Hedging Demand Deposits Interest Rate Margins Jean-Paul LAURENT, Mohamed HOUKARI [email protected] ; [email protected] Alexandre ADAM, BNP Paribas Asset and Liability Management Mohamed HOUKARI, ISFA, Université de Lyon, Université Lyon 1 and BNP Paribas ALM Jean-Paul LAURENT, ISFA, Université de Lyon, Université Lyon 1

PRESENTATION OUTLOOK „

Overview and Context

„

Modeling Framework, Objective and Optimal Strategy

„

Empirical Results

„

Conclusions

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

2

Demand Deposits in Bank Balance Sheet „

Demand Deposits involve huge amounts …

(Bank of America Annual Report – Dec. 2007; Source: SEC) Average Balance 2007

(Dollars in millions)

2006

Assets Federal funds sold and securities purchased under agreements to resell

$

155,828

$

175,334

Trading account assets

187,287

145,321

Debt securities

186,466

225,219

Loans and leases, net of allowance for loan and lease losses

766,329

643,259

All other assets

306,163

277,548

Total assets

$

1,602,073

$

1,466,681

$

717,182

$

672,995

Liabilities Deposits

253,481

286,903

82,721

64,689

Commercial paper and other short-term borrowings

171,333

124,229

Long-term debt

169,855

130,124

70,839

57,278

1,465,411

1,336,218

136,662

130,463

Federal funds purchased and securities sold under agreements to repurchase Trading account liabilities

All other liabilities Total liabilities Shareholders’ equity Total liabilities and shareholders’ equity

„

$

1,602,073

$

1,466,681

US Banks are monitored by the SEC as for Interest Rate Risk

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

3

Demand Deposit Interest Rate Margin – Definition „

Demand Deposit Interest Rate Margin for a given quarter: …

„

Income generated by the investment of Demand Deposit Amount on interbank markets while paying a deposit rate to customers

Risks in Interest Rate Margins: …

…

Interest Rate Risk: „

1. Investment on interbank markets

„

2. Paying an interest rate to customers (possibly correlated to market rates)

„

3. Demand Deposit amount is subject to transfer effects from customers, due to market rate variations

Non hedgeable Risk Factors on the Deposit Amount: „

Business Risk: Competition between banks, customer behavior independent from market conditions, etc.

„

Model Risk

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

4

We need to focus on Interest Rate Margins…

„

… according to the IFRS (International accounting standards) : …

The IFRS recommend the accounting of non maturing assets and liabilities at Amortized Cost / Historical Cost

„

Being studied: Recognition of related hedging strategies from the accounting viewpoint …

Interest Margin Hedge (IMH).

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

5

Why do not we use the Demand Deposit Fair Value? „

„

The fair value of Demand Deposits: …

is computed by Discounting future interest rate margins on the DD activity

…

Risk-neutral expectation of the related sum

Demand Deposits are a complex financial product! …

The fair-value involves some pricing of non-hedgeable risks „

… „

Business risk, customers’ behaviour, etc.

Which risk-neutral measure should we use?

Practical concern for banking establishments …

Fair Value-based hedging strategies lack of robustness as for model specification.

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

6

Risk Mitigation within Interest Rate Margins

„

Hedging Demand Deposit Interest Rate Margins: …

We mitigate risk using Interest Rate Derivatives such as Interest Rate Swaps

…

We include a risk premium on interest rate markets „

„

Investing in long-term assets financed by short-term liabilities is rewarding.

Return-Risk Tradeoff between: …

Risk Reduction: „

…

Using Interest Rate Swaps

Return Opportunities: „

Friday, 30th 2009

Taking advantage of long term investment risk premium.

