FeatURE The Hitchhiker’s Guide to Risk-adjusted Returns By Rex Macey, CIMA®, CFA® n The Hitchhiker’s Guide to the Galaxy, In The Hitchhiker’s Guide to the Galaxy, a computer—after working for 7.5 Imillion years—finds that 42 is the “a computer—after working for 7.5 million answer to “life, the universe, and every- years—finds that 42 is the answer to “life, thing.” By its very nature, a risk-adjusted return (RAR) promises to encapsulate the universe, and everything.” return and risk into a single performance ” measure. But just as simple answers to questions about the meaning of life tend the other half in the risk-free security they had assumed a level of risk equal to to be elusive, no single measure succinctly would earn a return of 9.5 percent. The that of the market. The MM risk-adjust- describes investment performance. With portfolio would have a standard devia- ed performance arrives at the score by that important caveat, here is a brief guide tion of 5 percent (because the standard leveraging or deleveraging the portfolio to risk-adjusted returns, including their deviation of the risk-free investment is so that its risk is identical to that of the uses and misuses. zero). The blend of the second invest- market (i.e., benchmark) portfolio. It ment and the risk-free security offers assumes that investors may borrow or The Sharpe Ratio more return for the same level of risk lend at a risk-free (standard deviation Due to its simplicity or to the pedi- as A, making the second investment = 0) rate. gree of its Nobel Laureate creator, the preferable. The Information Ratio Sharpe ratio is the most commonly Modigliani and Modigliani Ratio used RAR. William Sharpe introduced The information ratio (IR) is a more his measure more than 40 years ago Because the Sharpe ratio is the excess general version of the Sharpe ratio. To (Sharpe 1966). As it is used today, the return of a portfolio divided by its stan- use the IR one chooses a benchmark, Sharpe ratio is defined as the return of a dard deviation, the units are difficult such as the S&P 500 Index. Each period portfolio over and above a risk-free rate to interpret. The newest score on the (e.g., month) the difference between the divided by the standard deviation of the block, the Modigliani and Modigliani return of the portfolio and that of the portfolio. (MM) ratio, proposes a twist on the benchmark is computed. The IR is the Sharpe ratio to simplify interpretation average of the differences divided by the Sharpe r portfolio −rrisk- free = (Modigliani and Modigliani 1997). The standard deviation of the differences. ratio s portfolio MM ratio is the Sharpe ratio multiplied In other words, it is the excess return The Sharpe ratio and other RARs by the standard deviation of the market divided by the tracking error. demand a return differential, which portfolio. The rankings of a set of port- The Sharpe, MM, and IR measures also is called an excess return (Sharpe folios will be identical whether ordered all use standard deviation to represent 1994). An example will illustrate why by Sharpe or MM. MM provides a value risk. Standard deviation measures the using return alone, rather than a return that is the excess return that investors variability (i.e., uncertainty) of portfolio differential, might lead to the choice of would have achieved relative to the returns around an average. A criticism an inferior investment. market (i.e., the benchmark portfolio) if of standard deviation stems from the Consider investments A and B and TABLE 1 a risk-free security with the following characteristics (see table 1): Investment Return Std. Dev Ret/Std. Dev Sharpe While A offers more return per unit Risk-free security 5% 0% of risk (8/5 v. 7/5), B is the better invest- A 8% 5% 8/5 0.6=(8–5)/5 ment, a fact signaled by its superior B 14% 10% 7/5 0.9=(14–5)/10 Sharpe ratio. This is because an investor 1 1 /2 B + / 2 risk-free security 9.5% 5% 9.5/5 0.9=(9.5–5)/5 placing one-half of his assets in B and January/February 2008 5 © 2008 Investment Management Consultants Assn. Inc. Reprint with permission only. © 2008 Investment Management Consultants Assn. Inc. Reprint with permission only. FeatURE fact that observations above the mean compensated according to the capital between 1963 and 1990, high beta contribute as much to the statistic as asset pricing model (CAPM). portfolios did not outperform low observations an equal distance below beta portfolios, on average (Fama and = + the mean. Critics point out that inves- Rportfolio R risk-free French 1992). Entire papers, and prob- tors generally don’t mind unexpectedly β − ably books, could be written about (R market R risk-free ) high returns; they mind unexpectedly