While the subtraction in (1) determines a loss or gain in the budget variance, the sampled value for E determines the magnitude of the loss or gain.

4

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For USD transactions (Figure 1), except for a two month period near the end of FY 99/00, throughout 99/00 - 01/02 the budget rate was below the liquidated rate. This resulted in a negative realized variance or liquidated amounts in excess of those that had been budgeted. Interestingly, the reverse is largely true from the start of FY 03/04 through most of 05/06. In all cases, the large dollar amounts for Capital transactions take precedence in the variance calculations. For GBP transactions (Figure 2), until the start of FY 03/04, the budgeted rate was a reasonable approximation to the liquidated rate resulting in fairly minor variance perturbations. Poorer forecasting coupled with high rate volatility from then on resulted in large variances, mainly for the NP account. As for the USD, it is noted that a large delta between the budget and liquidated rates opens up at the start of FY 05/06. EUR transactions (Figure 3) were somewhat more limited in that zero dollars were exchanged between April 1999 and May 2000 for the NP account and only minor amounts between April 1999 and October 2001 for the Capital account. Considering that during the period between May 1999 and March 2001 any obligations would have been significantly overfunded, minor transactions such as these would have minimal affect on local budgets. A large delta between rates is also noted from the start of FY 05/06. Table 1 lists the annual realized variances by currency for both the national procurement and capital accounts. Negative variances reflect liquidated amounts in excess over budgeted amounts as defined by equation 1. Table 1: Annual Realized Variances in $CAD for NP and Capital (equipment) Accounts

FY

Account

USD

99/00

NP Capital

-1,834,932 -1,220,705

94,422 -38,314

N/Aa N/A

00/01

NP Capital

-8,628,290 -16,219,743

64,004 68,424

46,575 N/A

01/02

NP Capital

-9,736,531 -17,105,025

205,792 63,081

-380,264 -66,966

02/03

NP Capital

3,549,913 7,505,844

1,293,546 229,095

-1,120,261 -1,282,363

03/04

NP Capital

9,272,031 13,562,721

-1,774,706 -220,648

-380,237 -32,421

04/05

NP Capital

14,266,416 26,817,372

600,801 42,403

-478,562 -1,404,770

05/06

NP Capital

11,483,163 17,089,327

4,241,910 3,390,680

3,434,398 1,555,494

a Implies

GBP

EUR

zero or minor transactions

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6

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Capital Variance USD Monthly Rate (Average of Daily Rates)

April-05

April-04

April-03

April-02

April-01

April-00

April-99

CAD per USD

-$15,000,000

-$10,000,000

-$5,000,000

$0

$5,000,000

$10,000,000

Figure 1: Rates and Canadian Dollar Variance on U.S. Dollar Liquidated Obligations (NP and Capital (equipment) Accounts). Left-hand scale shows exchange rate; Right-hand scale shows variance.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

NP Variance USD Forecasted Budget Rate

Variance ($ CA)

DRDC CORA TR 2006–23

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Capital Variance GBP Monthly Rate (Average of Daily Rates)

April-05

April-04

April-03

April-02

April-01

April-00

April-99

-$1,500,000.00

-$1,000,000.00

-$500,000.00

$0.00

$500,000.00

$1,000,000.00

$1,500,000.00

Figure 2: Rates and Canadian Dollar Variance on U.K. Pound Sterling Liquidated Obligations (NP and Capital (equipment) Accounts). Left-hand scale shows exchange rate; Right-hand scale shows variance.

CAD per GBP

2.6

NP Variance GBP Forecasted Budget Rate

Variance ($ CA)

8

DRDC CORA TR 2006–23

Capital Variance EURO Monthly Rate (Average of Daily Rates)

April-05

April-04

April-03

April-02

April-01

April-00

April-99

CAD per EURO

-$1,500,000.00

-$1,000,000.00

-$500,000.00

$0.00

$500,000.00

$1,000,000.00

$1,500,000.00

Figure 3: Rates and Canadian Dollar Variance on EURO Liquidated Obligations (NP and Capital (equipment) Accounts). Left-hand scale shows exchange rate; Right-hand scale shows variance.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

NP Variance EURO Forecasted Budget Rate

Variance ($ CA)

2.2

Expenditure Data

Prior to forecasting the NP and Capital variance, expenditures need to be analyzed by first downloading the data from departmental financial web sites and filtering/manipulating it according to established rules. 2.2.1

The Rules for Data Filtering

Financial and Managerial Accounting Systems (FMAS) downloads were provided from FY 99/00 through 05/06 for the three accounts: C113 (NP), C513 (NP Betterment)4 and C503 (Capital). Data was provided for all transactions under the headings: Payment, Fund, Fund Centre, Cost Centre, Period and Currency; and for all SAP5 document types that included KR (Vendor Invoice) and SA (G/L Account Document) [16]. Filtering algorithms were written to analyze, extract and sum dollar amounts for the specified accounts according to the following rules that were applied in the specified order [17]: 1. Extract only KRs and SAs: Reason: Only these Document ID types account for purchases. 2. Use only positive KRs: Reason: They account for direct purchases. 3. Negative SAs are matched to Positive SAs only if: • •

Funds were the same; and, Currencies were the same.

Reason: On occasion a positive SA (indirect purchase transaction)) will be incorrectly entered, and a subsequent negative SA is entered at a later date to offset the original entry. The converse can also occur, therefore the period rule (see following item) is not enforced. 4. Negative SAs are matched to positive KRs only if: • • •

Funds were the same; Currencies were the same; and, Period6 for SA is ≥ Period for KR.

Reason: KRs that match negative SAs with the same fund means a direct purchase (KR) by a fund was incorrectly coded, and thus a subsequent entry (SA) was made to correct the original entry at a later date. A corresponding positive SA will be made to another fund to charge that fund for the purchase. Fund Centers and Cost Centers do not have 4 For

analysis purposes, data from C513, a sub-account of C113, was merged with C113 as both relate to

NP. 5 Founded in 1972 as Systemanalyse und Programmentwicklung (Systems Applications and Products in Data Processing) by five former IBM employees in Mannheim, Germany, SAP is the world’s largest interenterprise software company and the world’s fourth-largest independent software supplier, overall. As a consequence of its German roots, the data notation structure (KR, SA, etc.) follows the German nomenclature. 6 There are 15 periods in FMAS payments for any FY. Periods 1 through 12 represent the months of the standard FY. Periods 13 through 15 are payments captured beyond the FY for which invoices for goods and/or services were submitted prior to 31 March. The latter are normally rolled into period 12, which will tend to ”spike” towards an annual distribution at the end of the FY.

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9

to be matched since they identify only which ADM(Mat) divisions are charged, which is irrelevant since all divisions (cost centers and fund centers) have access to all funds. This study is only concerned with purchases made by specific funds, thus only the fund is relevant. 5. Include all remaining unmatched negative SAs: Reason: Negative SAs represent reductions of expenditures for specific funds. It can be the result of reversing a previous positive KR or positive SA. In a perfect world, we would be able to identify which KR and SA the negative SA referred to, however due to the non-uniform practices of the data entry system, this identification will not always be possible. However, it still stands that a negative SA correctly reduces expenditures for a fund and since this study is concerned with correctly identifying expenditures for funds, negative SAs must be included. 6. For 1999, remove negative SAs for first four periods of data: Reason: It is assumed that negative SAs in the first four periods of the 99/00 fiscal year could be matched to positive KRs towards the end of FY 98/99. 2.2.2

Data Integrity

Erroneous data errors are certainly possible when dealing with manual input of expenditures over a period of seven years. Inherent flaws for forecasting expenditures with SAP data are: •

There will be months where negative SAs (transfers) will outweigh positive KRs (direct purchases) and SAs (indirect purchases) due to the fact that negative and positive SAs are not necessarily entered in the same month as the corresponding KR. Negative SAs are entered in the period when the clerk enters the transaction in the system and not necessarily in the same month when the actual purchase transaction occurs. The algorithms correct for this as much as possible by searching forward until a match is found. If no match is found, the negative SAs will count against the sum for that period;

•

As specified in Rule 6, in the first few months of the first year (FY 99/00), there will be unmatched negative SAs for KRs that may have occurred in the prior year. Consequently, an assumption was made to correct for their usage; and,

•

Overall, there was only $4.3K in transactions where no currency was specified. This data was removed from further analysis.

2.2.3

Outliers

An outlier is an observation that lies an abnormal distance from other values in a random sample from a population. The Box-and-Whisker plot is a useful graphical display for describing the behaviour of the data in the middle as well as at the ends of the distributions and hence determine which points are outliers. For the expenditure data, Box-and-Whisker plots were drawn with all mild outliers defined as points beyond 3/2 the interquantile range from the edge of the box and extreme outliers defined as points beyond three times the interquantile range. With quantiles set at 10% and 90%, Figure 4 describe the outliers for the three currencies and two accounts.

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Observations show that each data set is non-normal skewed right with a total of seven extreme and six mild outliers. Since any period consists of the sum of individual payments, which may total to an extreme value, it remains to determine if any single payment, while not in themselves being outliers, would be large enough to skew the total and may be treated as an abnormal (non-reproducible) expense. Table 2 provides an extraction of all large payments that contribute to an outlier. After a detailed examination by DMG Compt staff, it was determined that the expenditures in Table 2 are in fact typical transactions that occur within the fund. They may be extraordinarily large because they represent advance payments on existing contracts (usually in an effort spend funds before they lapse at year-end). An advance payment is not the typical purchase transaction, but since they are transactions that will occur in the business environment, it was felt that removing them will result in a data set that doesn’t truly represent the normal activity of the applicable fund [18].

Table 2: USD, GBP and EUR Main Outlier Contributors for the NP and Capital Accounts

Currency

Account

Project Title

USD USD USD USD GBP GBP GBP GBP EUR EUR EUR

NP Capital Capital Capital NP Capital Capital Capital NP Capital Capital

Joint Strike Fighter CF-18 ECP 583 Challenger Acquisition Evolved Sea Sparrow Missle Sub Class Maintenance Strategic Air-to-Air Refuelling Sub Capability Life Extension Ltweight 155mm Towed Howitzer Ammunition/Land Strategic Air-to-Air Refuelling Strategic Air-to-Air Refuelling

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Amount ($CAD)

FY

Period

Outlier

88,480,000 52,445,847 91,837,500 124,321,772 20,428,393 5,395,152 4,321,250 3,484,960 5,155,734 30,799,000 9,511,200

01/02 01/02 01/02 00/01 02/03 02/03 04/05 05/06 01/02 04/05 04/05

12 12 13 12 13 5 12 13 13 12 9

Extreme Mild Mild Mild Extreme Mild Mild Mild Extreme Extreme Extreme

11

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5  106 0

2  107

0

0

5  107

1  108

1.5  10

8

0

1  106

2  106

3  106

4  10

6

5  10

6

6  106

GBP Capital Account

GBP NP Account

0

5  106

1  107

1.5  107

2  107

2.5  107

3  107

0

5  106

1  107

1.5  107

2  107

EUR Capital Account

EUR NP Account

Figure 4: USD, GBP and EUR Outliers for the National Procurement and Capital Accounts

USD Capital Account

1  107

4  107

7  106

1.5  107

6  107

2  108

2  107

8  107

USD NP Account

2.5  10

7

3  107

1  108

1.2  108

2.2.4

Final Expenditure Data

Figure 5 illustrates the distribution of all expenditure data used in this study. Except for the Euro where limited transactions required the data analysis to start when a steady state trend in expenditures was noted7 , all currencies started on 1 April 1999 and ended 31 March 2006. Expenses for periods 12-15 were summed under period 12. For NP, the seasonal aspect of the analysis becomes apparent with peaks in the data every 12 months, most notably for the USD. The large peak at period 12 (March) in 2002 is mainly due to the JSF project ($88.5M) with the influence of a number of smaller projects with several in the $5M range. The average expenditures at FY year-end for the three NP accounts were $55.5M, $13.8M and $8.2M for USD, GBP and EUR respectively; and $96.5M, $3.6M and $5.7M for the three capital accounts. For both GBP and EUR NP accounts, the peaking is not as clearly defined and only EUR was modelled with a seasonal trend. The large peak in March 2003 for GBP NP had one major expense of $20.4M for maintenance on the Victoria Class subs in period 13. For EUR transactions in NP, the large peak shown at the end of FY 01/02 was based on 74 transactions with the largest being $2.6M in period 12 and $3.7M in period 13. There were no discernible seasonal variations for Capital expenditures since peaks in the data may occur at any time period and not just end-of-year.

7 Euro

national procurement and capital expenditure analysis started on June 2000 and November 2001 respectively.

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13

USD Liquidated Obligations Dollars CAD

USD Liquidated Obligations Dollars CAD

0

5  107

1  108

1.5  108

2  108

0

2  107

4  10

7

6  107

8  107

1  10

8

1.2  108

0

1  106

2  106

3  106

4  106

5  106

6  106

7  106

Capital Account

9900 0001 0102 0203 0304 0405 0506 0607 Start of Fiscal Year

9900 0001 0102 0203 0304 0405 0506 0607 Start of Fiscal Year

0

5  106

1  10

7

1.5  107

2  107

2.5  10

7

3  107

National Procurement Account

0

5  106

1  107

1.5  107

2  107

2.5  107

3  107

0

5  106

1  107

1.5  107

2  107

National Procurement Account

9900 0001 0102 0203 0304 0405 0506 0607 Start of Fiscal Year

Capital Account

9900 0001 0102 0203 0304 0405 0506 0607 Start of Fiscal Year

Figure 5: USD, GBP and EUR Liquidated Obligations for the National Procurement and Capital Accounts

9900 0001 0102 0203 0304 0405 0506 0607 Start of Fiscal Year

Capital Account

9900 0001 0102 0203 0304 0405 0506 0607 Start of Fiscal Year

National Procurement Account GBP Liquidated Obligations Dollars CAD GBP Liquidated Obligations Dollars CAD

EUR Liquidated Obligations Dollars CAD EUR Liquidated Obligations Dollars CAD

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2.3

Foreign Exchange Data

For VaR applications, closing prices are normally used for assets trading on a local exchange, however, for foreign exchange markets that trade around the clock, the setting of a closing price for instruments trading in different time zones brings a non-synchronicity to the data that must be standardized for it to have any meaning [19]. Non-synchronicity of data can be addressed by several means and if it can’t be avoided, quotes could be collected in a single time zone with quotes from other time zones as broker quotes. The Bank of Canada derives its exchange rates from the USD/CAD exchange rate and from indicative wholesale market quotes. The closing rates used in this study are based on official parities or market rates and are updated at about 4:30 p.m. ET on the same business day [20]. Daily closing rates were extracted for the three currencies for all trading days from 01 April 1999 through 31 March 2006 (1757 data points). Figure 6 shows the currency trends over the last seven years. On average, in this period, there were 21 trading days per month ± 1 day8 . The trend in the last three years for each currency is downwards. Although conventional wisdom may suggest that the best available model for exchange rate movements is a random walk, it has been argued that traditional economic fundamentals of a country affect to a large extent the equilibrium value of a currency, whose movements are best forecast through more state-of-the-art econometric methods [21].

CAD per USD, GBP, EUR

2.4 2.2 2

USD GBP EUR

1.8 1.6 1.4 1.2 01 Apr 99

03 Apr 00

02 Apr 01

01 Apr 02 01 Apr 03 Date

01 Apr 04

01 Apr 05

01 Apr 06

Figure 6: USD, GBP and EUR Exchange Rates in Canadian Dollars

8 Note,

01 April 2000 and 2001 were non-trading days in Figure 6

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3

The Expenditure Models

Expenditures are modelled as discrete time series where all transactions during the month are assumed to accumulate at end-of-month. Modelling expenditure transactions on a daily basis, while certainly bringing a higher level of fidelity to the model, is not entirely feasible as there are 100’s of thousands of transactions that would need to be processed. Autoregressive moving average (ARMA) models are frequently used to model linear dynamic structures, to indicate linear relationships among lagged variables, and to serve as models for linear forecasting due to their flexibility in approximating stationary processes. For the FOREX model, each currency and account was modelled as either a pure autoregressive (AR) or pure moving average (MA) model or a combined ARMA model. In building each model, an initial approximation to the structure, through model identification, parameter estimation and diagnostic checking, was conducted through application of Brockwell and Davis’s ITSM (Interactive Time Series Modelling) software [22]. The final model parameter specification was conducted via maximum likelihood estimation within the FOREX model. In this section, a complete analysis is presented of the USD NP account with detailed analyses of the other currencies and accounts provided in Annex A. It assumes that the reader has some knowledge of time series processes including their prediction and validation. Two excellent references for further reading are the Brockwell and Davis text that accompanies a student version of ITSM [23] and a text on nonlinear time series by Fan and Yao [24].

