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.

Question

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.)

Answer 1

To estimate the maximum yearly return for tax-free municipal bonds, we need to find the value that corresponds to a certain percentile in a normal distribution. Specifically, we need to find the value that is greater than or equal to 99% of the values in the distribution.

To do this, we can use a z-score formula:

z = (x - μ) / σ

where z is the z-score, x is the value we want to find (maximum yearly return), μ is the mean return, and σ is the standard deviation.

For tax-free municipal bonds:

Mean return (μ) = 5.2%
Standard deviation (σ) = 8.6%

We want to find the value (x) that corresponds to a z-score of 99%. This means we are looking for the value that is greater than or equal to 99% of the values in the distribution.

Using a standard normal distribution table or a statistical calculator, we can find that the z-score corresponding to 99% is approximately 2.33.

Rearranging the z-score formula, we can solve for x:

x = z * σ + μ

x = 2.33 * 8.6 + 5.2

x ≈ 32%

So, the estimated maximum yearly return for tax-free municipal bonds is approximately 32%.

For part (b), we want to find the probability that the yearly return obtained by investing in common stocks will be higher than the maximum yearly return for tax-free municipal bonds (which is approximately 32%).

Since the yearly returns on common stocks are also normally distributed, we can use the z-score formula to calculate the probability.

For common stocks:

Mean return (μ) = 12.4%
Standard deviation (σ) = 20.6%

We want to find the probability that the return is higher than 32%. This is equivalent to finding the probability that the z-score is greater than the z-score corresponding to 32%.

Let's calculate the z-score:

z = (x - μ) / σ

z = (32 - 12.4) / 20.6

z ≈ 0.96

Now, we can use the z-score to find the probability using a standard normal distribution table or a statistical calculator. The probability will be the area under the curve to the right of the z-score.

The probability that the yearly return obtained by investing in common stocks will be higher than 32% is approximately 0.3321 (rounded to 4 decimal places).