Semi-deviation, downside deviation and the coefficient of variation

How do you measure only the bad volatility?

The standard deviation counts a surprise gain exactly like a surprise loss. Most investors do not see it that way. They worry about returns that fall short, either below the average or below a minimum they need. The semi-variance is the variance of the observations below the mean; its square root is the semi-deviation .

$$\sigma^2_{Semi} = \sum_{X_i < \bar{X}} \frac{(X_i - \bar{X})^2}{n}$$

The reading prints this definition with n, then says both semi-deviation measures divide by n − 1 because the data are a sample, and every worked number it gives uses n − 1 with n counting all observations. Follow the question; with no instruction, use n − 1.

For the eight returns below, the mean is 0.975% and the same four months fall below it: √(18.83 ÷ 7) = 1.64% (dividing by 8 instead gives 1.53%).

Downside risk is the risk of returns falling below a level you care about, such as a minimum return needed to meet a liability. The target semi-deviation , also called the target downside deviation , measures the spread of returns below a target B. Only observations below B enter the sum, but the divisor still counts every observation (n is the total number of observations, not just those below B).

$$s_{Target} = \sqrt{\sum_{\text{all } X_i \le B} \frac{(X_i - B)^2}{n - 1}}$$

Drag the target line and watch the bars that turn orange. More bars fall short as the target rises, and each shortfall grows.

Semi-variance (measured from the mean) is always lower than the variance of the same data, so the two cannot be compared directly: compare semi-deviation with semi-deviation.

With a target near the average, the downside deviation is usually lower than the standard deviation, because only part of the data counts. Raise the target far enough and it can exceed it.

What is the coefficient of variation?

Standard deviations are hard to compare across assets with very different average returns. A 10% standard deviation is mild for an asset that earns 20% and wild for one that earns 2%. The coefficient of variation (CV) fixes this by expressing the standard deviation relative to the mean. It is a measure of relative dispersion .

$$CV = \frac{s}{\bar{X}}$$

Turned upside down, mean ÷ standard deviation is the return per unit of risk, sometimes called the unit price of risk . Computed on excess returns (returns minus the risk-free rate), it compares the reward for risk across assets, the idea behind the Sharpe ratio in Portfolio Construction.

Two cases break the CV. If the mean (or the mean excess return) is negative, the ratio is hard to interpret. And daily or weekly data have tiny means and comparatively large standard deviations, so their CVs are huge and unstable; annualize the mean and the standard deviation first (lesson 7).

Dispersion of returnsfrom the previous lesson to this one

• Standard deviation Memorize all spread around the mean, up and down alike
• Semi-deviation Know the idea only observations below the mean
• Target semi-deviation Memorize only shortfalls below a target B
  • Divisor n − 1, with n counting every observation
  • Higher target more and larger shortfalls, larger result
• Coefficient of variation Memorize standard deviation ÷ mean, a relative measure
  • Reading it risk per unit of return, lower is better
  • Inverse return per unit of risk
  • Caution negative mean or high-frequency data: annualize first or avoid

Exam tip CV: lower is better. If a mean is zero or negative, do not use the CV at all.

Go deeper From downside deviation to the Sortino ratio

The Sortino ratio is the Sharpe ratio’s cousin: it divides the excess return over a target by the target downside deviation instead of by the standard deviation. It rewards a manager whose volatility comes mostly from gains. It appears in the performance measures of Portfolio Construction, and this lesson supplies its denominator.

Go deeper Why investors with a minimum return watch the downside

A pension fund that must earn at least 3% a year to pay benefits is not troubled by returns above 3%. Its pain comes from the months and years below that line. A manager with big upside swings and few shortfalls can have a high standard deviation but a low downside deviation, and the second number is the one that matches the fund’s problem.