3.1

An MA(4) Model for the USD National Procurement Account

Following the notation of Brockwell and Davis, letting X1 , . . . , Xn be n observations from a discrete time series, we define a zero-mean ARMA(p, q) process of order p and q by p

q

i=1

j=1

Xt − ∑ φi Xt−i = Zt + ∑ θ j Zt− j ,

(2)

where φ1 , φ2 , . . . , φ p and θ1 , θ2 , . . . , θq are real constants called autoregressive and moving average coefficients respectively. The right-hand side of (2) models the disturbance (stochastic) part as a linear combination of zero-mean, uncorrelated random variables or a zeromean Gaussian white noise process, Zt ∼ W N(0, σ 2 ). The ARMA(p, q) process is stationary if all the solutions of φ1 z+φ2 z2 +. . .+φ p z p = 1, with complex z, are outside the unit circle, |z| = 1. The process is said to be invertible if solutions of 1 + θ1 z + θ2 z + . . . + θq zq = 0 are also outside the unit circle. Box, Jenkins and Reinsel [25] describe the pioneering BoxJenkins methodology for the selection of appropriate ARMA models. The goal here is to determine the AR-order p and the MA-order q and to estimate the AR coefficients φ j , the MA coefficients θ j , and the variance, σ 2 , of the white noise process.

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3.1.1

USD NP: Transforming the Series

The objective of transforming the data is to produce a series with no apparent trend or seasonality and no apparent deviation from stationarity. The Box-Cox transformation [26] is a useful family of transformations where the variable X is transformed by (X λ − 1)/λ , where λ is the transformation parameter. For λ = 0, the natural log of the data is taken instead of using the above formula. This has the effect of re-scaling the data to “pull-in” the outliers and in effect, transform a multiplicative model into an additive one. Figure 7 (a) describes the original USD national procurement series. The series is of length 84 with sample mean and variance $18.06M and 2.6685 × 1014 respectively. All values are positive and none are zero, therefore no adjustments need be made before applying a log transformation. The process to remove seasonality is known as seasonal adjustment. Following the procedure of reference [25], we introduce the backshift operator B defined by ∗ B j Xt∗ = Xt− j,

(3)

where (∗ ) denotes the original series. Therefore, differencing a seasonal time series (Xt∗ ) with a period of 12 months would entail an operation of ∗ . (1 − B12 )Xt∗ = Xt∗ − Xt−12

To help identify any trend and/or seasonality, the sample autocorrelation function (ACF) ˆ plot, known as a correlogram, ρ(h) for lag-h, is used to observe the degree of dependence in the data. For example, the USD NP pre-transformed ACF, shown in Figure 7 (b), clearly shows (a) a decreasing trend and, (b) a seasonal component with period 12 consisting of the last payment in the fiscal year (sum of expenditures in periods 12–15). The two horizontal √ dashed lines are the 5% (two-sided) rejection bands defined by ±1.96/ n, where n is the sample size. The analysis proceeds by first performing a log transformation followed by a differencing at lag-12 and lag-1 according to equation (3), (1 − B)(1 − B12 ) ln Xt∗ = (1 − B − B12 + B13 ) ln Xt∗

∗ ∗ ∗ = ln Xt∗ − ln Xt−1 − ln Xt−12 + ln Xt−13 .

(4)

As a final step, once the apparent deviations from stationarity have been removed, the sample mean, μ, is subtracted from each observation to which we then fit a zero-mean stationary model [23], i.e., ∗ ∗ ∗ − ln Xt−12 + ln Xt−13 −μ. (5) Xt = ln Xt∗ − ln Xt−1 Equation (5) is plotted in Figure 7(d) as the log transformed, twice differenced and mean subtracted USD NP time series, Xt .

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Dollars CAD

Apr 02

Apr 04 Date

Apr 06

Apr 06

5 10 15 20 25 30 35 40 Lag

0

5 10 15 20 25 30 35 40 Lag

e ACF of Transformed USD NP

0

b ACF of USD NP

0

5 10 15 20 25 30 35 40 Lag

h ACF of USD NP Rescaled Residuals 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

0 2

0.1

0.2

0.3

0.4

0.5

0.6

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

5 10 15 20 25 30 35 40 Lag

5 10 15 20 25 30 35 40 Lag

1

0

1

i Histogram of Rescaled Residuals

0

f PACF of Transformed USD NP

0

c PACF of USD NP

2

Figure 7: (a)Time plot of monthly USD NP liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-12 and lag-1 differenced series. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from MA(4) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals.

Apr 02 Apr 04 Date

g USD NP Rescaled Residuals

Apr 02 Apr 04 Date

Apr 06

a USD NP Account

d Transformed USD NP

May 00

1.5 1 0.5 0 0.5 1 1.5

May 00

2

1

0

1

2

1.2  10 1  108 8  107 6  107 4  107 2  107 0 Apr 00

8

Density

3.1.2

USD NP: Fitting the ARMA Model

Both ACF and PACF (Partial Autocorrelation Function) plots describe important information on the correlation structure of time series as well as aid in model identification. For example, the ACF cuts off at q for an MA(q) process and the PACF cuts off at p for an AR process. For an ARMA(p, q) process, both the ACF and PACF exhibit decay. The ACF and PACF of the transformed series (5) are plotted in Figure 7 (e) and 7 (f) respectively. Close examination of the Figures suggest that we may fit an MA(4), an AR(3) or some combina√ ˆ ˆ ˆ tion model. Note that |ρ(h)| or |π(h)| ≥ 1.96/ n for h = 3 and 4, where π(h) is the partial autocorrelation function. Akaike’s Information Criterion (AICC) [27], modified by Hurvich and Tsai [28] to remove bias, and built into the ITSM software by Brockwell and Davis9 was used to determine the values of p and q for the fitted models that minimize the AICC statistic of the form AICC(p, q) = −2(maximized log likelihood) + 2

2(p + q + 1)n , n− p−q−2

(6)

where the second term on the right-hand side of (6) introduces a penalty for increasing values for p and q, i.e., increasing model complexity. In determining the AICC for values of p and q, there may be several models with AICC values close together. These are considered competitive models. Selection among the various competitive candidates may be based on interpretation, simplicity (low p and/or q), or diagnostic checking. We choose the latter and apply the Ljung-Box test for randomness of the residuals of the series once values for p and q are chosen. The Ljung-Box test [29, 30] is based on the autocorrelation plot and is perhaps the most widely used test for white noise residuals. Instead of testing for randomness at each lag, it 2 tests the “overall” randomness based on a number of lags. If the test statistic QLB > χ1−α;h , 2 where χ1−α;h is the 1 − α quantile of the chi-square distribution with h degrees of freedom, 2 the hypothesis of randomness is rejected. If QLB < χ1−α;h , the residuals are defined by a random process and the values of p and q define the zero-mean stationary ARMA process. Using ITSM, models were created for the USD NP expenditure data with 0 ≤ (p, q) ≤ 5. The AICC, Ljung-Box (h = 20) and p-value statistics were generated and sorted by minimum AICC. The low Ljung-Box statistic and high p-value (> 0.05) is an indication that there is no structure in the data (uncorrelated) up to lag 20. Table 3 displays the prioritized statistics for the first 20 candidates. The optimum model, according to Table 3 and based on AICC, is an MA(4) with an AICC value of 153.15. The estimated MA-coefficients θ1 , . . . , θ4 are − 0.6662, 9 See

−0.3197,

−0.1345

and

+ 0.3788 .

[23], section 5.5.2 for a detailed discussion.

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The ratios θ j /SE(θ j ) for j = 1, . . . , 4 are − 5.531,

−2.265,

−0.9081

and

3.118 ,

where SE(θ j ) is the standard error of θ j . Values of |θ j /SE(θ j )| σt+1 × Φ−1 , .05 where rt+1 is the value of the return on day t + 1 and σt+1 is the square root of the GARCH variance. The term Φ−1 .05 returns the inverse of the standard normal cumulative distribution with a mean of zero and standard deviation of one at probability p = 0.05, i.e., −1.645 for a one-tailed test or −1.96 for a two-tailed test. Kupiec [52] demonstrated that verification of the accuracy of tail estimates becomes substantially more difficult as the cumulative probability estimate becomes smaller, hence most financial institutions choose 95% as the confidence level. Figure 12 (a, c, e) show the excessions for the three currency returns based on 95% confidence levels bands above and below the return series. The sequence of excessions, according to equation 41, returns a 1 on day t + 1 if the loss on that day exceeds the previous day’s loss at the 95th percentile. If there was no excession, the sequence returns a 0. Therefore, the sequence of 1’s and 0’s N , indicating when past excessions occurred. Figure 12 (b, forms a set over N days, {It+1 }t=1 d, f) describe the sequence series for the three currencies. Using the period based on the GARCH variance (1755 returns) and a 95% confidence level, the results of applying equation (40) to the sequence series of Figure 12 (b, d, f) are given in Table 7 Table 7: Type I testing results of number of excessions found for each currency using the GARCH model, a 50 day observation period and a 95% confidence level

Return CAD/USD CAD/GBP CAD/EUR

Excessions % 5.24 4.79 4.27

Z-Score 0.465 -0.411 -1.396

Accept Model?

Accept Model?

Two-Tailed Test

One-Tailed Test

Yes Yes Yes

Yes Yes Yes

Table 7 shows the percentage of returns in excess of the calculated 95% loss. For each and this must be clearly distinguished. For example, a VaR taken at the 1% confidence level of the return distribution my result in 2 or 3 excessions. The validity of model through backtesting must then be satisfied at, say, the 5% level of significance.

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currency, a two-tailed Type I error test was used. Rejecting the model implies that we can be 95% certain that a valid model was not erroneously rejected. As can be seen, the volatility model was accepted as valid for all three currency returns.

a CADUSD 2

1

1

Sequence

Percentage Change

b CADUSD

2

3

0 1

0 1

2 3 99

00

01 02 03 04 Start of Fiscal Year

05

06

99

c CADGBP 2

01 02 03 04 Start of Fiscal Year

05

06

05

06

05

06

d CADGBP

2

3

1

1

Sequence

Percentage Change

00

0 1

0 1

2 3 99

00

01 02 03 04 Start of Fiscal Year

05

06

99

e CADEUR 2

01 02 03 04 Start of Fiscal Year f CADEUR

2

3

1

1

Sequence

Percentage Change

00

0 1

0 1

2 3 99

00

01 02 03 04 Start of Fiscal Year

05

06

99

00

01 02 03 04 Start of Fiscal Year

Figure 12: BackTesting GARCH models at the 95% confidence level

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4.5.2

Backtesting the GARCH models - Conditional Coverage Testing

To further confirm the analysis, we need to jointly test if the excessions are independently distributed over time and if the average number is correct. If the excessions exhibit some form of “clustering”, then the GARCH model may fail to capture the gain/loss variability under certain conditions. Christoffersen [50, 51] has devised a conditional coverage test to jointly test for independence and correct coverage   ˜ 1 ) ∼ χ22 . (42) LRcc = −2 L(p)/L(Π In (42), the numerator defines the likelihood function for unconditional coverage, i.e., N

L(p) = ∏(1 − p)1−It+1 pIt+1 = (1 − p)N0 pN1 ,

(43)

t=1

where N, p and It+1 were defined previously, and N0 and N1 are the number of 0’s and 1’s in the sequence. The denominator defines independence testing and gives us the ability to reject a model with clustered excessions. This term defines the likelihood function of a first-order Markov process as T01 T11 (1 − π11 )T10 π11 , (44) L(Π1 ) = (1 − π01 )T00 π01 where Ti j , i, j = 0, 1 is the number of observations with j following i; and Π1 is the transition probability matrix  1 − π01 π01 , Π1 = 1 − π11 π11 where the transition probabilities, π01 and π11 , specify the probability of tomorrow being an excession if today is not, i.e., π01 = Pr(It = 0 and It+1 = 1) and the probability of tomorrow being an excession if today is also an excession, i.e., π11 = Pr(It = 1 and It+1 = 1), respectively. In (42), as the number of observations, N, goes to infinity, the test will be distributed as a χ 2 with two degrees of freedom. Choosing a significance level of 10%, allows us to reject the model if the statistic is greater than 4.605220 . Table 8 gives the test elements and the test statistic, LRcc . For each currency LRcc < χ20.10 , and therefore we can confirm acceptance of each GARCH model.

20 In

the quest to manage risk, Type II errors can be very costly and increasing the significance level implies larger Type I errors but smaller Type II. Typically, 1%, 5%, or 10% are used.

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Table 8: Conditional Coverage testing for each currency

4.5.3

Test Element

CAD/USD

CAD/GBP

CAD/EUR

T0 T1 T00 T01 T10 T11 π01 π11 LRcc

1663 92 1576 87 86 6 0.0523 0.0652 -5.2929

1671 84 1589 82 81 3 0.0491 0.0357 -5.4593

1680 75 1608 72 71 4 0.0429 0.0533 -3.9794

Accept or Reject Model

Accept

Accept

Accept

In-Sample Check on the Autocorrelations

A final validation of the GARCH variance model relies on an in-sample check [53] where it is enough to determine whether or not the standardized returns (standardized by the GARCH variance) have no systematic autocorrelation patterns as do the squared returns, which show strong autocorrelation for short lags and decrease as the lag order increases. Figure 13 illustrates the effectiveness of the GARCH model in removing systematic patterns of autocorrelation from the squared returns of the three currencies. For CAD/USD (13 (a)), all strong positive autocorrelations have been removed as the standardized square returns fall within ± two standard errors. The CAD/GBP and CAD/EUR plots also show less autocorrelation albeit not as obvious as CAD/USD. The poorest result is shown by CAD/EUR where there are nine excessions (still < 5%) beyond the ± two standard error bands. There are other diagnostic procedures that could have been explored in validating the return distribution models. In addition to backtesting the standard risk quantile and the expected tail loss, we could also have been interested in backtesting how well the model predicts the entire distribution of losses and gains. This would have provided even stronger evidence for accepting or rejecting the model. However, in risk management it is usually more important that the model forecasts the tail of the distribution correctly and not its interior, which characterizes only small return fluctuations.

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a CADUSD 0.2 0.16 0.12

Autocorrelation

0.08 0.04 0 0.04 0.08 Squared Returns

0.12 Standardized Square Returns

0.16

1

10

20

30

40

50 Lag

60

70

80

90

100

80

90

100

80

90

100

b CADGBP 0.2 0.16 0.12

Autocorrelation

0.08 0.04 0 0.04 0.08 Squared Returns

0.12 Standardized Square Returns

0.16

1

10

20

30

40

50 Lag

60

70

c CADEUR 0.2 0.16 0.12

Autocorrelation

0.08 0.04 0 0.04 0.08 Squared Returns

0.12 Standardized Square Returns

0.16

1

10

20

30

40

50 Lag

60

70

Figure 13: Autocorrelation of CAD/USD, GBP and EUR standardized squared returns (dashed line) and squared returns (solid line).

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5

The VaR Model for DND

In the previous sections, models were built and validated for forecasting national procurement and capital expenditures as well as the conditional variances for the three currencies of interest. In this section, all the models are assembled to build a final VaR model for DND that allows a user to forecast the maximum expected loss from adverse exchange rate fluctuations over the budget year.

5.1

Background: VaR Methodologies

With the explosive growth of the derivatives market and the many well-publicized losses (e.g., Barings), regulatory standards have come under close scrutiny with the landmark Basel Capital Accord of 1988 providing the first steps towards firm risk management standards [49]. The accord sets minimum capital requirements that must be met by commercial banks to guard against credit and market risks. Therefore, being able to better quantify the risk becomes crucial for institutions to promote sound banking practices. Since its inception, VaR was being promoted as a good risk management practice and banks were free to develop and use their own VaR risk management models to publish, in an easy to understand number, their downside risk due to financial market variables. The Basel Committee, therefore, adopted VaR as the standard method to measure the market risk of a portfolio of financial assets. Consequently, VaR models have been implemented throughout the financial industry and the literature is full of VaR methodologies of varying degree of complexity, but no unanimous agreement on which VaR model should be preferred21 . The answer seems to be a function of the portfolio and the data set used to estimate the parameters [55, 56]. Notwithstanding the variability of methodologies, it is generally accepted that there are three basic methods for calculating VaR: Variance-Covariance, Historical Simulation and Monte Carlo [12, 57, 58, 59, 60]. Each has its advantages and disadvantages, and together can provide a comprehensive picture of risk. Without going into great detail, the Variance-Covariance method assumes percentage price changes are normally distributed, which in turn allows volatility to be expressed in terms of the standard deviation. Therefore, for a single asset portfolio, the negative percentage price change corresponding to 1.65 standard deviations provides a 95% confidence that a downward price change will not exceed this number, the VaR, in a certain period of time. For a portfolio of assets, the correlation between assets needs to be taken into consideration. The VaR of a portfolio is simply the volatility of the portfolio. Although the calculations are fast and simple, they are less accurate for skewed distributions such as currency returns. Simulation is the preferred method for calculating VaR as it is appropriate for all financial instruments, linear as well as non-linear. In terms of representing actual market behaviour, 21 The

GloriaMundi.org website is a comprehensive source for VaR literature as well as providing other VaR resources [54].

