YESIM TOKAT
Department of Economics, University of California, Santa Barbara, USA
SVETLOZAR T. RACHEV
Department of Statistics and Applied Probability, University of California, Santa Barbara, USA Institute of Statistics and Mathematical Economics, University of Karlsruhe, Germany e-mail: rachev@lsoe-4.wiwi.uni-karlsruhe.de
EDUARDO S. SCHWARTZ
Anderson School of Management, University of California, Los Angeles, USA
Contents
Abstract 510
1. Introduction 511
Part I: Review of the stochastic programming ALM literature 513
2. Stochastic programming ALM models 513
2.1. Chance-constrained model 513
2.2. Dynamic programming 515
2.3. Sequential decision analysis 516
2.4. Stochastic Linear Programming with Recourse (SLPR) 518
2.5. Dynamic generalized networks 520
2.6. Scenario optimization 521
2.7. Robust optimization 522
3. Multistage stochastic ALM programming with decision rules 523
4. Scenario generation 524
4.1. Discrete time series model 524
4.1.1. Multivariate approach 524
4.1.2. Cascade approach 525
4.2. Continuous time model 526
Part II: Stable asset allocation 528
5. Stable distribution 528
5.1. Description of stable distributions 529
5.2. Financial modeling and estimation 530
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
©2003 Elsevier Science B.V. All rights reserved
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5.2.1. Maximum likelihood estimation 531
5.2.2. Comparison of estimation methods 532
6. Multistage stable asset allocation model with decision rules 532
6.1. Scenario generation 534
6.2. Valuation of assets 536
6.3. Computational results 537
7. Conclusion 542
References 543
Abstract
Asset and liability management is the simultaneous consideration of assets and liabilities in strategic investment planning. In this chapter, asset and liability management models that use stochastic programming framework are reviewed. Most of these models describe the fi- nancial uncertainty by a set of representative scenarios. We propose to replace the classical assumption of Gaussian returns in the scenario generation with the stable Paretian distri- bution, which can capture the leptokurtic nature of financial data. A multistage stochastic asset allocation model with decision rules is analyzed. Optimal asset allocation under the Gaussian and stable Paretian returns are compared. Our computational results suggest that asset allocation may be up to 20% different depending on the utility function and the risk aversion level of the investor. Certainty equivalent return can be increased up to 0.13% and utility can be improved up to 0.72% by switching to the stable Paretian model.
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1. Introduction
Managing assets and liabilities is a concern for banks, pension funds and insurance compa- nies. Before the deregulation of interest rates, the market value of liabilities changed very little from year to year. However, after interest rates were deregulated in 1979, they showed much more volatility. This lead the institutional investors mentioned above to consider assets and liabilities simultaneously during their strategic planning. Strategic investment planning is the allocation of portfolio across broad asset classes such as bonds, stocks, cash and real estate considering the legal and policy constraints facing the institution. Em- pirical evidence by Brinson, Hood and Beebower (1986) suggests that asset allocation is the most important factor in determining investment performance.
Most of the early models in this field are either myopic or represent deterministic formu- lations of multiperiod problems. Hakansson (1971) show that solving a sequence of single period models optimizes investor’s long-run wealth or the expected utility of wealth.1They assume absence of transaction costs, market impact costs, and liquidity considerations.
However, these assumptions are not justifiable in many situations. Myopic models cannot capture long-term investment goals in the presence of transaction costs. There is consid- erable evidence of predictability in asset returns2and the myopic models do not take this empirical finding into account. These models tend to produce high portfolio turnovers and opportunistic asset trades.
