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# Coefficient of Variation

The coefficient of variation, CV, is a measure of spread that describes the amount of variability of data relative to
its mean. It has no units, and as such, we can use it as an alternative to the standard deviation to compare the variability of data sets that have different means.

## Coefficient of Variation Formula

$$\text{CV} = \cfrac {S}{\text x̄}$$

Where S is the standard deviation of a sample

And x̄ is the mean of the sample.

Note: the formula can be replaced with σ/μ when dealing with a population.

Below is the procedure to follow  when calculating the coefficient of variation:

1. compute the mean of the data;
2. calculate the sample standard deviation of the data set, S; and
3. find the ratio of S to the mean, x̄.

### Example: Calculating the Coefficient of Variation

Calculate the relative variability for the samples 40, 46, 34, 35, and 45 of a population.

Solution

Step 1: calculate the mean.

$$\text{Mean} =\cfrac {(40 + 46 + 34 + 35 + 45)}{5} =\cfrac {200}{5} = 40$$

Step 2: calculate the sample standard deviation. (Start with the variance, $$S^2$$.)

\begin{align*} S^2 & =\cfrac {{(40 – 40)^2 + … + (45 – 40)^2 }}{4} \\ &=\cfrac {122}{4} \\ & = 30.5 \\ \end{align*}

Note: since it is the sample standard deviation, and not the population standard deviation, we use n – 1 as the denominator.

Therefore,

$$S = \sqrt{30.5} = 5.52268$$

Step 3: calculate the ratio.

$$\text{Ratio} =\cfrac {5.52268}{40} = 0.13806 \text{ or } 13.81\%$$

(You can use these links to refresh your memory on calculation of the mean and standard deviation)

### Interpreting the Coefficient of Variation

In finance, the coefficient of variation is used to measure the risk per unit of return. For example, assume that the mean monthly return on a T-Bill is 0.5% with a standard deviation of 0.58%. Suppose we have another investment, say, Y with a 1.5% mean monthly return and standard deviation of 6%. Then,

$$\text{CV}_{\text T-\text {Bill}} =\cfrac {0.58}{0.5} = 1.16$$

$$\text{CV}_\text{Y} =\cfrac {6}{1.5} = 4$$

Interpretation: the dispersion per unit monthly return of T-Bills is less than that of Y. Therefore, investment Y is riskier than an investment on T-Bills.

## Question

If a security has a mean expected return of 10% and a standard deviation of 5%, its coefficient of variation is closest to:

1. 0.005
2. 0.5
3. 2

Solution

$$\text{CV} = \cfrac {S}{\text x̄} = \cfrac {0.05}{0.10} = 0.5$$

Where S is the standard deviation of a sample

And x̄ is the mean of the sample.

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