Performance, risk and measurement

The typical hedge fund investor is, like any other investor, risk-averse. There will be a preference for high returns but a dislike of the risk related to investment. All risk and performance risk measures attempt to satisfy the investor's need to be informed about the returns and about the related risks.

Some measures, like return analysis, focus on the return component. Others, such as extreme risk analysis, try to capture the riskiness of the hedge fund investment. Risk-adjusted performance measures and factor analysis combine both dimensions of any investment by relating return and risk to each other.

The literature on stocks and mutual funds has proposed an abundance of measures that can be used to measure the risk and return of these investments. The crucial question is whether these risk indicators can be similarly used for hedge fund portfolios.

The first work to analyse hedge fund risk and performance (including papers by Elton et al,1 Brown et al2 and Ackermann et al3) drew on these standard measures to determine whether hedge funds perform better than the market or not.

However, as subsequent work4,5,6,7 has shown, hedge funds exhibit some particularities that make them different from standard equity investments. First, hedge fund returns, as opposed to mutual fund returns, are not normally distributed.

Second, they are non-linear with respect to the standard market factors such as equity and bond markets. Hedge funds often modify their investment style so their exposure to risk factors is highly dynamic over time. These differences make the standard risk and return indicators, although still widely used, inappropriate for hedge funds.

The most important performance and performance risk indicators used in hedge fund reporting may shed some light on this.

Analysis of hedge fund returns

An initial assessment of hedge fund returns usually involves the analysis of past returns using descriptive statistics and statistical tests. Although simple to calculate, it is the basis of any hedge fund reporting and is by far the most informative to investors.

Past hedge fund returns are without doubt important information. Standard presentations include monthly and annual returns net of fees in absolute terms and relative to a benchmark.i Since investors are less concerned with past returns than with future returns, persistence measures are also important.

Gain frequency, calculated as the percentage of positive monthly returns, is an initial indicator of performance persistence. Closely related, but more complicated to calculate, is the Hurst coefficient.8

The volatility of monthly returns is an initial assessment of the riskiness of the hedge fund. Simpler, but equally informative measures of a hedge fund's volatility are minimum and maximum past monthly returns, or more generally their upper and lower deciles (or quartiles).

There are also some key figures that help to assess the downside risk of returns, an issue of particular interest to hedge fund investors. The maximum past drawdown is a simple but informative indicator of downside risk.

The skewness and kurtosis of fund returns are together important in assessing downside risk: a return distribution that is negatively skewed combined with positive (excess) kurtosis is a strong indicator of high downside risk.

Semi-deviation and other lower partial moments of hedge fund returns are similarly useful. Hedge fund returns during extremely negative equity markets are a good measure of a fund's ability to hedge downside stock market movements.

Information can also be gained by applying statistical tests to the hedge fund returns. Statistical tests such as Jarque-Bera9 make it possible to assess the normality of hedge fund returns.

If normality is rejected, the fund is likely to exhibit a larger downside risk than standard equity investments. Tests of auto-correlation, such as the Ljung-Box test,10 are generally used to detect the fraction of illiquid assets in the hedge fund portfolio. If the test of no auto- correlation is rejected, the portfolio is likely to contain a large fraction of illiquid and thus hard-to-value assets. Consequently, return figures must be treated with caution.

Extreme risk measures

Because of their non-normal return structure and so higher relatively higher downside risk, simple volatility measures underestimate a fund's riskiness. It is therefore crucial to assess the hedge fund's risk of extremely negative returns.

Value-at-risk (VaR) is perhaps the most important extreme risk measure. The VaR of a portfolio is the maximum amount of capital that can be expected to be lost within a specific time period (usually one month), given a specified confidence level (usually 95% or 99%).

There are several ways to calculate VaR. The simplest is to assume a normal distribution of returns that must be estimated to calculate the expected maximum loss. Since hedge fund returns are usually not normally distributed, an alternative is to use non-parametric estimation based on the historical distribution of hedge fund returns. In this way the non-normality is captured automatically.

Another approach uses Monte-Carlo simulation techniques to estimate the expected maximum loss. Such simulations can assume either normally distributed returns or more complex distributions that account for the asymmetry and fat tails of hedge fund returns.

