Question

Problem 13-16 Value-at-Risk (VaR) Statistic (LO4, CFA6)

Tyler Trucks stock has an annual return mean and standard deviation of 15.0 percent and 38 percent, respectively. Michael Moped Manufacturing stock has an annual return mean and standard deviation of 11.4 percent and 56 percent, respectively. Your portfolio allocates equal funds to Tyler Trucks stock and Michael Moped Manufacturing stock. The return correlation between Tyler Trucks and Michael Moped Manufacturing is −.5. What is the smallest expected loss for your portfolio in the coming month with a probability of 16.0 percent? (A negative value should be indicated by a minus sign. Do not round intermediate calculations. Round the z-score value to 3 decimal places when calculating your answer. Enter your answer as a percent rounded to 2 decimal places.)

Smallest Expected Loss=

Answer 1

The return of the portfolio is the weighted average return of the component assets

So, return of the portfolio = 0.5*15%+0.5*11.4% =13.2%

Monthly Return = 13.2%/12 = 1.1%

The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient beltween returns of security i and security j

So, standard deviation of portfolio returns =sqrt (0.5^2*0.38^2+0.5^2*0.56^2+2*0.5*0.5*0.38*0.56*(-0.5))

=sqrt(0.0613)

=0.247588 =24.76%

Monthly Standard deviation = 24.76%/ (12^0.5) = 7.15%

Z value corresponding to probability of 16% = 0.994458 or 0.994

So, Probability (Return <1.1%-0.994*7.15%) = 16%

Or Probability (Return < -0.060071) = 16%

So, the smallest expected loss is 6.01%