Does the choice of performance measure influence the evaluation of hedge funds, страница 4

Value at risk has also been proposed as an alternative risk measure in hedge fund performance measurement. Value at risk (VaR_i) describes the possible loss of an investment, which is not exceeded with a given probability of 1  a in a certain period. In case of normally distributed returns, the so-called standard value at risk can be computed by VaR_i ¼ ðr^d_i þ z_a  r_iÞ, where z_a denotes the a-quantile of the standard normal distribution. Instead of standard value at risk, the literature frequently considers expected loss under the condition that the value at risk is exceeded. This conditional value at risk is defined as CVaR_i = E[r_itjr_it 6 VaR_i]. The advantage of conditional value at risk is that it satisfies certain plausible axioms (see Artzner et al., 1999). If returns do not display a normal distribution pattern, the Cornish–Fisher expansion can be used to include skewness and kurtosis in computing value at risk. The modified value at risk based on the Cornish–Fisher expansion is calculated as MVaR_i ¼ ðr^d_i þ r_i  ðz_a þ ðz²_a  1Þ  S_i=6þ ðz³_a  3  z_aÞ  E_i=24  ð2  z³_a  5  z_aÞ  S²_i =36ÞÞ, where S_i denotes skewness and E_i excess kurtosis for security i (see Favre and Galeano, 2002).

The performance measures excess return on value at risk (see Dowd, 2000), the conditional Sharpe ratio (see Agarwal and Naik, 2004), and the modified Sharpe ratio (see Gregoriou and Gueyie, 2003) can be used when risk is measured by standard value at risk, conditional value at risk, or modified value at risk, respectively:

rdi  rf

             Excess return on value at risk_i ¼                ;                                                                     ð9Þ

VaR_i

rdi  rf

             Conditional Sharpe ratio_i ¼                 ;                                                                           ð10Þ

CVaR_i rdi  rf

            Modified Sharpe ratio_i ¼                  :                                                                               ð11Þ

MVaR_i

2.3. Classic performance measurement – Jensen’s measure and Treynor’s ratio

There are two classic performance measures explicitly constructed for situations in which only a small portion of the investor’s wealth is allocated to the hedge fund under consideration.

The Jensen measure considers the average return of the fund above that predicted by the capital asset pricing model. The beta factor (b) is generally calculated using the correlation between the returns of a market index and the returns of the investment fund. However, for the decision-making situation discussed in Section 4 (the hedge fund represents only a portion of the investor’s wealth), it is appropriate to compute b by replacing the market index with a so-called reference portfolio, which represents the investor’s portfolio without the hedge fund (see Breuer et al., 2004, pp. 374–396):

                      Jensen measurei ¼ r^d_i  r_f  r^d_rp  r_f                   b_i:                                                            ð12Þ

The Jensen measure is often criticized because it can be manipulated by leveraging the fund return. The Treynor ratio does not suffer from this defect. The Treynor ratio considers the excess return of the fund in relation to its beta factor: r^di  r^f

Treynor ratio_i ¼ :          ð13Þ b_i

Again, as with Jensen’s measure, in our empirical analysis the beta factor is estimated on the basis of the correlation between the hedge fund and the investor’s reference portfolio without the hedge fund.

3. Measuring performance when the fund represents the entire risky investment

3.1. Data and methodology