Besides the standard VaR, there are more sophisticated VaR measures that attempt to meet the needs of hedge fund reporting. The Cornish-Fisher VaR11 - also called modified VaR (MVaR) - is an extension of the standard VaR that incorporates the effect of skewness and fat tails of hedge fund returns.

Incremental VaR attempts to measure the change in VaR when a particular asset class is introduced to the portfolio. Closely related component VaR (described by Jorion12) indicates the contribution of a specific asset to the VaR of the overall portfolio. Conditional VaR (also called expected shortfall) is another extension of the VaR approach.

Compared to the VaR, it does not specify the maximum expected loss for a specific confidence level, but the average amount of loss if that significant loss actually occurs. Conditional VaR is important if the distribution of returns has fat tails since the loss might be much larger than that specified by VaR.ii

Applications of the conditional VaR to hedge funds have been made by Agarwal and Naik13 and De Souza and Gokcan.14

Shortfall probability can be considered the inverse of VaR. Instead of estimating the maximum loss given at a confidence interval, the shortfall probability indicates the probability that a given loss will actually occur.

Another method of calculating a VaR estimate relies on style analysis: the style VaR.15,16 This method first examines the relationships of the investment styles in the portfolio, then analyses the impact of the worst variation of each style on the portfolio.

Laporte17 proposes an extension of the style VaR to include liquidity risk borne by hedge fund investors through lock-up constraints.iii By adding an additional factor to the style VaR, this model facilitates analysis of the impact of such clauses on a fund's VaR.

Extreme value theory (EVT)16,18 is also important. It is not a variant of VaR but rather an important tool to calculate it.

EVT focuses on the modelling of the tails of a distribution, leaving the rest unspecified. However, since rare negative events are of great importance for hedge funds, this tool makes it possible to evaluate the risks related to such events. The distribution of the tail can then be used to estimate the VaR.

Stress tests

Besides the different VaR approaches, hedge funds often use stress tests to assess extreme risks. Unlike the VaR, stress-testing requires no assumptions on a fund's return probability distribution and is therefore essentially a non-statistical risk measure. It relies instead on Monte-Carlo techniques to evaluate the impact of extreme but probable situations on hedge fund performance.

The crucial difference between VaR and EVT is the stress test's ability to simulate shocks that have never occurred in the past or that are more likely to occur than historical evidence suggests.

Stress tests can also be used to analyse the impact of structural shocks to the financial system on hedge fund returns. The basic idea of stress tests is simple: evaluating the impact of sudden changes in the determinants of hedge fund returns on their performance.

Such stress tests can simulate either the consequences of changes in one particular key variable (also called sensitivity testing) on hedge fund performance or the impact of an extreme variation in many critical factors (called multi-dimensional scenario analysis).

Stress tests can simply assume a one-shot deviation from the variables under consideration or they can involve more complicated models that reflect the impact of so-called spiral effects (second-round effects) on hedge fund performance. (See Jorion12 for a detailed review of stress testing.)

Risk-adjusted performance measures

So far the emphasis has been on measures to analyse both hedge fund returns and their related risks. The most critical part of hedge fund reporting is, however, indicators than combine both elements of analysis because above-average returns are not a surprise when running high-risk strategies.

The real challenge is delivering good performance while keeping the risk exposure limited. Risk-adjusted performance measures are designed to detect investments that exhibit a good risk/return trade-off.

The Sharpe ratio19 is the most famous measure of risk/return. It is the ratio of the portfolio's expected excess return over the risk-free rate E (rp) - rf and its standard deviation s p:

Sp = E (rp) - rf (1)

s p

The higher the Sharpe ratio, the better the risk/return trade-off. Intuitively it can be interpreted as a fund's excess return per unit of risk. Apart from slight modifications, all risk-adjusted performance measures follow this principle by relating returns to units of risk.

Almost three decades after the Sharpe ratio was created, Sharpe20 proposed a generalisation of his original ratio: the information ratio (IR). This is the ratio of the portfolio's excess return over that of another portfolio (usually a benchmark) and the standard deviation of the return difference between both portfolios. It captures the fact that managers often try to outperform a benchmark while maintaining a low tracking error.

