Suppose the returns on long-term corporate bonds and T-bills are normally distri

Question

Suppose the returns on long-term corporate bonds and T-bills are normally distributed. Assume for a certain time period, long-term corporate bonds had an average return of 5.3% and a standard deviation of 8.1%. For the same period, T-bills had an average return of 3.8% and a standard deviation of 2.6%. Use the NORMDIST function in Excel® to answer the following questions: Required: (a) What is the probability that in any given year, the return on long-term corporate bonds will be greater than 10 percent? Less than 0 percent? (Do not round intermediate calculations. Enter your answers as a percentage rounded to 2 decimal places (e.g., 32.16).) Probability of return greater than 10 percent % Probability of return less than 0 percent % (b) What is the probability that in any given year, the return on T-bills will be greater than 10 percent? Less than 0 percent? (Do not round intermediate calculations. Enter your answers as a percentage rounded to 2 decimal places (e.g., 32.16).) Probability of T-bill return greater than 10 percent % Probability of T-bill return less than 0 percent % (c) In one year, the return on long-term corporate bonds was 4 percent. How likely is it that such a low return will recur at some point in the future? T-bills had a return of 10.72 percent in this same year. How likely is it that such a high return on T-bills will recur at some point in the future? (Do not round intermediate calculations. Enter your answers as a percentage rounded to 2 decimal places (e.g., 32.16).) Probability of return on long-term corporate bonds less than –4.00 percent % Probability of T-bill return greater than 10.72 percent

Explanation / Answer

Solution:

a)

For each of the questions asked here, we need to use the z-statistic, which is:

z = (X – µ)/

This z-statistic gives us the probability that the return is less than 10 percent, but we are looking for the probability the return is greater than 10 percent. Given the symmetry of the normal distribution, and the fact that the total probability is 100 percent (or 1), the probability of a return greater than 10 percent is 1 minus the probability of a return less than 10 percent. Using the cumulative normal distribution table, we get:

Probability of return greater than 10 percent:

z1 = (10% - 5.3%)/8.1%

= 0.5802

Pr(R=10%) = 1 – Pr(R=10%) = 1 – 0.7191 = 0.2809 28.09%

And the probability of return less than 0 percent:

z2 = (0% - 5.3%)/8.1%

= -0.6543

Pr(R0%) 25.65%

b)

The probability that T-bill returns will be greater than 10 percent is:

z3 = (10% – 3.8%)/2.6%

= 2.3846

Pr(R=10%) = 1 – Pr(R=10%) = 1 – 0.9915 0.85%

And the probability that T-bill returns will be less than 0 percent is:

z4 = (0% – 3.8%)/2.6%

= -1.4615

Pr(R=0) 7.19%

c)

The probability that the return on long-term corporate bonds will be less than –4.00 percent is:

z5 = (–4.00% – 5.3%)/8.1%

= -1.1481

Pr(R=–4.00%) 12.55%

And the probability that T-bill returns will be greater than 10.72 percent is:

z6 = (10.72% – 5.3%)/8.1%

= 0.6691

Pr(R=10.72%) = 1 – Pr(R=10.72%) = 1 – 0.7483 25.17%

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