Downside deviation, like semideviation, eliminates from the calculation of risk the returns that contribute to positive volatility. We calculate the downside deviation in the same way as semideviation, except we replace the mean return with the target return.
(10.4) To calculate downside deviation, we identify the fund returns less than the target and take the differences of these returns to the target. We then square the differences, add the squared differences, and divide by the total number of returns. This gives the downside variance, or below- target semivariance. Taking the square root of the downside variance yields a statistic measured in rate of return units.
Exhibit 10.7 shows the calculation of downside deviation with the result of 2.55% for the fund and 2.29% for the benchmark. We can annualize the downside deviation in the same way we annualize stan- dard deviation, by multiplying the downside deviation by the square root of the number of return observations per year.
(10.5) Downside deviation
∑
(RPi–T)2 where RPi<T---N
=
Annualized downside deviation RPi–T
( )2 where RPi<T
∑
---N × P
=
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EXHIBIT 10.7 Downside Deviation
whereP is the periodicity, or number of return observations in a year. If we are using monthly returns, we multiply by the square root of 12.
Given the target monthly return equal to 1.20%, the annualized downside deviation for the fund is 8.82% and 7.94% for the bench- mark. The fund has a higher propensity for downside returns than the benchmark, given the target return of 1.2% per month.
Like all of the statistics we calculate in the process of performance measurement, the downside deviation is calculated using the actual his- tory of fund returns experienced over the period. This fact makes down- side deviation very sensitive to the number of return observations and the time period selected. Basically, if market returns are generally posi- tive during the period sampled, then the downside deviation will be understated, potentially to a great degree. To deal with this problem, analysts have developed ways of calculating downside risk via the simu- lation of the true distribution of returns. One method, called bootstrap- ping, involves the generation of a return distribution by repeated sampling of the actual periodic returns. Bootstrapping allows us to con- sider a number of possible scenarios in the calculation of downside risk rather than relying on one scenario, actual past history. Determination of downside risk using simulations may be more useful for the estima- tion of forward-looking risk than that calculated using the actual histor- ical data. In this book we focus on the measurement of historical risk, and we have measured the downside risk inherent in the actual series of fund and benchmark returns.1
1For more on downside risk, see Frank Sortino and Stephen Satchell, Managing Downside Risk in Financial Markets (Oxford: Butterworth Heinemann, 2001).
CHAPTER 11
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Relative Risk
ne of the purposes of return measurement is to determine whether the manager added value over the benchmark. We quantify the por- tion of the returns that can be attributed to the actions of the portfolio manager by isolating the value added from the absolute returns to the fund over the period. Given the relationship between risk and return, we are also interested in whether the risk level of the portfolio differs from the benchmark. Many times the benchmark chosen guides the expected level of risk in the strategy. If the benchmark was set based on the inves- tor’s appetite for risk and the combination of the fund being evaluated with the other assets in the investor portfolio, it is the risk of the fund relative to that of the benchmark that we are interested in measuring. If the fund was riskier than the benchmark, then we would expect higher fund returns over time. For many situations, the manager will set an expected band of risk relative to the benchmark. We measure risk peri- odically to ensure that it is within acceptable limits. If the fund was less risky than the benchmark, the manager may be avoiding risks that are required to achieve the investor’s long-term goal. Taking the perspective of the manager, it is the benchmark relative return, benchmark relative risk, and risk-adjusted return that our performance is being measured by. So both investors and investment managers are interested in measur- ing the degree of benchmark-relative risks taken, along with the bench- mark-relative return.
Where absolute risk is proxied by the standard deviation of returns, relative risk is measured by looking at how the fund returns and the benchmark returns vary together. If the fund and benchmark move up and down together, there is a degree of absolute risk, as measured by standard deviation, but there would not be any relative risk. If the fund and benchmark returns vary in different ways, for example, if fund returns are generally more negative than the benchmark when the market
O
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is down, then the fund has a greater degree of benchmark-relative risk than the risk implied by the benchmark. The volatility of the periodic return differences to the benchmark measures the risk specific to the strategy being evaluated. In this chapter we look at ways to measure these benchmark relative risks.