Question

You are considering a long-term investment opportunity that is expected to yield 2% per quarter.

a What is the annual continuously compounded expected return on the investment?

b If the variance of annual continuous compounded return is 0.04, what is the variance of the continuous compounded return over the 10-year horizon?

c Assume that continuous compounded returns are normally distributed. What is the probability of loss after 10 years?

Answer 1

a. Annualcontinuously compounded expected return will be = 1.02^4 -1 = 8.24%. This has been calculated by the compounding formula for 4 quarters which is equal to one year.

b. Variance over the 10-year horizon will be = 0.04 x sqrt(10) = 0.1265

c. The sigma (std deviation) is going to be the square root of variance = root(0.1265) = 0.3556. The mean for the 10 year period will be = 1.0824^10 -1 = 1.2073 = 120.73%. (Calculated as in (a)).

Loss will result when we get a return of less than 0. So, that will be at the number of std deviations = 1.2073/0.1265 = 9.5438. That means the probability of loss will be calculated according to the normal distribution at -9.5438 standard deviations. Since this number is going to be very small, we assume it to be almost zero and hence we have a 0% probability of a loss.