This problem requires a basic understanding of the normal probability distribution. In- vestors are often interested in knowing the probabilities of poor returns. For example, for what cutoff return will the probability of the actual return falling below this cutoff value be at most 1%?
Consider the solution to the Markowitz portfolio problem given in Figure 8.9. The mean return of the portfolio is 10% and the standard deviation (calculated by taking the square root of the variance, which is the objective function value) is
s = 227.13615 = 5.209237
Assume that the portfolio scenario returns are normally distributed about the mean return. From the normal probability table, we see that less than 1% of the returns fall more than 2.33 standard deviations below the mean. This result implies a probability of 1% or less that a portfolio return will fall below
10 - (2.33)(5.209237) = -2.1375
Stated another way, if the initial value of the portfolio is $1, then the investor faces a probability of 1% of incurring a loss of 2.1375 cents or more. The value at risk is 2.1375 cents at 1%. This measure of risk is called the value at risk, or VaR. It was popularized by JPMorgan Chase & Co. in the early 1990s (then, just JP Morgan).
A table of normal probabilities appears in Appendix B, but they are also easily calcu- lated in LINGO and Excel. In LINGO the function @PSN(Z) and the equivalent function NORMDIST in Excel provide the probability that a standard normal random variables is less than Z.
a. Consider the Markowitz portfolio problem given in equations (8.10) through (8.19). Delete the required return constraint (8.18), and reformulate this problem to minimize the VaR at 1%.
b. Is minimizing the VaR the same as minimizing the variances of the portfolio? Answer Yes or No, and justify.
c. For a fixed return, is minimizing the VaR the same as minimizing the variances of the portfolio? Answer Yes or No, and justify.
Text Book: An Introduction to Management Science: Quantitative Approaches to Decision. By David Anderson, Dennis Sweeney, Thomas Williams, Jeffrey Camm, James Cochran.