358 CHAPTER 20 Incorporating CTAs into the Asset Allocation Process: A Mean-Modified Value at Risk Framework Maher Kooli V alue at risk has become a heavily used risk management tool, and an important approach for setting capital requirements for banks. In this study, we examine the effect of including a CTA in a traditional portfolio. Using a mean-modified value at risk framework, we examine the case of a Canadian pension fund and compute the optimal portfolio by minimizing the modified value at risk at a given confidence level. INTRODUCTION For the individual or the institutional investor who is simultaneously performance-oriented and risk-conscious, the key question is how best to achieve a higher overall rate of return with acceptable risk. The answer may be a diversified investment portfolio with some portion of the total assets invested in alternative investments. According to a survey by Nakakubo (2002), the alternative investment market reached $550 to $600 billion at the end of 2001. Pension funds also are increasing the proportion of alternative investments in their asset allocation. For many institutional investors, alternative investments are viewed largely as private, illiquid, alternative investments that include venture capi- tal, leveraged buyout, distressed securities, private equity, private debt, oil and gas programs, and timber or farmland. However, other alternative investment vehicles, such as hedge funds and commodity trading advisors (CTAs), also have observed a dramatic increase in investment and often provide access to c20_gregoriou.qxd 7/27/04 12:03 PM Page 358 Incorporating CTAs into the Asset Allocation Process 359 investment not easily available from traditional stock and bond investment. For instance, the Managed Accounts Reports (MAR) cites an increase in man- aged futures 1 from less than $1 billion in 1980 to almost $35 billion in 1999; hedge fund investment is now estimated to be over $300 billion. Further, Lint- ner (1983) uses the composite performance of 15 trading advisors and show that the return/risk ratio of a portfolio of trading advisors (or futures funds) is higher than a well-diversified stock/bond portfolio. Furthermore, he finds a low correlation between the returns of trading advisors and those of stocks, bonds, or a combined stock/bond portfolio. Lintner examines the 1979 to 1982 period. Schneeweis and Spurgin (1997) show that various CTA and hedge fund, energy-based investment provide risk and return opportunities not available from a wide range of traditional commodity investments or real estate investments. The Chicago Mercantile Exchange (1999) showed that for the 1980 to 1998 period, managed futures investments (as measured by the Barclay CTA Index) had a compound annual return of 15.8 percent. That compares very favorably with the 17.7 percent return that common stocks had during the same period, one of the strongest stock markets in U.S. his- tory. Further, it exceeded the 11.8 percent return on bonds. Moreover, during a similar period (1980 to 1997), analysis shows that a portfolio that com- prised some managed futures had similar profitability with far less risk. Liang (2003) finds that CTAs are good hedging instruments for hedge funds, fund of funds, and equity markets when the others are not well hedged. This is especially true in down markets. Schneeweis and Georgiev (2002), in exam- ining the benefits of managed funds, show that CTAs reduce portfolio volatil- ity risk, enhance portfolio returns in economic environments in which traditional stock and bond investment media offer limited opportunities, and participate in a wide variety of new financial products and markets not avail- able in traditional investor products. However, they note that for managed futures to grow as an investment alternative, individuals need to increase their knowledge and comfort level regarding the use of managed futures in their investment portfolios. For instance, there is still some confusion about the performance of CTAs as supply has expanded. In this study we first analyze the risk and return benefits of CTAs, as an alternative investment, using a more precise measure of risk. Then, we show how CTAs can be integrated into existing investment strategies and how to determine the optimal propor- tion of assets to invest in such products. 