Nội dung

CHAPTER 9 Portfolio Models rystal Ball is very useful for investigating different allocations of investment funds to a set of risky assets. This chapter demonstrates the use of Crystal Ball and OptQuest for determining the optimal allocation of funds in an investment portfolio based on the decision maker’s risk tolerance. We use Crystal Ball and OptQuest to find an optimal allocation for a situation where we know the true optimal allocation, and one where we do not. C SINGLE-PERIOD CRYSTAL BALL MODEL In this example, we consider investing in the five asset classes listed in Table 9.1. Figure 9.1 shows a segment of the single-period Crystal Ball model in Portfolio.xls. Cells B12:E88 contain annual rates of return in percent on four asset classes. These rates of return were calculated from the indices contained in the Indices worksheet of Portfolio.xls. The indices were constructed from data collected from various sources for use only in the examples presented in this book. For more specific data on asset returns available to investors during the period 1926–2002 (and more), see the Center for Research in Security Prices (www.crsp.com), Ibbotson Associates (2006), or Bodie, Kane and Marcus (2008). Overview. Assume that you have four asset classes from which to choose for an investment portfolio. These classes are listed in Table 9.1 along TABLE 9.1 Means and standard deviations for annual total returns, 1 + ri , during the period 1926–2002 for four asset classes. 132 Asset Class Name Mean Return Std. Dev. Large-Company Stocks Small-Company Stocks Corporate Bonds U.S. Government Bonds LCS SCS CB USGB 1.1212 1.1734 1.0595 1.0559 0.2052 0.3607 0.0794 0.0699 133 Portfolio Models FIGURE 9.1 Spreadsheet segment from model to simulate a portfolio. TABLE 9.2 Pearson correlation matrix for annual total returns during the period 1926–2002 for four asset classes. LCS SCS CB USGB LCS SCS CB 1 0.787 0.174 0.104 1 0.035 −0.032 1 0.948 USGB 1 with their historical mean total returns, and standard deviations. The Pearson correlations for the five asset classes are in Table 9.2. The data from which these parameters were estimated are in Cells B12:E88 of the Stochastic Model tab of Portfolio.xls. To keep things simple we will assume that you have $10,000 to invest (Cell A4) and wish to find the optimal percentage of your $10,000 to invest in each of the asset classes (B8:E8). We ignore the effects of inflation for now. Forecast. The forecast for this example is portfolio value in cell F8, the value of the portfolio in Year 1. Stochastic assumptions. The assumptions are defined in cells N13:Q13 by using Batch Fit to find the distributions and Spearman correlations. 134 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL TABLE 9.3 Spearman correlation matrix for annual total returns during the period 1926–2002 for four asset classes. LCS LCS SCS CB USGB 1 0.811 0.124 0.013 SCS 1 0.047 -0.045 CB USGB 1 0.936 1 The assumptions are referenced in cells B7:E7. Batch Fit was limited to considering only normal or lognormal distributions for the historical returns. The normal distributions used for LCS and SCS were truncated at zero to reflect the limited liability of investing in equities. Batch Fit found that lognormal distributions fit better to CB, and USGB. Lognormal distributions are bounded by zero from below by definition, so needed no truncation. The Spearman correlation matrix computed by Batch Fit is shown in Table 9.3. Decision variables. Each decision variable in cells B8:E8 represents the percent of the initial investment allocated to the corresponding asset class. Each decision variable is defined with a lower bound of 0 percent, an upper bound of 100 percent, and a step size of 1.0 percent. By assuming a lower bound of 0 percent we are precluding the possibility of selling short any of the asset classes. The upper bound of 100 percent precludes borrowing to buy on margin or selling short. The step size of 1 percent is specified to make the optimization converge on a solution more quickly than with a smaller step size. Summary. The results shown here were found when using OptQuest to maximize the mean of the total return forecast in cell F8, with the additional requirement that the standard deviation of total return be no more than $1,000. The optimal allocations are (27 percent, 11 percent, 0 percent, 62 percent) for (LCS, SCS, CB, USGB) as shown in cells B8:E8 in Figure 9.1. For these allocations, the mean portfolio value is $10,865 with a standard deviation of $998.87. By running OptQuest for longer than the 60 minutes used to obtain these results, one may be able to improve on the results slightly. SINGLE-PERIOD ANALYTICAL SOLUTION The worksheet Analytical Solution in Portfolio.xls shows the the optimal allocation to each asset class based on using Solver in Excel to maximize the mean return subject to the standard deviation of the portfolio remaining less than or equal to 10 percent. The optimal allocation is (.22, .13, .00, .65) for (LCS, SCS, CB, USGB). This is the solution to the following mathematical programming problem: 135 Portfolio Models max E(P) = α1 ,α2 ,α3 ,α4 4
