FO RM AT

Developing and Implementing a Stochastic Decision-Support Model Within an Organizational Context: AN Y

Part I—The Model

STEIN W. WALLACE

TH

IS

AR

TI

Insurance, and Gjensidige NOR Non-Life. The asset management company is a separate legal entity that is owned by the bank. The company manages a large part of the assets of the Life and Non-Life companies, as well as funds for clients outside the Gjensidige-NOR Group. Gjensidige NOR Asset Management employs about 90 people, of which the front office (market operations) employs about 30. The front office is split into equities, fixed income, and tactical asset allocation. The tactical asset allocation team (TAA team) employs seven people. This team runs an internal macro hedge fund. There has been real money behind the hedge fund since July 1999. The core tool for managing this fund is a tactical asset allocation model (TAA model). The model has been developed over a period of five to six years in co-operation with the Norwegian University of Science and Technology, and lately, Molde University College. This is the first in a series of three articles, the main purpose of which is to discuss the development and use of a stochastic-programming based decision-support model. The model has been adapted to the organization and to the investment philosophy of the TAA team. However, the organization of the TAA team has also been influenced by the development of the model. We will show how the model is used as a guide to collect input data, distribute responsibilities, and make investment decisions. The framework has been useful for introducing and orienting new employees to the

IT

IS

IL LE

GA L

TO

is a professor of quantitative logistics at Molde University College, Norway. [email protected]

CE

is senior vice president in charge of the Department of Asset and Risk Allocation at Gjensidige NOR Asset Management, Norway. [email protected]

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DU

ERIK RANBERG

nterprise risk management (ERM) has been defined as the strategy that aligns the firm’s business with the risk factors of its environment in the pursuit of strategic objectives. It concerns the totality of risks facing a firm, and covers everything from conceptual frameworks and organizational issues to tools that integrate market, credit, liquidity, operational, and business risks. To facilitate ERM, mathematical models can be used for many purposes. For example, asset and liability management models can be used to align the risks on both sides of the balance sheet to properly take into account correlations within assets and liabilities, as well as between the assets and liabilities. But for such a model to be useful in a firm, it must be properly aligned with the organization. Responsibilities must be clear and incentives well thought out. Otherwise, models will suffer from inadequate data and model results will not find their ways to where decisions are made. ERM is outlined by Rosen and Zenios [2001], and aspects of ERM are discussed in all articles in a special issue of The Journal of Risk Finance; see the editorial by Zenios [2001]. Gjensidige NOR Asset Management is one of the three largest private asset management companies in Norway with approximately 100 billion NOK under management. The company is a part of the Gjensidige NOR Group, which is one of Norway’s three biggest financial groups. The group consists of Union Bank of Norway, Gjensidige NOR Life

PR O

is first vice president in the Department of Asset and Risk Allocation at Gjensidige NOR Asset Management, Norway. [email protected]

RE

KJETIL HØYLAND

CL E

IN

KJETIL HØYLAND, ERIK RANBERG, AND STEIN W. WALLACE

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investment philosophy of the TAA team. The model framework is flexible enough to allow for differences in analytic styles among the members of the team, and has proven to be well suited for generating consensus in reaching decisions. Further, the quantitative framework creates a basis for an efficient learning process. The modeling framework also encourages the team to engage in a structured and transparent investment process. An internal model is also by itself valuable for improving the em-ployees’ understanding of the link between market data used as input and the investment policy recommendations resulting from the model. There are of course several challenges in using such a framework, for example in calibrating the market expectations of different members in the group. In particular, members of a team most likely have differences in their personal risk preferences, which can skew decisions regarding optimal portfolio allocation. Further, we will explain how the modeling framework is used in dynamic and fast-changing financial markets. In this first article, we describe the main characteristics of the model, briefly compare the model’s properties with the familiar Markowitz mean-variance model, and motivate the modeling framework. In part 2, we describe how the model is applied and discuss how the model has influenced the organization. We close in part 3 with a discussion of the performance and implementation of the model over the past four years. MOTIVATION AND MODEL DESCRIPTION

