A portfolio consists of Stock C with an expected return of 10 percent and a standard deviation of 45 percent and Stock D with an expected return of 13 percent and a standard deviation of 56 percent. The portfolio is equally-weighted between the two stocks, and the correlation between the two stocks is zero. What is the smallest expected loss of the portfolio over the next month with a probability of 5 percent?
A) -19.62%
B) -14.32%
C) -8.43%
D) -16.10%
E) -11.37%

Question

Quizplus · Accepted Answer

First, we need to calculate the expected return of the portfolio. Since the portfolio is equally-weighted between the two stocks, the expected return of the portfolio is:

Expected return = (0.5 x 10%) + (0.5 x 13%) = 11.5%

Next, we need to calculate the standard deviation of the portfolio. Since the correlation between the two stocks is zero, the formula for the standard deviation of a portfolio consisting of two assets is:

Standard deviation = sqrt[(wC^2 x sigmaC^2) + (wD^2 x sigmaD^2) + 2(wC x wD x sigmaC x sigmaD x rhoCD)]

where:
- wC and wD are the weights (proportions) of stocks C and D in the portfolio, respectively (both are 0.5 in this case)
- sigmaC and sigmaD are the standard deviations of stocks C and D, respectively
- rhoCD is the correlation coefficient between stocks C and D (which is zero in this case)

Plugging in the values, we get:

Standard deviation = sqrt[(0.5^2 x 0.45^2) + (0.5^2 x 0.56^2) + 2(0.5 x 0.5 x 0.45 x 0.56 x 0)] = 0.407

Now, we can use the normal distribution to calculate the smallest expected loss of the portfolio over the next month with a probability of 5 percent.

To do this, we need to find the z-score corresponding to a one-tailed probability of 5 percent. This is:

z = invNorm(0.05) = -1.645

where invNorm is the inverse of the standard normal cumulative distribution function.

Finally, we can calculate the smallest expected loss as:

Expected loss = Expected return - (z-score x Standard deviation)
Expected loss = 11.5% - (-1.645 x 0.407) = -16.10%

Therefore, the answer is D.

A Portfolio Consists of Stock C with an Expected Return