Work out the coefficient of variation and semi-deviation and explain what they mean
The idea
Standard deviation treats upside and downside surprises alike, but investors mostly fear the downside. Semi-deviation keeps only returns below the mean: square those shortfalls, sum them, divide by n - 1, where n counts all the observations and not just those below the mean, and take the square root. Target semi-deviation does the same using a chosen threshold, such as zero or a required return, in place of the mean. When semi-deviation is small relative to standard deviation, more of the dispersion lies above the mean. The coefficient of variation, CV = standard deviation / mean return, measures risk per unit of return, so it allows assets with different average returns to be compared; lower is better. It becomes unreliable when the mean is near zero.
A practice question
Fund A has a mean return of 6% and a standard deviation of 9%. Fund B has a mean return of 10% and a standard deviation of 13%. Which offers less risk per unit of return?
- Fund A, with a CV of 1.50Why not: Picked the fund with the higher coefficient of variation, which is more risk per unit of return.
- Fund A, with a CV of 0.67Why not: Inverted the ratio, dividing the mean by the standard deviation.
- Fund B, with a CV of 1.30 — correct
CV = standard deviation / mean. Fund A: 9 / 6 = 1.50. Fund B: 13 / 10 = 1.30. The lower CV means Fund B carries less risk for each unit of expected return.
A harder one
Five annual returns are 10%, -4%, 6%, 2% and 11%. Using n - 1 in the denominator, the semi-deviation is closest to:
- 6.16%Why not: Squared every deviation, above and below the mean, which gives the full standard deviation.
- 4.74% — correct
- 6.71%Why not: Divided by the two below-mean observations instead of by n - 1 = 4.
The mean is 5%. Only -4% and 2% lie below it, with squared deviations 81 and 9. Semi-deviation = sqrt(90 / 4) = 4.74%.
The hardest tier
An asset's returns over four years were 4%, 9%, -1% and 12%. Using the sample standard deviation, its coefficient of variation is closest to:
- 0.95 — correct
- 0.82Why not: Used the population standard deviation, dividing by n = 4 instead of n - 1 = 3.
- 1.05Why not: Inverted the ratio, dividing the mean return by the standard deviation.
Mean = 6%. Squared deviations sum to 4 + 9 + 49 + 36 = 98, so s = sqrt(98 / 3) = 5.72%. CV = 5.72 / 6 = 0.95.
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