## Measurement matters: Measuring trading performance

August 31, 2013 07:00 PM

Financial theory tells us that investors should expect a positive relationship between risk and return: Those who assume greater levels of market risk are expected to earn higher returns, while lower risk portfolios should earn lower returns. However, investors should not expect to earn higher returns by accepting higher levels of risk from single stocks or a concentrated portfolio, because the markets do not compensate investors for risks that easily can be diversified away.

Key to determining an investment is a measurement to estimate the magnitude of the risk-return tradeoff that can differ across investments and investment managers. Performance measures may show that a high return is not attractive given the extraordinarily high level of risk that is needed to earn that return, while lower levels of return may be quite attractive if they come with minimal risk.

There are two major types of performance measures. First, there are ratios of return to risk. Return can be expressed in several ways in the numerator, and risk can be expressed in numerous ways in the denominator. The denominator of the ratio can be any risk measure, although the most popular performance measures employ the most widely used risk measures, such as volatility (standard deviation) or beta. The risk measure may be an observed estimate of risk or the investor’s belief regarding expected risk. These ratios include those developed by William Sharpe, Jack Treynor and Frank Sortino.

A second method to measure performance is to generate the risk-adjusted return of an asset and compare that return to a standard. The alpha measure, developed by Michael Jensen, compares the return on an investment to the expected return on an investment of similar risk.

Sharpe Ratio

The most popular measure of risk-adjusted performance for investments is the Sharpe ratio (see “Measuring sticks,” below). When using annual returns and an annual standard deviation of returns, the Sharpe ratio may be interpreted as the annual risk premium that the investment earned per percentage point in annual standard deviation. In this case, the investment’s return exceeded the riskless rate by 35 basis points for each percentage point in standard deviation. In an analysis of past data, the mean return of the portfolio is used as an estimate of its expected return, and the historical standard deviation of the sample is used as an estimate of the asset’s true risk. Throughout the remainder of this analysis of performance measures, the analysis may be viewed as interchangeable between using historical estimates and expectations.

Obviously, both the numerator and denominator of the Sharpe ratio should be measured in the same unit of time, such as quarterly or annual values. But the resulting Sharpe ratio is sensitive to the length of the time period used to compute the numerator and the denominator. Note that the numerator is proportional to the unit of time, ignoring compounding. Thus the excess return expressed as an annual rate will be two times larger than a semiannual rate and four times larger than a quarterly rate, ignoring compounding. However, the denominator is linearly related to the square root of time, assuming that returns are statistically independent through time:

σT = σ1 √ T where σT is the standard deviation over T periods, σ1 is the standard deviation over one time period, such as one year, and T is the number of time periods.

This formula assumes that the returns through time are statistically independent. Thus, a one-year standard deviation is only √2 times a semiannual standard deviation, and a one-year standard deviation is only twice (√4) the quarterly standard deviation. Thus switching from quarterly returns to annualized returns roughly increases the numerator fourfold but increases the denominator only twofold, resulting in a twofold higher ratio.

For example, ignoring compounding for simplicity, and assuming statistically independent returns through time, the Sharpe ratios based on semiannual returns and quarterly returns are shown, using the same annual values as illustrated earlier:

Annual: (10% – 3%)/20% = 0.350

Semiannual: [(10% − 3%)/2]/ 20% √ 0.5= 0.247

Quarterly: [(10% − 3%) /4] / 20% √ 0.25= 0.175

Note that the Sharpe ratio declines from 0.350 to 0.175, which is a 50% decrease, as the time interval for measurement is reduced by 75% from annual to quarterly.

If returns were correlated perfectly through time, the Sharpe ratio would not be sensitive to the time unit of measurement; it would be dimensionless. However, in a perfect financial market, returns are expected to be statistically independent through time and, in practice, returns usually are found to be somewhat statistically independent through time. The point is that Sharpe ratio comparisons must be performed using the same return intervals.

Sharpe ratios should be computed and compared consistently with the same unit of time, such as with annualized data. They then can be easily intuitively interpreted and compared across investments.

However, Sharpe ratios ignore diversification effects and are primarily useful in comparing returns only on a stand-alone basis. This means that they typically should be used when examining total portfolios rather than evaluating components that will be used to diversify a portfolio. Of course, if the investments being compared are well-diversified portfolios, then the Sharpe ratio is appropriate because systematic risk and total risk are equal in well-diversified portfolios. It should be noted that a well-diversified portfolio is traditionally defined as containing only trivial amounts of diversifiable risk.

Finally, a Sharpe ratio is only as useful as volatility is useful in measuring risk. In the case of normally distributed returns, the volatility fully describes the dispersion in outcomes. But in many alternative investments with levels of skew and kurtosis that deviate from the normal distribution, volatility provides only a partial measure of dispersion. Thus, the Sharpe ratio is a less valuable measure of risk-adjusted performance for asset returns with non-normal distributions.

The Sharpe ratio should be used with caution when measuring the performance of particular investments, such as options and option-like strategies, which have return distributions that are skewed or contain the potential for non-linear payoffs.

The Treynor ratio

Another popular measure of risk-adjusted performance is the Treynor ratio. Unlike the Sharpe ratio, this ratio uses beta as the measure of risk in the denominator rather than standard deviation.

