Downside risk measures include target deviation, short-fall probability (covered in a different reading), and value at risk. A target return must be defined first.

Target Semi-Deviation

Downside risk assumes security distributions are non-normal and non-symmetrical. This is in contrast to what the capital asset pricing model (CAPM) assumes: that security distributions are symmetrical, and thus that downside and upside betas for an asset are the same.

Downside deviation is a modification of the standard deviation such that only variation below a minimum acceptable return is considered. It is a method of measuring the below-mean fluctuations in the returns on investment.

The minimum acceptable return can be chosen to match specific investment objectives.

Semi-deviation will reveal the worst-case performance to be expected from a risky investment.

The semivariance is not used in bond portfolio management extensively because of "ambiguity, poor statistical understanding, and difficulty of forecasting".

Coefficient of Variation

A direct comparison of two or more measures of dispersion may be difficult. For example, the difference between the dispersion for monthly returns on T-bills and the dispersion for a portfolio of small stocks is not meaningful because the means of the distributions are far apart. In order to make a meaningful comparison, we need a relative measure, to standardize the measures of absolute dispersion.

It is often useful to compare the relative variation in data sets that have different means and standard deviations, or that are measured in different units. Relative dispersion is the amount of variability present in comparison to a reference point or benchmark. The coefficient of variation (CV) is used to standardize the measure of absolute dispersion. It is defined as:

It gives a measure of risk per unit of return, and an idea of the magnitude of variation in percentage terms. It allows us direct comparison of dispersion across data sets. The lower the CV, the better; investments with low CV numbers offer less risk per unit of return. This measurement is also called relative standard deviation (RSD).

Note that because s and X-bar have the same units associated with them, the units effectively cancel each other out, leaving a unitless measure which allows for direct comparison of dispersions, regardless of the means of the data sets.

The CV is not an ideal measure of dispersion. What if the expected return is zero!? Generally, the standard deviation is the measure of choice for overall risk (and beta for individual assets).

Example

The mean monthly return on T-bills is 0.25% with a standard deviation of 0.36%. For the S&P 500, the mean is 1.09% with a standard deviation of 7.30%. Calculate the coefficient of variation for T-bills and the S&P 500 and interpret your results.

T-bills: CV = 0.36/0.25 = 1.44
S&P 500: CV = 7.30/1.09 = 6.70

Interpretation: There is less dispersion relative to the mean in the distribution of monthly T-bill returns when compared to the distribution of monthly returns for the S&P 500 (1.44 < 6.70).