###### Conditional Expectation in Investments

In the context of investments, conditional expectation refers to the expected value of... **Read More**

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.

$$ \text{CV} = \cfrac {S}{\text x̄} $$

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:

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

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)*

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.

QuestionIf a security has a mean expected return of 10% and a standard deviation of 5%, its coefficient of variation is

closestto:

- 0.005
- 0.5
- 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.