beta. I simply will state that beta must low returns. The expected return of a portfolio be used with extreme care. Because standard deviation mea- should be the risk-free rate plus the Fama-French Factor Models sures total variability, it is appropriate excess return of the market multiplied for measuring total portfolio return as by beta, which is the systematic risk of a Many professionals use what is known long as the distribution of returns is portfolio. Within the framework of this as the Fama-French three-factor model symmetric. If returns are symmetric, model, beta is the measure of risk. to compensate for the flaws associated above and below average observations Imagine two emerging market with the single beta used in the Jensen occur with the same frequency and investments with identical returns and measure. Whereas Jensen assumes that magnitude. If returns are asymmetric, standard deviations. While they may risk comes from one source—market then it makes sense to focus on down- have the same Sharpe ratio, an investor exposure as represented by beta— side risk. One downside risk statistic, would prefer the one with the lower Fama-French adds two factors.1 One is called semi-deviation, considers only correlation to the rest of his portfolio. an exposure to size, which they define the observations that are negative. Beta incorporates the correlation to the as the market capitalization, and the Similarly, one can choose to focus on rest of the portfolio. other is an exposure to value, defined observations that fall below a target or as book value divided by market value. Treynor Ratio required rate of return. In addition, if fixed income is included Like the Sharpe ratio, the Treynor ratio in the portfolio, they add two more fac- Sortino Ratio divides the excess return of the portfo- tors: One measures the time to maturity The Sortino ratio resembles the Sharpe lio by its risk. The difference is that the of the portfolio’s holdings and the other ratio, except that it uses downside risk Treynor ratio uses beta to represent assesses default risk. The general idea is in place of standard deviation. Use of risk. It also is useful for ranking invest- to measure the excess return of a port- this statistic instead of Sharpe makes ments. folio relative to what would have been sense when the distribution of returns earned by a benchmark portfolio taking − is asymmetric. r portfolio rrisk-free the same systematic risks. Treynor ratio = RARs that use standard deviation β portfolio Fama-French alpha are appropriate for well-diversified Jensen’s Measure = − [ + portfolios and for entire portfolios. R portfolio R risk- free A different measure should be used Also called Jensen’s alpha, this statistic β market (R market − R risk- free ) + for component styles or subasset measures a portfolio’s return in excess β (R − R )+ classes (e.g., emerging markets). The of that predicted by the CAPM. size large small fathers of modern portfolio theory, β ( − )] = − value R value R growth Harry Markowitz, Sharpe, and oth- Jensen’s alpha R portfolio Shapes Matter ers, recognized that investors should + β − [Rrisk-free (Rmarket Rrisk-free )] expect to be compensated for accept- Users of risk-adjusted returns should ing only systematic (i.e., market) risk. (The term in the square brackets is the consider probability distributions, They distinguish between systematic return predicted by the CAPM so that which define the likelihood of each pos- risk—that of the market, which cannot the expression is the portfolio return sible event. For example, the probability be diversified away—and unsystematic less the CAPM prediction.) that any one face of a die will appear risk, which can be reduced through di- While CAPM was enough to win when it is rolled is 0.1667 (1/6). This is versification. Thus, investors who take a Nobel Memorial Prize for Sharpe, called a uniform distribution. Distribu- the extra risk of holding a concentrated the model, particularly beta, has been tions can come in many shapes. Most portfolio can expect greater variability criticized on practical terms. Fama people are familiar with the normal in returns (i.e., risk) but not higher and French found that beta did not distribution, which is shaped like a average returns. explain differences in the returns of bell. This is a symmetric distribution. According to modern portfolio stocks. In their study of stocks on Security returns are not normally dis- theory, investors should expect to be the NYSE, AMEX, and NASDAQ tributed. A common assumption is that 6 Investments&Wealth MONITOR © 2008 Investment Management Consultants Assn. Inc. Reprint with permission only. © 2008 Investment Management Consultants Assn.