Hedging Demand Deposits Interest Rate Margins

7

PRESENTATION OUTLOOK „

Overview and Context

„

Modeling Framework, Objective and Optimal Strategy

„

Empirical Results

„

Conclusions

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

8

Setting the Objective Interest Rate Margin

IRM g (K T , LT ) = K T (LT − g (LT )) ⋅ ∆T

Deposit Amount at T Investment Market Rate during time interval [T,T+∆T] Customer rate at T

Mean-variance framework:

„

…

Including a return constraint – due to the interest rate risk premium

[

min E IRM g (K T , LT ) − S S

Friday, 30th 2009

]

2

under constraint

[

]

E IRM g (K T , LT ) − S ≥ r

Hedging Demand Deposits Interest Rate Margins

9

Dynamics for Market Rate „

Lt = L(t , T , T + ∆T )

Libor Market Model for Investment Market Rate

dLt = µ L dt + σ L dWL (t ) Lt

µL ≠ 0 „

Ex.: Brace, Gatarek, Musiela (1997)

Long-Term Investment Risk Premium

Coefficient specification assumptions: … Our model: µ L , σ L constant (and can be easily extended to time-dependent framework) …

‘Almost Complete’ framework „

Friday, 30th 2009

H. Pagès (1987), Pham, Rheinländer, Schweizer (1998), Laurent, Pham (1998) W µ L , σ L bounded and adapted to F L Hedging Demand Deposits Interest Rate Margins

10

Deposit Amount Dynamics „

Diffusion process for Deposit Amount

[

]

dK t = K t µ K dt + σ K d WK (t )

(US marketplace) …

Sensitivity of deposit amount to market rates „

…

Money transfers between deposits and other accounts

Interest Rate partial contingence.

680

4

660

3,5

640

3

620 2,5 600 2 580 1,5 560 1

540

Friday, 30th 2009

juil-07

avr-07

janv-07

juil-06

oct-06

avr-06

janv-05

juil-04

oct-04

avr-04

janv-04

juil-03

oct-03

avr-03

janv-03

juil-02

dWK (t ) = ρdWL (t ) + 1 − ρ 2 dWK (t )

oct-02

avr-02

janv-02

juil-01

oct-01

avr-01

oct-00

0 janv-01

500 janv-06

Incomplete market framework

juil-05

„

0,5

US Demand Deposit Amount US M2 Own Rate

520

oct-05

Business risk, …

avr-05

„

−1 < ρ < 0

Hedging Demand Deposits Interest Rate Margins

11

97

Euro Overnight Deposits

EuroZone − µˆ K = 10.19%, σˆ K = 6.56%

Friday, 30th 2009 Turkey - M1-M0

Hedging Demand Deposits Interest Rate Margins sept-07

mai-07

janv-07

sept-06

mai-06

janv-06

sept-05

mai-05

janv-05

sept-04

mai-04

janv-04

sept-03

mai-03

janv-03

sept-02

mai-02

janv-02

0

mai-01

0

sept-01

100

janv-01

500

mai-00

1000

sept-00

300

janv-00

400

mai-99

1500

sept-99

30000

2500 600

15000 200

200

0

UAH Bln.

500

janv-99

2000

mai-98

US and Euro Zone

sept-98

700

TRY Bln.

3000

USD bln.

800

janv-98

EUR bln. 3500

sept-97

19 -0 9 9 19 8-0 98 1 19 -0 9 5 19 8-0 99 9 19 -0 9 1 19 9-0 99 5 20 -0 0 9 20 0-0 00 1 20 -0 0 5 20 0-0 01 9 20 -0 0 1 20 1-0 01 5 20 -0 0 9 20 2-0 02 1 20 -0 0 5 20 2-0 03 9 20 -0 0 1 20 3-0 03 5 20 -0 0 9 20 4-0 0 1 20 4-0 04 5 20 -0 0 9 20 5-0 05 1 20 -0 0 5 20 5-0 06 9 20 -0 0 1 20 6-0 0 5 20 6-0 07 9 20 -0 0 1 20 7-0 07 5 -0 9

19

Deposit Amount Dynamics – Examples

dK t = K t (µ K dt + σ K dWK (t )) Emerging Markets (Turkey, Ukraine) 400

25000 350

20000 300

250

10000 150

5000 100

50

0

US Demand Deposits

Ukraine - M1-M0

Turkey − µˆ K = 51.74%, σˆ K = 37.38% 12

Modeling Deposit Rate – Examples „

We assume the customer rate to be a function of the market rate. …

Affine in general (US) / Sometimes more complex (Japan)

g (LT ) = α + β ⋅ LT

g (LT ) = (α + β ⋅ LT ) ⋅ 1{LT ≥ R}

United States

Japan

3.00% M2 Own Rate

0,9 JPY Libor 3M

0,8

2.50%

Japanese M2 Own Rate

0,7 2.00%

0,6

Affine Dependance

0,5

1.50%

0,4 1.00%

0,3

Quasi Zero Rates !