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41

historical simulation is markedly superior to the Variance-Covariance method. Historical simulation calculates the profits and losses for each day based solely on historical price changes. For this it requires a significant amount of daily rate history so that it captures all possible values of the historical distribution of price returns, in particular the tail events critical to VaR calculations. However, sampling too far back could be problematic if the data is irrelevant to current conditions. Furthermore, as pointed out by Christoffersen [47], if the sample size is too large, the most recent observations, which are most relevant for tomorrow’s distribution, will carry little weight when calculating the VaR which will tend to smooth-out over time. Bodoukh et al. proposed an adjustment to the simple historical simulation technique by assigning weights to the returns whereby recent events were assigned more weight and past events less weight [61]. Regardless, there are issues with historical simulation, not least of which both methods still require a significant amount of historical data, which is not easily obtainable, to be a viable choice for risk management [47, 62, 63]. Monte Carlo simulation, on the other hand, involves artificially generating a large set of price changes from which a VaR is calculated. A large set of random numbers are drawn, usually from, but not limited to, a standard normal distribution to generate random hypothetical scenarios. Collecting the results and sorting them into percentiles, the VaR may be determined from the 5th percentile result. Unlike historical simulation which makes no distributional assumptions, a Monte Carlo simulation will quantify the tail risk accurately only if the scenarios are generated from the appropriate distribution. While historical simulation imposes no structure on the distribution of returns (model-free) and Monte Carlo simulation assumes a distribution of standardized returns, Barone-Adesi et al. proposed a combined approach known as Filtered Historical Simulation (FHS) which takes into account the changes in past and current volatilities of historical returns with the least number of assumptions about the statistical properties of future price changes [64, 65]. Not without limitations, e.g., accounting for time-varying correlations in historical data and the appropriate length of historical sample period [63], the FHS methodology shows the most promise and its implementation forms the VaR modelling structure for this study.

5.2

Filtered Historical Simulation For Returns

Filtered Historical Simulation (FHS)22 , is non-parametric in the sense that the simulation imposes no structure on the distribution of returns. There is no need to make any distributional assumptions, whether normal or t˜(d), on the standardized returns of the currency exchanges. Following [66], we start the process by considering the set of past returns {rt+1−τ : τ = 1, 2, . . . , T } where T = 1756, to account for the period between 01 April 1999 and 31 March 0623 . From (36), we can write the one-day ahead return as the product of the estimated 22 For further reading on the FHS process, Christoffersen [66] has written a simple, easy-to-follow procedure for implementation. 23 While 01 April 1999 – 31 March 2006 is the time period used in this study to develop the initial modelling conditions, the model will always be evolving in time. When a new month elapses and actual expenditures and rates are available, the user must update the model on the supplied forms and re-optimize. See Annex B

42

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standard deviation and the error term, i.e., rt+1 = σt+1 z t+1 ,

(45)

where σt+1 is defined through the GARCH variance equation (37), already calibrated using seven years of historical data, to be 1/2  , σt+1 = ω + αrt2 + β σt2

(46)

with parameters (ω, α, β ) defined in Table 6. Using the data set {rt+1−τ : τ = 1, 2, . . . , T } we can now estimate the model parameters and calculate the set of realized standardized returns, {ˆzt+1−τ : τ = 1, 2, . . . , T }, defined by zˆ t+1−τ = rt+1−τ /σt+1−τ ,

for τ = 1, 2, . . . , T

(47)

Therefore, given actual returns up to time t (31 March 2006), we can immediately evaluate the GARCH variance and equation (46) for time t + 1 (01 April 2006). To compute hypothetical returns for tomorrow, 01 April 2006, we draw with replacement from the set of past standardized residuals, {ˆzt+1−τ : τ = 1, 2, . . . , T }, through sampling a discrete uniform distribution of elements consisting of the τ = 1, 2, . . . , T standardized returns defined by (47). The estimated exchange rate, Pt+1 , on 01 April 2006 is then defined through (32) to be Pt+1 = e rt+1 Pt . (48) To illustrate the process for the next 264 trading days ending 31 March 200724 , consider the algorithm described in Figure 15. The return and conditional variance on the last day of actual data (31 March 2006) starts the simulation. After each 22-day trading period, the estimated exchange rate at that time is captured for each iteration and used in a subsequent calculation for the VaR based on equation (1). As depicted in Figure 14, days 22, 44, etc., correspond to 30 April 2006, 31 May 2006, etc., respectively. Therefore, the end result is 10,000 sequences of hypothetical daily returns for day t + 1 through day t + 264. Figure 14: Extraction of monthly exchange rates



P1,1 P1,2 · · · P1,22 · · · P1,44 · · · P1,264



P2,1 P2,2 · · · P2,22 · · · P2,44 · · · P2,264

. .. .. ..

.. . . .

. . . .

. .. .. ..

.

P 10k,1 P10k,2 · · · P10k,22 · · · P10k,44 · · · P10k,264

24 12

↓

↓

VaR

VaR

···



















↓ VaR

months @ 22 trading days per month

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43

44

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⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2 )→r (ˆz2,2 , σ2,2 2,2 → P2,2 .. . .. .

2 )→r (ˆz2,1 , σ2,1 2,1 → P2,1 .. . .. .

··· .. . .. .

···

2 (ˆz2,264 , σ2,264 ) → r2,264 → P2,264 .. . .. .

2 (ˆz1,264 , σ1,264 ) → r1,264 → P1,264

2 2 2 (ˆz10k,1 , σ10k,1 ) → r10k,1 → P10k,1 (ˆz10k,2 , σ10k,2 ) → r10k,2 → P10k,2 · · · (ˆz10k,264 , σ10k,264 ) → r10k,264 → P10k,264

2 )→r (ˆz1,2 , σ1,2 1,2 → P1,2

→

2 )→r (ˆz1,1 , σ1,1 1,1 → P1,1

Days

6. Repeat Steps 1 through 5 for each of 10,000 iterations.

5. Repeat Steps 1 through 4 for each of 264 trading days per year, i.e., t + 2, t + 3,. . . , t + 264.

4. Calculate exchange rate for day t + 1.

3. Calculate return for day t + 1.

2. Draw with replacement from set of past standardized residuals.

1. Calculate GARCH variance for day t + 1 based on return and variance for day t.

2 (r31/03 , σ31/03 )⇒

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

↓

Iterations

Figure 15: The FHS process for returns

5.3

Filtered Historical Simulation For Expenditures

As the FHS for returns sampled a set of past standardized residuals, so does the FHS for expenditures sample the set of past residuals specified by each ARMA model. For example, let {Zˆt+1−τ : τ = 1, 2, . . . , M} be the set of past residuals for the USD national procurement account where the residual at time t is defined by (7) to be Zt = Xt + 0.6976Zt−1 + 0.3623Zt−2 − 0.3161Zt−4 ,

(49)

where Xt is the zero-mean stationary model defined by (5). The process to determine the estimated expenditure is simpler then that for the returns as there are no intermediate calculations. Simply choosing a τ from 1, 2, . . . , M will yield the current residual as input to equation (8) for the estimated expenditure, where all other values are found as linear combinations of past expenditures and past residuals. Also, rather then calculating the expenditure on a daily basis, since the set of Zˆt+1−τ is based on monthly data, the calculation of expenditures is also done monthly for each iteration. Therefore, for the next 264 trading days, an expenditure is matched to an exchange rate as in Figure 14, i.e., every 22 trading days. Figure 16 describes the process whose end result is 10,000 sequences of hypothetical expenditures for day t + 22, t + 44, . . . , t + 264. Figure 16: The FHS process for expenditures

Iterations ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

Days

↓

→

∗ Zˆ 1,22 → X1,22 ∗ Zˆ 2,22 → X2,22

∗ Zˆ 1,44 → X1,44 ∗ Zˆ 2,44 → X2,44

.. . .. .

.. . .. .

··· ··· .. . .. .

∗ Zˆ 1,264 → X1,264 ∗ Zˆ 2,264 → X2,264

.. . .. .

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ∗ ⎩ Zˆ 10k,22 → X ∗ ˆ ˆ 10k,22 Z10k,44 → X10k,44 · · · Z10k,264 → X10k,264

1. Draw with replacement from set of past residuals. 2. Calculate expenditure for day t + 22. 3. Repeat Steps 1 and 2 for each of 22 trading days per month for 12 months, i.e., t + 44, t + 66,. . . , t + 264. 4. Repeat Steps 1 through 3 for each of 10,000 iterations.

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5.4

Building the VaR Model

In section 3, expenditure models were built as linear combinations of past expenditures and current and previous values of white noise disturbance terms. Changing the notation slightly to fit equation (1), the forecast expenditures are ∗ Ec,k a,t+22n = fc,a (Zˆt+22n , θi Zt−i , φ j Xt− j) ,

(50)

where the subscripts c, a denote the currency and account respectively; k = 1, . . . , 10, 000, the number of iterations in the FHS process; n = 1, . . . , 12, the number of months; i = 1, . . . , q, and j = 1, . . . , p, the number of moving average and autoregressive terms respectively. Similarly, based on the results of section 4, the forecast exchange rates can be written as pkc,t+22n = fc (ˆzt+22n , σt+22n , rt+22n ) ,

(51)

where c is the currency and k and n were previously defined. Given that the budget rates are also forecast on a monthly basis, i.e., bt+22n , we can write the relationship that defines the overall VaR model as a variation on equation (1) 12  , Vc,k a, n = Ec,k a,t+22n × (bc,t+22n − pkc,t+22n ) , k = 1, . . . , 10, 000 n=1

(52)

where Vc,k a, n is the variance for currency c, account a, iteration k and month n, and b, the budget rate, is fixed for each n. The VaR is therefore defined by the 5th percentile of (52), i.e.,   k VaRc,0.05 a, n = Vc, a, n , k = 1, . . . , 10, 000

0.05

.

(53)

The forecast budget rates as of 09 March 200625 are found in Table 9 where n = 1, 2, . . . , 12 refer to 30 April 2006, 31 May 2006, . . . , 31 March 2007 respectively [67]. We conclude this section and confirm that equation (53) finalizes our initial tasking where we have applied VaR techniques to forecast the potential impact of foreign currency fluctuations and “... determine the maximum expected loss from adverse exchange rate fluctuations over the remaining periods of the budget year”. In the following sections, equation (53) is applied to the USD NP account to provide a forecast of the 5th percentile VaR and what additional expenditures may need to be requested so that budgets can be more successfully managed.

25 As

46

of writing, the model was developed for historical data valid to 31 March 2006.

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Table 9: Forecast budget rates as of 09 March 2006

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n

CAD/USD

CAD/GBP

CAD/EUR

1 2 3 4 5 6 7 8 9 10 11 12

1.1430 1.1440 1.1440 1.1450 1.1460 1.1470 1.1490 1.1520 1.1540 1.1570 1.1580 1.1590

1.9840 1.9860 2.0000 2.0050 2.0060 2.0070 2.0030 2.0270 2.0280 2.0620 2.0800 2.0830

1.3840 1.3860 1.3870 1.3990 1.3990 1.4000 1.4090 1.4100 1.4100 1.4210 1.4210 1.4220

47

6 6.1

Simulation Results: Running the FOREX Model The Simulation

The methodologies described in the preceding sections are combined into a risk simulation called the FOREX model. The simulation uses Excel with a special add-on software called @Risk [68] to conduct filtered historical simulation with Latin Hypercube stratified sampling to ensure good representation of actual variability. The simulation forecasts per month for a 12 month period, which may not coincide with a fiscal or calendar year. Each currency expenditure account is forecasted per month for the following 12 months (starting from the last month of actual data) using a uniform distribution to sample the expenditure data set as shown in Figure 16. Each currency return is forecasted per day for 22 trading days per month for the following 12 months (matching expenditure 12 month period) using a uniform distribution to sample the data set of standardized returns as shown in Figure 15. For each iteration, k, currency, c and account, a, the simulation goes through the following steps: 1. For the next 12 months (n = 1, . . . , 12), draw with replacement from the set of past expenditure residuals (see section 5.3); 2. Convert back to forecasted expenditure for each month ({Ec,k a,t+22n } (see equation (8) k for example, where Xt∗ is replaced by the expression for Ec,a,... ); 3. For the next 12 months, draw with replacement from the set of past standardized residuals for returns (See section 5.2); 4. Generate GARCH(1,1) model and forecast (next day) return and convert to exchange rate; 5. Collect end-of-month (every 22 trading days) exchange rate ({pkc,t+22n }); 6. Subtract rate from forecasted budget rate ({bc,t+22n }); 7. Multiply by forecasted expenditure (item 2) to get variance ({Vc,k a, n }). 8. Collect 0th, 5th, 50th and 95th and 100th percentile results.

6.2

Forecasting Value-at-Risk

The simulation was run for 10,000 iterations or until convergence (see section 6.5) signified by a tolerance of less than 1.5% difference in each percentile value specified by the software (0% to 100% in 5% steps). Tolerances were also applied to the mean and standard deviation of each output. Convergence was tested every 100 iterations. The results per month for six months ahead (relative to 31 March 2006) are given in Table 10, partitioned by 5th (VaR), 50th (median) and 0th (maximum loss) percentiles of a distribution of 10,000 results of

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equation (52)26 . For example, Figure 17 illustrates the output for CAD/USD forecasted national procurement and capital transactions for one month ahead (30 April 2006). The red areas to the left and right of average correspond to the lower and upper 5% of the results respectively. Since we are mainly interested in the VaR, the value at the 5th percentile is reported in the upper portion of Table 10. The median (50th percentile) of the distribution, which could be a loss or a gain, is reported in the middle portion of the table. Values close to zero imply a budget rate that is close to the forecasted exchange rate. The maximum loss (0th percentile) is reported at the bottom of the table. In Figure 17, each distribution is skewed left with a long tail that is sparsely populated. Clearly extreme values can be reported as, unlike historical simulation, FHS can forecast large losses even if a large loss was never recorded in the historical data set. For example, from equation (47), if the sampling chooses a large negative zˆ and combines it with a highvariance day in the simulation period, the resulting hypothetical loss will be large [66]. Table 10: 5th, 50th and 0th percentiles for national procurement and capital accounts

5th Percentile Loss (Value-at-Risk) Months Ahead 1 2 3 4 5 6

NP Account (Dollars CAD) USD GBP EUR -397,018 -490,649 -354,408 -440,661 -679,247 -367,856

-1,101,114 -893,218 -1,674,138 -1,635,166 -953,256 -1,466,679

-102,336 -236,554 -87,780 -61,347 -82,967 -46,356

Capital Account (Dollars CAD) USD GBP EUR -3,287,886 -2,726,042 -1,932,770 -2,086,494 -3,323,651 -3,818,306

-534,952 -583,373 -582,379 -636,433 -645,523 -734,837

-116,195 -170,770 -202,268 -193,508 -192,919 -245,487

-66,323 -50,581 -26,185 -16,069 -11,546 -6,758

91 231 354 1,888 2,111 1,910

50th Percentile Gain/Loss 1 2 3 4 5 6

-89,850 -79,978 -45,732 -43,616 -43,565 -12,965

-120,372 -69,590 -69,741 -44,622 -19,744 -15,887

4,266 9,350 3,817 5,251 6,502 2,359

-444,067 -271,615 -152,612 -111,726 -101,714 -66,771

0th Percentile (Expected Maximum Loss) 1 2 3 4 5 6 26 In

-2,142,328 -2,594,218 -2,558,785 -3,274,104 -7,456,064 -3,440,768

-7,641,469 -9,920,118 -19,015,012 -30,189,302 -13,261,287 -24,513,950

-1,740,517 -3,547,616 -896,529 -927,661 -1,051,550 -1,404,730

-25,982,948 -27,577,604 -22,292,008 -32,797,102 -47,655,204 -82,265,440

-1,831,942 -2,587,456 -3,492,113 -3,999,430 -6,292,994 -6,389,739

-2,923,322 -7,728,662 -7,055,717 -7,903,703 -33,362,848 -10,976,101

reporting VaR results, it was felt that losses were more important then gains for a VaR model.

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49

6.3

Forecasting Expenditures

Expenditures derived from FHS sampling were also an output of the simulation. The results per month for six months ahead (relative to 31 March 2006) are given in Table 12, partitioned by 0th (minimum expenditure), 5th, 50th (median), 95th and 100th (maximum expenditure) percentiles of a distribution of 10,000 results based on the algorithm depicted in Figure 16. As a result of the limiting behaviour of Euro transactions (zero dollar transactions in certain months), the minimum and 5th percentile results are either zero or a few dollars. At the other extreme, the 95th percentile values, while reasonable expenses, are significantly less than the 100th percentile (maximum) results implying a distribution strongly skewed right. Figure 18 illustrates the cumulative distribution of expenditures for USD forecasted national procurement and capital transactions for one month ahead (30 April 2006). Also shown as an insert to each figure are the frequency distributions for each account. The skewness of the distributions is clearly evident in each plot as approximately 70% of the total forecasted dollar expenditures take place in the last 10% of the cumulative distribution as a result of the extreme values specified at the far right in each frequency distribution. 6.3.1

Forecasted Expenditure Validation

Depending on the level of differencing required in the construction of each ARMA model, the number of points available for FHS could range from a low of 52 for EUR capital to a high of 85 for USD capital. In comparison, the standardized return sampling for the currency GARCH models consisted of 1757 points and consequently a much lower standard error. Notwithstanding the small sample size, Table 11 displays the results of “out-of-sample” testing of expenditure forecasting accuracy. In other words, data prior to 1 April 2006 was used to fit the model (the fit period), and data post 31 March 2006 (the test period) was reserved to assess the model’s forecasting accuracy. Raw FMAS data for months 1 through 4 (April 06 through July 06) was filtered according to rules established in section 2.2.1. For each actual expenditure, the corresponding forecasted percentile was interpolated from the forecasted expenditure cumulative distributions. Inspection of Table 11 for national procurement shows the actuals randomly distributed as expected, with each currency showing values on both sides of the median. For capital expenditures, randomness is also experienced, however, the very nature of capital introduces a complexity to the model. Capital costs are based on contracts which must introduce a new capability or modernize an existing capability rather then maintaining an existing capability, as in national procurement. The annual (1 April - 31 March) capital spending pattern is observed to be non-linear with increasing trend in the monthly frequency of payments and their corresponding magnitude as the fiscal year progresses. This occurs because capital contracts are of a fixed duration often with flexible payment and delivery schedules. It is observed that large payments occur in the final quarter of the fiscal year leaving a significantly smaller payment for the first quarter of the new fiscal year as the cycle repeats itself.