1 The term “managed futures” describes an industry made up of professional money managers known as commodity trading advisors. These trading advisors manage client assets on a discretionary basis using global futures markets as an investment medium. c20_gregoriou.qxd 7/27/04 12:03 PM Page 359 360 PROGRAM EVALUATION, SELECTION, AND RETURNS MEAN-MODIFIED VALUE AT RISK FRAMEWORK Investment decisions are made to achieve an optimal risk/return trade-off from the available opportunities. To meet this objective, the portfolio man- ager has to identify the set of assets that are the most efficient, in the sense of providing the lowest level of risk for a desired level of expected return, and then to select one combination that is consistent with the risk aversion of the investor. Mean-variance analysis has been increasingly applied to asset allocation and is now the standard formulation of the investment deci- sion problem. Although the principle of identifying portfolios with the required risk and return characteristics is clear, the proper definition of risk is vague. Risk may be defined differently according to the sensibility and the objectives of the portfolio manager. One manager might define risk as the probability of underperformance relative to some benchmark level of return, while another may be more sensitive to the overall magnitude of a loss. In a mean-variance framework, risk is defined in terms of the possible variation of expected portfolio returns. The focus on standard deviation as the appropriate measure for risk implies that investors weigh the probabil- ity of negative returns equally against positive returns. However, it is highly unlikely that the perception of investors to downside risk faced on invest- ments is the same as the perception to the upward potential. Thus, investors needed a more precise measure of downside risk. With the value at risk (VaR) approach, it is possible to measure the amount of portfolio wealth that can be lost over a given period of time with a certain probability. VaR has become a widely used risk management tool. The Basel Accord of 1988, for example, requires commercial banks to com- pute VaR in setting their minimum capital requirements (see Jorion 2001). One of the main advantages of VaR is that it works across different asset classes such as stocks and bonds. Further, VaR often is used as an ex-post measure to evaluate the current exposure to market risk and determine whether this exposure should be reduced. Our objective consists in drawing the efficient frontiers based on the VaR framework. We also use the Cornish-Fisher (1937) expansion to adjust the traditional VaR with the skewness and kurtosis of the return dis- tribution, which often deviates from normality. 2 We call the VaR with the Cornish-Fisher expansion modified VaR. Favre and Galeano (2002b) show that risk measured only with volatility will be lower than risk measured 2 Mina and Ulmer (1999) provide four methods—Johnson transformations, Cornish- Fisher expansion, Fourier method, partial Monte-Carlo—to compute the VaR for nonnormally distributed assets. c20_gregoriou.qxd 7/27/04 12:03 PM Page 360 Incorporating CTAs into the Asset Allocation Process 361 with volatility, skewness, and kurtosis. Thus, results with modified VaR will be less biased. For details on obtaining the normal VaR, the Cornish-Fisher expansion to VaR, and other VaR methods, see Christoffersen (2003). CHARACTERISTICS OF CTA Before we engage in a detailed analysis of the risk-return properties of the CTA, a word of caution is necessary: Unlike traditional asset classes (bonds and equity), where performance data and benchmarks are readily and reli- ably available, the infrastructure and reliability of performance data for alternative investments, in general, and CTAs, in particular, are still rather underdeveloped. In this chapter, the CTA Qualified Universe index 3 (CTA QU) is used to give an overall picture of CTA, as it is more representative of the performance of trading advisors as a whole and cannot be criticized as having selection bias. The sample portfolio is made up of CTA, Canadian, U.S., and interna- tional equities as well as domestic bonds. Canadian equities are represented by the Standard & Poor’s (S&P)/Toronto Stock Exchange index, the CTA by the CTA QU Index (from CISDM database), the U.S. equities asset by the S&P 500 Index, the international equities asset by the Morgan Stanley Capital Index for Europe, Asia, and the Far East (MSCI EAFE), and the bonds by the Scotia McLeod universe bond index. We use monthly data from January 1990 to February 2003. Within the assets considered (see Table 20.1), the CTA index is less risky than the S&P 500, the S&P/TSX, and the MSCI EAFE indices. In addition, CTA QU index possesses a higher Sharpe ratio than equity indices, indicating that CTAs offer superior risk-adjusted returns. These estimates may understate true risk, so monthly modified Sharpe ratios (using VaR instead of standard deviation) is also presented and confirms the advantage of the CTA QU index. Using VaR and modified VaR to measure risk, the CTAs are still less risky than equity indices. For instance, a one percent VaR of −5.3 percent for CTA QU index means that there is a 1 percent chance that the loss will be greater that 5.3 percent next month (or a 99 percent chance that it will be less than 5.3 percent). Besides very attractive risk adjusted return characteristics, one of the most important features of CTAs is their favorable correlation structure to traditional assets classes (see Table 20.2). By including CTAs in their port- folios, traditional asset managers are given the opportunity to produce more consistent returns with lower levels of risk in their global portfolio by 3 See www.cisdm.org for data and description of CTA Qualified Universe Index. c20_gregoriou.qxd 7/27/04 12:03 PM Page 361 362 PROGRAM EVALUATION, SELECTION, AND RETURNS means of diversification. CTA QU index has negative correlation to equity markets (−0.19 correlation to MSCI EAFE, −0.13 correlation to the S&P 500, and −0.12 correlation to the TSX/S&P). Furthermore, CTAs demon- strate remarkably low correlation with the bond market (0.20). Thus, including CTAs in a diversified asset portfolio may provide additional diversification benefits. TABLE 20.1 Characteristics of CTA and Traditional Asset Classes, January 1990 to February 2003 Annual Annual Excess Assets Mean Volatility Skewness Kurtosis CTA QU Index 11.8% 10.2% 0.7 2.2 SCM Bond Index 9.8% 5.5% −0.2 0.7 S&P/TSX Index 7.3% 15.7% −0.7 2.3 S&P 500 Index 11.1% 15.2% −0.5 0.6 MSCI EAFE Index 1.9% 16.4% −0.5 0.8 Monthly Monthly Monthly Normal Modified Sharpe Modified Assets VaR VaR Ratio Sharp Ratio CTA QU Index −5.9% −5.3% 0.18 0.10 SCM Bond Index −2.9% −3.4% 0.23 0.11 S&P/TSX Index −9.9% −13.9% 0.04 0.01 S&P 500 Index −9.3% −11.2% 0.11 0.04 MSCI EAFE Index −10.8% −13.1% −0.05 −0.02 TABLE 20.2 Correlations Across CTA and Traditional Asset Classes, January 1990 to February 2003 CTA QU S&P/ S&P MSCI Index SCM TSX 500 EAFE CTA QU Index 1 SCM Bond Index 0.20 1 S&P/TSX −0.12 0.32 1 S&P 500 −0.13 0.26 0.75 1 MSCI EAFE −0.19 0.20 0.66 0.70 1 c20_gregoriou.qxd 7/27/04 12:03 PM Page 362 Incorporating CTAs into the Asset Allocation Process 363 INCORPORATING CTA TO THE ASSET ALLOCATION PROCESS In this section, we show the results obtained by applying the mean-VaR framework explained previously. We compute the efficient frontier and the optimal portfolio allocation for a Canadian pension fund assuming that the portfolio manager has a VaR limit, that is, the manager does not want to lose more than a specified amount each month, with a specified probability (typically 1 or 5 percent). The individual asset classes can vary within specific limits. As a result, a relatively conservative asset allocation was chosen to match the alloca- tions of conservative investors, pension funds, and institutions. The weightings of individual asset classes are then changed within the permit- ted margins to minimize the normal VaR (see Table 20.3). This first step permits us to examine the effect of including a CTA in a traditional port- folio. In the second step, modified VaR values are used to measure risk more precisely. Table 20.4 shows that CTAs take the place of U.S equities. Once the weights of the tangent portfolios are obtained, we compute the monthly returns that each portfolio would have yielded from January 1990 to Feb- ruary 2003. Based on these monthly returns, we compute the average return over the period and the modified VaR. We obtain the results shown in Table 20.5, which shows that while the average return of the portfolio with 10 percent CTA is less than the one with 0 percent CTA, the level of risk, meas- ured with the modified VaR, is decreased by adding CTA. The modified Sharpe ratio is also improved by adding CTA investments in the traditional portfolio. TABLE 20.3 Upper and Lower Limits for Individual Asset Classes Asset Class Minimum Maximum Commodity trading advisors 0% 10% Canadian equities 10% 40% U.S. equities 0% 30% International equities 0% 30% Canadian bonds 25% 