0,2 0.50%

0,1 USD 3M Libor Rate

Hedging Demand Deposits Interest Rate Margins

mars-07

sept-06

mars-06

sept-05

mars-05

sept-04

mars-04

sept-03

mars-03

sept-02

mars-02

sept-01

mars-01

sept-00

mars-00

sept-99

6.00%

5.00%

4.00%

3.00%

2.00%

1.00%

0.00%

Friday, 30th 2009

mars-99

0

0.00%

13

Sets of Hedging Strategies …

1st case: Investment in FRAs contracted at t=0

… is contained in …

H S 1 = {S = θ (LT − L0 ) ; θ ∈ R} …

2nd case: Dynamic self-financed strategies taking into account the evolution of market rates only

HS2

… is contained in …

…

T ⎧ ⎫ L L L = ⎨S = ∫ θ t dLt ; θ ∈ Θ ⎬ 0 ⎩ ⎭

Set of admissible investment strategies adapted to

F WL

3rd case: Dynamic strategies taking into account the evolution of the deposit amount T ⎧ ⎫ H D = ⎨S = ∫ θ t dLt ; θ ∈ Θ⎬ 0 ⎩ ⎭

z Friday, 30th 2009

Set of admissible investment strategies adapted to

F WL ∨ F WK

‘Admissible strategies’ are such that each of the sets above are closed Hedging Demand Deposits Interest Rate Margins

14

Variance-Minimal Measure „

Martingale Minimal Measure / Variance Minimal Measure T ⎛ 1T 2 ⎞ dP ⎜ = exp⎜ − ∫ λ dt − ∫ λdWL (t )⎟⎟ … Martingale Minimal Measure: dP 0 ⎝ 20 ⎠ „ Föllmer, Schweizer (1990)

…

In ‘almost complete models’, it coincides with the variance minimal measure:

„

…

⎡ dQ ⎤ P ∈ Arg min E P ⎢ Q∈Π RN ⎣ dP ⎥⎦

2

Delbaen, Schachermayer (1996)

N.B.: In our case, the Variance Minimal Measure density is a power λ − function of the Libor rate. dP ⎛ LT ⎞ σ ⎛1 ⎞ = ⎜ ⎟ exp ⎜ ( λ 2 − λσ L ) T ⎟ L

dP

Friday, 30th 2009

⎝ L0 ⎠

⎝2

Hedging Demand Deposits Interest Rate Margins

⎠

15

Optimal Dynamic Hedging Strategy – Case #2 T ⎡ ⎤ P minL E ⎢ IRM g (K T , LT ) − ∫ θ t dLt ⎥ θ ∈Θ 0 ⎣ ⎦

„

In Case #2, we determine:

„

The projection theorem applies …

Delbaen, Monat, Schachermayer, Schweizer, Stricker (1997)

…

In case #2, the solution consists in replicating

where …

2

ϕ S 2 (LT )

ϕ S 2 ( x ) = E P [IRM g (K T , LT ) LT = x ]− E P [IRM g (K T , LT )]

Under the “almost complete” assumption, this payoff can be replicated on interest rate markets. …

Friday, 30th 2009

N.B.: The latter payoff is a function of

LT

Hedging Demand Deposits Interest Rate Margins

16

Optimal Dynamic Hedging Strategy – Case #3 2 T ⎡ ⎤ P „ We recall the related problem: min E ⎢ IRM g (K T , LT ) − ∫ θ t dLt ⎥ θ ∈Θ 0 ⎣ ⎦ „