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-2 6, 00 -2 0,0 4, 0 47 0 -2 0,0 2, 0 94 0 -2 0,0 1, 0 41 0 -1 0,0 9, 0 88 0 -1 0,0 8, 0 35 0 -1 0,0 6, 0 82 0 -1 0,0 5, 0 29 0 -1 0,0 3, 0 76 0 -1 0,0 2, 0 23 0 -1 0,0 0, 0 70 0 0 -9 ,00 ,1 0 70 -7 ,00 ,6 0 40 -6 ,00 ,1 0 10 -4 ,00 ,5 0 80 -3 ,00 ,0 0 50 -1 ,00 ,5 0 20 ,0 00 10 , 1, 000 54 0, 3, 000 07 0, 4, 000 60 0, 6, 000 13 0, 7, 000 66 0, 9, 000 19 0 10 ,00 ,7 0 20 12 ,00 ,2 0 50 13 ,00 ,7 0 80 15 ,00 ,3 0 10 16 ,00 ,8 0 40 18 ,00 ,3 0 70 ,0 00

Frequency -2 ,2 00 -2 ,00 0 ,0 81 -1 ,20 0 ,9 62 -1 ,40 0 ,8 43 -1 ,60 0 ,7 24 -1 ,80 0 ,6 06 -1 ,00 0 ,4 87 -1 ,20 0 ,3 68 -1 ,40 0 ,2 49 -1 ,60 0 ,1 30 -1 ,80 0 ,0 12 ,0 -8 00 93 ,2 -7 00 74 ,4 -6 00 55 ,6 -5 00 36 ,8 -4 00 18 ,0 -2 00 99 ,2 -1 00 80 ,4 0 -6 0 1, 60 0 57 ,2 00 17 6, 00 29 0 4, 80 41 0 3, 60 53 0 2, 40 65 0 1, 20 77 0 0, 00 88 0 8, 8 0 1, 00 0 7, 60 0

Frequency 0.04

(a) CAD/USD National Procurement Account

0.03

0.03

0.02

0.02

0.01

0.01

0.00

Expenditure in Dollars CAD

0.09

(b) CAD/USD Capital Account

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Expenditure in Dollars CAD

Figure 17: Value-at-Risk distributions for CAD/USD national procurement (a) and capital accounts (b) for one month ahead from 31 March 2006. Red areas to left and right of average correspond to the lower and upper 5% of results respectively.

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For example, the Month 12 (March, FY 05/06) payout for the GBP capital account was $7,240,564; the Month 1 (April, FY 06/07) payout was $0. In seven years of data there was never a zero payout for GBP capital transactions and therefore was never forecastable by the MA(1) model for the GBP capital account (section 3.4). In the case of EUR Month 2, a zero was built into the MA(1) model for the Euro capital account and was therefore forecastable, albeit at a low probability of occurring which would necessarily increase as the frequency of zero payments increase as well. The results of Table 11 provide a useful diagnostic of the expenditure models. There are no observable trends in the percentiles with a reasonable distribution of values on both sides of the median. It is noted however, that the observation of zero payments will need to be monitored for model adjustments if necessary. Increasing the size of the dataset would also improve the forecasting ability of the model. Going from monthly to weekly payments would solve two problems: removing outliers that are aggregates of multiple transactions in any month, but usually end-of-year; and, increase the dataset by a factor of four thereby opening the door to volatility forecasting models which introduce economic factors missing from the purely statistical time series approach. Table 11: Results of interpolation of actual expenditures to the forecasted cumulative distribution

National Procurement Account Months Ahead 1 2 3 4

USD Actual Value

Perc.

GBP Actual Value

Perc.

EUR Actual Value

Perc.

2,516,656 6,742,831 4,629,105 6,504,982

16 78 77 84

711,779 5,280,800 4,064,819 643,727

16 81 56 11

494,060 1,335,048 698,761 741,535

18 41 62 77

0 18 46 7

105,241 0 519,777 1,518,540

22 0.01 55 78

Capital Account 1 2 3 4

52

4,041,456 17,887,159 9,385,801 16,999,278

3 51 42 65

0 492,625 1,303,910 92,995

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Table 12: Expenditure percentiles for national procurement and capital accounts

0th Percentile (Minimum) Expenditure Months Ahead 1 2 3 4 5 6

NP Account (Dollars CAD) USD GBP EUR 1,232,908 862,642 592,605 680,984 713,970 360,323

30,442 18,376 17,387 15,727 10,670 5,017

110,023 48,140 11,692 3,328 5,206 800

Capital Account (Dollars CAD) USD GBP EUR 603,875 433,278 155,167 130,720 117,495 97,992

385 125 120 77 89 144

0 0 0 0 0 0

79,650 76,136 76,354 74,118 71,751 73,737

2 2 2 2 2 2

1,364,431 1,383,435 1,424,181 1,436,992 1,453,091 1,499,062

400,237 419,100 420,004 425,743 438,549 444,578

7,145,899 7,656,412 7,901,995 8,270,300 8,631,826 8,997,945

11,386,293 11,503,094 11,321,159 12,030,179 12,575,327 13,247,488

5th Percentile Expenditure 1 2 3 4 5 6

1,714,478 1,723,870 1,151,482 1,397,959 1,894,559 916,752

219,100 168,995 331,166 315,779 182,666 230,933

269,234 269,294 84,739 48,678 55,443 15,006

6,195,646 4,649,158 2,810,817 2,470,009 3,285,409 3,863,909

50th Percentile Expenditure 1 2 3 4 5 6

4,034,063 4,362,163 2,982,829 3,613,120 5,189,283 2,626,121

2,417,874 1,886,251 3,389,361 3,289,827 1,894,090 2,542,889

1,232,609 1,687,345 515,293 343,161 383,311 157,686

22,401,024 17,467,242 11,241,523 11,566,225 16,668,820 20,010,132

95th Percentile Expenditure 1 2 3 4 5 6

13,796,353 14,300,339 9,832,291 11,672,454 16,909,999 8,744,563

13,599,761 14,233,591 22,642,940 22,162,491 12,901,706 18,843,171

4,766,372 7,641,043 2,386,653 1,688,051 1,993,589 1,345,343

100,081,232 83,940,760 56,011,271 61,217,996 92,993,288 106,419,136

100th Percentile (Maximum) Expenditure 1 2 3 4 5 6

22,799,958 43,872,308 30,559,678 32,307,882 65,280,100 33,287,076

DRDC CORA TR 2006–23

41,026,340 47,015,188 102,440,064 117,697,096 65,027,760 172,354,816

6,640,927 25,298,498 9,111,101 8,891,987 11,075,930 9,594,034

316,566,368 292,127,392 273,505,952 436,374,208 660,167,360 762,068,928

9,723,584 13,576,950 18,569,404 19,411,846 30,375,230 30,224,586

26,519,748 39,482,452 51,768,920 62,358,400 75,570,240 109,070,248

53

60 0 13 ,00 0 ,4 24 26 ,00 0 ,2 48 39 ,00 0 ,0 72 51 ,00 0 ,8 96 64 ,00 0 ,7 20 77 ,00 0 ,5 44 90 ,00 0 ,3 6 10 8,0 00 3, 19 11 2,0 0 6, 01 0 12 6,0 0 8, 84 0 14 0,0 0 1, 66 0 15 4,0 0 4, 48 0 16 8,0 0 7, 31 0 18 2,0 0 0, 13 0 19 6,0 0 2, 96 0 20 0,0 0 5, 78 0 21 4,0 0 8, 60 0 23 8,0 0 1, 43 0 24 2,0 0 4, 25 0 6 , 25 00 7, 08 0 26 0,0 0 9, 90 0 28 4,0 0 2, 72 0 29 8,0 0 5, 55 0 30 2,0 0 8, 37 0 32 6,0 0 1, 20 0 0, 00 0

54 50%

40%

30%

20%

10%

Frequency

1, 23 0, 00 2, 09 0 2, 80 2, 95 0 5, 60 3, 81 0 8, 40 4, 68 0 1, 20 5, 54 0 4, 00 6, 40 0 6, 80 7, 26 0 9, 60 8, 13 0 2, 40 8, 99 0 5, 20 9, 85 0 8, 00 10 0 ,7 20 , 80 11 0 ,5 83 12 ,600 ,4 46 13 ,400 ,3 09 14 ,200 ,1 72 15 ,000 ,0 34 15 ,800 ,8 97 16 ,600 ,7 60 17 ,400 ,6 23 18 ,200 ,4 86 19 ,000 ,3 48 20 ,800 ,2 11 21 ,600 ,0 74 21 ,400 ,9 37 22 ,200 ,8 00 ,0 00

30%

20% 1, 23 2, 0,0 09 0 2, 2,8 0 95 0 3, 5,6 0 81 0 4, 8,4 0 68 0 5, 1,2 0 54 0 6, 4,0 0 40 0 7, 6,8 0 26 0 8, 9,6 0 13 0 8, 2,4 0 99 0 9, 5,2 0 8 10 58 00 ,7 ,0 11 20 00 ,5 ,8 12 83 00 ,4 ,60 13 46 0 ,3 ,4 14 09 00 ,1 ,2 15 72 00 ,0 ,0 15 34 00 ,8 ,80 16 97 0 ,7 ,6 17 60 00 ,6 ,4 18 23 00 ,4 ,2 19 86 00 ,3 ,0 20 48 00 ,2 ,8 21 11 00 ,0 ,6 21 74 00 ,9 ,4 22 37 00 ,8 ,20 00 0 ,0 00

Percentage 40%

Frequency

50%

60%

6 13 00, ,4 00 26 24, 0 ,2 00 39 48, 0 ,0 00 51 72, 0 ,8 00 64 96, 0 ,7 00 77 20, 0 ,5 00 90 44, 0 , 0 10 368 00 3, ,0 11 192 00 6, ,0 12 016 00 8, ,0 14 840 00 1, ,0 15 664 00 4, ,0 16 488 00 7, ,0 18 312 00 0, ,0 19 136 00 2, ,0 20 960 00 5, ,0 21 784 00 8, ,0 23 608 00 1, ,0 24 432 00 4, ,0 25 256 00 7, ,0 26 080 00 9, ,0 28 904 00 2, ,0 29 728 00 5, ,0 30 552 00 8, ,0 32 376 00 1, ,0 20 00 0, 00 0

Percentage

100%

(a) CAD/USD National Procurement Account

90%

80%

70%

60%

14.0% 12.0% 10.0% 8.0% 6.0% 4.0% 2.0% 0.0%

10%

Expenditure in Dollars CAD

0%

Expenditure in Dollars CAD

100% (b) CAD/USD Capital Account

90%

80%

70%

25.0%

20.0%

15.0%

10.0%

5.0%

0.0%

Expenditure in Dollars CAD

0%

Expenditure in Dollars CAD

Figure 18: Cumulative expenditure distribution for USD national procurement (a) and capital accounts (b) for one month ahead from 31 March 2006.

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6.4

Forecasting Performance for Currency Returns

There is really no reliable method to forecast exchange rates and we have not attempted to do so here. Models for exchange rate movements are largely driven by changes in macroeconomic factors like unexpected economic or political events, interest rates, the pattern of trade between one country and another and what is known as absolute purchasing power parity (PPP) which holds that goods-market arbitrage will tend to move the exchange rate to equalize prices between countries [69]. Currently, DND uses time series methods for short-term prediction of exchange rates [70]. Simple Autoregressive Integrated Moving Average (ARIMA) models attempt to isolate trends in past data to predict future values. While much simpler then economic models that rely on explanatory variables, they only rely on past data and ignore causal relations that influence future expectations. The FOREX VaR model is designed to forecast expected foreign exchange risk and not expected returns. Nevertheless, in calculating the VaR from equations (52, 53) a return distribution from the FHS process is given as a product of the sampled standardized return and the modelled GARCH variance as in equation (45). Figure 19 illustrates the return distribution of each currency exchange forecasted one month ahead from 31 March 2006. Table 13 displays the “out-of-sample” testing of return forecasting accuracy. Actual returns were calculated by applying equation (32) to the Bank of Canada rates for end-of-months: April-August 2006 inclusive [20]. For each actual return, the corresponding percentile was interpolated from the forecasted returns distribution. For example, the data for Figure 19 would be used to interpolate the one month ahead percentile from the actual value. As for expenditures, Table 13 shows the actual returns to be randomly distributed on both sides of the median. Table 13: Results of interpolation of actual returns to the forecasted cumulative distribution

Months Ahead 1 2 3 4 5

CAD/USD

CAD/GBP

CAD/EUR

Actual Value

Perc.

Actual Value

Perc.

Actual Value

Perc.

-0.0409 -0.0157 0.0110 0.0142 -0.0217

2 24 77 81 16

0.0055 0.0121 -0.0021 0.0247 -0.0038

47 73 50 86 47

-0.0018 0.0006 0.0066 0.0131 -0.0194

82 53 63 70 26

The analysis of Table 13 is based on forecasting daily returns from 31 March 2006. The question is: How accurate are the forecasts for the five month horizon specified in Table 13? The answer, according to Christoffersen and Diebold [71], is not very if the horizon of interest is more than 20 days, since volatility is effectively not forecastable beyond that limit. Therefore, forecasts up to one quarter should be treated with varying degrees of

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56

0 0.0126

0.0077

0.0028

-0.0020

-0.0069

-0.0117

-0.0166

-0.0215

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0.012 0.0609

0.014 0.0556

(c) CAD/EUR Return Distribution

0.1068

0.016 0.0271

0 0.0503

0.001

0.0968

0.002 0.0223

0.003

0.0451

0.004

0.0869

0.005 0.0174

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0.0770

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(b) CAD/GBP Return Distribution

0.0292

0.0239

0.0186

0.0134

0.0081

0.0028

-0.0025

-0.0078

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-0.0183

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Frequency 0.01

0.0571

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Frequency

Frequency 0.01

(a) CAD/USD Return Distribution

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

Figure 19: Return Distributions for (a) CAD/USD (b) CAD/GBP and (c) CAD/EUR exchanges for one month ahead from 31 March 2006. Red areas to left and right of average correspond to the lower and upper 5% of results respectively.

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confidence. The FOREX model has been developed such that the GARCH models get updated every 22 trading days. Consideration was also given to testing the forecasting ability of the model in terms of evaluating the proximity of forecasted returns to the actuals through calculating the Mean Error (ME), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE)27 . As a benchmark comparison, we assume a pure random walk with zero forecast as the expected value and no imposition of past trends on future values. The average of all 10,000 iterations at each month was used as the forecasted value in the ME, MAE and RMSE calculations. The results are displayed in Table 14 as well as the absolute difference between the results. The forecasting performance of the monthly returns is quite good for all three currencies but especially for CAD/USD and CAD/GBP. Each statistic shows a negligible difference between the benchmark and forecasted result which is significant in itself considering that the returns were compared on a monthly basis even though they were generated daily. This only gives further justification for the GARCH variance model with standardized t˜(d) error. Table 14: Forecasting performance of CAD/USD, CAD/GBP and CAD/EUR monthly returns

Measure

Forecast

CAD/USD Benchmark

Abs. Diff

ME MAE RMSE

-0.0085 0.0207 0.0231

-0.0106 0.0207 0.0233

0.0021 0.0000 0.0002

Measure

Forecast

ME MAE RMSE Measure ME MAE RMSE

0.0076 0.0089 0.0136 Forecast 0.0064 0.0136 0.0158

CAD/GBP Benchmark Abs. Diff. 0.0073 0.0096 0.0127

0.0003 0.0007 0.0009

CAD/EUR Benchmark Abs. Diff. -0.0002 0.0083 0.0109

0.0066 0.0053 0.0049

27 Mean Error is the average of the forecast error (the difference between the actual value and the forecast at a

given point in time); Mean Absolute Error or Mean Absolute Deviation, is the average of the absolute errors. It is similar to the RMSE but is less sensitive to large forecast errors; Root Mean Square Error weighs the effects of large forecast errors proportionally more than smaller forecast errors. This type of evaluation technique is useful if the cost of large forecast errors is substantially greater than the cost of small forecast errors.

