# How to Calculate Beta using Correlation and Volatility. Overview and Explanation

In this article, I want to show you how to calculate the beta of a stock using the correlation and the stock returns with the returns of the market. We're going to take the correlation of the returns of stock Ri, let's say Walmart stock, and the returns of the market. We take that correlation and multiply it by the standard deviation of the stock returns for firm Ri. So the standard deviation of company stock returns as we could also call that volatility. We take the volatility of the stock and multiply it by the correlation of the returns of the firm, with the return of the market and we divide all of that by the volatility of the market return. I'm going to show you how to do all these calculations by hand instead of just giving the excel formula.

Just one note, the correlation to get that what we're first going to do is we're going to have to calculate the covariance of the returns of a stock (Ri) and the market returns (Rm) and that covariance divided by the cross volatilities. When we said cross volatility just means standard deviation of the returns of the firm (Ri) and the standard deviation of the market return (Rm). So that'll allow us to figure out the correlation and then once we know the correlation then we plug in the standard deviation or the volatility for each one and we'll be able to get the beta. If you're in a class they might just give you the correlation, you wouldn't have to calculate it but I'm going to show you how to do all of this by hand.

So let's say we've got the monthly stock returns for the past six months. We've got six observations here for a company called Bubsy so that'll be firm Ri and then we've got a market index could be the S&P 500 there's some market index and we've got the returns, we've got 11% for Bubsy in January, 17% February and so forth. Then we've got the returns of the market index. Now we can calculate the average return for both Bubsy and the market index. So we have got 7.83% for Bubsy, and 5.67% for the market index. So to get Bubsy's average return we just add all the returns and divide by 6 because there are six data points.

Now once we have these monthly returns, what we're gonna be trying to do is get the covariance which we need to calculate the correlation. We calculate the deviation from the mean and what we're going to do is take Bubsy 11% minus Bubsy's mean which is 7.83% that'll give us 3.2%, it's not the exact case there's a little rounding here but here we're just taking each observation. (17% - 7.83%) that gives us 9.2% and so forth. That's how we get the deviations from the mean. Then for the market index, we take that 8% minus the market Index's mean, which is 5.67%. It gives us 2.3% and so forth. So that gives us the deviations from the mean.

Then we multiply these deviations together. So like 3.2% times 2.3%, we'll get 0.0007 that's the product of the deviations and again there's some rounding there and then we add all the deviations together. We sum all these products of deviations and we get 0.0547. We multiply by 1/n minus 1, where (n) is the number of observations. So we have 6 minus 1, which is 1/5 which is 0.2. So 0.2 times 0.0547 that's going to give us our covariance which is 0.0109333.

So now we have the covariance, according to our formula to get the correlation we've got the covariance, we've got the numerator but now we have to calculate the volatility for both the Bubsy and for the market index. So we're gonna calculate the volatility which is the standard deviation of their returns to get that we're going to calculate the variance of each period. Now to get the variance which we need to calculate the standard deviation, to get the variance we're going to take the squared deviations for Bubsy. So Bubsy the firm Ri, we'd take 3.2% that's the deviation from the mean, so we take that but square it and that gives us 0.001002778. Now for the market 2.3%, we square that, and that gives us 0.000544444. Again for Bubsy here 9.2% squared which gives us 0.008402778 and so forth. So we get all these squared deviations but then we take the sum of the squared deviations divided by (n-1) and again n is six. So divide the sum by 5 and then we've got the sum for the market we divide that by (n-1) which is 5. That's going to give us the variance.

So variance for Bubsy and we just get again the sum of the squared deviations divide it by 5 that's our variance. Now to get the standard deviations, remember we got the covariance already to calculate covariance before, and then we said we need the standard deviation of each of these variances to get the correlation. To get the standard deviation we just take the square root of the variance. So to get the volatility or the standard deviation for Bubsy stock returns we just take these variations here and take the square root of them. So the square root of the variance will give us the standard deviation.

Now here I'll just reproduce the same formula as I had up above. Now we know the covariance and standard deviations. So we could plug that in this formula. So we got covariance which is 0.0109333 and then we divide that by the cross volatilities. Again cross volatilities is the standard deviation of Ri times the standard deviation of the market index (Rm) that's our denominator. We take the covariance and divide it by these cross volatilities it gives us a correlation of 0.9985481. So this is the correlation of the stock returns for Bubsy and the market index, we have this correlation very high, and a positive correlation means they move together.

## Calculation of Beta using Correlation and Volatility

Now that we have the correlation we go back to our original formula for the beta to calculate with the correlation. We've got the correlation already and then we're gonna multiply it by the volatility for Bubsy that's firm Ri in this case and then we're gonna divide that by the volatility for the market return. So here I'll just plug in this correlation 0.9985481 and then we've got the standard deviation for Bubsy so we plug that in and then divide all of them by the standard deviation for the market returns. So if you take this and do the math what you end up with is a beta of 1.851.

So our beta is 1.851 what does that mean for example if the market return were to increase by 1% we would expect that Bubsy's return would go up by 1.851%.