The M2 measure, owing its name to the authors Modigliani and Modigliani,21 also focuses on how to evaluate performance compared to a benchmark. They suggest adjusting the portfolio to have the same risk as the benchmark before comparing them.

The M2 is hence equivalent to the return the fund would have achieved if it had had the same risk as the benchmark, often the market index. In essence it is equal to the Sharpe ratio multiplied by the standard deviation of the benchmark.

Closely related to the Sharpe ratio is the Treynor ratio22 which divides the expected excess return of a portfolio by its beta, where the beta is calculated on the basis of the CAPM.23,24 The advantage of the Treynor ratio is that it focuses only on the systematic component of risk, not on total risk.

Along with the evolution of multi-factor models, such as the Fama and French25,26 three-factor model, a generalisation of the Treynor ratio has been proposed, including the portfolio's sensitivities to more than one factor.27

Although the Sharpe ratio is easy to calculate, it is not very appropriate to hedge funds, for reasons explained above. The Sortino ratio28 offers a slightly different measurement by replacing the standard deviation with the downside deviation and is consequently more appropriate.

Another variant of the Sharpe ratio is the modified Sharpe ratio, also called return over VaR, which has been proposed by Gregoriou and Gueyie.29 The modified Sharpe ratio replaces the standard deviation in the denominator of the Sharpe ratio with the above-mentioned modified VaR (or Cornish-Fisher VaR). This replacement is motivated by the fact that the MVaR takes the skewness and kurtosis of the return distribution into account.

Besides the Sharpe ratio and modifications thereof, some other risk-adjusted performance measures are explicitly tailored for the non-normal distribution of hedge fund returns. The Stutzer index30 tries to capture a behavioural element of investors who are assumed to seek to minimise the probability that the return of their investment will be negative over a long time horizon. Since a great likelihood of severe losses increases the likelihood of negative excess returns, the Stutzer index punishes hedge funds with strongly negative skewness and high kurtosis.

The Omega ratio has been put forward by Keating and Shadwick.31 Its advantage is that it also includes its third and fourth moment while requiring no assumption on a fund's return distribution.

The Omega is a relative gain-to-loss function using an exogenously (and arbitrary) return threshold. The Omega is then the quotient of the (expected) excess return over a threshold and the expected loss below the same threshold. Hence, the higher the Omega, the better.

In an attempt to create a measure that combines the positive aspects of the Omega ratio (reflecting higher moments) and the Sharpe ratio (easy interpretation), Kazemi et al32 proposed a Sharpe-Omega ratio. While identical in terms of ranking to the Omega ratio, it uses the expected excess return in the numerator of the ratio, similar to the Sharpe ratio.

Compared to the previous ratios, the alternative investment risk-adjusted performance (AIRAP) measure by Sharma33 is an indicator that incorporates an individual risk-aversion parameter when comparing hedge fund performance.

Based on a constant relative risk-aversion utility function, it also makes it possible to capture the effect of the third and fourth moments of hedge fund returns on an investor's expected utility.

Two conceptually simple ratios to determine return relative to downside risk are the Calmar and Sterling ratios. The Calmar ratio34 is the annual average (expected or past) excess return divided by the maximum drawdown. The aim is to capture downside risk (captured by the maximum drawdown) since some funds provide high returns only by bearing extremely high drawdown risk. Although easily calculated for past returns, the projection of expected returns and drawdowns can be cumbersome.

The Sterling ratio35 is closely related to the Calmar ratio. It is the ratio of the average annualised excess return to the average maximum drawdown per year over the last three years, minus (an arbitrary) 10%.

Beta analysis: factor models

All previous indicators focus on measuring the risk-adjusted performance of hedge fund returns. However, they neglect a crucial part of risk/return considerations: the source of hedge fund returns and their related risks.

The aim of beta or correlation analysis is to detect a fund's underlying return drivers by explaining the hedge fund returns through the fund's exposure to various risk factors. Factor analysis is especially important for investors who consider hedge fund investment only as a part of their overall portfolio, such as institutional investors.

