An investment broker reports that the yearly returns on common stocks are approximately normally distributed with a mean return of 12.4 percent and a standard deviation of 20.6 percent. On the other hand, the firm reports that the yearly returns on tax-free municipal bonds are approximately normally distributed with a mean return of 5.2 percent and a standard deviation of 8.6 percent.
(a)
Use the investment broker’s report to estimate the maximum yearly return that might be obtained by investing in tax-free municipal bonds. (Round your answer to the nearest whole percent.)
Maximum yearly return 32%
(b)
Find the probability that the yearly return obtained by investing in common stocks will be higher than the maximum yearly return that might be obtained by investing in tax-free municipal bonds. (Round your answer to 4 decimal places.)
0.1707
To estimate the maximum yearly return that might be obtained by investing in tax-free municipal bonds, we need to find the value on the normal distribution with a mean of 5.2% and a standard deviation of 8.6% that corresponds to a certain percentile. Since we want the maximum possible return, we need to find the value that corresponds to a high percentile, such as the 99th percentile.
To find this value, we can use the formula:
Maximum Return = Mean + (Z-Score * Standard Deviation)
First, let's find the Z-Score for the 99th percentile. The Z-Score represents the number of standard deviations a value is from the mean.
Using a standard normal distribution table or a Z-Score calculator, we find that the Z-Score for the 99th percentile is approximately 2.33.
Now we can calculate the maximum return:
Maximum Return = 5.2 + (2.33 * 8.6) = 5.2 + 20.038 = 25.238%
Rounding to the nearest whole percent, the estimated maximum yearly return for tax-free municipal bonds is 25%.
For part (b), we need to find the probability that the yearly return from investing in common stocks will be higher than the maximum yearly return obtained from tax-free municipal bonds.
To find this probability, we need to calculate the Z-Score for the maximum return on tax-free municipal bonds and then find the probability of the values exceeding that Z-Score in the normal distribution with a mean of 12.4% and a standard deviation of 20.6%.
First, let's calculate the Z-Score for the maximum return on tax-free municipal bonds:
Z-Score = (Maximum Return - Mean) / Standard Deviation
Z-Score = (25.238 - 12.4) / 20.6 = 0.623
Now we can find the probability using a standard normal distribution table or a Z-Score calculator. The probability of the yearly return obtained by investing in common stocks being higher than the maximum yearly return from tax-free municipal bonds is approximately 0.2661.
Rounding to 4 decimal places, the probability is 0.2661.