αi E(1 + ri ) i=1 subject to 4
αi = 1 i=1 √ α T Sα = σ (P) ≤ 10% 0 ≤ ri ≤ 1 for all i, where E(P) is the expected return on the portfolio, σ (P) is the standard deviation of the portfolio return, αi is the portfolio weight allocated to asset i = 1, 2, 3, 4 for the ordering of assets (LCS, SCS, CB, USGB) as shown in cells J18:J21, and ri is the mean rate of return for asset i. The 4 × 4 matrix S is the covariance matrix shown in cells H4:K7. For the optimal allocation, the expected return is 8.6 percent with a standard deviation of 10 percent. It is not surprising, but is reassuring that this allocation agrees with the OptQuest allocation. For this simple problem, we can be certain that the deterministic solution gives us the optimal allocation for the given values of the means, standard deviations, and correlations. Because OptQuest is a heuristic technique subject to sampling variation, there is no guarantee that it will find the globally optimal solution. However, the fact that the OptQuest solution is so close to the known deterministic solution in this simple case encourages us to believe that OptQuest will also find solutions that are very close to the global optimum in problems that are too complicated for deterministic solutions to be used. MULTIPERIOD CRYSTAL BALL MODEL For investment advisors, a major consideration in planning for a client in retirement is the determination of an appropriate asset allocation that will enable the client to withdraw funds necessary to maintain his or her desired standard of living. If a client withdraws too much or if investment returns fall below expectations, there is a danger of either running out of funds or reducing the desired standard of living. In the model presented in this section we assume that the client is a woman. As women have slightly longer life expectancies than men, our results are conservative when applied to the retirement portfolio planning problem for a man of the same age. The sustainable retirement withdrawal is the amount a client can withdraw periodically from her retirement funds for a selected planning horizon. This amount cannot be determined with complete certainty because of the stochastic nature of investment returns. In practice, the sustainable retirement withdrawal is determined by limiting the probability of running out of funds to some specified level, such as 136 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL 5 percent. The sustainable retirement withdrawal amount is typically expressed as a percentage of the initial value of the assets in the retirement portfolio, but is actually the inflation-adjusted monetary amount that the client will use each year for living expenses. Suppose that at the end of 2002, a 60 year-old woman has $1 million in a tax-deferred retirement account, and that she would like to withdraw $40,000 per year in 2002 dollars. Assume that she has a life expectancy of 30 years, and that the inflation rate will be 3.12 percent. Her withdrawal in each year is $40,000 adjusted by the inflation rate. That is, her withdrawal at the end of 2003 will be $41,248, at the end of 2004 will be $42,535, and so on. In this scenario, her retirement withdrawal, or ‘‘spending rate’’ is specified at 4 percent based on the initial balance of her total retirement funds. Her retirement portfolio planning problem is to allocate her initial $1 million to the asset classes available to her for investment, while maximizing her spending rate without running out of funds before she dies. An optimal choice of spending rate and allocations can be defined as one that limits to 5 percent the chance that she spends all of her accumulated wealth at the end of a deterministic, 30-year planning horizon. As a secondary issue, she may also be concerned with the value of her estate that is bequeathed to her heirs when she dies. For this model, the data in Portfolio.xls were used to parameterize the Crystal Ball model by using Crystal Ball’s Batch Fit tool. Table 9.1 shows the means and standard deviations of the annual returns (in percent) during the period 1926–2002 for four asset classes: large-company stocks (LCS), small-company stocks (SCS), corporate bonds (CB), and U.S. government bonds (USGB). Table 9.3 shows the Spearman correlation matrix for these four asset classes during the same period. The annual rate of inflation during 1926–2002 averaged 3.12 percent. The model in SustainableRetirementWithdrawals.xls generates stochastic returns in cells I11:L71 for the assets LCS, SCS, CB, and USGB in years 2003–2063. The four returns in each row are correlated with each other, but each row is statistically independent of the other rows. The Spearman correlations, in cells G78:J81, were computed by the