What is the Coefficient of Variation (CV)?

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Definition:

The coefficient of variation (CV) helps you understand the amount of risk you take in comparison to the return you are expecting from your investment.

🤔 Understanding coefficient of variation

In statistics, the coefficient of variation (CV) measures how widely different data points are distributed in a series around the mean (average) of those data points. The CV helps an investor understand the amount of risk they are taking on compared to the amount of return they are expecting on an investment. You often see it as a ratio of the standard deviation (amount of variation or dispersion) to the mean return. For example, if you don't like risk, you may want to choose investments with lower volatility (price swings or standard deviation from the mean price) and a better risk/reward ratio. If you have a higher risk tolerance, then you may choose an asset with higher volatility, or price movements, in the hopes of higher overall returns.

Example

For example, let's compare two investments for someone who is relatively risk-averse and wants a lower-volatility investment with a better risk/reward ratio. For that, we will calculate and compare the coefficient of variation (CV) for each.

As of December 2019, The 10-year return for the S&P Index is 13.34%, with a standard deviation of 12.42%, giving it a CV of 0.93.

Compare that to the Dow Jones. It has a 10-year return of 13.47% and a standard deviation of 12.02%, giving it a CV of 0.89.

Even though these indexes have similar 10-year historical returns, Dow Jones has a lower CV, so you are theoretically taking less risk for the reward of those returns.

Takeaway

The coefficient of variation is like choosing which games you play at the state fair…

You may want to play the riskiest games to take home the giant stuffed animal, or you may want to play the safest games and settle for a smaller reward. By knowing your risk tolerance, you can choose the games and investments that give you the risk/reward level that keeps you comfortable.

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What is the difference between the coefficient of variation (CV) and variance?

The coefficient of variation (also known as relative variability) and variance both help an investor understand the volatility and risk involved with their investment, but CV takes things one step further.

Variance helps compare the performance of different assets in a portfolio. It analyzes how far each asset is from the average return of the portfolio and from each other. The variance can help an investor determine how they could be allocating assets in a portfolio.

Having a significant variance means the asset is far away from the average return and other assets –- It can also mean more risk. A smaller variance means the asset is closer to the average returns of the portfolio and may carry lower risk.

Sometimes the variance can be difficult to interpret, and it gives added weight to outliers (numbers that are far from the sample mean). Another drawback is that it uses units in its measurement, so it can be challenging to compare assets in different units (such as different currencies). Frequently, investors will take the square root of the variance to get the standard deviation –- This helps you understand volatility and risk and may be easier to interpret.

The coefficient of variation (CV) also helps measure volatility and risk. Still, it pulls in another element – the average returns – to give an investor a more useful risk vs. reward picture. It also gets rid of the unit problem of variance, so it makes it easier to compare different assets across different units – such as euro and dollar-based assets. Standard deviation is the bridge between variance and CV. You need variance to arrive at standard deviation, and you need standard deviation to arrive at CV.

While both variance and CV can help you measure volatility, only the coefficient of variation can help you understand the risk you are taking versus the reward you hope to gain in comparison to other assets.

What is the difference between the coefficient of variation and standard deviation?

Standard deviation measures how spread out (dispersed) a set of data is from the average of those numbers. Standard deviation is typically a measurement of volatility or risk of an asset. It can help you with your portfolio allocation by understanding how volatile different assets are when compared to the average of the portfolio.

You calculate standard deviation by taking the square root of the variance (another measurement of volatility). A lower number indicates less volatility or movement away from the average. A higher number means there is more volatility or more significant potential price swings.

While standard deviation measures risk, coefficient of variation (also known as relative variability) measures the risk/reward trade-off, you can expect with different assets. It takes standard deviation one step further by creating a ratio between standard deviation and average returns (standard deviation/average).

A lower CV means a better risk/reward for the asset. It doesn’t mean it will have a higher return. It simply means it will have a better return based on the amount of risk you are taking to achieve that return.

What does the CV tell us?

The coefficient of variation gives you a clearer picture of the risk you are taking versus the reward you can expect. It is especially useful when comparing several different investment opportunities. If you understand your risk tolerance — the amount of risk or volatility you can handle — then you can look for investments that fit that profile.

When you have a lower ratio of standard deviation to average return, you end up with a better risk/reward trade-off. The higher the ratio, the worse the trade-off is. It’s important to point out that CV does not tell you that the returns of one investment may be better or worse than another. It merely compares the volatility or risk of the investment to the average historical returns.

Let’s compare two different investments to see which one historically offered a more favorable risk/reward trade-off. Investment 1 has an average 10-year return of 17% and a standard deviation of 14%, giving it a CV of 0.82. Investment 2 has a 10-year average return of 12% and a standard deviation of 6%, which yields a CV of 0.5.

Even though Investment 1 had a higher overall return, Investment 2 historically had a lower coefficient of variation, meaning it offered a more favorable risk/reward trade-off for this time period.

Of course, historical returns do not predict future returns. All investments carry risk.

What is the CV formula?

The coefficient of variation formula has two key variables:

  • Standard deviation
  • Mean

The formula then looks like this:

The standard deviation measures how spread out or dispersed the numbers are from the average or mean. To get the standard deviation, you need to:

  1. Find the mean of the data (average) by adding up all the data points in the data series and dividing by the total number of points in the data series.
  2. Next, find the variance by subtracting the value of each data point from the average. Then, you need to square (^2) each of those values and add them all together. Then, divide that number by the total number of data points, minus 1.
  3. Last, you use the variance you calculated in step 2 and take the square root (). This number gives you the standard deviation.

Now you have the standard deviation of the data series. After that, you divide that number by the mean of the data set. This number gives you the coefficient of variation (relative variability). It’s also important to note that if the number in the denominator is a negative value or zero, then the CV may be misleading.

As you can see, to arrive at CV, you need to understand other measures of volatility, such as variance and standard deviation. You build on each of these measures of volatility and risk. Then you arrive at the risk/reward trade-off of the investment.

How to calculate coefficient of variation in Excel

The coefficient of variation measures the risk/reward trade-off for different assets. This means you need to understand the risk (standard deviation) and the reward (the sample mean) within the set of data.

To calculate the CV in Excel, you need to first calculate the standard deviation and average for the data points. From there, you can easily calculate the coefficient of variation. Check out this PDF to see the Coefficient of Variation in action.

Now you can see the coefficient of variation for three separate investments to determine which one may offer the best risk/reward trade-off for your risk tolerance.

Additional disclosures:

Indexes are unmanaged, do not incur fees or expenses, and cannot be invested in directly. The S&P 500 Index is a market-capitalization-weighted index comprising 500 widely traded stocks

The Dow Jones Industrial Average is an index that tracks 30 large, publicly-owned companies that trade on the major stock exchanges.

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