**Introduction: a background in economics**

These impressive theoretical frameworks dominate the world of applied corporate finance and investment decision making. This is despite the evidence of Mandelbrot and others that most financial data cannot be adequately described as Gaussian normal. Nor does CAPM work particularly well empirically. But MPT and CAPM are still widely used because they offer convenient and relatively easy ‘recipes’ to follow.

**The Sharpe ratio and its descendants**

Out of the mean variance framework, one performance measure came to dominate investment appraisal, the Sharpe ratio, which divides the excess return of a portfolio (usually relative to the risk free rate) by its standard deviation, as a proxy for risk:

_{p}is the return on the portfolio; R

_{f}is the risk-free rate and σp is the standard deviation of the returns of the portfolio. This ratio shows the ‘price’ of excess return in terms of risk (volatility). Sharpe revisited the ratio in 1994 and reinterpreted it as the return of a portfolio relative to some other portfolio (equivalent to being long one and short the other). This revised version is known as the information ratio.

_{p}, the volatility of the portfolio, with a measure of systematic risk. The intuition here is from the CAPM approach, namely that investors should only expect risk compensation for exposure to nondiversifiable or systematic risk.

Another set of developments was begun by Modigliani and Modigliani (the Nobel laureate Franco and his granddaughter). Known as M^{2}, their measure compares portfolios by leveraging or deleveraging them to the point where they have the same volatility (normally chosen to be the market volatility). This allows the portfolios to be compared simply by looking at the resulting returns. The fund with the highest M^{2} will have the highest return for a given amount of risk. There is a further extension of this, developed by Muralidhar and called M^{3}, which additionally looks at differences in the correlations of the various portfolios being compared (see Muralidhar’s contribution in this issue).

The Sharpe ratio is still the bread and butter tool of investment management. It is easy to compute, relatively easy to understand, and has a fine theoretical pedigree behind it. The higher the Sharpe ratio for a portfolio, the better. (Unfortunately it is often quoted without any measure of statistical significance, even though this may render the numbers useless).

**Downside Risk Measures**

Many of the early hedge fund investors were high net worth individuals for whom capital preservation was paramount. So low volatility was less important than low downside volatility. This has given rise to two sorts of measures based on the downside risk: the Sorting ratio and Maximum Drawdown (MDD).

_{DD}is the downside deviation of return. Sortino’s ratio appealed to the practitioners in the hedge fund world but was initially criticised for not being rooted in sound market theory, although that turned out to be a false criticism1.

Maximum Drawdown (MDD) measures are often quoted in standard tables of hedge fund performance and describe the worst peak- totrough fall in asset values experienced by the fund to date. Although widely quoted by practitioners, MDD was not widely analysed by academics until fairly recently. Some limitations are relatively clear: MDD can only be compared for funds with the same time scale and similar reporting frequency. One perverse result is that a very new fund almost certainly has a smaller MDD than a long established one, but it hardly follows that one should only invest in new funds.

Moreover MDD is a very inefficient statistic for describing a fund’s performance and carries a high potential level of error i.e. an investor should not reject or accept a fund on MDD alone. A measure such as the worst five trading days might be a superior measure because it uses more information about the historic returns.

MDD can be derived analytically as a formula for some return distributions but researchers have tended to preferred to use a Monte Carlo numerical approach. In other words, the estimated parameters of the return distribution (typically only the mean and variance) are used to generate a very high number of possible outcomes, of which the actual outcome is only one. Then the researcher (or investor) can pick a confidence level, say 95% or 99%, and see what the MDD would be. This should be a better guide to the underlying downside risk compared with the actual MDD.

**Limitations of Mean Variance**

ii) The investment returns are normal; or

iii) The risks are ‘small’ in the sense that a second order Taylor approximation to the utility function is satisfactory.

^{2}emphasises that this is a false assumption. But for much economic work the assumption of ‘small risks’ has justified the use of mean variance analysis. Paul Samuelson, one of the greatest of economists, argued that in this case, ‘the mean-variance result is a very good approximation’

^{3}.

**Dealing with higher moments**

The biggest problem with the Sharpe ratio and its spin offs remains their failure adequately to capture the higher moments of the distribution. If we reject the assumptions that investors don’t care about higher moments and that investment data can be characterised as Gaussian normal, then we need a new approach. Sharma (2005) shows that the Sharpe ratio can be extended in a useful way by replacing the denominator by the value at risk (VaR) at say the 99% level. VaR, the expected return in a defined statistical set of worst cases, is typically based on a mean variance normal distribution but can be modified to incorporate skew ness and kurtosis using the Cornish-Fisher expansion to yield:

_{1}and z

_{2}.

_{i}is the fund total return of the ith period and p

_{i}is the probability of that return; c is the coefficient of relative risk aversion, which captures the shape of the utility function, and can take a number of values, so long as the same value is used for comparing different fund returns (see Sharma article in this issue).

**Nonparametric approaches:**

**Omega**

The appeal of the Gaussian normal distribution is that it can be described by just two parameters, the mean and standard deviation, which simplifies computation enormously. Other parametric distributions, such as the Weibull or Gumbel, have the same usefulness. But financial return data don’t necessarily fit any of these distributions. Choosing parameters may amount to excessive simplification. The parameters may not even be well defined

^{4}.

^{5}.

The starting point is the cumulative distribution function, which plots the ordered returns for an investment in a cumulative way. Figure 2 shows a density function for some daily hedge fund returns. This looks a bit like a normal distribution but with negative skew and high kurtosis ie a lot of extreme results but particularly on the downside. Perhaps unsurprisingly, this is data from a merger arbitrage fund.

The Omega function uses the information in the cumulative distribution to compare, for each chosen threshold return, the distribution weighted returns above that threshold versus the distribution weighted returns below. That ratio is then plotted graphically against the threshold returns. The Omega ratio is:

**Factor analysis**

^{6}Fung and Hsieh (2002) and Naik and Aggarwal (2004), among others, have extended the Fama-French work into the world of alternative assets. Fund returns are regressed on a range of factors capturing equity, fixed income, currency, commodity and interest rate returns. Momentum and trend following effects can also be captured, the latter by using the concept of a ‘lookback straddle’. This is an option strategy that pays the maximum difference between the highest and lowest prices of an asset over the maturity period. It therefore pays out what a trend follower would achieve with perfect foresight.