In this article, we will compute the equilibrium returns for the hedge fund strategies for the next 3 to 5 years using a model that accounts for non-linearity in hedge fund indices.

We have learned at the university, in our finance books, in the press or from our financial advisor that the asset returns depend on its exposure to the market, the so-called beta. An asset with a high beta has, in the case of positive market returns, an expected return near or higher than the market. An asset with a low beta has a low expected return near the risk free level.

Let's take an example with two hedge funds for the period May 1997 to May 2005. We name these Hedge Fund A (equity hedge) and Hedge Fund B (statistical arbitrage). The former is risky whereas the latter is less risky. We define the market as a portfolio composed of 50% equity (MSCI World Equity Index), 30% bond (JPM Global Bond Index), 10% hedge funds (HFR Weighted Composite Index) and 10% cash (3-month T-bills). Hedge Fund A has a high historical return and a high beta. Hedge Fund B has a lower historical return and almost a zero beta. The portfolio has a historical annualised return of 5.6%. How is it possible that Hedge Fund B generates a higher return than the portfolio, which is by definition, the market with a beta of only 0.05? According to finance theory, the expected annual return for Hedge Fund B should be 3.5% and for Hedge Fund A 8.2%.

Part of this difference between the realised return of 11.5% (27.9%) and the expected return of 3.5% (8.2%) could be explained by:

- A misspecification of the classical finance model that does not account for exposure to extreme events.

- Some abilities of both funds to generate the so-called "Alpha".

Hedge Fund A | Hedge Fund B | Portfolio | |
---|---|---|---|

Historical annualised return | 27.9% | 11.5% | 5.6% |

Historical annualised volatility | 46.9% | 2.7% | 8.8% |

Beta to the portfolio | 2.23 | 0.05 | 1.00 |

Skewness | -0.11 | 0.79 | -0.40 |

Kurtosis | 6.53 | 0.01 | -0.22 |

Source: AlternativeSoft AG. Data from May 1997 to April 2005.

**Misspecification of the Classical Finance Model**

To correct for this model misspecification, we enhance the 2-Moment CAPM to the 4-Moment CAPM. For more on the 4-Moment CAPM, see Jurcenzko and Maillet, "The Four-Moment Capital Asset Pricing Model: Some Basic Results", 2002, or Ranaldo and Favre, "How to Price Hedge Funds: From Two- to Four-Moment CAPM", 2005 (to be published). The Four-Moment CAPM increases the asset expected returns when they have high beta, high contribution to portfolio negative skewness and high contribution to portfolio kurtosis.

We apply this equation to the HFR indices and present the results in table 2, column 4.

In table 2, the second column exhibits the historical annualised returns for each index. The third column presents the expected return using the beta of each index with respect to the portfolio. The fourth column shows the expected return using a model that account for non-linear relations between each index and the portfolio. The portfolio is composed of 50% equity, 40% bond and 10% cash.

Table 2: Hedge fund indices expected returns

Historical annualised return | Expected annualised return with traditional model | Expected annualised return with 4-Moment CAPM | |
---|---|---|---|

HFR Weighted Composite | 10.1% | 6.4% | 6.7% |

HFR Relative Value | 9.0% | 3.8% | 4.5% |

HFR Equity Hedge | 13.3% | 7.3% | 6.9% |

HFR Convertible Arbitrage | 9.1% | 3.5% | 3.2% |

HFR Distressed | 11.5% | 4.7% | 6.2% |

HFR Macro | 9.6% | 4.9% | 5.1% |

Barclay CTA | 4.8% | 2.8% | 0.7% |

Portfolio (50% equity, 40% bond, 10% cash) | 5.4% | 6.4% | 6.4% |

Risk free | 2.9% | 3.0% | 3.0% |

Source: AlternativeSoft AG, HFOptimizer software, Barclay, HFR data from May 1997 to April 2005. The portfolio is assumed to be the world market portfolio. Equity is MSCI World Equity Index. Bond is JPM Global Bond Index. Cash is US 3-month T-bills. For expected returns for the traditional indices, we use equity 8.2% (see Dimson, Marsh, Stauton, "Risk and return in the 20th and 21st centuries", Business Strategy Review, 2000), bond 5% and cash 3%. All data are in USD. The third column assumes that the relation between the indices and the market portfolio depends only on beta. The fourth column assumes that the relation between the indices and the market portfolio is non-linear. The results are sensitive the choice of the portfolio, to the cash, bond and MSCI world equity index expected returns.

We see that the biggest difference between history (9.1%) and expectation (3.2%) is in convertible arbitrage. This is due to the low linear and non-linear exposure of convertible arbitrage index with respect to the portfolio. The indices with the highest expected returns are equity hedge (6.9%) and MSCI world equity index (8.2%).

To summarise, first the investor should add CTA and convertible in his portfolio for diversification purpose and he should not expect high returns from them in the next 3 to 5 years, if he believes in a model that takes into account not only beta, but systematic skewness (ie exposure to market volatility) and systematic kurtosis (ie exposure to market extreme events). Second, given the expected returns in table 2, the equity investor should invest in hedge funds for diversification purpose and not for return enhancing.

*Note:
The HFOptimizer platform from AlternativeSoft AG delivers the hedge fund index expected returns using the Four-Moment Capital Asset Pricing Model presented above. With this and many other features, HFOptimizer is well suited for fund of funds construction.*