Question

Here are the results of a monthly index model of stock returns for FinCorp Stock: r(Fincorp) = .001 + .8 r(S&P500) + e The standard deviation of the monthly return on the S&P 500 was .05. The standard deviation of the monthly value of the residual, e, was .03.

What was the variance of the return on FinCorp stock? Remember that e and the return on the S&P 500 are essentially uncorrelated. What was the standard deviation of the return on FinCorp stock?

Answer 1

The results of the monthly index model here are expressed in the form below:

This form of the Capital Asset Pricing Model essentially differentiates between systematic and non systematic risk:

Systematic risk - indicated by the segment   is also known as "undiversifiable" risk - meaning it is the risk the investor is undertaking by putting money into that market and is impossible to avoid

Unsystematic risk - indicated by the segment is known as "diversifiable risk" is the risk taken on by investing in a particular stock or company and can be removed by diversifying the investment portfolio

Now to solve the given question:

a) Since we are directly using the monthly returns of the S&P Index, the market risk is essentially captured in that and we do not need to worry about the risk free rate or (by calculating the standard deviation for the S&P index, we are already accounting for all the risk, the investor is taking by investing in this market)

So, our formula now reduces to

Now, we have already been told that return on S&P 500 and e (or as per our formula) are uncorrelated. This means that Covariance of and = 0.

Therefore, taking the variance on both sides of the above formula, we get :

b) By the definition of Risk Free rate (), it is the return that the investor shall definitely get in case money is invested in a stock that gives such return. Which means that variance or volatility in the return of the stock is 0

Therefore, our variance formula now reduces to:

c) The measure of variance is obtained simply by taking the square of the standard deviation of the variable. Hence, the above formula would become:

Where refers to the Standard deviation of the variable.

Now, putting in the values:

= 0.05, = 0.03 and = 0.8

So, the Variance of the Return on FinCorp Stock is 0.0025

Standard deviation is obtained by taking the square root of the Variance number, so in this case,

or the Standard Deviation of the Fincorp Stock is 5%