Question

Jennifer invests her money in a portfolio that consists of 70% Fidelity Spartan 500 Index Fund and 30% Fidelity Diversified International Fund. Suppose that, in the long run, the annual real return X on the 500 Index Fund has mean 10% and standard deviation 15%, the annual real return Y on the Diversified International Fund has mean 9% and standard deviation 19%, and the correlation between X and Y is 0.6.

a) The return on Jennifer’s portfolio is R = 0.7X + 0.3Y. What are the mean and standard deviation of R?

I understand how to find the mean but not the standard deviation. My understanding is that you need to find the variance to find the standard deviation. I don't really understand the formula for variance, however. I think I remember my professor telling my class that there is a lot of room for error in using the formula and so she showed us how to do it on a calculator. Unfortunately, I don't remember what she said. If you could explain how to find the standard deviation/variance using a function on the calculator (I have a TI 84 Plus CE) it would be much appreciated!

b) The distribution of returns is typically roughly symmetric but with more extreme high and low observations than a Normal distribution. The average return over a number of years, however, is close to Normal. If Jennifer holds her portfolio for 20 years, what is the approximate probability that her average return is less than 5%?

c) The calculation you just made is not overly helpful because Jennifer isn’t really concerned about the mean return Rbar. To see why, suppose that her portfolio returns 12% this year and 6% next year. The mean return for the two years is 9%. If Jennifer starts with $1000, how much does she have at the end of the first year? At the end of the second year? How does this amount compare with what she would have if both years had the mean return, 9%? Over 20 years, there may be a large difference between the ordinary mean Rbar and the geometric mean, which reflects the fact that returns in successive years multiply rather than add.

Answer 1

C) Since returns for the first year is 12%, amount at the end of year 1 will be 1000(1+i) where i is 0.12 = 1120.

Similarly, Amount at the end of the second year will be 1120(1.06) = 1187.2

Using the mean return of 9%, amount at the end of year 2 would be 1000(1.09)^2 = 1188.1

So our amounts are almost same using the mean returns.

However over 20 years, the mean return would give a value of 1000(1.09)^20 = 5604.41 but the returns in succesivse years may be very different and their mean may not necessarily be 9% so multiplying them successively would give a large difference.