There has been a growing interest in the development of multiperiod stochastic models for asset and liability management (ALM). Kusy and Ziemba (1986) developed a mul- tiperiod stochastic linear programming model for Vancouver City Savings Credit Union for a 5-year planning period. Their work suggests that their stochastic ALM model is supe- rior to 5-year deterministic models. Another successful application of multistage stochastic programming is the Russell–Yasuda Kasai model by Carino et al. (1994). The investment strategy suggested by the model resulted in extra income of $79 million during the first two years of its application (1991 and 1992). An ALM model designed by Mulvey (1994) has been implemented by the Pacific Financial Asset Management Company. Boender (1997) reported the success of a hybrid simulation/optimization scenario model for ALM of pen- sion funds in the Netherlands. The application of the model to a particular pension fund lead to a reduction of the yearly expected contributions of $100 million.
The ALM models that have gained applicability are based on stochastic programming with or without decision rules. In these models, the future economic uncertainty is modeled using discrete scenarios. Most of the models assume that the variables or the innovations of these variables follow normal distribution or the continuous time counterpart, Brownian
1 Merton (1969) and Samuelson (1969) show that a constant relative risk aversion investor chooses the same asset proportions independent of the investment horizon if the market is frictionless and returns are independent over time.
2 See for example Hodrick (1992), Bekaert and Hodrick (1992), Kandel and Staumbaugh (1996), and Brandt (1999).
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motion. In response to the empirical evidence about the heavy tails, high peak and possi- ble skewness in financial data, Fama (1965) and Mandelbrot (1963, 1967) propose stable Paretian distribution3as an alternative model. Among the alternative non-Gaussian distri- butions in the literature,4stable distribution has unique characteristics that make it an ideal candidate. The stable laws are the only possible limit distributions for properly normal- ized and centered sums of independent identically distributed random variables (Embrechts et al., 1997; Rachev and Mittnik, 2000). If a financial variable can be regarded as the result of many microscopic effects, then it can be described by a stable law. Stable distributions are leptokurtotic. When compared to normal distribution, they typically have fatter tails and higher peak around the center. Asymmetry is allowed. Due to its flexibilities, stable model fits the empirical distribution of the financial data better [see Mittnik et al. (2000)].
Gaussian distribution is a special case of stable distribution. In fact, it is the only distribu- tion in the stable family with a finite second moment. Although autoregressive conditional heteroskedastic models driven by normally distributed innovations imply unconditional distributions that possess heavier tails, there is still considerable kurtosis unexplained by this model. Mittnik, Paolella and Rachev (2000) present empirical evidence favoring sta- ble hypothesis over the normal assumption as a model for unconditional, homoskedastic conditional, and heteroskedastic conditional distributions of several asset return series.
The purpose of this chapter is to review the stochastic programming models in the ALM literature and to analyze an asset allocation problem in the presence of heavy tails. In the first part of the chapter, we review ALM models that utilize stochastic programming methodology. In the second part, a multistage asset allocation model with decision rules is analyzed under the Gaussian and stable returns scenarios. Our computational results suggest that if the investor has very high or low risk aversion, then the normal and sta- ble scenarios result in similar asset allocations. However, when the risk aversion level is between the two cases, the two distributional assumptions may result in considerably dif- ferent asset allocations depending on the utility function and the risk aversion level of the decision maker. The investor may reduce his equity allocation up to 20%, increase his cer- tainty equivalent wealth up to 0.13% and improve his utility by 0.72% by switching to stable model.
Section 2 reviews the stochastic programming ALM models without imposing any de- cision rules. Section 3 describes stochastic programming models that assume the investor uses the same decision rule to update the asset allocation every period. In Section 4, we present the scenario generation methods available in the ALM literature. In the second part of the chapter, a stable asset allocation model is described. Section 5 states the rea- sons for desirability of the stable model, describes the distribution and presents estimation methods. In Section 6, we set up the asset allocation model and report the computational results. Section 7 concludes.
3 We will call it stable distribution from now on.
4 A well known alternative to stable model is Student-tdistribution. A major drawback of Student-tdistribution is its lack of stability with respect to summation, i.e., a portfolio of Student-t distributed asset returns does not have Student-tdistribution. It is not supported by a central limit theorem. Student-t distribution is a symmetric distribution and it cannot capture the possible skewness in financial data.
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