Downside deviation and 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. 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 downside deviation , also called the target semideviation , measures the spread of returns below a target B. Only observations below B enter the sum, but the divisor still counts every observation.

$$s_{target} = \sqrt{\frac{\sum_{X_i < B} (X_i - B)^2}{n - 1}}$$

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

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 .

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

Dispersion of returnsfrom the previous lesson to this one

• Standard deviation Memorize all spread around the mean, up and down alike
• Target downside deviation Memorize only shortfalls below a target B
  • Divisor n − 1, with n counting every observation
  • Higher target more observations count, bigger shortfalls, larger result
• Coefficient of variation Memorize standard deviation ÷ mean, a relative measure
  • Reading it risk per unit of return, lower is better
  • Caution only meaningful with a positive mean not near zero

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 portfolio management performance-measures material, 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.