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6.5 6.5.1

Assessing Convergence Convergence of Results

A final validation of the model involves testing for simulation accuracy, or what is normally called convergence. For example, when running a simulation, an analyst should always ensure that the number of iterations is sufficient to yield an estimate within an acceptable range of the true solution. If the number of iterations is too small, the simulation may yield an imprecise result, and in this study, erroneously predicting the potential losses due to market risk would significantly affect operational budgets and the proper management of considerable public funds. The FOREX simulation was run for 10,000 iterations using Latin Hypercube stratified sampling. The simulation either ran for 10,000 iterations or until convergence signified by a tolerance of less than 1.5% difference in each percentile value, the mean and standard deviation of each output. The results were tested for convergence every 100 iterations, i.e., 100 times. Each simulation run generates results for each month in a year (12), each currency (3) and each account (2) for a total of 72 outputs at 10,000 iterations each, or 72,000 output results. There are also four types of output: two for expenditures and VaR that each generate 72k results; and two for currency returns and variance that each generate 36k results (NP and Capital accounts are not a factor in simulating currency returns). Rather than test all output results, we have chosen to test convergence on one of each type of output at the VaR percentile (5%) and for only one currency (USD), one account (NP) and one month ahead (end April 2006). Figure 20 shows the convergence results obtained by calculating the 5th percentile of each output sample as sample size increases with iteration. Each plot shows a period of oscillatory behaviour as the sample size increases and the percentile eventually stabilizes. Except for Figure 20 (b), VaR convergence, the strongest oscillations are stabilized within 1000 iterations and long-term stability occurs at approximately 4000 iterations. For 20 (a), expenditures convergence, the long-term 5th percentile, $1,714,478 (see Table 12) stabilizes at 3510 iterations. The expenditure test exhibits no noise as a consequence of the uniform FHS sampling of the historical dataset. For Figure 20 (c), returns convergence, the difference between iteration 2000 and iteration 10000 is a increase of 2.2%. The simulation therefore stabilizes at a return of -0.0317 at the 5th percentile. Similarly, the rate variance convergence test, 20 (d), shows a difference of -0.57% from 4000 to 10,000 iterations. Examination of the VaR convergence test (Figure 20 (b)) seems to indicate a somewhat stabilized VaR as late as 9000 iterations. While it takes approximately 2000 iterations to “stabilize” the strongest oscillations, the VaR still shows a increasing trend from 3500 to 9000. However, the difference in VaR is only 5.6%. From 9000 to 10,000 iterations, the difference is 0.57%.

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0.045

0.04

0.035

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1

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c CADUSD Returns Convergence Test

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a USD NP Expenditure Convergence Test

9000

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d CADUSD Variance Convergence Test

2000

b USD NP VaR Convergence Test

8000

8000

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Figure 20: Convergence results for 5th percentile of CAD/USD national procurement account on April 30 2006.

1.2  106

1.4  106

1.6  106

1.8  106

2  106

2.2  106

2.4  106

Dollars CAD

Dollars CAD

6.5.2

Test for VaR Convergence

While Figures 20 (a, c, d) clearly show convergence for expenditures, returns and rate variance respectively, it isn’t sufficient to state that the VaR has converged without further testing through multiple simulation runs whether, using an appropriate level of significance, the simulation has in fact reached convergence. Consider the results in Table 15 as a sample of 20 runs at 10,000 iterations each and starting from the same initial conditions for each run. Each VaR result represents the 5th percentile CAD/USD VaR for one month ahead and should be compared to the simulation result specified in Table 10, i.e., -$397,018. In this case, we wish to test whether in fact the results in Table 10 are estimates of the mean of a normally distributed population of results, i.e., H0 : μ = −397, 018, and the simulation has converged H1 : μ = −397, 018, and the simulation has not converged , where H0 defines the null hypothesis and H1 is the alternate hypothesis. Given that the mean and standard deviation of the 20 runs are -395,533 and 4632 respectively, a two-tailed t-test was used to test the hypothesis. Under the hypothesis H0 , t=

X¯ − μ √ −395, 533 − (−397, 018) √ N −1 = 20 − 1 = +1.40 , s 4632

where N − 1 are the degrees of freedom. Since t = +1.40 lies inside the interval −t0.975 to t0.975 , which for 19 degrees of freedom is the interval −2.093 to 2.093, we cannot reject H0 at the 0.05 level of significance and conclude that μ = −$397, 018 and the simulation has in fact converged. Table 15: Simulation results for CAD/USD 5th percentile one month ahead VaR

60

Run

VaR

Run

VaR

Run

VaR

Run

VaR

1 2 3 4 5

-$394,030 -$398,295 -$395,888 -$393,371 -$390,109

6 7 8 9 10

-$402,643 -$390,346 -$393,118 -$395,083 -$400,217

11 12 13 14 15

-$394,758 -$391,189 -$403,134 -$392,812 -$399,433

16 17 18 19 20

-$398,591 -$386,012 -$402,210 -$392,219 -$397,197

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7

Discussion on Modelling

The times series analysis for expenditures was not developed without concerns. Size of the data samples, non-seasonal peaks in the capital data, and zero values for Euro transactions made fitting the sample (or subset, in certain cases) problematic. In many cases, significant effort was spent fitting a model that to all extents and purposes was ideal, only to have it fail at diagnostics. In the final model selection, interpretability, simplicity and feasibility played important roles. For example, a model should be simple enough to extract the salient features from the observed data and not so large that the defining parameters can no longer be estimated with reasonable accuracy. The results also have to make sense. Models with low to mid-range percentiles that forecast sizeable expenditures beyond historical values are inconsistent and need to be redesigned such that extreme values fall where they should, in the tails of the distribution. The underlying goal in this study has always been to first find a model that provides a good fit to the data, admits proper interpretation and is equally good for forecasting. For the expenditures, the results would only improve with more data, and an extension of this study might wish to consider using weekly, if not daily, data. Going from a data set of 84 to 364 would definitely allow a more aggressive approach to model definition and provide for an expanded set of models from which to choose. Also, with an increase in the number of payments, the volatility should increase as sizeable payments and outliers, which currently aggregate into a monthly extreme, would now differentiate into weekly payments. This would open the door to volatility forecasting models, e.g., the GARCH family, which are more easily interpretable, bring in the economic content currently lacking with purely statistical time series constructs, and have proven to be extremely useful for financial risk management. All that being said, converting from low frequency to high frequency datasets introduces temporal de-aggregation effects on structural inference that would also need to be investigated. Questions such as: What kind of structural model results when the data is temporally de-aggregated? Does the observed structure correspond to the underlying causal relationship among the variables or does the observed structure differ from the original causal structure? In forecasting a stationary time series, we can only predict the future if the underlying process sustains stability over time. We assume that both the form of the model and the coefficients in the model are known and remain unchanged for whatever forecasting horizon we desire. In truth, the forecasting accuracy decreases with time and while we may be able to forecast expenditures for the next 12 months, it is reasonable to expect that the accuracy of the prediction is limited to a quarter or possibly less, and that once actual values become available, the coefficients of each model should be updated through maximum likelihood estimation. The FOREX model does this automatically. Unlike the linear ARMA models for expenditures, to model non-linear behaviour such as dependence beyond linear correlation, we need to expand the analysis to non-linear models and specifically, to the non-constant conditional variance functions – a phenomenon called conditional heteroscedasticity. Non-linear models such as the GARCH family are

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61

particularly suited for financial time series as they specifically take into consideration the dependency of the conditional second moment (variance). In this study we have concentrated on the simple GARCH(1,1) model with coefficients specified through the conditional maximum likelihood estimator defined on the standardized t˜(d) distribution. While MLE is considered the benchmark estimator and is widely used throughout the banking industry, it is an L2 -estimator and not as robust as L1 -estimators, such as Peng and Yao’s least absolute deviations estimator (LADE) [72], for very heavytailed distributions. In this study, it is doubtful that the MLE would have underperformed against LADE, nevertheless, if this study is to be expanded to other exchanges, preliminary analysis into the shape of the return distribution is crucial when specifying the best estimator. Consideration should also be given to extensions and/or modifications of the GARCH model itself. For example, Engle and Lee [73] specified a higher-order, GARCH(2,2), model, called a “component” model with additional lags to allow for both fast and slow decay of information, a possibly better approach to modelling the conditional variances of foreign exchanges [74]. Alternatively, there are a number of asymmetric GARCH models that could be used to test the results of this study including the direction of the returns, which are ignored by GARCH models. Models such as Nelson’s Exponential GARCH (EGARCH) [41, 42], Rabemananjara and Zakoian’s Threshold ARCH (TARCH) [75] and Glosten et al.’s GJR − GARCH [43] are several that are useful, depending on the portfolio of interest. Finally, this analysis has been built on a univariate approach. While we know the expenditure transactions are univariate, we cannot be absolutely sure that the volatility of one currency will not influence the volatility of another. For example, the amplitude of return movements on the United States exchange markets may be a response to the volatility observed earlier in the U.K. markets. Also, as pointed out by one reviewer of this report, “One could argue that the equipment and supplies purchases in different currencies have a common component (e.g. Canada’s overall military engagement) and the univariate modeling of the purchases in each currency misses such a common component.” [76]. The modelling of conditional correlations is non-trivial as the covariance matrix must remain positive-semidefinite and it is unclear how the GARCH parameters, originally specified per currency model, must now vary across currencies. Based on Engle [77] and Engle and Sheppard’s [78] multivariate GARCH analyses, Christoffersen [79] has modelled the conditional correlations rather than covariances and provides a simple procedure whereby the single currency GARCH(1,1) model now becomes a 2 × 2 matrix equation for a twocurrency model. For estimating the model parameters, the approach uses a Quasi-Maximum Likelihood Estimation procedure which, for three currencies, relies on the log likelihood of the three-dimensional normal distribution function. Regardless, as also pointed out in reference [76], “... the univariate time series perspective taken is arguably the most parsimonious and thus likely to yield the best out of sample results going forward.”. This discussion, nevertheless, opens the door to multivariate modelling where not only the volatilities but also the correlations need to be examined. While not as simple as univari-

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ate modelling, it does provide the link for currency to currency response and ultimately may provide an better explanation for volatility than available via univariate models. This analysis could be the subject of a follow-on version to FOREX.

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8

Conclusions and Recommendations

Given the historical and continued importance of foreign exchange exposure to the department, this study focused primarily on the measurement and evaluation of this form of risk. Value-at-Risk, as applied to foreign exchange exposure, is an estimation of the probability of likely losses that could arise from changes in exchange rates. It has become very popular in financial risk management because it is easily understood and condenses a vast amount of information into a single, summary statistical measure of market risk under normal or expected market conditions. As such, it is an estimate of risk that is based on historical data that relies on the concept that the future will be like the past with extreme events predictable, based on likelihood of occurrence. And that might be its one weakness, it says nothing about the size of losses once the VaR limit (5% in this study) has been exceeded, and which reside in the tail of the distribution. While no risk measure is perfect, VaR is certainly better than the alternative, which is no risk measure at all. Nevertheless, the model developed and documented in this report accounts for extreme losses by not only reporting the 5th percentile VaR, but also the maximum expected loss of the 0th percentile (or any other level). Measuring the risk of exposure is recognized as the first step any organization must take before considering risk mitigation strategies to reduce and hopefully eliminate foreign transaction exposure. Currently, DND has made no allowance for foreign currency hedging (although certain financial instruments have been proposed) for the reduction and possible elimination of foreign exchange losses. Without such instruments in place, there is great uncertainty as to what the expected foreign exchange transaction losses will be in the future. Budget rates, although based on viable time series analysis, provide only point estimates of what the future exchange rate may be, without any direction as to how this estimate will vary. The end result is that procurement/budget managers within capital equipment projects and in-service equipment management teams must provide a “best guess” as to how much funds they need to hold back (reserves) to account for unforecasted losses due to exchange. VaR analysis helps by providing a quantitative, statistical aid to support them in their estimates and ultimately reduce the dependency of holding more money than is necessary for foreign currency losses that may or may not materialize. Therefore, quantifying and managing exchange rate exposure properly means managers can now exercise proper responsiveness to foreign exchange volatility. This study illuminates certain policy implications for functional finance and performance/risk management specialists in the department. In particular, the VCDS Group through the Director Force Planning and Programme Coordination (DFPPC) and ADM(Fin CS) through Director Budget and Director Strategic Finance and Costing (DSFC) may want to examine the possibility of adjusting corporate budget allocations (quarterly) based on the results of the VaR model. In fact, these groups should consider adopting the VaR methodology as part of the department’s integrated risk management framework for managing the budgetary risk attributed to exposure to foreign currency fluctuations. Under this scenario, the quarterly foreign exchange reports produced by DSFC would continue to provide a benchmark forecast exchange rate for all anticipated foreign currency denominated expenditures

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in the department from which the VaR model would measure the risk of loss on exchange. By extension, the department should also examine opportunities to apply the VaR analytical approach to quantifying the financial risk in other budget expenditure areas subject to market/price risk such as bulk fuels, energy/hydro, and certain commodities (e.g., steel, ballistic materials, etc.) where expenditure amounts warrant. As the department embarks on large multi-year capital acquisitions and continues to be engaged in sizeable, complex overseas deployments, the need to measure and accurately assess financial risk has never been greater. Moreover, should the department decide to seek central government agency concurrence to implement (or pilot) a financial hedging strategy to limit foreign exchange risk (as is the case in the UK and proposed by Essaddam et al. [10]), the ability to measure and report exchange rate risk would be fundamental for successful hedging. Notwithstanding, this study does illustrate the practical application of the VaR method to arguably the largest department financial risk area, foreign currency exposure, and it is hoped that it will contribute to a better understanding of this risk parameter and how it can be more consistently and accurately measured, reported and ultimately controlled.

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15. Email, Mr. André Deschenes, ADM(Fin CS)/DSFC 7-4 (29 December 2005, 0850 EST). 16. Central Michigan University Office of the Comptroller (2006). Accounting Services (Online). http://www.controller.cmich.edu/Accounting/doc_type.htm. 17. Email, Mr. Spencer C. Lee, ADM(Mat)/DMG Compt 5-4-2 (07 April 2006, 1357 EST). 18. Email, Mr. Spencer C. Lee, ADM(Mat)/DMG Compt 5-4-2 (11 July 2006, 1138 EST). 19. Lo, Andrew W. and MacKinlay, A. Craig (1999). A Non-Random Walk Down Wall Street, Princeton University Press. Chapter 4: An Econometric Analysis of Nonsynchronous Trading. 20. Bank of Canada (2006). Rates and Statistics: Exchange Rates (Online). http://www.bankofcanada.ca/en/rates/exchange.html. 21. Macdonald, Ronald (1999). Exchange Rate Behaviour: Are Fundamentals Important?. The Economic Journal, 109(459), 673–691. 22. ITSM2000 ver. 7.3 (Professional). B & D Enterprises Inc. October 2005. 23. Brockwell, Peter J. and Davis, Richard A. (2002). Introduction to Time Series and Forecasting, Second ed. Springer. 24. Fan, Jianqing and Yao, Qiwei (2005). Nonlinear Time Series - Nonparametric and Parametric Methods, First ed. Springer. 25. Box, George E.P., Jenkins, Gwilym M., and Reinsel, Gregory C. (1994). Time Series Analysis - Forecasting and Control, Third ed. Prentice-Hall. 26. Box, G.E.P. and Cox, D.R. (1964). An Analysis of Transformations. Journal of the Royal Statistical Society, B 26, 211–252. 27. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petroc, B.N. and Caski, F., (Eds.), Second International Symposium in Information Theory, pp. 267–281. Akademiai Kiado, Budapest. 28. Hurvich, C.M. and Tsay, C.L. (1989). Regression and time series model selection in small samples. Biometrika, 76(2), 297–307. 29. Ljung, G.M. and Box, G.E.P. (1978). On a measure of lack of fit in time series models. Biometrika, 65(2), 297–303. 30. Tsay, Ruey S. (2005). Analysis of Financial Time Series, Second ed. Wiley. 31. Email, Mr. Spencer C. Lee, ADM(Mat)/DMG Compt 5-4-2 (17 August 2006, 1322 EST).

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32. J.P. Morgan/Reuters (1996). RiskMetricsT M – A Practical Guide, Fourth ed. Morgan Guaranty Trust Company. 33. Cont, Rama (2001). Empirical properties of asset returns: stylized facts and statistical issues. In Quantitative Finance, Vol. 1, pp. 223–236. Institute of Physics Publishing. 34. Taylor, Stephen J. (2005). Asset Price Dynamics, Volatility, and Prediction, Princeton University Press. Chapt. 4, pp. 51–96. 35. Liu, Yanhui, Gopikrishnan, Parameswaran, Cizeau, Pierre, Meyer, Martin, and Peng, Chung-Kang (1999). Statistical properties of the volatility of price fluctuations. Physical Review E, 60(2), 1390–1400. 36. Kaizoji, Taisei (2004). Intermittent chaos in a model of financial markets with heterogeneous agents. Solitons and Fractals, 20(2), 323–327. 37. Rachev, Svetlozar T., Fabozzi, Frank J., and Menn, Christian (2005). Fat-Tailed and Skewed Asset Return Distributions : Implications for Risk Management, Portfolio Selection, and Option Pricing, Wiley. 38. Engel, James and Gizycki, Marianne (1999-04). Value at Risk: On the Stability and Forecasting of the Variance-Covariance Matrix. Reserve Bank of Australia. 39. Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31(3), 307–327. 40. Engle, R.F. (1982). Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4), 987–1007. 41. Nelson, D.B. (1991). Conditional heteroskedasticity in asset pricing: A new approach. Econometrica, 59(2), 347–370. 42. Bollerslev, T., Engle, R.F., and Nelson, D.B. (1994). ARCH Models. In Engle, R.F. and McFadden, D.L., (Eds.), Handbook of Econometrics, Vol. IV, Chapter 49, Elsevier Science. 43. L.R. Glosten, R. Jagannathan and Runkle, D. (1993). Relationship between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48(5), 1770–1801. 44. Nelson, D.B. (1990). Stationarity and Persistence in the GARCH(1,1) Model. Econometric Theory, 6(3), 318–334. 45. Christoffersen, Peter F. (2003). Elements of Financial Risk Management, Academic Press. Chapter 4. 46. Email, Prof. Peter F. Christoffersen, Associate Professor of Finance, McGill University (29 November 2006, 1339 EST). 47. Christoffersen (2003). Elements of Financial Risk Management. p. 102.

