# 5 Implementing the measures: a fund-of-funds perspective.

This chapter provides a hypothetical application of the techniques used to evaluate a given set of portfolios. The main goal is to illustrate the required steps for avoiding the pitfalls associated with data biases, performing the statistical tests, and interpreting the results. The application is developed from the perspective of a funds-of-funds (FOF) hedge fund manager. Evaluating the performance of hedge funds is a major responsibility for FOF managers. This task is complicated by the voluntary reporting of historical returns and the lack of transparency about the underlying strategies.5.1 Evaluating a Set of Individual Hedge Funds

Imagine that in early 2006, an FOF manager is invited to invest with the following three hedge funds: Lucky Capital, Merger LP, and Murky LLC. These funds have passed an initial due diligence screen that includes background checks, manager interviews, and other qualitative analysis. Available data for these funds include the organizational characteristics and historical returns reported to a hedge fund database. The characteristics for three actual funds (aside from a relabeling of fund names) are reported in Table 5.1.

For example, Lucky Capital is a Long/Short Equity hedge fund that was founded on September 21, 1993. Its performance data begin in the following month, resulting in 147 monthly return observations at the end of 2005. Other data about the fund are also provided. For example, Lucky Capital is an offshore fund with a six month lockup, a minimum investment amount of $5 million, a 1.5% management fee, and a 20% performance ("incentive") fee subject to a high-water mark. The high-water mark field is a one or a zero, indicating the presence or absence of a high water mark.

The monthly net returns have already been reduced by the management and performance fees paid by investors to the fund manager, and by the costs of trading. Table 5.2 presents the annual net returns for each fund, along with the value-weighted return on NYSE/AMEX stocks and the return on 30-day Treasury Bills. Although we show the annual figures in the table, the actual analysis is based on the monthly returns. We want to use all of the available data.

A careful performance measurement for these funds must account for several important features of the hedge fund industry. First, the voluntary nature of the reported data suggests that a fund might choose to report only the most optimistic data available. For example, Table 5.1 shows that Lucky Capital was added to the database in September 2001. However, there may be other, less successful funds managed by the same advisor that will never be reported to the database, thereby upward-biasing the inferences about an advisor's skill. It would be wise, therefore, for the FOF to distinguish between the full sample of returns and the sub-sample of returns (the shaded cells) that follow the date the fund decided to report. Section 6.1.2 provides a detailed discussion of the "backfilling bias."

The FOF must also account for the fact that a hedge fund's systematic risk exposure might change within the evaluation period, either through derivatives usage or dynamic trading strategies. For example, Merger LP reports Event Driven as its primary category. This category includes the merger arbitrage strategy, in which the shares of a merger target are purchased with the proceeds from selling short the shares of the proposed acquirer. Mitchell and Pulvino (2001) show that the returns from merger arbitrage are similar to the returns from writing put options on the stock market, because merger deals are less likely to succeed in depreciating markets. Chapter 3 discusses performance measures that accommodate time-varying systematic risk exposures.

Finally, a fund's illiquid asset holdings make it difficult for the fund to produce reliable estimates of net asset value (NAV). For example, the primary category of Murky LLC involves fixed income securities. If some of the debt securities are rarely traded, the available price quotes will be stale, thereby introducing a downward bias in estimates of a fund's systematic risk. Asness et al. (2001) show that stale NAV can have an important effect on estimates of hedge fund performance.

As an illustration, the FOF can use the following fund-level regression model to account for the backfilling bias, dynamic risk, and illiquid asset exposure:

[r.sub.pt+1] = ap + [b.sub.p][r.sub.mt+1] + [b.sub.1p][r.sub.mt] + [b.sub.2p][r.sub.mt-1] + [b.sub.3p][r.sub.mt-2] + [[LAMBDA].sub.p]Max([r.sub.mt+1],0) + [u.sub.t+1], (5.1)

where E[[u.sub.pt]] = E[[u.sub.pt]([r.sub.mt],[r.sub.mt-1],[r.sub.mt-2],[r.sub.mt-3])] = 0. The above model nests the traditional Jensen's alpha model in which [[LAMBDA].sub.p] = [b.sub.1p] = [b.sub.2p] = [b.sub.3p] = 0. The backfilling bias can be addressed by comparing the estimate of the Jensen's alpha using the full sample of returns (Model 1) with the alpha estimate after excluding the monthly returns that precede the date the fund was added to the database (Model 2). The above equation also nests the Merton-Henriksson (1981) market-timing model (Model 3), in which [b.sub.1p] = [b.sub.2p] = [b.sub.3p] = 0. The option payoff in the timing model allows the fund's beta to change based upon market performance, and therefore allows for a specific form of dynamic risk exposure. Finally, the lagged market model (Model 4) advocated by Scholes and Williams (1977) is nested within the constraint that [[LAMBDA].sub.p] = 0. Lagged market returns will have explanatory power to the extent that a fund's reported returns do not fully reflect the true economic return in a given month. Models 3 and 4 are estimated without backfilled data. In this illustration we use the US stock market index to form the OE portfolio. This may be appropriate for Lucky Capital and Merger LP. However, it is less appropriate for Muddy LLC, which is a fixed income fund. We would run the analysis with some fixed income returns in place of the US stock index and see how the results are affected.

