An investment portfolio contains stocks of a large number of companies. Over the past\ year the rates of return on these corporate stocks followed a normal distribution with-As mean 12.5% and standard deviation 7.4%. Assuming that the rates of return for different companies are independent of each other, select the appropriate statement below.
- A. The probability of a rate of return above 15% for these companies is 0.4801, a level of return of 82.59% was exceeded by the top 10% of stocks, and the average rate of return over any 4 selected stocks in this portfolio has a distribution that is normally distributed with a mean of 12.5% and a standard deviation of 3.7%.
- B. The probability of a rate of return above 15% for these companies is 0.3669, a level of return of 21.97% was exceeded by the top 10% of stocks, and the average rate of return over any 4 selected stocks in this portfolio has a distribution that is normally distributed with a. mean of 50% and a standard deviation of 14.8%.
- C. The probability of a rate of return above 15% for these companies is 0.3669, a level of return of 21.97% was exceeded by the top 10% of stocks, and the average rate of return over any 4 selected stocks in this portfolio has a distribution that has an unknown form but with a mean of 12.5% and a standard deviation of 3.7.4.
- D. The probability of a rate of return above 15% for these companies is 0.2709, a level of return of 23.4% was exceeded by the top 10% of stocks, and the average rate of return over any 4 selected stocks in this portfolio has a distribution that is normally distributed with a mean of 12.5% and a standard deviation of 3.7%.
- E. The probability of a rate of return above 15% for these companies is 0.3669, a level of return of 21.97% was exceeded by the top 10% of stocks, and the average rate of return over any 4 selected stocks in this portfolio has a distribution that is normally distributed with a mean of 12.5% and a standard deviation of 3.7%.
I don´t write the percentage sign in my calculations.
$\textrm{... a level of return of x% was exceeded by the top 10% of stocks,}$
The equation is $P(X\geq x)=1-P(X\leq x)=0.1$
$$1-\Phi\left(\frac{x-12.5}{7.4} \right)=0.1$$
$$0.9=\Phi\left(\frac{x-12.5}{7.4} \right)$$
$$\Phi\left(0.9\right)^{-1}=\frac{x-12.5}{7.4} $$
$$1.282=\frac{x-12.5}{7.4} $$
$$x=21.9868$$
So the answer is approximately $21.97\%$. The differences in decimal digits are not so important. You can exclude the options $A$ and $D$.
$\textrm{ and the average rate of return over any 4 selected stocks in this portfolio}$ $\textrm{ has a distribution that is normally distributed with a mean of y% and a standard deviation of z%}$
Let $X_1, X_2, ..., X_n$ be n independent, identically and normally distributed random variables as $X_i\sim \mathcal N(\mu, \sigma^2)$.
And let $\overline X_n=\frac{X_1+ X_2+ ...+ X_n}{n}$. Then the distribution is $\overline X_n\sim \mathcal N\left(\mu, \frac{\sigma^2}n\right)$. That means in your case $$\overline X_4\sim \mathcal N\left(12.5, \frac{ 7.4^2}4\right)$$
Therefore the standard deviation of $\overline X_4$ is $\sqrt{\frac{ 7.4^2}4}=\frac{7.4}{2}=3.7$