# Coefficient of Variation

## Meaning of the Coefficient of Variation

Coefficient of Variation (CV) is a statistical measure that helps to measure the relative variability of a given data series. Or, we can say it measures the distribution of data points in accordance with the mean. Since the key factors involved in the calculation are standard deviation and mean values, hence, it can also be referred to as a ratio of standard deviation to the mean. Such a measure helps to compare the level of deviation between two or more data sets. However, the means of those data series are different from each other. CV or relative standard deviation must always be seen in relation to the mean.

## Coefficient of Variation (CV) in Finance

In the financial world, the coefficient of variation helps to determine the volatility in comparison to the expected return on investment. Another application of CV is that it helps compare the results of different tests or surveys. Suppose the CV of two surveys – A and B – is 5% and 10%, respectively, we could say that Survey B has more variation in relation to its mean.

So, we can say that the lower the CV value is, or the lower the ratio of the standard deviation to mean, the better it is. A lower ratio suggests a superior risk-return trade-off.

A point to note is that we must apply CV only on data with a ratio scale. The CV does not hold any value for data points on the interval scale. For example, temperature data, such as Celsius or Fahrenheit, is the interval scale. Kelvin scale, however, is a ratio scale. Or, we can say if the mean or expected return is either 0 or negative, then the CV number may not be accurate.

## How to Find the Coefficient of Variation?

### Formula

The following formula is used for the calculation of CV.

CV = Standard Deviation / Expected Return

The above formula is a general one. For financial purposes, the formula for CV is – Volatility/ Expected Return.

To get the answer in percentage terms, we can multiply the resultant number by 100.

We also have a mathematical formula to compute the CV. And this formula is:

In the formula, Xi is the ith random variable, X is the mean of the data series, and N is the number of variables.

Following are the steps to calculate CV using the mathematical formula:

First, determine the random variables which are part of the data series. These variables will be Xi.

Second, find the number of variables. This number will be N.

Third, measure the mean of the series. To calculate the mean, take a sum of all the random variables, and divide it by the number of variables. Denote the mean by X.

Fourth, now calculate the standard deviation. For the calculation of SD, we need to determine two things – the number of variables in the given data series. And the second one is the variation of each variable from the mean value.

Last, now put the above values of mean and SD in the formula, or divide the SD by the mean.

If Standard Deviation is given, then it becomes simpler, and the formula would be:

Standard Deviation / Mean of the Data Series

For calculation, you can use the Coefficient of Variation Calculator.

### Example

Let’s understand this with the help of a simple example. Mr. X, an investor, wish to compare three stocks: α, β, and λ, for making an investment for his portfolio. He has the following information about three stocks:

Mr. X decided to evaluate all three stocks on the basis of the coefficient of variation.

### Interpretation

A smaller coefficient of variation is good than one which is high. In the example above, β has the highest CV among the three, and that of λ is the lowest. This simply means that the stock λ is the best-performing stock instead of the fact that it has the lowest return out of all. Same way, β having the highest return of all, is the least performing among the three

## Coefficient of Variation – Applications

CV is a simple, quick, and efficient measure to compare varying sets of data. Because of this, the Coefficient of variation is of use in several fields, such as:

### Probability Analysis

CV is very useful for probability analysis. For instance, one can use it in renewal theory, queuing theory, and reliability theory. In an exponential distribution, the standard deviation is about the same as the mean, and this results in a CV equaling 1. So, if the CV of a distribution is less than 1, then it is seen to be having a low-variance. And, if the CV is more than 1, then the distribution has a high variance.

### Finance

In finance, the CV helps to select between several investment alternatives. The CV in finance gives a risk-to-reward ratio, where the volatility represents the risk, and the mean represents the reward or return on the investment.

To arrive at a decision, one needs to calculate the CV of different securities or investment options. This way, an investor gets the risk-to-reward ratio for different securities to arrive at an investment decision.

Usually, an investor goes for an investment with a lower CV. This is because it offers less volatility but more return. But, if the average expected return is less than zero, then going for security with a lower CV is not beneficial.

## Limitations of the coefficient of variation

This statistical tool has a few limitations as well.

### Unreliable in volatile situations –

The returns from an investment option might have varied significantly from the mean over the last few periods or years. It can be due to unfavorable economic conditions or some other uncontrollable factors. The situation may have improved in favor of the investment option and can be an excellent return opportunity in the present times.

The standard deviation will be high for the stock when calculated, and hence, the coefficient of variation will be high too. If an investor solely relies on this figure for making his investment decision, he might not go for this option. Hence, he may miss out on an opportunity to get high returns.

### Misleading in case of negative values or zero-

This statistical measure may be misleading or incorrect if the mean annual returns from an investment option are negative or zero.

## Coefficient of Variation vs. Standard Deviation

Both Standard Deviation and coefficient of variation help to measure the dispersion of a given dataset. Talking about SD, it tells an average distance of a data point from a mean. The SD is zero if all data points are same in a dataset. A value of SD is impacted by the high and low data points in the data set. For instance, if the data points are more spread out, then the value of SD will be more.

CV, on the other hand, is useful when a person knows nothing about the data set except for mean and SD. Basically, it helps to take a decision when comparing two or more datasets. Or, we can say CV gives a value that helps us to easily compare two or more datasets accurately.

SD is an absolute measure of dispersion for a distribution, while CV is relative measure of dispersion. We can calculate CV if we know SD (using formula discussed above).

Let’s take an example to better understand the difference between the two. Suppose Player A has a mean of 60 and SD of 18, and player B has a mean of 48 and SD of 12. Just looking at these stats, one may conclude that player A is better because he has more goals and difference in SD between the two is also not much.

But, if we calculate CV for both players, we get 30% for A and 25% for B. This means dispersion is higher for A, or B is better.

## Conclusion

The coefficient of variation is an essential statistical measure to protect a rational investor from volatile investment options. It can also help in predicting returns from any investment as it takes into account data from several periods.

It is not solely based on risk and returns data from just one single period or instance. Hence, it helps in making wise and correct investment decisions and to achieve a proper balance between risk and returns. It is for these reasons that portfolio managers and analysts widely use this statistical tool in their reports and analysis. However, as explained earlier, this also needs to be considered in conjunction with other similar indicators for a better view. 