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48. RiskMetrics Group (2006). Backtesting VaR. 2006 RiskServer Release Notes. 49. Bank for International Settlements (2005). International Convergence of Capital Measurement and Capital Standards: A Revised Framework. CH–4002 Basel, Switzerland. Updated. 50. Christoffersen (2003). Elements of Financial Risk Management. Chapt. 8. 51. Christoffersen, P. (1998). Evaluating Interval Forecasts. International Economic Review, 39(4), 841–862. 52. Kupiec, Paul H. (1995). Techniques for Verifying the Accuracy of Risk Measurement Models. Journal of Derivatives, 3, 73–84. 53. Christoffersen (2003). Elements of Financial Risk Management. pp. 30–31. 54. GloriaMundi.org. All About Value at RiskT M (Online). http://www.gloriamundi.org/. 55. Beder, T.S. (1995). VAR: Seductive but Dangerous. Financial Analysts Journal, 51(5), 12–24. 56. Hendricks, D. (1996). Evaluation of value-at-risk models using historical data. Federal Reserve Bank of New York Economic Policy Review, pp. 39–69. 57. Holton, Glyn A. (2003). Value-at-Risk Theory and Practice, First ed. Academic Press. 58. Best, Philip (1999). Implementing Value at Risk, John Wiley & Sons. 59. Linsmeier, Thomas J. and Pearson, Neil D. (2000). Value at Risk. Association for Investment Management and Research, pp. 47–67. 60. Benninga, Simon and Wiener, Zvi (1998). Value-at-Risk (VaR). Mathematica in Education and Research, 7(4), 1–8. 61. Bodoukh, J., Richardson, M., and Whitelaw, R. (1998). The Best of Both Worlds. Risk, 11, 64–67. 62. Finger, Christopher C. (2006). How historical simulation made me lazy. RiskMetrics Group Research Monthly. 63. Pritsker, M. (2001). The Hidden Dangers of Historical Simulation. Manuscript, Federal Reserve Board. 64. Barone-Adesi, G., Giannopoulous, K., and Vosper, L. (1999). VaR without Correlations for nonlinear Portfolios. Journal of Futures Markets, 19, 583–602. 65. Barone-Adesi, G., Giannopoulous, K., and Vosper, L. (2000). Filtering Historical Simulation. Backtest Analysis. Manuscript. 66. Christoffersen (2003). Elements of Financial Risk Management. pp. 110–112.

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67. Email, Mr. André Deschenes, ADM(Fin CS)/DSFC 7-4 (10 March 2006, 1201 EST). Foreign Exchange Quarterly Update As of 09 March 2006. 68. Palisade Corp. (2004). @Risk – Advanced Risk Analysis for Spreadsheets. Palisade Corporation, Newfield, New York. Version 4.5.5, http://www.palisade.com. 69. Meese, Richard A. and Rogoff, Kenneth (1983a). Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?. Journal of International Economics, 14(1/2), 3–24. 70. Orok, Bruce (2003). Exchange Rate Forecasting. For internal purposes only. 71. Christoffersen, Peter F. and Diebold, Francis X. (1998). How Relevant is Volatility Forecasting for Financial Risk Management?. (NBER Working Papers 6844). National Bureau of Economic Research, Inc. 72. Peng, Liang and Yao, Qiwei (2003). Least absolute deviations estimation for ARCH and GARCH models. Biometrica, 90(4), 967–975. 73. Engle, Robert and Lee, Gary G. J. (1999). A Permanent and Transitory Component Model of Stock Return Volatility. In Engle, Robert F. and White, Halbert, (Eds.), Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive W. J. Granger, pp. 475–497. Oxford University. Oxford University Press. 74. P. Christoffersen, Private Communication (21 July 2006). 75. Rabemananjara, R. and Zakoian, J. M. (1993). Threshold ARCH Models and Asymmetries in Volatility. Journal of Applied Econometrics, 8(1), 31–49. 76. Christoffersen, Peter F. (2006). Referee Report. 77. Engle, R. (2002). Dynamic Conditional Correlation: A Simple Case of Multivariate GARCH Models. Journal of Business and Economic Statistics, 20, 339–350. 78. Engle, Robert F. and Sheppard, Kevin (2001). Theoretical and Empirical properties of Dynamic Conditional Correlation Multivariate GARCH. (NBER Working Papers 8554). National Bureau of Economic Research, Inc. 79. Christoffersen (2003). Elements of Financial Risk Management. Chapter 3.

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Annex A Fitting the Expenditure Models In section 3, a complete analysis was given for the USD NP account with only final details presented for the other five expenditure models. This annex provides complete analysis details for the remaining two currency-NP accounts (GBP NP and EUR NP) and all three currency-capital accounts. The reader is referred to section 3 for notation and definitions.

A.1

USD Capital Account

A.1.1 USD Capital: Transforming the Series

Figure A.1 (a) describes the original USD Capital (equipment) series. The series is of length 84 with sample mean and variance $29.32M and 1.1981 × 1015 respectively. All values are positive and none are zero, therefore no adjustments need be made before applying a log transformation. To help identify any trend and/or seasonality, the sample ACF plot was used to observe the degree of dependence in the data. The USD Capital pre-transformed ACF, shown in Figure A.1 (b), shows (a) a decreasing trend and, (b) a weak seasonal component with period 12 that disappears after lag 24 and which is also within the 5% (two-sided) rejection √ bands defined by ±1.96/ n, where n is the sample size. The analysis proceeds by first performing a log transformation followed by a differencing at lag-1 using the backshift operator, B, ∗ (1 − B) ln Xt∗ = ln Xt∗ − ln Xt−1 , (A.1) where X ∗ denotes the original (untransformed) series. Once the apparent deviations from stationarity have been removed, the sample mean, μ, is subtracted from each observation and a zero-mean stationary model is fit to the resulting series, i.e., ∗ −μ. Xt = ln Xt∗ − ln Xt−1

(A.2)

Equation (A.2) is plotted in Figure A.1 (d) as the log transformed, differenced and mean subtracted USD Capital time series, Xt .

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Figure A.1: (a)Time plot of monthly USD Capital liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-1 differenced series. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from ARMA(2,3) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals.

Apr 01

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d Transformed USD Capital

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5  107

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Density

A.1.2 USD Capital: Fitting the ARMA Model

The ACF and PACF of the transformed series (A.2) are plotted in Figure A.1 (e) and A.1 (f) respectively. Close examination of the Figures suggest that we may fit models with 0 ≤ (p, q) ≤ 3. Using the Ljung-Box statistic and AICC for model selection, models were created for the USD Capital expenditure data. The AICC, Ljung-Box (h = 20) and p-value statistics were generated and sorted by minimum AICC. The low Ljung-Box statistic and high p-value (> 0.05) is an indication that there is no structure in the data (uncorrelated) up to lag 20. Table A.1 displays the prioritized statistics for all candidates. Table A.1: USD Capital model selection statistics

(p,q)

AICC

(2,3) (2,1) (1,1) (3,1) (2,2) (0,2) (3,0) (1,3) (1,2) (3,2) (3,3) (0,1) (2,0) (0,3) (1,0)

246.173 247.976 248.947 250.062 250.186 250.275 250.513 251.142 252.081 252.566 254.83 255.485 256.633 257.223 277.487

Ljung-Box

p-value

19.499 23.429 21.922 25.012 25.271 22.449 22.232 25.707 25.174 25.347 24.706 21.959 29.629 27.003 46.827

0.48963 0.26821 0.34475 0.20097 0.19127 0.31666 0.32805 0.17571 0.19485 0.18846 0.21291 0.34272 0.07609 0.13519 0.00062

The optimum model, according to Table A.1 and based on AICC, is an ARMA(2,3). The estimated AR and MA-coefficients φ1 , φ2 , θ1 , θ2 , θ3 are respectively φ1 , φ2

   +0.4804, −0.8145,

 −1.516,

θ1 , θ2 , θ3

 +1.566,

 −0.8236 ,

with ratios φ j /SE(φ j ) for j = 1, 2 7.132

and

− 12.232 ,

and θ j /SE(θ j ) for j = 1, 2, 3 − 24.124,

35.830

and

− 13.109 ,

where, for example, SE(θ j ) is the standard error of θ j . Since the absolute value of each

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ratio is >> 1.96, all φ and θ coefficients are non-zero. The final model is therefore Xt = Zt + 0.4804Xt−1 − 0.8145Xt−2 − 1.516Zt−1 + 1.566Zt−2 − 0.8236Zt−3 ,

(A.3)

where Zt ∼ W N(0, 0.9123). Solving (A.2) for Xt∗ and substituting (A.3) converts the time series back to the original expenditures. The model, prior to coefficient substitution is Xt∗ = exp [Zt + θ1 Zt−1 + θ2 Zt−1 + θ3 Zt−3

∗ ∗ ∗ +(1 + φ1 ) ln Xt−1 + (φ2 − φ1 ) ln Xt−2 − φ2 ln Xt−3 + μ(1 − φ1 − φ2 ) ,



(A.4)

and after coefficient substitution, Xt∗ = exp [Zt − 1.516Zt−1 + 1.566Zt−2 − 0.8236Zt−3

 ∗ ∗ ∗ +1.4804 ln Xt−1 − 1.2949 ln Xt−2 + 0.8145 ln Xt−3 + 1.3341μ .

(A.5)

A.1.3 USD Capital: Testing for Invertibility and Stationarity

To test for invertibility and stationarity, we need to express the process defined by equation (A.3) in terms of the backshift operator defined in equation (3). Equation (A.3) thus takes the form Xt − 0.4804Xt−1 + 0.8145Xt−2 = Zt − 1.516Zt−1 + 1.566Zt−2 − 0.8236Zt−3 (1 − 0.4804B + 0.8145B2 )Xt = (1 − 1.516B + 1.566B2 − 0.8236B3 )Zt φ (B)Xt = θ (B)Zt ,

(A.6)

where φ (B) = 1 − 0.4804B + 0.8145B2 and θ (B) = 1 − 1.516B + 1.566B2 − 0.8236B3 . If φ (B) and θ (B) have no common factors and if their roots lie outside the unit circle, Xt is defined by an ARMA(2,3) process which is both stationary28 and invertible. Solving φ (B) = 0 and θ (B) = 0, the roots are, respectively {0.294905 − 1.06807 i}, {0.294905 + 1.06807 i}, and {0.345941 − 0.940305 i}, {0.345941 + 0.940305 i}, {1.20953}. which all lie outside the unit circle. Hence, the ARMA(2,3) model for USD Capital is both stationary and invertible. 28 Some

authors refer to this condition as the causality condition, i.e., an ARMA model is causal if all the zeros of its AR polynomial lie outside the unit circle.

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A.1.4 USD Capital: Testing the Residuals

If it is assumed that the ARMA(2,3) model is a true representation of the data, then the rescaled residuals, obtained by dividing the residuals by the estimate of the white noise standard deviation, should resemble a realization of a white noise sequence with variance one. The rescaled residuals obtained from the ARMA(2,3) model that was fitted to the transformed USD Capital data are plotted in Figure A.1 (g). The graph shows no strong indication of nonzero mean or nonconstant variance. On this basis, there are no indications to doubt the compatibility of the series with unit variance white noise. From the ACF plot of USD NP residuals in Figure A.1 (h), if more then 5% of the 40 lags fall outside the bounds, the white noise hypothesis is rejected. As seen from the graph, no lags lie outside the 5% rejection bands and therefore there is no reason to reject the ARMA(2,3) model on the basis of the autocorrelations. Compatibility of the residuals with a normal distribution can be decided by inspecting a histogram of the residuals. If the fitted model is appropriate, the histogram should show a mean close to zero and an approximate shape to a normal distribution. The histogram of the transformed, differenced and mean-corrected USD Cap data shown in Figure A.1 (i) suggests the assumption of Gaussian white noise is not unreasonable, although the distribution is slightly skewed left. The final test has already been specfied by the Ljung-Box statistic test for randomness. With a test statistic of 19.499 and p-value of 0.48963, the conclusion is that, at the 5% level, the data are uncorrelated and support the compatibility of the residuals with white noise.

A.2

GBP NP Account

A.2.1 GBP NP: Transforming the Series

Figure A.2 (a) describes the original GBP NP series. The series is of length 84 with sample mean and variance $3.424M and 2.4379 × 1013 respectively. All values are positive and none are zero, therefore no adjustments need be made before applying a log transformation. To help identify any trend and/or seasonality, the sample ACF plot was used to observe the degree of dependence in the data. The GBP NP pre-transformed ACF, shown in Figure A.2 (b), shows (a) no trend and, (b) a seasonal component with period 12. The analysis initially proceeded by first performing a log transformation followed by a differencing at lag-12 and lag-1. However, due to the large component at end-of-year FY 02/03 due to the Victoria Class sub maintenance program (Figure A.2 (a)), a differencing at lag-12 only highlighted the discrepancy in expenditures. Further testing of order selection with the Innovations and Hannan-Rissanen algorithms followed by maximum likelihood estimation did provide valid models which later failed during diagnostic testing of the residuals. Consequently, the analysis was redone with a log transformation followed by only a regular differencing at

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lag-1 using the backshift operator, B, ∗ (1 − B) ln Xt∗ = ln Xt∗ − ln Xt−1 ,

(A.9)

where X ∗ denotes the original (untransformed) series. Once the apparent deviations from stationarity have been removed, the sample mean, μ, is subtracted from each observation and a zero-mean stationary model is fit to the resulting series, i.e., ∗ −μ. Xt = ln Xt∗ − ln Xt−1

(A.10)

Equation (A.10) is plotted in Figure A.2 (d) as the log transformed, once differenced and mean subtracted GBP NP time series, Xt . A.2.2 GBP NP: Fitting the ARMA Model

The ACF and PACF of the transformed series (A.10) are plotted in Figure A.2 (e) and A.2 (f) respectively. Close examination of the Figures suggest that we may fit models with AR(5) or possibly MA(12). Using the Ljung-Box statistic and AICC for model selection, models in the range 0 ≤ (p, q) ≤ 5 were created for the GBP NP expenditure data. The AICC, Ljung-Box (h = 20) and p-value statistics were generated and sorted by minimum AICC. The low Ljung-Box statistic and high p-value (> 0.05) is an indication that there is no structure in the data (uncorrelated) up to lag 20. Table A.2 displays the prioritized statistics for the first 20 candidates. Table A.2: GBP NP model selection statistics

76

(p,q)

AICC

(2,5) (0,1) (3,3) (3,1) (3,4) (5,3) (4,3) (4,4) (4,1) (3,2) (3,5) (1,2) (1,1) (4,2) (0,2) (4,0) (5,4) (4,5) (2,3)

291.206 292.909 293.409 296.494 297.503 297.517 297.897 298.017 298.488 298.784 299.326 300.229 300.702 300.912 301.277 301.362 301.727 302.511 302.814

Ljung-Box

p-value

32.171 27.156 29.612 32.602 36.72 24.497 20.791 27.984 32.06 33.475 33.611 40.998 44.471 33.667 33.979 34.151 33.733 31.53 52.31

0.04151 0.13093 0.07639 0.03729 0.01264 0.22137 0.40955 0.10978 0.04266 0.02991 0.02888 0.00373 0.0013 0.02847 0.02627 0.02512 0.02799 0.04858 0.0001

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Figure A.2: (a)Time plot of monthly GBP national procurement liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-1 differenced series. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from ARMA(4,3) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals.

