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 nonlinearity 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 socalled 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 (3month
Tbills). 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 socalled "Alpha".
Table 1: Beta exposures of the
two hedge funds

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 2Moment CAPM to the 4Moment CAPM.
For more on the 4Moment CAPM, see Jurcenzko and
Maillet, "The FourMoment Capital Asset Pricing
Model: Some Basic Results", 2002, or Ranaldo
and Favre, "How to Price Hedge Funds: From
Two to FourMoment CAPM", 2005 (to be published).
The FourMoment 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 nonlinear 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 4Moment 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 3month Tbills.
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 nonlinear. 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
nonlinear 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 FourMoment Capital Asset Pricing Model
presented above. With this and many other features,
HFOptimizer is well suited for fund of funds construction.
