Solution to Question 3b) from UoL exam 2018, Zone A
I thought that after all the work we've done with my students the answer to this question would be obvious. It was not, so I am sharing it.
Question. Consider a position consisting of a $20,000 investment in asset and a $20,000 investment in asset . Assume that returns on these two assets are i.i.d. normal with mean zero, that the daily volatilities of both assets are 3%, and that the correlation coefficient between their returns is 0.4. What is the 10-day VaR at the critical level for the portfolio?
Solution. First we have to work with returns and then translate the result into dollars.
Let be the daily returns on the two assets. We are given that ,
Since the total investment is $40,000, the shares of the investment are Therefore the daily return on the portfolio is see Exercise 2.
It follows that
These figures are for daily returns. We need to make sure that is normally distributed. The sufficient condition for this is that the returns are jointly normally distributed. It is not mentioned in the problem statement, and we have to assume that it is satisfied.
Let denote the return on day Under continuous compounding the daily returns are summed: if we invest initially, after the first day we have after the second day we have and so on. So the 10-day return is
Since the daily returns are independent and identically distributed, by additivity of variance we have
is normally distributed because the daily returns are independent. It remains to apply the VaR formula
for normal distributions. From the table of the distribution function of the standard normal Thus, This translates to the minimum loss of Thus, with probability 1% the loss can be $7362 or more.