Question

Suppose the returns on long-term government bonds are normally distributed. Assume long-term government bonds have a mean return of 6.0 percent and a standard deviation of 9.9 percent. a. What is the approximate probability that your return on these bonds will be less than -3.9 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? (A negative answer should be indicated by a minus sign. Enter your answers from lowest to highest. Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) c. What range would you expect to see 99 percent of the time? (A negative answer should be indicated by a minus sign. Enter your answers from lowest to highest. Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) Probablility less than |-3.90% Expected range of returns Expected range of returns

Answer 1

Given information:

mean = 6%

standard deviation (sigma )= 9.9%

Sub part a:

Probability that return is less than -3.9% in a given year = P[R<-3.0%] = NORM.DIST(-3.9%, 6%,9.9%, True) this is the excel function that we need to use.

The value from excel comes to be 0.158655 = 15.8655% probability

A normal distribution is symmetric around the mean. The total area under the normal curve is 100%. The returns -3.9% will be on the left side of the mean. Here we need to find the area less than this point to -infinity, hence we will use the cumulative argument as True in the excel function.

Sub part b:

Range of returns that we see 95% of the time. The returns will be closer to the mean. We need to determine 42.5% on the left and 42.5% on the right. In case of a standard normal distribution where mean is 0 and standard deviation is 1, the range is between -1.96 to +1.96.

We can convert our normal distribution to standard normal distribution by defining a variable z = (x-mean)/sigma

and then using the range -1.96, 1.96 around z to determine x

z= (x-6%)/9.9%

-1.96<=(x-6%)/9.9% <= +1.96

(-1.96*9.9%) + 6% <=x<=1.96*9.9% + 6%

-13.404 <=x<=25.404

Range of returns: (-13.40%, 25.40%)

Sub part c:

Range of returns that we see 99% of the time. The returns will be closer to the mean. We need to determine 44.5% on the left and 44.5% on the right. In case of a standard normal distribution where mean is 0 and standard deviation is 1, the range is between -2.58 to +2.58.

We can convert our normal distribution to standard normal distribution by defining a variable z = (x-mean)/sigma

and then using the range -2.58 , 2.58 around z to determine x

z= (x-6%)/9.9%

-2.58 <=(x-6%)/9.9% <= +2.58

(-2.58 *9.9%) + 6% <=x<=2.58 *9.9% + 6%

-19.542% <=x<=31.542%

Range of returns: (-19.54%, 31.54%)