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# Common Univariate Random Variables

#### malhotsu

##### New Member
Subscriber
Assume the annual returns of Fund A are normally distributed with a mean and standard deviation of 30%. The annual returns of Fund B are also normally distributed, but with a mean and standard deviation of 40%. The returns of both funds are independent of each other. What is the mean and standard deviation of the difference of the returns of the two funds, Fund B minus Fund A? At the end of the year, Fund B has returned 80%, and Fund A has lost 12%. How likely is it that Fund B outperforms Fund A by this much or more?

Answer: Because the annual returns of both funds are normally distributed and independent, the difference in their returns is also normally distributed: ~( − , + ) The mean of this distribution is 10%, and the standard deviation is 50%. At the end of the year, the difference in the expected returns is 92%. This is 82% above the mean, or 1.64 standard deviations. Using Excel or consulting the table of confidence levels in the chapter, we see that this is a rare event. The probability of more than a 1.64 standard deviation event is only 5%.

Can someone explain how 50% s.d is calculated in the answer?

#### Matthew Graves

##### Member
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$VAR(A-B)=VAR(A)+VAR(B)-2COV(A,B)$

In this case, since the returns of Fund A and B are independent the covariance of A and B is zero. Leaving:

$VAR(A-B)=VAR(A)+VAR(B)=0.4^2+0.3^2=0.25=0.5^2$

Therefore $\sigma_{A-B}=0.5$ or 50%