The Treynor ratio may be interpreted as the risk premium that the investment earns per unit of beta.

The Treynor ratio depends on the unit of time used to express returns. Generally, the beta of an asset, the denominator of the ratio, would be expected to be quite similar, regardless of the unit of time used to express returns. However, ignoring compounding, the quarterly returns would be expected to be one-quarter the magnitude of annual returns, and monthly expected returns to be one-twelfth the magnitude of annual returns. Thus the numerator is proportional to the time unit, and the denominator is roughly independent of the time unit, meaning that the ratio is proportional to the unit of time.

The Treynor ratio is easily intuitively interpreted as excess return earned by bearing systematic risk. Unlike the Sharpe ratio, the Treynor ratio should not be used on a stand-alone basis. Beta is a measure of only one type of risk, systematic risk. Therefore, selecting a stand-alone investment on the basis of the Treynor ratio might tend to maximize excess return per unit of systematic risk but not maximize excess return per unit of total risk unless each investment were well-diversified.

Beta does serve as an appropriate measure of the marginal risk of adding an investment to a well-diversified portfolio. Therefore, the Treynor ratio is designed to compare well-diversified investments and to compare investments that are to be added to a well-diversified portfolio. But the Treynor ratio should not be used to compare poorly diversified investments on a stand-alone basis. It is less frequently applied in alternative investments, because beta is not an appropriate risk measure for many alternative investment strategies.

The Sortino ratio

A measure of risk-adjusted performance that tends to be used more in alternative investments than in traditional investments is the Sortino ratio. Unlike the Sharpe ratio, the Sortino ratio subtracts a benchmark return, rather than the riskless rate, from the asset’s return in the numerator. Also, it uses downside standard deviation, rather than standard deviation, as the measure of risk in the denominator. Therefore, the Sortino ratio can be used for investments with skewed returns, especially those where the downside risk seems larger than the upside potential.

As a semistandard deviation, the target semistandard deviation (TSSD) focuses on the downside deviations. As a target semistandard deviation, TSSD defines a downside deviation as the negative deviations relative to the target return rather than a mean return or zero. Thus, the Sortino ratio uses the concept of a target rate of return both in expressing the return in the numerator and the risk in the denominator.

Even if the target return is set equal to the riskless rate, the Sortino ratio is not equal to the Sharpe ratio. Although they would share the same numerator, the denominator would be the same only for perfectly symmetrical distributions and where the mean return of the asset equaled the riskless rate.

The point is that the emphasis of the Sortino ratio is the use of downside risk rather than the use of a target rate of return. To the extent that a return distribution is nonsymmetrical and that the investor is focused on downside risk, the Sortino ratio can be useful as a performance indicator.

Jensen’s Alpha

Jensen’s alpha follows directly from the single- factor market model, which links the expected return of an investment to the amount of beta risk incurred.

Jensen’s alpha is a direct measure of the absolute amount by which an asset is estimated to outperform, if positive, the return on efficiently priced assets of equal systematic risk in a single-factor market model. It is tempting to describe the return in the context of the capital asset pricing model (CAPM), but strictly speaking, no asset offers a nonzero alpha in a CAPM world, because all assets are priced efficiently. In practice, expected returns on the asset and the market, as well as the true beta of the asset, are unobservable. So Jensen’s alpha is typically estimated using historical data as the intercept (a) of the following regression equation:

Rt- Rf = a + b(Rm,t- Rf)+εt

where Rt is the return of the portfolio or asset in period t, Rm,t is the return of the market portfolio in time t, a is the estimated intercept of the regression, b is the estimated slope coefficient of the regression and εt is the residual of the regression in time t.

The error term εt estimates the idiosyncratic return of the portfolio in time t, b is an estimate of the portfolio’s beta and a is an estimate of the portfolio’s average abnormal or idiosyncratic return. Because the intercept, a, is estimated, it should be interpreted subject to levels of confidence.  Positive levels of alpha show outperformance, meaning that the manager has earned a greater amount of return than justified by the amount of risk undertaken.  Conversely, negative alpha measures underperformance, where the return on the investment was lower than expected for the amount of risk incurred.

Other popular performance measures exist, and some firms use those unique to the firm. In practice, a variety of performance measures should be explored, each of which is selected to view performance from a relevant perspective.

Mark J.P. Anson, Ph.D., CFA,  has headed up several asset management firms, including Nuveen Investments, Hermes Pension Mgmt. and British Telecom Pension Scheme, as well as was CIO of the California Public Employees’ Retirement System. He also is on the board of the Chartered Alternative Investment Analyst (CAIA) Association. Donald R. Chambers, Ph.D., is the associate director of programs at CAIA and a professor of finance at Lafayette College. Keith H. Black, Ph.D., CFA, is the director of curriculum for CAIA. Hossein B. Kazemi, Ph.D., CFA, is a professor of finance at the Isenberg School of Management at the University of Massachusetts, Amherst. He also is a CAIA managing director.

This piece is an exerpt from “CAIA Level I: An Introduction to Core Topics in Alternative Investments, Second Edition,” Wiley, 2012.