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This problem involves the Capital Asset Pricing Model (CAPM), where returns are assumed to be normally distributed.

### Given:
- Mean (average) return (\(\mu\)): 14.7% (or 0.147)
- Standard deviation (\(\sigma\)): 33% (or 0.33)
- A return of 0% means no change in portfolio value.

### (a) What percent of years does the portfolio lose money (i.e., have a return less than 0%)?

We need to find the probability that the portfolio return is less than 0%. This involves calculating the z-score and using the standard normal distribution.

\[
Z = \frac{X - \mu}{\sigma}
\]
Where:
- \( X = 0\% \) (the return threshold),
- \( \mu = 0.147 \),
- \( \sigma = 0.33 \).

Thus, the z-score is:

\[
Z = \frac{0 - 0.147}{0.33}
\]

### (b) What is the cutoff for the highest 15% of annual returns?

For this part, we need to find the value of \(X\) such that 85% of the returns are below this value. Using the z-score corresponding to 85% from the standard normal distribution, we can find the return cutoff.

Let me calculate the z-scores and probabilities.### Results:

#### (a) What percent of years does the portfolio lose money (return less than 0%)?
Approximately **32.8%** of the years, the portfolio will have a return less than 0%, meaning it loses money.

#### (b) What is the cutoff for the highest 15% of annual returns?
The cutoff for the highest 15% of annual returns is approximately **48.9%**.

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How would changing the standard deviation affect the probability of a loss?
2. What is the probability of the portfolio return being exactly 0%?
3. How would a different mean return impact the cutoff for the top 15% returns?
4. Can you calculate the cutoff for the highest 5% of annual returns?
5. What is the probability of achieving a return of more than 30%?

**Tip:** Understanding z-scores helps to quickly find probabilities in a normal distribution, which is widely applicable in finance and statistics.

The Capital Asset Pricing Model (CAPM) is a financial model that assumes returns on a portfolio are normally distributed. Suppose a portfolio has an average annual return of 14.7% with a standard deviation of 33%. A return of 0% means the value of the portfolio doesn’t change, a negative return means that the portfolio loses money, and a positive return means that the portfolio gains money. What percent of years does the portfolio lose money (i.e., have a return less than 0%)? What is the cutoff for the highest 15% of annual returns with this portfolio?

Solve a CAPM problem where a portfolio has an average annual return of 14.7% and a standard deviation of 33%. Learn how to calculate the percentage of years the portfolio loses money and find the cutoff for the top 15% of annual returns.

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