50% c20_gregoriou.qxd 7/27/04 12:03 PM Page 363 364 PROGRAM EVALUATION, SELECTION, AND RETURNS Further, Figure 20.1 shows the degree to which the sample portfolio with a CTA portion of maximum 10 percent is represented too positively if we do not take into account the skewness and kurtosis of the return distri- butions—in other words, if we do not use modified VaR. It is assumed that the investor is seeking an annual return of 7.2 percent with this sample portfolio. Our calculation using the Cornish-Fisher expansion shows that the investor will underestimate the risk by 14.28 percent if he or she is look- ing to achieve this return with normal VaR. The crucial question for an investor is whether including CTAs as an alternative investment makes sense for his or her portfolio. To assess this, we use both normal and modified VaR with traditional and nontraditional portfolios (with CTA). The arrows in Figure 20.2 show the shift in efficiency lines or, rather, the positive effect on including CTA QU index in a traditional portfolio. Figure 20.3 shows the added value of CTAs if skewness and kurtosis are taken into account (by using modified VaR as a risk measurement). The two TABLE 20.4 Portfolio Weights from Mean-VaR Optimization No CTA CTA Investment Asset Class Available Limit of 10% CTA QU Index 0% 10% MSCI EAFE 6% 8% S&P 500 29% 17% SCM Bond Index 50% 50% S&P/TSX 15% 15% Total 100% 100% TABLE 20.5 Average Return, Modified VaR, and Modified Sharpe Ratio Average Modified Modified Return VaR Sharpe Ratio Portfolio with 0% CTA 0.593% 5.93% 0.100 Portfolio with 10% CTA 0.581% 4.56% 0.128 c20_gregoriou.qxd 7/27/04 12:03 PM Page 364 Incorporating CTAs into the Asset Allocation Process 365 figures show the classic picture, as can be seen in a mean-variance dia- gram. It is obvious that including CTAs with high negative skewness and kurtosis values in a portfolio does bring a benefit in the sense of better risk-adjusted returns. 0.45% 0.50% 0.55% 0.60% 0.65% 0.70% 0.75% 0.80% 0.85% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00% Normal and Modified VaR Monthly Returns Efficient frontier with modified VaR. Efficient frontier with normal VaR. Annual return of 7.20% a b a b FIGURE 20.1 Pension Fund Portfolio with 10% CTA 0.40% 0.45% 0.50% 0.55% 0.60% 0.65% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% Normal VaR Monthly Returns with CTA without CTA FIGURE 20.2 Pension Fund Portfolio with and without 10% CTA c20_gregoriou.qxd 7/27/04 12:03 PM Page 365 366 PROGRAM EVALUATION, SELECTION, AND RETURNS CONCLUSION Nowadays it is clear that a traditional strategy that divides investments into asset classes is no longer sufficient. The results of this study provide impor- tant information to the investment community about the benefits of CTAs. We show that an efficiently allocated portfolio consisting of CTAs and tra- ditional assets should provide a better reward/risk ratio than an investment in traditional assets. We showed, as did Favre and Galeano (2002), that it is possible to use modified VaR risk measure to build a portfolio composed of traditional and alternative assets and that has the lowest probability of losing more than the modified VaR at a defined confidence level. However, investors must be very cautious in CTA selection. There are various CTAs with different characteristics and strategies. These differences need to be a major consideration, perhaps even more important than the decision of whether to invest in the asset class itself. Finally, analysis of alternative methods of measuring risk for alternative investments, in general, and CTA and hedge funds, in particular, is, of course, required. 0.40% 0.45% 0.50% 0.55% 0.60% 0.65% 0.70% 0.75% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% Modified VaR Monthly Returns without CTA with CTA FIGURE 20.3 Pension Fund Portfolio with and without 10% CTA c20_gregoriou.qxd 7/27/04 12:03 PM Page 366 . (bonds and equity), where performance data and benchmarks are readily and reli- ably available, the infrastructure and reliability of performance data for alternative investments, in general, and. private equity, private debt, oil and gas programs, and timber or farmland. However, other alternative investment vehicles, such as hedge funds and commodity trading advisors (CTAs), also have. true risk, so monthly modified Sharpe ratios (using VaR instead of standard deviation) is also presented and confirms the advantage of the CTA QU index. Using VaR and modified VaR to measure risk,