Question

g If the economy improves, a certain stock stock will have a return of 23.4 percent. If the economy declines, the stock will have a loss of 11.9 percent. The probability of the economy improving is 67 percent while the probability of a recession is 33 percent. What is the standard deviation of the returns on the stock

Answer 1

`E(X) = 23.4* 0.67 -11.9*0.33= 11.759 \%`

Now we can find the second central moment with this formula:

`E(X^2) = \sum_(i=1)^n X^2_i P(X_i)`

And replacing we got:

`E(X^2) = (23.4)^2* 0.67 +(-11.9)^2*0.33= 413.5965`

And the variance is given by:

`Var(X) = E(X^2) - [E(X)]^2`

And replacing we got:

`Var(X) = 413.5965 -(11.759)^2 =275.5105`

And finally the deviation would be:

`Sd(X) = √(275.5105)= 16.599 \%`

Explanation:

We can define the random variable of interest X as the return from a stock and we know the following conditions:

`X_1 = 23.4 , P(X_1) =0.67` represent the result if the economy improves

`X_2 = -11.9 , P(X_1) =0.33` represent the result if we have a recession

We want to find the standard deviation for the returns on the stock. We need to begin finding the mean with this formula:

`E(X) = \sum_(i=1)^n X_i P(X_i)`

And replacing the data given we got:

`E(X) = 23.4* 0.67 -11.9*0.33= 11.759 \%`

Now we can find the second central moment with this formula:

`E(X^2) = \sum_(i=1)^n X^2_i P(X_i)`

And replacing we got:

`E(X^2) = (23.4)^2* 0.67 +(-11.9)^2*0.33= 413.5965`

And the variance is given by:

`Var(X) = E(X^2) - [E(X)]^2`

And replacing we got:

`Var(X) = 413.5965 -(11.759)^2 =275.5105`

And finally the deviation would be:

`Sd(X) = √(275.5105)= 16.599 \%`