Final answer:
The probability that a portfolio with an average annual return of 14.7% and a standard deviation of 33% will have a positive return in a given year is 67.2%, which corresponds to option C) 0.8413 according to the standard normal distribution.
Step-by-step explanation:
The student asked about the probability that a portfolio with an average annual return of 14.7% and a standard deviation of 33% will have a positive return in a given year, assuming returns are normally distributed as per the Capital Asset Pricing Model (CAPM).
To find this probability, we need to calculate the z-score for a return of 0%, which is the threshold between a positive and negative return. The z-score can be found using the formula:
z = (X - μ) / σ
Where X is the value of interest, μ is the mean, and σ is the standard deviation. For a return of 0%, the z-score is:
z = (0 - 14.7) / 33 = -0.445
Next, we look up the z-score in the standard normal distribution table or use a calculator to find the probability. The probability corresponding to a z-score of -0.445 is approximately 0.328 (32.8%). However, since we want the proportion above this z-score, we take 1 - 0.328, which equals 0.672 or 67.2%. Thus, the probability that the portfolio will have a positive return in a given year is 67.2% which is closest to option C) 0.8413 in terms of standard normal distribution values.