This ratio measures the return of a portfolio in excess of the risk-free rate, also called the risk premium, over the total risk of the portfolio, measured by its standard deviation. It is drawn from the capital market line, and not the Capital Asset Pricing Model (CAPM). It does not refer to a market index and is not therefore subject to Roll’s (1977) criticism that the market portfolio is not observable.

This ratio has been subject to generalisations since it was initially defined. It thus offers significant possibilities for evaluating portfolio performance, while remaining simple to calculate. One of the most common variations on this measure involves replacing the risk-free asset with a benchmark portfolio. The measure is then called the information ratio (Sharpe 1994). Since this measure is based on the total risk, it makes it possible to evaluate portfolios that are not well diversified, as the unsystematic risk taken by the manager is included in this measure. On the other hand, the Treynor ratio, which measures the relationship between the excess return on the portfolio and its systematic risk, is applicable to well diversified portfolios, since it takes only the systematic risk of the portfolio into account. Another well known indicator is the Sortino ratio. It is similar to the Sharpe ratio. However, the minimum acceptable return (MAR) replaces the risk-free rate and the standard deviation of the returns below the MAR replaces the standard deviation of all returns. VaR measures the maximum amount of loss in the portfolio for a given level of confidence.

The Sharpe ratio is the most widely used measure, used, as it is, by approximately four-fifths of the 222 respondents who identify the absolute performance measures they use. Fewer than half of those responding use the average return in excess of the risk-free rate as an absolute performance measure. The downside and loss measures—the Sortino ratio and VaR—are each used by less than one-third of respondents. Only 11.26% of respondents use the Treynor ratio and only 4.5% report using other measures such as return in excess of a consumer price index (CPI), return to drawdown2, or Omega ratios3. These results are shown in Figure 1.

The clear dominance of the Sharpe ratio, used by four out of five respondents, comes as no surprise. After all, this measure, which has been in existence for more than forty years, is well-known, easy to compute and does not require any information other than the series of portfolio returns. None of the other performance measures come close to rivalling the popularity of the Sharpe ratio, despite its well known drawbacks. Indeed, this measure assumes that the distribution of portfolio returns is normal, ie. that the moments with an order that is strictly greater than two are null. So this measure makes it impossible to take into account the asymmetry of the distribution, measured by the third moment (skewness), or the possible existence of fat tails, and consequently of extreme returns, which are evaluated by the fourth moment (kurtosis). It beggars belief that the average return in excess of the risk-free rate is used as a performance measure by more than 40% of respondents. In fact, this makes it the second most widely-used performance measure. But the simple average return is not a performance measure at all, as it fails to account for risk. If more than 40% of respondents still use such simple measures, a greater focus on risk-adjusted performance measurement is clearly called for.

We then identify the measures used to gauge relative performance. Relative risk measures evaluate a portfolio’s risk-adjusted returns using a reference to a benchmark. The possibilities we listed in the questionnaire included Modigliani and Modigliani’s M-squared measure, the Graham Harvey measures, Jensen’s alpha, the information ratio, adapted information ratios, and average return difference with a broad market index. Before turning to the results, a short description of these measures is in order. The information ratio is the residual return of the portfolio compared to its residual risk.

In the standard information ratio, risk is defined as tracking error volatility, but other risk definitions related to tracking error may also be used. Jensen’s alpha is the difference between the excess return on the portfolio and the excess return explained by the market model. Modigliani and Modigliani’s M-squared, expressed in percentage terms, evaluates the annualised risk-adjusted performance (RAP) of a portfolio in relation to themarket benchmark.

The Graham Harvey measures are two. The first (GH1) is based on a leveraged/deleveraged benchmark portfolio that has the same volatility as the managed portfolio over the evaluation period. GH1 is the difference between the portfolio return and the return on the volatility-matched benchmark portfolio. The second, GH2, indicates the difference between the return on the leveraged/deleveraged managed portfolio that has the same volatility as the benchmark and the return on the benchmark.

214 survey respondents identify the relative performance measures they use. As indicated in Figure 2, the information ratio, used by seven in ten respondents, prevails. It should be noted, however, that 7% of respondents, apparently realising the limitations of the standard information ratio, replace the tracking error volatility in the information ratio with a measure of asymmetric tracking error. More than one-third (36.45%) of respondents use Jensen’s alpha. The average return difference with a broad market index is used by one-third of respondents. M-squared and Graham Harvey measures are used by only a small minority of respondents. This finding suggests that industry practitioners see no advantages to these more complex measures, possibly because these measures rely essentially on comparing portfolio returns and volatility with benchmark returns and volatility, when it may be simpler and just as informative to compare Sharpe ratios. A further 3% of respondents mention other measures, the most popular being peer group comparison.

It is clear that for gauging relative performance the information ratio dominates, much as the Sharpe ratio dominates when it comes to absolute performance measurement. Another similarity with the results for absolute performance measurement is that many respondents choose a measure that does not properly adjust for risk. Similar to the average return in absolute performance measurement, the average return difference with a broad market index is a popular measure of relative performance. It should be noted that this measure assumes that the beta of the portfolio with respect to the broad market index is one, in which case the average return difference would be an estimate of the alpha of a simple single factor model. Obviously, such a comparison will result in distortions of performance when comparing portfolios with different betas. In addition, comparing a portfolio to a broad market index may not be relevant if this market index does not accurately represent the asset selection and weighting of the portfolio, which is why it is important to choose a good benchmark for the portfolio. For the minority who compare results in a peer group, the issue is the constitution of this peer group. It may be relatively difficult to be sure that the portfolios included in a peer group really are a homogeneous group with the same kind of allocation and the same risk exposition. If this is not the case, the comparison of portfolios in the peer group becomes irrelevant.

The return to drawdown compares the portfolio return with the largest drop from peak to valley during the period considered.

The Omega function captures all of the higher- moment information in the returns distribution. For any return level r, defined as the minimum acceptable return, the number Ω (r) is the probability-weighted ratio of gains to losses, relative to the threshold r (Keating and Shadwick 2002).

Biography

Felix Goltz is a senior research engineer and co-head of the indices and benchmark research programme with the EDHEC Risk and Asset Management Research Centre. Goltz holds a PhD. in Finance from the University of Nice Sophia-Antipolis, and has studied economics and business administration at the University of Bayreuth, the University of Nice Sophia-Antipolis and EDHEC.

**Explore Categories**- Commentary
- Event
- Manager Writes
- Opinion
- Profile
- Research
- Sponsored Statement
- Technical

## Commentary

## Issue 132

## The Convexity of Trend Following

## Protecting your assets but perhaps not as much as you would like!