The solution is dynamically determined as follows: P ∂ E λ ** t [IRM (K T , LT )] θt = + EtP IRM g (K T , LT ) − Vt x** , θ ** ∂Lt σ L Lt

[ [

Delta term

+

Hedging Numéraire

] (

×

)]

Feedback term -

Shift between the RN anticipation of the margin and the present value of the hedging portfolio

Investment in some Elementary Portfolio which verifies This portfolio aims at some fixed return while minimizing the final quadratic dispersion. Friday, 30th 2009

2

⎡ λ ⎤ ⎡ ⎤ EP ⎢∫ dLt − (− 1)⎥ = min E P ⎢ ∫ θ t dLt − (− 1)⎥ θ ∈Θ ⎣ 0 σ L Lt ⎦ ⎣0 ⎦ T

T

Hedging Demand Deposits Interest Rate Margins

2

17

Optimal Dynamic Hedging Strategy – Some Remarks „

Case of No Deposit Rate: g (LT ) = 0 …

Explicit solution (Duffie and Richardson (1991)):

[

] , L )] ⎛ ρσ = ⎜1 +

EtP IRM g (K T , LT ) = K t Lt exp[(T − t )(µ K − ρσ K λ + ρσ K σ L )]

[

∂EtP IRM g (K T ∂Lt „

T

⎜ ⎝

K

σL

⎞ ⎟⎟ K t exp[(T − t )(µ K − ρσ K λ + ρσ K σ L )] ⎠

The model works for ‘almost complete models’ …

The Hedging Numéraire remains the following: t

HN t = 1 + ∫ 0

Friday, 30th 2009

λ σ L Lt

2

dLt

or

T T ⎡ ⎤ ⎡ ⎤ λ P P E ⎢∫ dLt − (− 1)⎥ = min E ⎢ ∫ θ t dLt − (− 1)⎥ θ ∈Θ ⎣ 0 σ L Lt ⎦ ⎣0 ⎦

Hedging Demand Deposits Interest Rate Margins

2

18

PRESENTATION OUTLOOK „

Overview and Context

„

Modeling Framework, Objective and Optimal Strategy

„

Empirical Results

„

Conclusions

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

19

Comparing Strategies in Mean-Variance Framework „

Efficient Frontiers …

Dynamic Efficient Frontier vs. Other Strategies at minimum variance point … More discrepancies between strategies when the deposit rate escapes from linearity Mean-Variance Framework - No Deposit Rate

Mean-Variance Framework - Barrier Deposit Rate Barrier Threshold = 3,00% - L(0) = 2,50% Deposit Rate = a. L(T) + b if L(T) > Threshold; a = 30% ; b = -0,50%

3,45

3,20

3,40 3,35 Expected Return

Expected Return

3,15

3,10

3,05

3,30 3,25 3,20 3,15

3,00 3,10 2,95 0,20

0,22

0,24

0,26

0,28

0,30

0,32

0,34

0,36

0,38

0,40

3,05 0,15

0,20

0,25

0,30

0,35

Standard Deviation

Blue: Unhedged Margin

Green: Delta-Hedging at t=0 only

Red: Optimal Dynamic Strategy following only market rates

Purple: Dynamic Delta-Hedging

„

0,40

0,45

0,50

0,55

0,60

0,65

Standard Deviation

The performances of other hedging strategies strongly depend upon the specification of the deposit rate.

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

20

Dealing with Deposits’ ‘Specific’ Risk „

Comparing the optimal dynamic strategy following only market rates (blue) and the optimal dynamic strategy following both rates and deposits (pink): …

„

At minimum variance point (risk minimization)

As expected, the deposits’ ‘specific’ risk is better assessed using a dynamic strategy following both rates and the deposit amount Risk Reduction and Correlation Total Deposit Volatility = 6.5% - K(0) = 100 0,35

Hedged Margin Standard Deviation

0,30 0,25 0,20 0,15 0,10 Optimal Dynamic Hedge (Rates)

0,05

Optimal Dynamic Hedge (rates + deposits)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Deposit / Rates Correlation Parameter

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

21

Robustness towards Risk Criterion „

The mean-variance optimal dynamic strategy (following deposits and rates) behaves quite well under other risk criteria …