The motivation for the modeling framework comes from what the TAA team believes is the two key sources of potential excess return (i.e., return in excess of a benchmark return): • Our own market expectations and • Portfolio construction.1 There are of course other sources of excess return, such as market timing, arbitrage opportunities, and dynamic utilization of risk limits (as well as luck and chance), but we believe that our own market expectations and the portfolio construction are the two most important elements. Most portfolio managers are subjective/judgmental on both elements. Hence, the decisions made are a mixture of judgmental views on the market and a judgmental view on how to create a portfolio consistent with the market expectations. Within the modeling framework we are about to describe, we are still judgmental on market expectations, but 56

we run a tactical asset allocation model to create an optimal portfolio given our own market expectations. Description of Market Expectations

To be able to run a portfolio optimization model, we must elicit and quantify the market expectations. The number of asset classes in the TAA model is typically between 20 and 30. It is a comprehensive task to specify distributional properties for, say, a 25-dimensional joint distribution. Several methodologies have been developed over the last few years. Many rely on regression procedures to reduce the number of random variables that need to be identified and modeled explicitly. This modeling is based on the development of stochastic processes or sampling (see for example Mulvey [1996]). The choice will depend on analytical preferences as well as the availability of data and understanding of the stochastic processes. We have chosen to consider the asset classes directly since that is what the decision-makers relate to. This is therefore an example of where the organization has affected the modeling. For tactical asset allocation, the TAA team believes that it is best to describe the return of an asset class by its marginal distribution, and its interrelation with the other asset classes. This raises the question of which properties of a distribution are in fact important to capture correctly in order to have a good decision support model. Høyland and Wallace [2001] discuss why the selection of important statistical properties is problem dependent. The Markowitz mean-variance model, for example, reacts only to the first two moments of a distribution (including the crossmoments). Any efforts to obtain correct descriptions of higher moments are a waste of time since the model does not react to them. To find a portfolio that has the best risk/reward balance, it is necessary to estimate the expected portfolio return and some measure of uncertainty. Hence, expected value and standard deviation are obvious statistical properties that need to be specified. However, as we shall show later, the shape of the probability distribution can influence the optimal portfolio choice to a larger degree than the magnitude of the uncertainty.2 Skewness and kurtosis give us the ability to express our subjective views on how the uncertainty is distributed around the mean. The motivation for introducing the skewness and the kurtosis comes from two basic facts: First, asset class returns are usually not normal (see for example Jackwerth and Rubinstein [1996]) and, second, the optimal portfolio is highly influenced by these distribution properties.

DEVELOPING AND IMPLEMENTING A STOCHASTIC DECISION-SUPPORT MODEL WITHIN AN ORGANIZATIONAL CONTEXT It is illegal to reproduce this article in any format. Email [email protected] for Reprints or Permissions.

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EXHIBIT 1 Scenario Generation Process Market expectations specified by marginal moments and correlations

Scenario genereration model

Scenario tree

EXHIBIT 2 Aggregation of Scenario Trees

The portfolio managers find the possibility of expressing views on the asymmetry and the degree of fat tails on the return distributions very useful. In many cases, return distributions are obviously asymmetric. The zero level in interest rates, for instance, naturally creates asymmetry. For equities, asymmetry is less obvious, but the pricing of the market relative to future expected earnings and the macroeconomic environment often indicate whether potential large moves are to the upside or to the downside. Investors are always faced with the possibility of a market crash. We consider it to be crucial to be able to express crash scenarios and assign them probabilities. Both the expected magnitude of a potential crash and the probability of its happening change over time, and portfolio managers seek to have an active view on these factors. Scenario Generation

