While the mean absolute deviation is a functional description of the variability in a series of returns, it is not commonly used in performance analysis. This is because there are measures of variability with better sta- tistical properties than the mean absolute deviation that also convey the same information. We use the deviations from the mean in the calcula- tion of the most commonly used statistic representing return dispersion, the standard deviation of returns. Standard deviation is a measure of how widely the actual returns were dispersed from the average return.
When calculating standard deviation, instead of taking the absolute Mean absolute deviation ABS(RPi–RP) 1
N---
∑
×=
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value of each deviation, we square each of the deviations. This has the same effect as taking the absolute value in that it turns all of the devia- tions into positive numbers. The squared deviations are summed and then divided by the number of returns to give the variance. We typically avoid using variance as a measure of ex-post risk because it is measured in squared returns, rather than returns. In other words we cannot directly compare the variance to the return in order to assess reward to risk. We can, however, compare the standard deviation to the return to make a direct comparison. Standard deviation is the square root of the variance. To calculate the standard deviation of a return series:
1. Square each difference between the periodic returns and the arithmetic mean return.
2. Sum the squared differences.
3. Divide the sum of the squared differences by the number of returns.
4. Take the square root of the result.
(9.5) Equation (9.5) is the formula for standard deviation, where RPi is the individual portfolio return observations, is the arithmetic mean return, and Nis the count of returns. Note that because we are squaring the deviations, the standard deviation is affected more by outliers than the mean absolute deviation. Exhibit 9.6 illustrates the calculation of standard deviation for our fund, equal to 4.13%. The standard devia- tion for the benchmark is lower, 3.65%.
Standard deviation is the primary statistic used to describe the vari- ability in a pattern of returns. Because this variability is a proxy for risk, standard deviation of the periodic returns is the chief proxy for risk used in the management and analysis of investments. A higher standard deviation indicates a wider dispersion of returns around the mean return. A portfolio that has twice the standard deviation of another fund has twice the volatility as that fund.
There are a few considerations we should keep in mind when using the standard deviation and related measures of risk. These include the number of observations used to form the statistic and the underlying periodicity of the data used.
Standard deviation
∑
(RPi–RP)2 ---N=
RP
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EXHIBIT 9.6 Standard Deviation
Length of Time Period
The examples in this section of the book use 13 months of monthly returns as inputs. We use a small number of observations to facilitate the study of the calculation and interpretation of the various statistics.
A practical analysis, however, will usually involve longer time periods.
But how many return observations should we use? Most commonly, practitioners will use three years of monthly data, or 36 monthly obser- vations, to analyze the historical risk of the investment strategy. The choice of the time period has implications as to both the number of observations used and the exposure to all phases of the market cycle. If we have a short time period, the statistics may be highly unstable as they are sensitive to the addition or deletion of additional periods. The results will have a low statistical validity. In addition, if we are making inferences based on, say, the standard deviation, these inferences may be invalid if the time period does not represent different phases of market cycles, in our case periods of both high and low volatility. Here we are also implicitly equal weighting each return observation when calculat- ing the risk statistics. For example, if we are calculating a 3-year trailing standard deviation, the return from 36 months ago contributes equally to the calculation as last month’s return. There are alternatives to equal weighting, for example, by using weighted moving averages, where the most recent time periods would have a higher weighting in the calcula-
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tion. We use the weighting techniques more in the estimation of ex-ante, or future risk, than in the measurement of historical risk.
A related issue is the use of the sample vs. population version of the standard deviation. When deriving averages used in the risk statistics, such as the average of the return deviations used in the standard devia- tion, we divided by the total number of returns in the data set, or N.
Dividing by N gives the average when we are calculating risk statistics for the entire populationof returns we want to describe. When we are using statistics such as the standard deviation to identify the characteris- tics of a complete data set, we use the population version of these statis- tics. Sometimes we calculate the risk statistics using a part of the population, or a sample set of returns. When we calculate risk statistics using a sample, but intend to use the statistics to make judgments about the entire population, we divide by N−1 instead of N. If we were to recalculate the examples in this chapter by dividing by N−1 instead of N wewouldget meaningfully different absolute results. This is because we chose to use only 13 observations to facilitate the study of each mea- sure. The adjustment for samples is not a major consideration in the analysis of risk, because we usually use a large enough number of return observations that, dividing by either N orN−1, gives approximately the same result. Regardless, none of the relative rankings and inferences made based on differences between the fund and benchmark risk would change. If we were taking risk statistics calculated using two different sources and then using them in a comparison, it is useful to know whether the population or sample method was used.
Measurement Frequency
Assume we are calculating daily single period time-weighted returns. To calculate a 1-year cumulative return, we could link either the approxi- mately 250-trading-day returns or the twelve monthly returns. In either case, we would get the same 1-year return because time-weighted returns can be compressed—the daily returns for one month equal the one month return and so on. This property does not extend to risk measures.
The periodicity of the underlying returns impact the risk calculations.
For example, because returns usually fluctuate less on a daily basis than on a monthly basis, the standard deviation of a series of daily returns is usually smaller than the standard deviation of monthly returns, over the same time period. Risk statistics are commonly computed using daily, weekly, monthly, or quarterly periodic return frequencies. It is inappropriate to compare risk statistics that were calculated using a dif- ferent underlying periodicity. Choice of the return frequency is depen- dent on the availability of the underlying fund returns, which is in turn
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dependent on the valuation frequency. Traditionally, valuation was only available on a monthly basis for most investment vehicles, except mutual funds. As daily valuation of investment portfolios has become more prevalent outside of the mutual fund industry, daily frequency risk mea- surement is now possible. Assuming that we experience risk on a daily basis, risk statistics calculated using monthly inputs might not represent the risk implied by the daily returns. In addition, it is possible that the relative rankings of portfolios or strategies would change with the use of more or less frequent data. Using daily observations allows for the mea- surement of the true volatility experienced by the investor.
Arithmetic versus Geometric Risk Statistics
So far we have calculated the standard deviation of the periodic returns around the arithmetic mean return. We could have instead calculated the standard deviation of the returns around the geometric mean return. To do this we first calculate the natural logs of the growth rates of the single period returns, and then take the standard deviation of these growth rates. In practice most analysts use the arithmetic methodology to calcu- late historical risk statistics, but the geometric statistics are also valid.