Question

You manage a U.S. core equity portfolio that is sector-neutral to the S&P500 Index (its industry sector weights approximately match the S&P 500's). Taking a weighted average of the projected mean returns on the holdings, you forecast a portfolio return of 12%. You estimate a standard deviation of annual return of 22%, which is close to the long-run figure for the S&P 500. For the year-ahead return on the portfolio, assuming Normality for portfolio returns, you are asked to do the following:

(1) Calculate and interpret a two-standard deviation confidence interval for the portfolio returns. (2) You can buy a one-year T-bill that yields 5%. What is the probability that your portfolio return will be equal to or less than the risk-free rate?

Answer 1

1) As the portfolio returns are assumed to be normally distributed

two-standard deviation confidence interval for the portfolio returns

= (mean return -2* standard deviation , mean return + 2 *standard deviation)

= (12%-2*22%, 12%+2*22%)

= (-32%,56%)

As the normal probability distribution interval of  two-standard deviation implies a probability of 95.45% . This implies that for 95.45% of the time, the portfolio returns will lie between -32% and 56%

2) Z value corresponding to 5% = (5%-12%)/22% = -0.31818

The probability corresponding to Z = -0.31818

Using Normal Distribution table or NORMSDIST function in Excel

z = 0.375174

So, probability that the portfolio return will be equal to or less than the risk-free rate is 37.5174%