Batch Fit tool. Crystal Ball’s Correlation Matrix tool was used to create the upper triangular matrix in cells K78:N81, which references the Spearman correlations. These are the values used by Crystal Ball during the simulation trials. The portfolio weights of the asset classes in cells I8:L8 are defined as decision variables in the range [0, 1] (i.e., no short sales nor margin purchases are allowed) in steps of 1 percent. At the end of each simulated year, a constant real amount (in 2002 dollars) is withdrawn for living expenses. The withdrawal amount is defined as both a decision variable and a forecast variable in Crystal Ball. The portfolio is assumed to be composed entirely of tax-deferred dollars and the effects of taxes on the amounts withdrawn are not considered. We consider two different planning horizons: (1) a deterministic 30-year horizon, and (2) a stochastic horizon equal to the remaining lifetime of the woman Portfolio Models 137 FIGURE 9.2 Custom distribution representing the death age of a 60-year-old female as given by the 2001 CSO mortality table. characterized by the 2001 Commisioner’s Standard Ordinary (CSO) mortality table. The distribution of the death age of a 60 year-old woman is shown in Figure 9.2. In OptQuest, the percentage allocation to each asset class and the withdrawal rate are specified as decision variables. The percentage allocations are bounded by 0 and 100 and are constrained to sum to 100 percent. An indicator variable is defined as a Crystal Ball forecast for the event that wealth at the end of each planning horizon is positive. The withdrawal rate forecast variable in cell H4 is specified as the objective to be maximized with an additional requirement that the mean of the positive-wealth indicator variable have a lower bound of 0.9540. To account for sampling error in the estimates used by OptQuest in its optimization algorithm, this lower bound exceeds 0.9500 by approximately two standard errors of the mean of the positive-wealth indicator variable resulting from 4,000 trials. Table 9.4 shows the allocations obtained for two different planning horizons: 1. A deterministic horizon of 30 years. 2. A stochastic horizon equal to the woman’s remaining lifetime. From a final run of 10,000 trials of the Crystal Ball model for the deterministic, 30-year horizon, the estimated median value of the woman’s wealth is $3.79 million, with a 95.30 percent probability of being solvent (wealth greater than zero) at the end of 30 years. Figure 9.3 shows the distribution of wealth at the end of 30 years. For the stochastic, remaining lifetime horizon, the estimated median value of her 138 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL TABLE 9.4 Optimal asset allocations and sustainable withdrawal rates for two planning horizons. Horizon LCS SCS CB USGB Spend 30-year Rem. life .25 .23 .12 .13 0 0 .63 .64 4.08% 4.53% estate is $1.78 million, with a 95.25 percent probability of leaving to her heirs an estate greater than zero. Figure 9.4 shows the distribution of her estate. Her spending rate is 4.08 percent with a fixed 30-year horizon, and 4.53 percent with a stochastic horizon. With an initial investment of $1 million, the difference between the spending rates amounts to an additional $4,500 in 2002 dollars to spend each year. This differential stems primarily from the fact that the retiree has a high likelihood of dying before she reaches age 90. For the assumptions stated above, our analysis quantifies the risk of dying broke if one chooses to withdraw more to live better during retirement. From comparison of the results in Table 9.4 and Figures 9.3 and 9.4, it is evident that very large stock allocations are not necessary for sustainability of withdrawals. In fact, allocation of a majority of the portfolio to equities will increase the likelihood of depleting the retiree’s funds during her lifetime. For both planning horizons, the split between debt and equity in the retirement portfolio is roughly 60–40. FIGURE 9.3 Distribution of wealth after 30 years with asset allocations listed in Table 9.4 for the 30-year planning horizon. 139 Portfolio Models FIGURE 9.4 Distribution of the retiree’s estate with asset allocations and withdrawal rate listed in Table 9.3 for the remaining-lifetime planning horizon. This example is intended only to demonstrate the use of Crystal Ball and OptQuest for financial planning. A more thorough analysis would include analyses of other potential investments, such as real estate or international equities, and more specific and recent data on the components of the asset classes used in this chapter. However, the analysis presented here does serve to inform individuals who are facing retirement about the tradeoffs involved in the retirement portfolio planning problem, and gives financial planners an idea of how to use Crystal Ball and OptQuest to demonstrate to their clients the risks involved in retirement planning.