Assume that the equity risk premium is normally distributed with a population
mean of 6 percent and a population standard deviation of 18 percent. Over
the last four years, equity returns (relative to the risk- free rate) have averaged
-2.0 percent. You have a large client who is very upset and claims that results
this poor should never occur. Evaluate your client’s concerns.
What is the probability of a -2.0 percent or lower average return over a
four- year period?
You know the distribution follows the normal distribution with a mean of 6 and a standard deviation of 18. Your next step is to find the z-value corresponding to an equity premium of -2.0% and then figure out the area under the curve to the left (hint, hint) of that.
Thanks for your reply. i understand that need to use z-value to find out the answer. However, the solution show that z-value is calculated ( observation - mean) / ( standard deviation / the square root of number of years). Can you explain more about why the denominator is ( standard deviation / the square root of number of years) instead of only using the value of standard deviation? Thank you!
The mean of 6 and standard deviation of 18 is for a single year. The observed equity risk premium is over 4 years, so what you’re looking at is a sample mean. The variance of any sample mean is
\sigma^2 /n
so the sample standard deviation is
\sigma / \sqrt n
In this case, the sample standard deviation is 18/2 = 9.
