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.)
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).