5.1.1 Lucky Capital

Table 5.3 shows that Lucky Capital has positive alpha for the full sample. Its monthly risk-adjusted performance, at 0.77%, is significant. However, Model 2 shows that the performance is negative and insignificant after excluding backfilled data. Therefore, the positive performance can be explained by a backfilling bias. The R-squared in Model 2 implies that the market model captures almost 70% of the variation in the fund's returns. Incorporating the nonlinear market excess return in Model 3 adds no explanatory power. If we found significance of the nonlinear term we would explore the fund's total performance in the presence of market timing ability, as described in Section 2.1. The significant one and three month lagged market returns in Model 4 suggests that there is some illiquidity in Lucky Capital's portfolio. However, the intercept, at -0.16% per month, is negative and insignificant. We conclude that Lucky Capital does not add value to investors.

5.1.2 Merger LP

Table 5.4 shows that Merger LP has insignificant systematic risk using the sample of non-backfilled returns (Model 2). The estimated performance is neutral--the intercept is 0.07% per month and insignificant. In addition, stale NAV does not appear to be an issue because the lagged observations of the market return have no explanatory power in the fund return regressions. In Model 3, we estimate a significant positive market beta ([b.sub.p] = 0.115) in down markets, but a flat relation between Merger LP's return and the market return when the excess market return is positive. (The up-market beta is [b.sub.p] + [[LAMBDA].sub.p] = 0.115 - 0.174= - 0.059.) (1) This is consistent with Mitchell and Pulvino's (2001) finding of a negative relation between deal failure and market returns. Accounting for this nonlinearity also reveals a positive intercept of 0.33% per month. The payoff of Merger LP resembles the payoff from writing put options on the market. The issue is whether the fund is able to achieve this payoff cheaper than an OE benchmark. Section 2.2.1 shows that the appropriate OE for Model 3 consists of cash, a weight of [b.sub.p] in the market index, and a weight [[LAMBDA].sub.p][P.sub.0] in a one-period call option written on the gross return of the market index with a strike price equal to the gross return on the riskless rate, where [P.sub.0] denotes the option price. (2) The difference between the excess return of the fund and that of the OE portfolio may be computed as [[alpha].sub.p] = [a.sub.p] + [[LAMBDA].sub.p][P.sub.0][R.sub.f]. Under Black and Scholes (1973), we can estimate [P.sub.0] using the average risk-free rate and sample volatility of returns on the market index. Using the sample standard deviation of monthly market returns of 4.1% and average monthly risk-free rate of 0.15%, this gives

[[alpha].sub.p] = 0.33% - 0.174 x 0.0164 x 1.0015 = 0.0442%.

In other words, Merge LP delivers an additional 4.4 basis points per month relative to a stock/options option strategy that replicates the dynamic risk exposure of Merger fund's portfolio.

5.1.3 Murky LLC

Table 5.5 shows that Murky LLC has very little market exposure (market beta of 0.087 or 0.067) according to Model 1 or 2. Moreover, the table reveals a positive alpha of 0.19% per month. However, the R-squared is only 7%, suggesting that the Jensen's alpha model is misspecified. Accounting for dynamic risk exposure in Model 3 also reveals a positive alpha, but the coefficient on the up-market variable is insignificant. Model 4 shows that lagged observations of the market index excess return are significant. Summing the betas reveals that the "true" market beta is approximately 0.18. An F-test shows that the change in R-squared from 0.07 to 0.17 is significant. More importantly, whereas in Model 2 we could reject the null hypothesis of zero intercept at the 7.5% significance level, Model 4 shows that the intercept, at 0.11% per month, is insignificant after accounting for a stale price bias in the fund's reported returns. We conclude that Murky LLC does not add value to investors.

(1) Note that these are not conditional betas given public information, as discussed in Chapter 3, because the market return at time t + 1 is not known at time t. We do not illustrate the CPE approach here for simplicity.

(2) Equivalently, by put-call parity, the OE consists of cash, a weight of [b.sub.p] + [[LAMBDA].sub.p] in the market index, and put options on the gross market return with an exercise price equal to the gross return on the riskless asset.

