Question

Suppose the returns on an asset are normally distributed. The historical average annual return for the asset was 5.9 percent and the standard deviation was 10.5 percent. a. What is the probability that your return on this asset will be less than –7.3 percent in a given year? Use the NORMDIST function in Excel® to answer this question. (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) b. What range of returns would you expect to see 95 percent of the time? (Enter your answers for the range from lowest to highest. A negative answer should be indicated by a minus sign. Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) c. What range of returns would you expect to see 99 percent of the time?

Answer 1

According to the problem,

X = -7.3%

u = 5.9% and

SD = 10.5 %

Using the formula, we get,

-7.3 - 5.9 / 10.5

= - 1.26

If we refer to the standard normal distribution table, we see that 1.26 corresponds to 89.62% area to it's left-hand side, which means that we will be left with 10.38% to the right hand side of 1.26

Since normal distributions are symmetrical, the area to the right of 1.26 should be same as the area to the left of -1.26, giving us our required probability of 10.38% that the return n our asset would fall below -7.3%

ii) For calculating the range of returns we shall get 95% of the time, we need to find the confidence interval

The Z Score that corresponds to a 95% confidence interval is 1.96 ( Can be again found from the standard normal distribution chart)

= x + z score * S.D and

x - z score * S.D

= 5.9 + 1.96*10.5 = 26.48%

and 5.9 - 1.96*10.5 = 14.68%

Thus, it can range from 14.68% to 26.48%

iii) We can find it similarly for the 99% confidence interval

The Z Score that corresponds to a 99% confidence interval is 2.58 ( Can be again found from the standard normal distribution chart)

= x + z score * S.D and

x - z score * S.D

= 5.9 + 2.58*10.5 = 32.99%

and 5.9 - 2.58*10.5 = 21.19%

Thus, it can range from 21.19% to 32.99%