## The Coefficient-of-Variation

One measure of an investment's absolute performance is its variance around the mean of past returns. In this context, the smaller the variance, the more stable the series of returns and consequently the lower the risk associated with such an investment. However, one difficulty encountered in its application, is that higher returns often coincide with higher risks.

A common device employed to overcome this difficulty is the statistical measure known as the coefficient-of-variation.^{4} When applied to an investment, the coefficient-of-variation is defined as follows:

where *μ*_{α}, is the mean of investment α and *σ*_{α} is the standard deviation of investment *α*.

This presents two problems. The average return over the period in question may be negative. Given that a poor return coupled with high volatility is assumed to be dividing a negative number by a denominator unattractive, has the opposite effect to that required, as the negativity of the result is reduced when it should increase.

The second problem is the impact that the standard deviation has on the result. Over

a reasonable period of time, returns are usually higher than the associated risk measure. The standard deviation therefore has a greater influence on the measure due to the range of standard deviations being higher than the range of average returns. In particular, the existence of extreme outliers creates instability, which results in excessive changes in the coefficient-of-variation ranking, when compared to the returns-only ranking.

The *t*-statistic is a measure of statistical significance, derived by dividing the coefficient-of-variation by its standard error. The standard error of the coefficient-of-variation is one over the square root of the number of years in the period. Thus, the *t*-statistic equals the coefficient-of-variation multiplied by the square root of the number of years. For example, 36 years of annual data would be required in order to test whether a fund manager's performance, with an average return of two percent p. a. above his benchmark and a standard deviation of six percent p.a., was statistically significant. Therefore, in order to test the significance of a measure, one often needs a long time series which may be prohibitive.

The coefficient-of-variation is thus a measure of relative dispersion used to compare the risk of assets with differing expected returns. However, there is a theoretical defect with the coefficient-of-variation which is not mentioned above. It accentuates the relationship between risk and return because it takes no account of a risk-free rate-of- return always being available.

## Excess Return to Variability Measure

This is one of the measures first utilised in portfolio evaluation, and is alternatively known as the Sharpe measure or Sharpe ratio (Sharpe [1966]). It is simply the coefficient-of-variation with an investments mean return entering the equation net of

the risk-free rate. All combinations of any portfolio and the risk-free asset are assumed to lie on a line that intersects the vertical axis at *R*_{f}. The preferred portfolio, according to the Sharpe measure, is that which lies on the ray passing through *R*_{f} which lies furthest in the counter-clockwise direction. This is equivalent to stating that the slope of the ray is the highest.

The slope of the line is given by:

where *R*_{f} is a proxy for the risk-free rate of interest,
the arithmetic mean return, and *σ*_{p} the standard deviation of the excess return on portfolio *p* during the period under consideration.

The Sharpe measure - equation (4.1) - is usually compared to a similar ratio:

for a passive index, such as the Financial Times All-Share Index. It attempts to measure whether the sample return of the investment (above the risk-free rate) was sufficient for the total risk being borne. Further discussion of the Sharpe measure is undertaken in the next section.

However, a number of theoretical problems remain unresolved in this area of risk analysis. These include:

- the assumed continuous - as opposed to discrete - distribution of variables;
- the possibility of a skewed distribution (further discussed in Chapter 6); and/or

- serial correlation between cash flows of the various investment opportunities.
^{5}

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^{4}For an example of this measure being applied in the analysis of real estate risk and return, see MacGregor & Nanthakumaran [1992].

^{5}For a detailed discussion see Baum & Crosby [1988].