Normal tree + Crash tree

=

Tree used in TAA model

For the TAA problem at hand, we concluded that for the marginal distributions, the first four moments, (i.e., expected value, standard deviation, skewness, and kurtosis) are needed to create a stable decision support model (see Høyland and Wallace [2001] for the meaning of a stable model). For describing the relationship among asset classes, correlations were found to be enough. The first four moments of a marginal distribution can either be derived from specifications of percentiles of the marginal return distributions, or they can be specified directly. As the experience of the TAA team has increased, the latter approach has become standard. Hence, for the TAA team, describing market expectations is equivalent to describing the necessary moments and correlations. It is a well-known fact that in stressed market conditions the longer-term average correlation structures tend to break down (which was the main reason for the failure of Long Term Capital Management; see Jorion [2000]). It can also be shown that crash scenarios to a large extent influence the optimal portfolio structure. To capture the crash scenarios in an appropriate way, we split the future into two states, one “normal” state and one “crash” state, and then specify the necessary marginal distributions and correlations separately for the two states. The two are then combined using an estimated probability of the two states occurring. The next subsection gives more details on how this is done.

Having estimated the four moments for the returns of all asset classes and the correlations between them, in both the “normal” and the “crash” state of the world, we need to transform these expectations to a format suitable for the quantitative TAA model. Being a stochastic programming model (see Kall and Wallace [1994] for an overview), it needs discrete distributions. Each outcome in this discrete distribution consists of returns on all asset classes. In order to be able to solve our TAA model, the number of outcomes has to be limited. We create the outcomes by using an iterative procedure that combines simulation, Cholesky decomposition, and various transformations; see Høyland, Kaut, and Wallace [2003] for details. The scenario generation process is illustrated in Exhibit 1: The scenario generation model constructs a limited set of outcomes, in the academic literature known as a scenario tree, which is consistent with the specified moments and correlations; i.e., the tree has exactly the required properties. In order to capture the extreme events of a market crash, two scenario trees are generated this way (one for normal market conditions and one for crashes) and these are aggregated to a larger tree. The large tree is used as input to the TAA model; see Exhibit 2. The Model: Objective Function and Constraints

When creating an asset allocation mix, the goal is to find the optimal balance between risk and reward. Hence, the objective function in the TAA model reflects

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EXHIBIT 3 A Shortfall Cost Function with Quadratic Penalty shortfall cost 1.00 0.80 0.60 0.40 0.20

-4 %

-2 %

-0.20

0%

4% 6% 8% 2% Shortfall relative to target return

10 %

12 %

EXHIBIT 4 The Utility Function Consistent with the Shortfall-Cost Function in Exhibit 3

to determine the degree of utilization of the risk limit, which is what is done in the risk constraint. • Liquidity constraints. For a relatively large player in futures and options, liquidity is a constraint in certain markets, in particular in domestic markets (Norway). • Subjective constraints. When solving the model we often find that there are many different portfolios that have fairly equal risk/reward profiles. Subjective constraints are sometimes used in order to push the solution to a portfolio that best fits the portfolio managers’ judgmental views on the portfolio composition. Examples are constraints that limit the allocations directly, or constraints that limit the amount of risk taken on a single asset class relative to the total risk.

Utility

The Single Period Framework 0.10

Utility

0.05 -4 %

-2 %

-0.05 0 %

2%

4%

6%

8%

-0.10 -0.15 -0.20

Excess return

a trade-off between expected return and some measure of risk. As for a portfolio manager who is measured relative to a benchmark, the risk is to underperform the benchmark. We therefore use a shortfall risk concept where outcomes that give negative excess return (i.e., lower return than the benchmark) are penalized. A shortfall cost function measures the (subjective) cost of shortfalls of different magnitudes relative to the benchmark return. We let the shortfall cost function be convex; that is, we assume an increasing marginal penalty. A simple quadratic version of such a shortfall cost function is illustrated in Exhibit 3. The objective of the optimization is to maximize the expected return net of expected shortfall costs. The objective function will be linear for returns above the target return and strictly concave for returns below the target return. The utility function consistent with the shortfall cost function in Exhibit 3 is shown in Exhibit 4. The model has three types of constraints that limit positioning: • Risk constraints. A portfolio manager who is measured relative to a benchmark will usually have a limit on the maximum risk level. A key decision then is 58