Table 5.1 Fund characteristics at the end of 2005. Name Lucky Capital Merger LP Primary style Long/Short Equity Event Driven Inception date September 21, 1993 January 31, 1986 Performance start October 31, 1993 January 31, 1986 date Date added to September 30, 2001 September 30, 2001 database Minimum 5,000,000 100,000 investment ($$) Management fee 1.5 1.5 (percent) Incentive fee 20 20 (percent) High-watermark 1 0 Subscription Monthly Monthly frequency Redemption Monthly Monthly frequency Redemption notice 60 10 (days) Lockup period 6 0 (months) Domicile country Virgin Islands Virgin Islands (British) (British) Fund identifier 478 464 Name Murky LLC Primary style Fixed Income Arbitrage Inception date November 30, 1993 Performance start November 30, 1993 date Date added to July 19, 1995 database Minimum 250,000 investment ($$) Management fee 1 (percent) Incentive fee 15 (percent) High-watermark 0 Subscription Monthly frequency Redemption Quarterly frequency Redemption notice 45 (days) Lockup period 0 (months) Domicile country Jersey Fund identifier 1030 Note: The table reports organizational characteristics for each hedge fund. Data are available from the database at the end of 2005. Table 5.2 Annual fund net returns over the sample period. Year Lucky Merger Murky Stocks T-Bills 1986 -- 0.05 -- 0.18 0.06 1987 -- -0.03 -- 0.05 0.05 1988 -- 0.59 -- 0.17 0.06 1989 -- 0.09 -- 0.31 0.08 1990 -- -0.02 -- -0.03 0.08 1991 -- 0.16 -- 0.31 0.06 1992 -- 0.05 -- 0.08 0.03 1993 0.06 0.30 0.02 0.10 0.03 1994 0.06 0.04 -0.12 0.01 0.04 1995 0.53 0.34 0.05 * 0.38 0.06 1996 0.25 0.10 0.11 * 0.23 0.05 1997 0.08 0.20 0.12 * 0.34 0.05 1998 0.15 0.08 -0.05 * 0.29 0.05 1999 1.00 0.11 0.15 * 0.21 0.05 2000 0.34 0.12 0.07 * -0.08 0.06 2001 0.04 * -0.01 * 0.05 * -0.12 0.04 2002 -0.24 * -0.01 * 0.04 * -0.22 0.02 2003 0.34 * 0.03 * 0.08 * 0.29 0.01 2004 -0.05 * 0.04 * 0.03 * 0.11 0.01 2005 0.08 * 0.08 * 0.05 * 0.05 0.03 Note: The table reports annualized net fund returns computed from the raw monthly return series. Annual returns are also reported for the value-weighted return on NYSE/AMEX stocks, and for the return on 30-day Treasury Bills. * Shaded cells correspond to fund return observations that are not backfilled. Table 5.3 The performance of lucky capital. Lucky Capital Coefficient 1 2 3 4 [b.sub.p] 0.725 0.853 0.940 0.83 (0.080) ** (0.100) ** (0.153) ** (0.076) ** [[LAMBDA].sub.p] -0.185 (0.304) [b.sub.1p] 0.204 (0.060) ** [b.sub.2p] -0.109 (0.057) [b.sub.3p] 0.197 (0.067) ** [a.sub.p] 0.77 -0.12 0.16 -0.16 (0.36) * (0.32) (0.49) (0.29) Observations 147 52 52 52 R-squared 0.34 0.69 0.69 0.77 Robust standard errors in parentheses. * Significant at 5% level; ** Significant at 1% level. Note: The table reports the estimated coefficients from various market model regressions. The dependent variable is the monthly excess fund return. The intercept is reported in percent per month. Significance levels correspond to a two-tailed test of the null hypothesis that the coefficient equals zero. Table 5.4 Performance of merger LP. Lucky Capital Coefficient 1 2 3 4 [b.sub.p] 0.348 0.034 0.115 0.032 (0.143) * (0.030) (0.056) * (0.027) [[LAMBDA].sub.p] -0.174 (0.081) * [b.sub.1p] 0.204 (0.060) [b.sub.2p] 0.041 (0.023) [b.sub.3p] 0.015 (0.016) [a.sub.p] 0.31 0.07 0.33 0.06 (0.28) * (0.79) (0.10) ** (0.08) Observations 240 52 52 52 R-squared 0.18 0.06 0.21 0.20 Robust standard errors in parentheses. * Significant at 5% level; ** Significant at 1% level. Note: The table reports the estimated coefficients from various market model regressions. The dependent variable is the monthly excess fund return. The intercept is reported in percent per month. Significance levels correspond to a two-tailed test of the null hypothesis that the coefficient equals zero. Table 5.5 Performance of Murky LLC. Lucky Capital Coefficient 1 2 3 4 [b.sub.p] 0.087 0.067 0.165 0.069 (0.043) * (0.040) (0.082) * (0.039) [[LAMBDA].sub.p] -0.201 (0.120) [b.sub.1p] 0.055 (0.027) [b.sub.2p] 0.058 (0.024) [b.sub.3p] 0.012 (0.020) [a.sub.p] 0.04 0.19 0.55 0.11 (0.14) * (0.11) (0.18) ** (0.11) Observations 146 126 126 126 R-squared 0.06 0.07 0.13 0.17 Robust standard errors in parentheses. * Significant at 5% level; ** Significant at 1% level. Note: The table reports the estimated coefficients from various market model regressions. The dependent variable is the monthly excess fund return. The intercept is reported in percent per month. Significance levels correspond to a two-tailed test of the null hypothesis that the coefficient equals zero.

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Title Annotation: | Portfolio Performance Evaluation |
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Author: | Aragon, George O.; Ferson, Wayne E. |

Publication: | Foundations and Trends in Finance |

Geographic Code: | 1USA |

Date: | Apr 1, 2006 |

Words: | 2719 |

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