Apr 01

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2 1 0 1 2 3 4

4 2 0 2 4 6 8 May 99

3  107 2.5  107 2  107 1.5  107 1  107 5  106 0 Apr 99

Density

According to Table A.2, the best model based on AICC and the Ljung-Box statistic is an ARMA(5,3)29 . A competitive model with lower order is still considered a better choice. Therefore, the optimum model based on AICC and Ljung-Box is an ARMA(4,3) with an AICC value of 297.897. The estimated AR and MA-coefficients φ1 , . . . , φ4 , θ1 , θ2 , θ3 are respectively  −0.8596,

φ1 , φ2 , φ3 , φ4

 −1.2590, −0.6035,

 −0.2981, −0.1924,

θ1 , θ2 , θ3

 +0.7060,

 −0.6466 ,

with ratios φ j /SE(φ j ) for j = 1, . . . , 4 − 6.1171,

−7.5171,

−3.5423

and

− 2.3149 ,

and θ j /SE(θ j ) for j = 1, 2, 3 − 1.6219,

13.0943

and

− 5.4519 ,

where, for example, SE(θ j ) is the standard error of θ j . The final model is therefore Xt = −0.8596Xt−1 − 1.2590Xt−2 − 0.6035Xt−3 − 0.2981Xt−4 +Zt − 0.1924Zt−1 + 0.7060Zt−2 − 0.6466Zt−3 ,

(A.11)

with Zt ∼ W N(0, 1.5853). Solving (A.10) for Xt∗ and substituting (A.11) converts the time series back to the original expenditures. The model, prior to coefficient substitution is  ∗ Xt∗ = exp Zt + θ1 Zt−1 + θ2 Zt−2 + θ3 Zt−3 + (1 + φ1 ) ln Xt−1 ∗ ∗ ∗ +(φ2 − φ1 ) ln Xt−2 + (φ3 − φ2 ) ln Xt−3 + (φ4 − φ3 ) ln Xt−4  ∗ −φ4 Xt−5 + μ(1 − φ1 − φ2 − φ3 − φ4 ) ,

(A.12)

and after coefficient substitution, Xt∗ = exp [Zt − 0.1924Zt−1 + 0.7060Zt−2 − 0.6466Zt−3

∗ ∗ ∗ +0.1404 ln Xt−1 − 0.3994 ln Xt−2 + 0.6555 ln Xt−3  ∗ ∗ +0.3054 ln Xt−4 + 0.2981Xt−5 + 4.0202μ .

29 Earlier

(A.13)

models fail diagnostic testing with high Ljung-Box and low p-values indicating structure to the

residuals.

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A.2.3 GBP NP: Testing for Invertibility and Stationarity

To test for invertibility and stationarity, we need to express the process defined by equation (A.11) in terms of the backshift operator defined in equation (3). Equation (A.11) thus takes the form Xt + 0.8596Xt−1 + 1.2590Xt−2 + 0.6035Xt−3 + 0.2981Xt−4 = Zt − 0.1924Zt−1 + 0.7060Zt−2 − 0.6466Zt−3 (1 + 0.8596B + 1.2590B2 + 0.6035B3 + 0.2981B4 )Xt = (1 − 0.1924B + 0.7060B − 0.6466B3 )Zt φ (B)Xt = θ (B)Zt ,

(A.14)

where φ (B) = 1 + 0.8596B + 1.2590B2 + 0.6035B3 + 0.2981B4 and θ (B) = 1 − 0.1924B + 0.7060B − 0.6466B3 are the characteristic equations. If φ (B) and θ (B) have no common factors and if their roots lie outside the unit circle, Xt is defined by an ARMA(4,3) process which is both stationary and invertible. Solving φ (B) = 0 and θ (B) = 0, the roots are, respectively {0.917227 − 1.31443 i}, {−0.917227 + 1.31443 i}, {−0.0950177 − 1.13875 i}, {−0.0950177 + 1.13875 i}, and {−0.227201 − 0.973942 i}, {−0.227201 + 0.973942 i}, {1.54627}. which all lie outside the unit circle. Hence, the ARMA(4,3) model for GBP NP is both stationary and invertible. A.2.4 GBP NP: Testing the Residuals

The rescaled residuals obtained from the ARMA(4,3) model that was fitted to the transformed GBP NP data are plotted in Figure A.2 (g). The graph shows some divergence from a zero mean and constant variance, particularly at Apr-02 and Apr-05. From the ACF plot of GBP NP residuals in Figure A.2 (h), if more then 5% of the 40 (2) lags fall outside the bounds, the white noise hypothesis is rejected. As seen from the graph, only two lags lie outside the 5% rejection bands and therefore there is no reason to reject the ARMA(4,3) model on the basis of the autocorrelations. Compatibility of the residuals with a normal distribution can be decided by inspecting a histogram of the residuals. If the fitted model is appropriate, the histogram should show a mean close to zero and an approximate shape to a normal distribution. The histogram

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of the transformed, differenced and mean-corrected GBP NP data shown in Figure A.2 (i) suggests the assumption of Gaussian white noise, although not ideal, is not unreasonable. The final test has already been specified by the Ljung-Box statistic test for randomness. With a test statistic of 20.791 and p-value of 0.40955, the conclusion is that, at the 5% level, the data are uncorrelated and support the compatibility of the residuals with white noise.

A.3

GBP Capital Account

A.3.1 GBP Capital: Transforming the Series

Figure A.3 (a) describes the original GBP Capital (equipment) series. The series is of length 84 with sample mean and variance $1.161M and 1.9552 × 1012 respectively. All values are positive and none are zero, therefore no adjustments need be made before applying a log transformation. To help identify any trend and/or seasonality, the sample ACF plot was used to observe the degree of dependence in the data. The GBP Capital pre-transformed ACF, shown in Figure A.3 (b), shows (a) no trend and, (b) no seasonality. The analysis proceeds by first performing a log transformation followed by a differencing at lag-1 using the backshift operator, B, ∗ (1 − B) ln Xt∗ = ln Xt∗ − ln Xt−1 , (A.17) where X ∗ denotes the original (untransformed) series. Once the apparent deviations from stationarity have been removed, the sample mean, μ, is subtracted from each observation and a zero-mean stationary model is fit to the resulting series, i.e., ∗ −μ. Xt = ln Xt∗ − ln Xt−1

(A.18)

Equation (A.18) is plotted in Figure A.3 (d) as the log transformed, differenced and mean subtracted GBP Capital time series, Xt . A.3.2 GBP Capital: Fitting the ARMA Model

The ACF and PACF of the transformed series (A.18) are plotted in Figure A.3 (e) and A.3 (f) respectively. Close examination of the Figures suggest a strong MA(1) component and a possible AR(3) as candidate models. Using the Ljung-Box statistic and AICC for model selection, models were created for the GBP Capital expenditure data. The AICC, LjungBox (h = 20) and p-value statistics were generated and sorted by minimum AICC. The low Ljung-Box statistic and high p-value (> 0.05) is an indication that there is no structure in the data (uncorrelated) up to lag 20. Table A.3 displays the prioritized statistics for all candidates in the range 0 ≤ (p, q) ≤ 2.

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Apr 03 Date

Apr 05

Apr 01

Apr 03 Date

Apr 05

Dollars CAD

Apr 03 Date

Apr 05

5 10 15 20 25 30 35 40 Lag

0

5 10 15 20 25 30 35 40 Lag

e ACF of Transformed GBP Capital

0

b ACF of GBP Capital

0

5 10 15 20 25 30 35 40 Lag

h ACF of GBP Capital Rescaled Residuals 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 0

5 10 15 20 25 30 35 40 Lag

c PACF of GBP Capital

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

5 10 15 20 25 30 35 40 Lag

8

6

4

2

0

2

4

i Histogram of Rescaled Residuals

0

f PACF of Transformed GBP Capital 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

Figure A.3: (a)Time plot of monthly GBP Capital liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-1 differenced series. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from MA(1) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals.

Apr 01

g GBP Capital Rescaled Residuals

8 May 99

6

4

2

0

2

Apr 01

a GBP Capital Account

d Transformed GBP Capital

May 99

7.5 5 2.5 0 2.5 5 7.5

7  106 6  106 5  106 4  106 3  106 2  106 1  106 0 Apr 99

Density

Table A.3: GBP Capital model selection statistics

(p,q)

AICC

(0,1) (0,2) (1,2) (2,2) (1,1) (2,1) (2,0) (1,0) (0,0)

313.772 315.896 318.236 321.459 322.560 322.712 327.842 335.204 363.889

Ljung-Box

p-value

15.310 13.801 16.631 20.142 18.717 17.160 21.593 31.830 51.898

0.75840 0.84046 0.67678 0.44907 0.54029 0.64256 0.36298 0.04514 0.00012

The optimum model, according to Table A.3 and based on AICC, is an MA(1). The estimated MA-coefficient is θ = −0.8410 with θ /SE(θ ) = −16.829 where SE(θ ) is the standard error of θ . The final model is therefore Xt = Zt − 0.8410Zt−1 ,

(A.19)

where Zt ∼ W N(0, 2.4053). Solving (A.18) for Xt∗ and substituting (A.19) converts the time series back to the original expenditures. The model prior to coefficient substitution is   ∗ +μ , (A.20) Xt∗ = exp Zt + θ Zt−1 + ln Xt−1 and after coefficient substitution,   ∗ +μ . Xt∗ = exp Zt − 0.8410Zt−1 + ln Xt−1

(A.21)

A.3.3 GBP Capital: Testing for Invertibility and Stationarity

The MA(1) model is already stationary by definition. To test for invertibility, we need to express the process defined by equation (A.19) in terms of the backshift operator defined in equation (3). Equation (A.19) thus takes the form Xt = Zt − 0.8410Zt−1 Xt = (1 − 0.8410B)Zt Xt = θ (B)Zt ,

(A.22)

where θ (B) = 1 − 0.8410B. Solving θ (B) = 0, the root, 1.1891, lies outside the unit circle and, hence, the MA(1) model for GBP Capital is both stationary and invertible.

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A.3.4 GBP Capital: Testing the Residuals

The rescaled residuals obtained from the MA(1) model that was fitted to the transformed GBP Capital data are plotted in Figure A.3 (g). The graph shows a fairly strong deviation from zero mean at Apr-00. From the ACF plot of GBP Capital residuals in Figure A.3 (h), if more then 5% of the 40 (2) lags fall outside the bounds, the white noise hypothesis is rejected. As seen from the graph, only two lags lie outside the 5% rejection bands and therefore there is no reason to reject the MA(1) model on the basis of the autocorrelations. Compatibility of the residuals with a normal distribution can be decided by inspecting a histogram of the residuals. If the fitted model is appropriate, the histogram should show a mean close to zero and an approximate shape to a normal distribution. The histogram of the transformed, differenced and mean-corrected GBP Capital data shown in Figure A.3 (i) is not ideal for the Gaussian white noise assumption even though the mean is close to zero. The final test has already been specified by the Ljung-Box statistic test for randomness. With a low test statistic of 15.310 and p-value of 0.75840, the conclusion is that, at the 5% level, the data are uncorrelated and support the compatibility of the residuals with white noise.

A.4

EUR NP Account

A.4.1 EUR NP: Transforming the Series

Figure A.4 (a) describes the original EUR NP series. As a result of nil expenditures in the first 14 months, a subset of the series starting at 01 June 2001 was used. The subset is of length 70 with sample mean and variance $1.975M and 1.08730 × 1013 respectively. All values are positive and none are zero, therefore no adjustments need be made before applying a log transformation. To help identify any trend and/or seasonality, the sample ACF plot was used to observe the degree of dependence in the data. The EUR NP pre-transformed ACF, shown in Figure A.4 (b), shows (a) no trend and, (b) a seasonal component with period 12. Consequently, the analysis proceeded by first conducting a log transformation followed by a lag-12 and lag-1 differencing using the backshift operator, B, (1 − B)(1 − B12 ) ln Xt∗ = (1 − B − B12 + B13 ) ln Xt∗

∗ ∗ ∗ = ln Xt∗ − ln Xt−1 − ln Xt−12 + ln Xt−13 .

(A.23)

where X ∗ denotes the original (untransformed) series. Once the apparent deviations from stationarity have been removed, the sample mean, μ, is subtracted from each observation and a zero-mean stationary model is fit to the resulting series, i.e., ∗ ∗ ∗ − ln Xt−12 + ln Xt−13 −μ. Xt = ln Xt∗ − ln Xt−1

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Equation (A.24) is plotted in Figure A.4 (d) as the log transformed, twice differenced and mean subtracted EUR NP time series, Xt . A.4.2 EUR NP: Fitting the ARMA Model

The ACF and PACF of the transformed series (A.24) are plotted in Figure A.4 (e) and A.4 (f) respectively. Close examination of the Figures suggest that we may fit models with AR(12) or possibly MA(6). Using the Ljung-Box statistic and AICC for model selection, models in the range 0 ≤ (p, q) ≤ 7 were created for the EUR NP expenditure data. The AICC, Ljung-Box (h = 20) and p-value statistics were generated and sorted by minimum AICC. The low Ljung-Box statistic and high p-value (> 0.05) is an indication that there is no structure in the data (uncorrelated) up to lag 20. Table A.4 displays the prioritized statistics for the first 20 candidates. Table A.4: EUR NP model selection statistics

84

(p,q)

AICC

(2,6) (2,3) (2,4) (3,3) (3,6) (3,7) (1,1) (2,0) (3,4) (4,0) (0,6) (6,0) (2,5) (1,2) (3,0) (5,0) (1,3) (2,1) (2,2) (1,4)

167.447 170.106 171.904 172.714 172.909 173.709 173.958 174.383 174.733 175.083 175.168 175.170 175.197 175.714 176.453 176.609 177.043 177.121 177.364 178.061

Ljung-Box

p-value

18.383 27.610 30.132 34.722 16.615 17.890 39.467 32.244 30.631 31.829 31.091 25.512 21.509 43.770 37.488 32.324 33.959 31.830 35.183 34.778

0.56221 0.11895 0.06774 0.02164 0.67784 0.59467 0.00583 0.04077 0.06026 0.04516 0.054 0.18254 0.36774 0.00162 0.01022 0.03997 0.02641 0.04514 0.01915 0.02132

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Apr 05

Apr 05

Apr 03 Date

Apr 05

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0

5 10 15 20 25 30 35 40 Lag

e ACF of Transformed EUR NP

0

b ACF of EUR NP

0

5 10 15 20 25 30 35 40 Lag

h ACF of EUR NP Rescaled Residuals 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

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5 10 15 20 25 30 35 40 Lag

5 10 15 20 25 30 35 40 Lag

2

1

0

1

2

i Histogram of Rescaled Residuals

0

f PACF of Transformed EUR NP

0

c PACF of EUR NP

Figure A.4: (a)Time plot of monthly EUR national procurement liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed, lag-12 and lag-1 differenced subset starting 01 June 2000. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from ARMA(2,6) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals.

Jul 01

2

1

0

1

Apr 03 Date

d Transformed EUR NP

Apr 01

a EUR NP Account

g EUR NP Rescaled Residuals

Jul 01

3

2

1

0

1

2

0 Apr 99

5  106

1  107

1.5  107

7

Density

According to Table A.4, the best model based on AICC and the Ljung-Box statistic is an ARMA(2,6). The estimated AR and MA-coefficients φ1 , φ2 , θ1 , . . . , θ6 are respectively φ1 , φ2

   −1.4171, −0.9979, +1.0553,

θ1 , ..., θ6

+0.2982

 − 0.7375, +0.1353,

+0.5582,

 +0.5169 ,

with ratios φ j /SE(φ j ) for j = 1, 2 − 162.330,

and

− 114.528 ,

and θ j /SE(θ j ) for j = 1, . . . , 6 9.264,

1.815,

−4.392,

0.806,

3.397

4.538 ,

and

where, for example, SE(θ j ) is the standard error of θ j . Values of |θ j /SE(θ j )| 0.05) is an indication that there is no structure in the data (uncorrelated) up to lag 20. Table A.5 displays the prioritized statistics for all candidates.

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Apr 04 Date

Apr 06

Apr 03 Date

Apr 05

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Date

Apr 05

5 10 15 20 25 30 35 40 Lag

0

5 10 15 20 25 30 35 40 Lag

e ACF of Transformed EUR Capital

0

b ACF of EUR Capital

0

5 10 15 20 25 30 35 40 Lag

h ACF of EUR Capital Rescaled Residuals 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 0

5 10 15 20 25 30 35 40 Lag

c PACF of EUR Capital

0

0.05

0.1

0.15

0.2

5 10 15 20 25 30 35 40 Lag

10

5

0

5

10

i Histogram of Rescaled Residuals

0

f PACF of Transformed EUR Capital 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8

Figure A.5: (a)Time plot of monthly EUR Capital liquidated obligations over a period of seven years. (b) and (c) are ACF and PACF plots of the time series displayed in (a). (d) Log transformed and lag-1 differenced subset starting 01 November 2001. (e) and (f) are ACF and PACF plots of the differenced series displayed in (d). (g) Rescaled residuals from MA(1) model. (h) ACF of rescaled residuals. (i) Histogram of rescaled residuals.

Apr 03

g EUR Capital Rescaled Residuals

Dec 01

5 2.5 0 2.5 5 7.5 10 12.5

Apr 02

a EUR Capital Account

d Transformed EUR Capital

Dec 01

15

10

5

0

5

10

5  106 0 Apr 00

1.5  107 1  107

2  107

2.5  107

3  107

Density

Table A.5: EUR Capital model selection statistics

(p,q)

AICC

(0,1) (0,2) (1,1) (2,0) (1,2) (2,1) (1,0) (2,2) (0,0)

292.185 296.807 299.199 299.415 299.907 302.147 306.533 317.904 318.442

Ljung-Box

p-value

12.536 16.589 18.157 20.208 21.704 23.580 25.443 93.337 26.755

0.89638 0.67948 0.57709 0.44500 0.35678 0.26122 0.18500 0.00000 0.14229

The optimum model, according to Table A.5 and based on AICC, is an MA(1). The estimated MA-coefficient is θ = −0.8984 with θ /SE(θ ) = −11.8437 where SE(θ ) is the standard error of θ . The final model is therefore Xt = Zt − 0.8984Zt−1 ,

(A.33)

where Zt ∼ W N(0, 14.4075). Solving (A.32) for Xt∗ and substituting (A.33) converts the time series back to the original expenditures. The model prior to coefficient substitution is   ∗ +μ , (A.34) Xt∗ = exp Zt + θ Zt−1 + ln Xt−1 and after coefficient substitution,   ∗ +μ . Xt∗ = exp Zt − 0.8984Zt−1 + ln Xt−1

(A.35)

A.5.3 EUR Capital: Testing for Invertibility and Stationarity

The MA(1) model is already stationary by definition. To test for invertibility, we need to express the process defined by equation (A.33) in terms of the backshift operator defined in equation (3). Equation (A.33) thus takes the form Xt = Zt − 0.8984Zt−1 Xt = (1 − 0.8984B)Zt Xt = θ (B)Zt ,

(A.36)

where θ (B) = 1 − 0.8984B. Solving θ (B) = 0, the root, 1.11309, lies outside the unit circle and, hence, the MA(1) model for EUR Capital is both stationary and invertible.