In the literature we find many studies that argue for dynamic, multi-period models; see for example Carino and Ziemba [1998], Consigli and Dempster [1998], Consiglio, Cocco, and Zenios [2001], and Kouwenberg and Zenios [2001]. Despite the possible advantages of a multiperiod framework, the TAA team uses a single period model (technically a two-stage stochastic programming model). A multi-period framework radically increases the number of distributions that must be estimated or specified. Even in a single period framework the estimations are extensive and time consuming, so moving to a multi-period setting would be a severe challenge from a practical point of view. Clearly, if the necessary distributions are too extensive, the quality of the input data will decrease. Going forward, though, we will consider moving into a multi-period framework and investigate the pros and cons. In particular, we believe that the multi-period setting is useful when options are allowed as investment vehicles. Further, we hope that we can achieve a better understanding of the dynamics of portfolio management by analyzing the solutions from a multi-period decision model. Comparison with the Mean-Variance Model

The model differs from the standard mean-variance framework in at least four aspects: • • • •

transaction costs treatment of risk asymmetric and fat-tailed distributions sensitivity to changes in input parameters

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EXHIBIT 5 Marginal Central Moments

Money Market Bonds Equities E(return)

5.0%

6.6%

9.0%

Std. dev.

0.0%

8.0% 20.0%

Skewness

0

0

0

Kurtosis

3

3

3

EXHIBIT 6 Correlation Estimates

Money Market

Bonds

Equities

Money Market

1

0

0

Bonds

0

1

0

Equities

0

0

1

EXHIBIT 7 A Distribution with Skewness of +1 0.12 0.1 0.08 0.06 0.04 0.02

-3 8.

92 -2 % 8. 00 -1 % 7. 07 % -6 .1 4 % 4. 78 % 15 .7 1 % 26 .6 4 % 37 .5 7 % 48 .4 9 % 59 .4 2 % 70 .3 5 %

0

In the standard mean-variance model, transaction costs are not taken into account. We shall see in the next section that derivatives are used as investment vehicles. When trading futures, the transaction costs in terms of broker fees are low. The transaction costs are, however, larger if we take into account the bid-offer spread and the potential market impact. With these costs included, the transaction costs do influence the optimal asset allocation mix. We saw earlier how risk is defined in the TAA model. Only downside risk is penalized in the objective function. This is in contrast to the mean-variance framework where

variance—i.e., deviations from the mean, both down- and upward—is treated equally. With symmetric return distributions (as the mean-variance framework assumes) this is only a conceptual problem, since using semi- (downside) variance or variance as risk measures will lead to the same solution. However, with asymmetrical distributions this is not only a conceptual problem. Let us illustrate and quantify the effect by an example. Consider a case with three asset classes, cash, bonds, and equities. Expectations for the returns on these asset classes are given in Exhibits 5 and 6. To simplify the presentation, no crash scenarios are included. Note that the expectations in Exhibits 5 and 6 are consistent with a multi-variate normal distribution. We now run the scenario generation model and create 1,000 outcomes with statistical properties as in Exhibits 5 and 6. We then solve with the meanvariance model and the model proposed in this article while targeting equal risk levels. With a reasonable shortfall cost function, the optimal solutions obtained from the two approaches are very similar. Now, let us assume the return distribution for equities is skewed to the right as in Exhibit 7. Solving the mean-variance model leads to exactly the same solution as before, since this model only cares about the two first moments of the distribution. With the asymmetrical risk measure, however, the optimal holding of equities rises from 17.5% to 26.4%. (In addition to the market expectations, the only other inputs are the parameters defining the shape of the shortfall cost function. Changing these parameters within reasonable bounds does not significantly alter the result.) The equity holding increased because the (positive) asymmetry in the return distribution of equities fits the asymmetry of the utility function. As explained earlier, the objective is to maximize the expected return net of expected shortfall costs, where the marginal shortfall cost is increasing with respect to the shortfall. Since outcomes with relatively large negative excess returns give large negative contributions to the objective function value, the model will seek to avoid such outcomes. Since a rightskewed return distribution has less probability mass in the left tail of the distribution relative to a symmetrical return distribution, the right-skewed return distribution will be preferred relative to a symmetric return distribution (all other factors being equal). The above example leads us to the final difference between the two approaches. The mean-variance framework has been criticized for the results being oversensitive to changes to the input parameters, in particular to

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the expected returns. With our framework, the sensitivity is lower. This is because there are more properties than the mean, variance, and the correlations that influence the solution. As argued above, the model will for instance prefer asset classes that are positively skewed as they fit the utility function, making the optimal holdings less sensitive to changes in other parameters such as the mean. At this point, however, this is only an empirical and intuitive result which may warrant future research.