Example of Expected Shortfall (99.5%) and VaR (99.95%). ES (99.5%)

Standard Deviation Barrier Deposit Rate

Expected Return Level

Risk Reduction

Level

VaR (99.95%)

Risk Reduction

Level

Unhedged Margin

3.16

0.39

Static Hedge Case 1

3.04

0.28

-0.11

-2.34

-0.32

-2.26

-0.36

Static Hedge Case 2

3.01

0.23

-0.16

-2.26

-0.24

-2.04

-0.14

Jarrow and van Deventer

3.01

0.24

-0.15

-2.35

-0.33

-2.25

-0.35

Optimal Dynamic Hedge

3.01

0.22

-0.17

-2.38

-0.36

-2.29

-0.39

The optimal dynamic strategy features better tail distribution than for other strategies …

Blue: Optimal Dynamic Strategy (following rates) … Pink: Optimal Dynamic Strategy (following both deposits and rates)

-1.90

Probability Densities Hedging Following Rates vs. Hedging Following Deposits and Rates 1,6 1,4

Hedging Following Rates

1,2

Hedging Following Rates and D it

1,0 Density

„

-2.02

Risk Reduction

0,8 0,6 0,4 0,2 -

-

0,50

1,00

1,50

2,00

2,50

3,00

3,50

4,00

4,50

5,00

Interest Rate Margin Level (incl. Hedge)

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

22

Dealing with Massive Bank Run „

Introducing a Poisson Jump component in the deposit amount:

[

]

dK t = K t µ K dt + σ K d WK (t ) − dN (t )

(N (t ))0≤t ≤T is assumed to be independent from WK and „

WL

∂EtP [IRM (KT , LT )] λ + EtP IRM g (K T , LT ) − Vt x** , θ ** Then, we have: θ = ∂Lt σ L Lt ** t

[ [

] (

)]

EtP ⎣⎡ IRM g ( KT , LT ) ⎦⎤ = e −γ (T −t ) × (Previous conditional expectation term) …

„

Due to independence, the jump element can be put out the conditional expectations

N.B.: When a bank run occurs, the manager keeps investing the current hedging portfolio’s value in the Hedging Numéraire

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

23

PRESENTATION OUTLOOK „

Overview and Context

„

Modeling Framework, Objective and Optimal Strategy

„

Empirical Results

„

Conclusions

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

24

Conclusions (1) „

A dynamic strategy to assess risk in mean-variance framework …

„

Results about Mean-variance hedging in incomplete markets yield explicit dynamic hedging strategies

Practical Conclusions: …

Better assessment of deposits’ ‘specific’ risk with a dynamic strategy taking into account both deposits and rates;

…

Lack of stability for other strategies towards the deposit rate’s specification;

…

Robustness towards risk criterion

…

No negative consequences as for tail distribution

…

Additivity of Optimal Dynamic Strategies „

Applicable to various balance sheet items

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

25

Conclusions (2)

„

We use some mathematical finance concepts: …

For Financial Engineering problems

…

with the aim of providing applicable strategies

…

And improve risk management processes

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

26

Technical References …

Duffie, D., Richardson, H. R., 1991. Mean-variance hedging in continuous time. Annals of Applied Probability 1(1).

…

Gouriéroux, C., Laurent, J.-P., Pham, H., 1998. Mean-variance hedging and numéraire. Mathematical Finance 8(3).

…

Hutchison, D., Pennacchi, G., 1996. Measuring Rents and Interest Rate Risk in Imperfect Financial Markets : The Case of Retail Bank Deposits. Journal of Financial and Quantitative Analysis 31(3).

…

Jarrow, R., van Deventer, D., 1998. The arbitrage-free valuation and hedging of demand deposits and credit card loans. Journal of Banking and Finance 22.

…

O’Brien, J., 2000. Estimating the value and interest risk of interest-bearing transactions deposits. Division of Research and Statistics / Board of Governors / Federal Reserve System.

Friday, 30th 2009

Hedging Demand Deposits Interest Rate Margins

27