The Beta equal to 1 indicates that the degree of covariance in the fund and benchmark returns equals the amount of variance in the benchmark returns. Notice that even though Beta is 1, the fund and benchmark returns are different for each individual period. The tracking risk statis- tic quantifies the amount of this fund-specific difference between the fund and benchmark returns, if we assume that the benchmark is a meaningful benchmark for the portfolio. That is, tracking risk is most useful when the Beta of the portfolio is close to 1. If it is not, the track- ing risk statistics will lose their meaning.
Many investment strategies are designed to minimize return differ- ences to the market benchmark. A fund or strategy that perfectly tracked its benchmark would have a correlation equal to one. This fund would have a zero tracking risk. Other strategies are managed to a par- ticular expected tracking risk target. Tracking risk, also called tracking error, or active risk, is a measure of the magnitude of departures in fund returns from benchmark returns over time. The term tracking error has its roots in the management of index funds. The minimization of expected tracking error at an acceptable cost is a key factor in the man- agement of index funds. But the estimation of expected tracking risk is also a key consideration in the construction of many types of portfolios.
Some investment policies include a particular fixed boundary around the allocation of assets to particular asset classes or other portfolio seg- ments in order to minimize the expected tracking error to benchmarks.
Risk budgeting and other portfolio management techniques also serve to minimize tracking error. There are tools for estimating future track- ing error, for example, by using a multifactor risk model. Here we are concerned with measuring historical tracking error, which provides
178 RISK MEASUREMENT
information both as to the success of the manager in meeting the track- ing error targets and the benchmark relative risk taken by the manager.
The historical tracking risk of a fund or strategy equals the standard deviation of the difference between the periodic fund and benchmark returns:
(11.8) WhereDiare the periodic differences between the fund and benchmark returns and is the mean of the return differences. To calculate track- ing risk, take the periodic differences between the returns of the fund and the returns of the benchmark. Then take the standard deviation of these differences. This will yield a tracking risk with the same periodic- ity as the underlying returns. The tracking error statistic is useful because it is expressed in units of return. Exhibit 11.12 illustrates the calculation of the tracking risk, 1.95% per month, for our sample fund.
The tracking risk is a function of the standard deviation of the fund returns and the correlation of the fund and benchmark returns. In fact, we also can calculate tracking error with the following calculation:
(11.9)
EXHIBIT 11.12 Tracking Risk
Tracking risk
∑
(Di–D)2 ---N=
D
Tracking risk = Stdevp(RPi)× 1–(correlation)2
Relative Risk 179
We annualize tracking risk by multiplying it by the square root of the number of periodic return observations in a year.
(11.10) WhereDi are the periodic differences between the fund and bench- mark returns, is the mean of the return differences, and P is the fre- quency of the return observations. Exhibit 11.12 illustrates the calculation of the annualized tracking risk for our sample fund equal to 6.76%. Passive funds have a very low tracking risk. For an actively managed strategy, tracking risk provides a measure of active, rather than total, risk. When the fund and the benchmark are equally risky and returns are highly correlated, there will be a low tracking risk. It is pos- sible for a fund and benchmark to have similar average returns and standard deviations of return, but imperfect correlation of returns. The tracking risk statistic can identify this situation. The lower the tracking risk, the closer you should be to the benchmark return. One issue with tracking error is the same problem exhibited by standard deviation: It provides no information as to the direction of the deviation. A manager who consistently outperformed the benchmark could exhibit the same tracking risk as the manager who trailed his benchmark.
The risk statistics we have calculated here are only some of the pos- sibilities at our disposal for evaluating a particular situation. For example, one of the problems with using tracking risk as a measure of benchmark relative risk is that it is based on the standard deviation, so it weights upside deviations equally with downside deviations. If the benchmark deviations are normally distributed, this is not a problem.
But if the manager can produce returns with more upside risk than downside, tracking risk will not capture this fact. We can measure this instead by calculating the downside tracking risk, which is the stan- dard deviation of the returns below some target difference. For exam- ple, we can calculate a downside tracking risk with a target value added equal to zero. There are many other possible adaptations of the basic calculations.
We could also calculate tracking risk as the relative difference between the fund and benchmark returns, or
(11.11) Annualized tracking risk
∑
(Di–D)2---N × P
=
D
Relative tracking risk Standard deviation RPi RMi ---
=
180 RISK MEASUREMENT
EXHIBIT 11.13 Risk Statistics Using Different Benchmarks
Benchmark Selection
Tracking risk and other relative risk statistics are very sensitive to the benchmark selected for the comparison. We need to keep this in mind when ranking portfolios based on benchmark relative or risk adjusted return statistics. We demonstrate the reason for this in Exhibit 11.13.
Here we calculated the Alpha, Beta, and R-squared for the same fund versus several different benchmarks to show how much the results can differ depending on the benchmark used.
Absolute, downside, and relative risk statistics are valuable tools for measuring risk. But these statistics still leave some of our performance measurement questions unanswered. These questions include the ques- tion as to whether the return earned is high enough to justify the risks quantified given the standard deviation, downside deviation, or tracking error exhibited by the strategy. In the Part III of the book we address the issue of relating risk to return.
three PART
Measuring Risk-Adjusted
Performance
CHAPTER 12
183
Absolute Risk-Adjusted Return
he standard deviation and downside risk measures quantify the dis- persion of returns earned over time, which is our primary proxy for absolute risk. The Beta and tracking error provide us with measures of the benchmark relative risk of a portfolio. Given a measure of risk, our next task is to address the question of whether the return was sufficient given the risks taken. One way to do this is to compare the combined risk and return earned by several peer group portfolios and our portfo- lio to a benchmark. We can make a visual comparison by creating a chart plotting the combination risk/return observations for each portfo- lio. Exhibit 12.1 plots the 3-year annualized return and 3-year annual- ized standard deviations for ten large company stock funds.
EXHIBIT 12.1 Risk versus Return
T
184 MEASURING RISK-ADJUSTED PERFORMANCE
The Y-axis represents the standard deviation of returns over the period and the X-axis represents the measure of return, here the arithmetic aver- age return, for the period. We can see that there is a general relationship between the risks taken and the returns earned. But there are some excep- tions. For example, Fund B exhibited the highest standard deviation, but earned a lower return than funds F and J. Funds C and H had approxi- mately the same risk, but Fund C had a higher return. So it is important to relate risk and return in order to evaluate the performance of a fund.
In addition to the graphical representation, it would be worthwhile to have numerical measures of the combined risk and return exhibited by a portfolio. We can adjust the returns earned over time by the stan- dard deviation of return and other statistical descriptions of risks taken in order to derive measures of risk-adjusted return. Risk-adjusted returns are composite risk-return measures that are used to help deter- mine whether or not the returns earned were sufficient compared to those earned by similar portfolios and benchmarks exhibiting a similar level of risk. There are several ways to determine the risk-adjusted returns to a portfolio. In this chapter, we consider measures of risk- adjusted return calculated in the context of Modern Portfolio Theory (MPT). MPT-based statistics evaluate risk using the Capital Asset Pric- ing Model, a theoretical model of risk and return. In Chapter 13 we look at some measures of risk-adjusted return useful, where risk is mea- sured as the downside or benchmark relative risk taken by the manager.
Chapter 14 reviews how we can use the return history, along with the composite risk and return statistics, to quantify the consistency, or skill, of an investment manager.