Consiglio, A., F. Cocco, and S.A. Zenios. “The Value of Integrative Risk Management for Insurance Products with Guarantees.” The Journal of Risk Finance, Spring 2001, pp. 6-16. Høyland, K., M. Kaut, and S.W. Wallace. “A Heuristic for Moment-Matching Scenario Generation.” Computational Optimization and Applications, 24 (2003), pp. 169-185. Høyland, K., and S.W. Wallace. “Generating Scenario Trees for Multistage Decision Problems.” Management Science, 47 (2001), pp. 295-307.

CONCLUSION—PART 1

In this first article in a series of three, we have described the model itself and its relation to other similar models. In part 2 we will focus on how the model has affected the organization, and the organization the model. In part 3, we will discuss our experience with the model, including the presentation of its track record.

Jackwerth, J.C., and M. Rubinstein. “Recovering Probability Distributions from Option Prices.” Journal of Finance, 51 (1996), pp. 1611-1631. Jorion, P. “Risk Management Lessons from Long-Term Capital Management.” European Financial Management, 6 (2000), pp. 277-300. Kall, P., and S.W. Wallace. Stochastic Programming. Chichester, UK: Wiley, 1994.

ENDNOTES The methodologies presented in this article are largely based on the doctoral dissertation of Kjetil Høyland, who holds a Ph.D. in stochastic programming from the Norwegian University of Science and Technology. The work was performed under the supervision of Stein W. Wallace, formerly a professor of business administration at that school. Much of the work presented in this article was done while Stein W. Wallace visited the Centre for Advanced Study at the Norwegian Academy of Science and Letters, as group leader during the academic year 2000/01. 1 With this motivation for the modeling framework, the efficient market theory is abandoned. In particular, we believe markets are not efficient across asset classes, since investors in general put too little effort into constructing a portfolio of different assets in a consistent way. For a comprehensive discussion on market efficiency, see Malkiel [1990] and the references therein. 2 This is under the assumption of an asymmetrical utility function, for which we shall argue later.

Kouwenberg, R., and S.A. Zenios. “Stochastic Programming Models for Asset Liability Management.” Working Paper 0101, HERMES European Center of Excellence on Computational Finance and Economics, University of Cyprus, 2001. Forthcoming in Handbook of Asset and Liability Management, in the series Handbooks in Finance, North-Holland, publishers. Malkiel, B.G. A Random Walk Down Wall Street. New York: W.W. Norton & Company, 1990. Mulvey, J.M. “Generating Scenarios for the Towers Perrin Investment System.” Interfaces, 26 (1996), pp. 1-13. Rosen, D., and S.A. Zenios. “Enterprise-wide Asset and Liability Management: Issues, Institutions and Models.” Working Paper 01-18, HERMES European Center of Excellence on Computational Finance and Economics, University of Cyprus, 2001. Forthcoming in Handbook of Asset and Liability Management, in the series Handbooks in Finance, North-Holland, publishers.

REFERENCES Carino, D.R., and W.T. Ziemba. “Formulation of the Russell-Yasuda Financial Planning Model.” Operations Research, 46 (1998), pp. 443-449. Consigli, G., and M.A.H. Dempster. “Dynamic Stochastic Programming for Asset-Liability Management.” Annals of Operations Research, 81 (1998), pp. 131-161.

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Zenios, S.A. “Elements of Enterprise Risk Management.” Editorial, The Journal of Risk Finance, Fall 2001.

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DEVELOPING AND IMPLEMENTING A STOCHASTIC DECISION-SUPPORT MODEL WITHIN AN ORGANIZATIONAL CONTEXT It is illegal to reproduce this article in any format. Email [email protected] for Reprints or Permissions.

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