Pension funds or insurance companies must monitor their overall portfolio risk, not only a hedge fund's specific risks. Fund of hedge fund (FoHF) investors are in the same situation when they make decisions to allocate assets to funds.

Linear factor models

Most factor models used to analyse hedge fund returns are linear models:

K

rt = a + S bk Fk,t + et (2)

k = 1

where rt is the monthly return of the hedge fund, a its abnormal performance, bk the sensitivity of the hedge fund to the risk factor k, Fk,t the return of risk factor k and et the disturbance term. The primary advantage of linear models is their simplicity: their calculation is straightforward and easy to understand. In addition, since they are commonly used, comparisons of hedge funds are simplified.

Most well-known factor models try to explain portfolio returns by stock return factors. The most prominent model is the CAPM,23,24 which uses the return of the market portfolio as a single factor to account for portfolio returns.

Three- and four-factor extensions have been devised by Fama and French25,26 and Carhart.36 They include the return spreads between value and growth stocks, small and large stocks, and the differences in the returns of past winners compared to past losers.

Another well-known variant of the linear factor model is the so-called mutual fund-style analysis by Sharpe.37,38 This variant accounts for portfolio returns by benchmark returns of different standard asset classes (bills, bonds, equities), thereby attributing the portfolio returns to different investment styles or categories.

Although popular for traditional investment universes, these linear factor models using standard asset classes as factors tend to have poor explanatory power when it comes to analysing hedge fund returns. The primary reason for this relative impotence is the non-linear and non-normal structure of hedge fund returns,5,39,40 which cannot be captured by simple equity factor models. Several extensions or adaptations of previous models have been proposed; the aim is to be more suitable for hedge fund factor analysis.

An initial approach, proposed by Fung and Hsieh41 is to include in the set of factors the returns of other asset classes, such as commodities,42,43 exchange rates4 or hedge fund indices.15 Also called hedge fund-style analysis, these additional factors then capture the non-linear and dynamic trading strategies as used by hedge funds in the linear factor model. A second possibility has been proposed by Agarwal and Naik.40 who use equity option portfolios as factors in their model to account for non-linearities.

Non-linear and dynamic models

Instead of using factors in the linear model that exhibit non-linear features, an alternative is to model the non-linearity or dynamics in the estimation. There are many ways to model these non-linearities: higher moment adjusted models of the CAPM,44,45 conditional regression models46,47 or regime-switching models.48

Finally, truly dynamic models either use Kalman filter techniques to incorporate the dynamic betas of hedge funds49,50 or explicitly model the time-varying structure of a hedge fund's risk exposure.51,52

Alpha analysis

Alpha analysis is the flip side of beta analysis. In fact, it is simply the intercept of the factor model presented in equation (2): the residual or abnormal return that cannot be explained by the fund's risk exposure. Accordingly, alpha can be calculated with any of the factor models outlined above - and the value of alpha depends very much on the chosen estimation method. Alpha is basically what investors most prefer, since it reflects (positive) returns without risk.iv

- This article is drawn from the EDHEC Hedge Fund Reporting Survey53 conducted with the support of the prime brokerage group at Newedge. Lhabitant,16 Amenc et al54 and Gehin55 provide more comprehensive surveys on most of the measures currently used by hedge funds. Le Sourd56 reviews the measures aimed at traditional investment universes, but used by hedge funds.

- David Schroder, business analyst, and Felix Goltz, head of applied research, EDHEC Risk and Asset Management Research Centre.

FOOTNOTES

i. The calculation of past returns is not as simple as it seems since it is preceded by the determination of the hedge fund's value. There are several ways to calculate the returns. For more details see chapter 2 of Lhabitant.16

ii. Conditional VaR and expected shortfall are two different concepts. Conditional VaR calculates the expected shortfall given a pre-specified confidence level, whereas the expected shortfall can also be calculated using another loss limit.

iii. This is an important difference to the general notion of liquidity risk which usually denotes the liquidity risk within the hedge fund.

iv. Since the main purpose of alpha analysis is to calculate risk-adjusted returns, it is often classified as a risk-adjusted performance measure.

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