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A.5.4 EUR Capital: Testing the Residuals

The rescaled residuals obtained from the MA(1) model that was fitted to the transformed EUR Capital data are plotted in Figure A.5 (g). The graph shows fairly strong deviations from zero mean at Apr-03 and Apr-05. From the ACF plot of EUR Capital residuals in Figure A.5 (h), if more then 5% of the 40 (2) lags fall outside the bounds, the white noise hypothesis is rejected. As seen from the graph, no lags lie outside the 5% rejection bands and therefore there is no reason to reject the MA(1) model on the basis of the autocorrelations. Compatibility of the residuals with a normal distribution can be decided by inspecting a histogram of the residuals. If the fitted model is appropriate, the histogram should show a mean close to zero and an approximate shape to a normal distribution. The histogram of the transformed, differenced and mean-corrected EUR Capital data shown in Figure A.5 (i) is not ideal for the Gaussian white noise assumption even though the mean is close to zero. The final test has already been specified by the Ljung-Box statistic test for randomness. With a low test statistic of 12.536 and p-value of 0.89638, the conclusion is that, at the 5% level, the data are uncorrelated and support the compatibility of the residuals with white noise.

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Annex B The FOREX Model B.1

Installation

Installation of the FOREX model is straightforward. The application and the two output files need to be copied to a folder on the user’s hard drive (Figure B.1). The main file, FOREX model.xls, is just under 10 MB in size and the OutputRpt.xls file is a small temporary file for @Risk reports. The overall size of the application is determined by the Output.xls file which is slightly under 50 MB to accommodate all output from 10,000 iterations. Double-clicking on the file FOREX Model.xls will launch the application and link to the current values in the Output.xls file. The FOREX model is run by coded Visual Basic for Applications (VBA) subroutines and macros. When prompted by Excel, the user must press the “Enable Macros” button for full functionality.

Figure B.1: The FOREX Model

B.2

The Risk Analysis Software

At this stage, unless the user has previously installed the risk analysis software, @Risk [68], the model will fail at the compile stage. At minimum, the standard version of this software must be installed before continuing. The @Risk software is crucial for conducting the random sampling for the discrete uniform probability distribution functions necessary for Filtered Historical Simulation of expenditures and standardized returns. There are 12 samplings (one per month) for each type of expenditure for a total of 72; and, 792 for each currency (22 samplings per month from 01 April 2006 to 31 March 2009) for a total of 2,376 for all three currencies. Each iteration therefore requires 2,448 samplings of the discrete uniform distribution.

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B.3

Starting the Model

B.3.1 The SplashScreen

Double-clicking on the application icon will bring up the splash screen shown in Figure B.2. A splash screen is an image that appears while a program or operating system is loading. Most large applications use splash screens to provide feedback to the user, and as advertising information of version and developer. This screen will disappear after a few seconds.

Figure B.2: The FOREX Model Splash Screen

B.3.2 The SwitchBoard

Following the splash screen, the Switchboard (Figure B.3) appears, displaying a series of six buttons with the following functionality: moving counterclockwise from L to R, Run FOREX for starting the simulation, Edit FOREX for unhiding all worksheets for editing, What is FOREX? for providing a brief description of the model, Reset Forex for debugging purposes (not enabled), Minimize or Maximize Switchboard for resizing the switchboard. Minimize will resize the switchboard to a rectangle 1.2 × 0.8 inches and move it to the top-right of the screen. Maximize will reverse the operations. The button acts as a toggle. The last button, Exit FOREX , exits the model after prompting the user to save any changes. Clicking the button Run FOREX starts the Filtered Historical Simulation using pre-selected settings and Latin Hypercube sampling. Execution of the simulation is indicated by the window shown in Figure B.4, where, for example, 2850 iterations (28.5%) of a total of 10000 iterations have already been completed. The simulation may be halted at any time by clicking the ‘Cancel’ button. Output results are then displayed for the iterations that have finished, however, convergence of the displayed output distributions may not have been achieved unless enough iterations have completed.

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Figure B.3: The SwitchBoard

Figure B.4: @Risk Simulating

B.4

Input Data Options

Clicking the button Run FOREX causes the message shown in Figure B.5 to be displayed. The FOREX Model is the only Excel application that can be open. Clicking ‘Yes’ will halt the simulation and return the user to the Windows environment to close all other Excel applications. Clicking ‘No’ will continue the simulation where the Enter Actuals form, shown in Figure B.6, is displayed.

Figure B.5: Close Excel Applications

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B.4.1 The Enter Actuals Form

The Enter Actuals form (Figure B.6) is meant to capture all current data as sources are updated. The dates in the first column are augmented as soon as new information is input. For example, at writing, the model is current until 31 March 2006 for both Account Expenditures and Foreign Exchange Rates. Once new data is entered, the date will automatically adjust to the following month, i.e., ‘May-06’. There are three sections to the form. The first section specifies the new expenditures for the three currencies and two accounts as of, for the example shown., end-of-April 2006. As there is significant data filtering required in gathering the actual expenditures, enough time should be allocated for this process to take place. After a value is entered and the user tabs to the next cell, the entry is automatically formatted for the model. The second section on the form specifies the actual liquidated exchange rates for the three currencies. Already listed is the forecasted rate given as a function of the standardized return (sampled via FHS) and the GARCH variance. The user just needs to overwrite the values and hit the ’Enter’ button for model acceptance. The third and final section lists the forecasted DND budget rates as specified by ADM(Fin CS) staff [67]. As for the liquidated rates, the user just needs to overwrite the values and hit the ’Enter’ button for model acceptance. After either accepting the defaults or making changes to the cells, the user must click the Enter button for the simulation to continue. The cells on the form are then copied to the applicable Excel worksheet for further processing by the model. Clicking the Cancel button immediately terminates the run with any changes ignored.

B.5

Model Output

B.5.1 The Report Sheet

Figure B.7 shows a portion of the main output worksheet from which all results can be accessed. This was for a run of 10000 iterations with outputs and discussion as specfied in section 6 . As noted at the top-left, the run took 00:09:42 (hrs:mins:secs) to complete. Also at the top-left, a ‘Print Report’ icon allows the user to print the entire report (4 pages of output) to the default printer. Just below the print icon there are two buttons that allow the user to either ‘Save Report to File’ or ‘Save All Results to File’. In the case of the former, the Report worksheet is saved to a new workbook that is named by the user. In the case of the latter, the Report worksheet and all output graphs are saved to a new workbook.

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Figure B.6: Input form to enter actual expenditures, liquidated foreign exchange rates and budget rates

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Figure B.7: The FOREX Model Report Sheet (partial)

B.6

FOREX Graphical Output

To the right of the icon there are six buttons that allow the user to navigate the VaR output graphs for each currency and account. For example, pressing ‘USD NP’ simply plots the 5th and 50th percentile VaR results of the model as in Figure B.8. The 5th percentile columns are shaded blue and the 50th are shaded red. Both show the plotted data value at the end of each column. Using the buttons on the chart, the graph can either be printed or saved to a file. Clicking the button ‘View Report’ will navigate the user back to the Report worksheet.

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Figure B.8: FOREX Model VaR Graphical Output (USD NP)

List of symbols/abbreviations/acronyms/initialisms Abs. Diff. ACF ADM(Fin CS) ADM(Mat) AICC AR ARCH ARIMA ARMA B CORA DB DCC DMG Compt DND DRDC ECP EGARCH EM ET EUR EWMA FHS FMAS FOREX FY GARCH GBP GJR-GARCH H0 H1 IBM ID ITSM K KR LADE LR

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Absolute Difference Autocorrelation Function Assistant Deputy Minister(Finance & Corporate Services) Assistant Deputy Minister(Materiel) Akaike’s Information Criterion (Modified) Autoregressive Autoregressive Conditional Heteroskedastic Autoregressive Integrated Moving Average Autoregressive Moving Average Billion Centre for Operational Research and Analysis Director Budget Defence Comptrollership Council Director Materiel Group Comptroller Department of National Defence Defence Research and Development Canada Engineering Change Proposal Exponential Generalized Autoregressive Conditional Heteroskedastic Economic Model Eastern Time Euro Exponentially Weighted Moving Average Filtered Historical Simulation Financial and Managerial Accounting Systems FOReign EXchange Fiscal Year Generalized Autoregressive Conditional Heteroskedastic Great Britain Pound Sterling Gosten, Jagannathan and Runkle - Generalized Autoregressive Conditional Heteroskedastic Null Hypothesis Alternate Hypothesis International Business Machines Corporation Identification Interactive Time Series Modelling Thousand Vendor Invoice (German) Least Absolute Deviations Estimator Likelihood Ratio

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M MA MAE ME MLE NP PACF Perc. PPP QQ RMSE S&P SA SAP SE TARCH USD VaR VBA WN Z-Score

Million Moving Average Mean Absolute Error Mean Error Maximum Likelihood Estimation National Procurement Partial Autocorrelation Function Percentile Purchasing Power Parity Quantile-Quantile Root Mean Square Error Standard & Poor’s G/L Account (German) Systems Applications and Products in Data Processing Standard Error Threshold Generalized Autoregressive Conditional Heteroskedastic United States Dollar Value at Risk Visual Basic for Applications White Noise Standard (or Normal) Score

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1630-1 (DMGOR) January 2007 Distribution List

Estimating Foreign Exchange Exposure in the Department of National Defence Reference: P. E. Desmier, Estimating Foreign Exchange Exposure in the Department of National Defence, DRDC CORA TR 2006–23, January 2007 (enclosed). 1. Like any institution engaged in international trade, the Department of National Defence (DND) is often required to make payment in foreign currency when acquiring new equipment and supplies to support military operations at home and abroad. Foreign currency exposure refers to the sensitivity of an organization’s cash flows to changes in the exchange rates. Given the high volatilities of various currencies, it is generally accepted that exchange rate exposure, if not quantified and managed properly, may expose the organization to significant budgetary consequences due to poor responsiveness to foreign exchange volatility. 2. This report documents the theory and application of a model that is built on forecasting expenditures for the ADM(Mat) National Procurement and Capital (equipment) accounts and the time-varying volatilities of foreign currency returns. These diverse methodologies are then combined into an overall risk model to determine the maximum expected loss from adverse exchange rate fluctuations over the budget year. This is recognized as the first step any organization must take before considering risk mitigation strategies to reduce and hopefully eliminate foreign transaction exposure. 3. This study also highlights certain policy implications for functional finance and performance/risk management specialists in the department. In particular, the VCDS Group through the Director Force Planning and Programme Coordination (DFPPC), and ADM(Fin CS) through Director Budget and Director Strategic Finance and Costing (DSFC), may want to examine the possibility of adjusting corporate budget allocations (quarterly) based on the results of the model. The department should also examine opportunities to apply the analytical approach to quantifying the financial risk in other budget expenditure areas subject to market/price risk such as bulk fuels, energy/hydro, and certain commodities (e.g., steel, ballistic materials, etc.) where expenditure amounts warrant. As the department embarks on large multi-year capital acquisitions and continues to be engaged in sizeable, complex overseas deployments, the need to measure and accurately assess financial risk has never been greater.

4. Questions or comments are welcome and should be addressed to the author, Dr. P.E. Desmier at (613) 996-8440, CSN 846-8440 or by email at [email protected]. Electronic copies of this report are also available upon request from [email protected].

R.G. Dickinson Director Joint & Strategic Analysis for Director General Centre Operational Research & Analysis Enclosures: 1 Distribution List COS ADM(Mat) C Prog DG Fin Mgt DGMPD DGIIP DFPPC (2 copies) DSFC (2 copies) DB (2 copies) DDSM DMG Compt DMGSP DMGPI CEFCOM J8 DRDC CORA//DG CORA/DOR(Joint)/DOR(MLA)/Chief Scientist (1 copy on circulation) DMG Compt 5-4 DRDC CORA Library (2) DRDKIM (2) Spares (8)

DOCUMENT CONTROL DATA (Security classification of title, body of abstract and indexing annotation must be entered when document is classified)

1.

ORIGINATOR (the name and address of the organization preparing the document. Organizations for whom the document was prepared, e.g. Centre sponsoring a contractor’s report, or tasking agency, are entered in section 8.)

2.

DRDC – Centre for Operational Research and Analysis NDHQ, 101 Col By Drive, Ottawa ON K1A 0K2 3.

SECURITY CLASSIFICATION (overall security classification of the document including special warning terms if applicable).

UNCLASSIFIED

TITLE (the complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S,C,R or U) in parentheses after the title).

Estimating Foreign Exchange Exposure in the Department of National Defence 4.

AUTHORS (Last name, first name, middle initial. If military, show rank, e.g. Doe, Maj. John E.)

Desmier, P.E. 5.

DATE OF PUBLICATION (month and year of publication of document)

6a.

January 2007 7.

NO. OF PAGES (total containing information. Include Annexes, Appendices, etc).

122

6b.

NO. OF REFS (total cited in document)

79

DESCRIPTIVE NOTES (the category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report, e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered).

Technical Report 8.

SPONSORING ACTIVITY (the name of the department project office or laboratory sponsoring the research and development. Include address).

DRDC – Centre for Operational Research and Analysis NDHQ, 101 Col By Drive, Ottawa ON K1A 0K2 9a.

PROJECT OR GRANT NO. (if appropriate, the applicable research and development project or grant number under which the document was written. Specify whether project or grant).

10a. ORIGINATOR’S DOCUMENT NUMBER (the official document number by which the document is identified by the originating activity. This number must be unique.)

9b.

CONTRACT NO. (if appropriate, the applicable number under which the document was written).

10b. OTHER DOCUMENT NOs. (Any other numbers which may be assigned this document either by the originator or by the sponsor.)

DRDC CORA TR 2006–23 11.

DOCUMENT AVAILABILITY (any limitations on further dissemination of the document, other than those imposed by security classification)

( X ) Unlimited distribution ( ) Defence departments and defence contractors; further distribution only as approved ( ) Defence departments and Canadian defence contractors; further distribution only as approved ( ) Government departments and agencies; further distribution only as approved ( ) Defence departments; further distribution only as approved ( ) Other (please specify):

12.

DOCUMENT ANNOUNCEMENT (any limitation to the bibliographic announcement of this document. This will normally correspond to the Document Availability (11). However, where further distribution beyond the audience specified in (11) is possible, a wider announcement audience may be selected).

13.

ABSTRACT (a brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual).

Quantifying foreign exchange risk is not a trivial process although it is generally accepted that the standard method for reporting financial risk today is the “Value-at-Risk” or VaR method. Simply put, VaR is defined as the predicted worst-case loss at a specific confidence level over a certain period of time. Thus VaR provides a quantitative measure of the downside risk of exposure in all foreign currency transactions. This report documents the theory and application of a model that is built on forecasting expenditures for the ADM(Mat) National Procurement and Capital (equipment) accounts and the time-varying volatilities of foreign currency returns. These diverse methodologies are then combined into an overall VaR model to determine the maximum expected loss from adverse exchange rate fluctuations over the budget year. This is recognized as the first step any organization must take before considering risk mitigation strategies to reduce and hopefully eliminate foreign transaction exposure. This study also illuminates certain policy implications for functional finance and performance/risk management specialists in the department. In particular, the VCDS Group through the Director Force Planning and Programme Coordination (DFPPC), and ADM(Fin CS) through Director Budget and Director Strategic Finance and Costing (DSFC), may want to examine the possibility of adjusting corporate budget allocations (quarterly) based on the results of the VaR model. The department should also examine opportunities to apply the VaR analytical approach to quantifying the financial risk in other budget expenditure areas subject to market/price risk such as bulk fuels, energy/hydro, and certain commodities (e.g., steel, ballistic materials, etc.) where expenditure amounts warrant. As the department embarks on massive multi-year capital acquisitions and continues to be engaged in sizeable, complex overseas deployments, the need to measure and accurately assess financial risk has never been greater. 14.

KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus-identified. If it not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title).

AICC Akaike’s Information Criterion ARMA Autocorrelation Function Autoregressive FHS Filtered Historical Simulation Foreign Exchange Exposure FOREX GARCH Generalized Autoregressive Conditional Heteroskedastic Interactive Time Series Modelling ITSM Maximum Likelihood Estimation MLE Moving Average National Procurement Quantile-Quantile Plots Time Series Value at Risk VaR

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DRDC CORA

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