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Chapter 4: Problem 8

CAPM. 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 \(14.7 \%\) (i.e. an average gain 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. (a) What percent of years does this portfolio lose money, i.e. have a return less than \(0 \% ?\) (b) What is the cutoff for the highest \(15 \%\) of annual returns with this portfolio?

Short Answer

Expert verified
(a) 32.85% of years lose money. (b) The cutoff for the top 15% returns is 48.89%.

Step by step solution

01

Understand the problem

We have a portfolio with an average annual return \[ \mu = 14.7\% \] and a standard deviation \[ \sigma = 33\% \]. We need to (a) find the percentage of years the portfolio loses money (returns < 0%) and (b) identify the cutoff for the highest 15% of annual returns.
02

Convert problem to standard normal distribution

Standardize the normal distribution using the formula for the z-score:\[ Z = \frac{X - \mu}{\sigma} \]For part (a), we find \( Z \) when \( X = 0\% \). For part (b), we find \( X \) such that the cumulative probability is 0.85.
03

Step 3-a: Calculate z-score for part (a)

Using the z-score formula:\[ Z = \frac{0\% - 14.7\%}{33\%} \approx \frac{-14.7}{33} \approx -0.445 \]
04

Step 4-a: Find the probability for part (a) from z-table

Using the standard normal distribution table, find the probability for \( Z = -0.445 \). This probability represents the percent of years with returns less than 0.\[ P(Z < -0.445) \approx 0.3285 \]
05

Step 5-a: Conclusion for part (a)

The portfolio loses money approximately 32.85% of the years.
06

Step 3-b: Find z-score for the top 15%

To find the cutoff for the highest 15% of returns, we need to find \( Z \) such that \[ P(Z > Z_{0.85}) = 0.15 \]Thus, find \( Z \) for \( P(Z < Z_{0.85}) = 0.85 \). From z-tables, \( Z_{0.85} \approx 1.036 \).
07

Step 4-b: Calculate X for part (b) using z-score

Plug into the z-score formula to find the corresponding \( X \):\[ Z = 1.036 \]\[ 1.036 = \frac{X - 14.7\%}{33\%} \]\[ X = 1.036 \times 33\% + 14.7\% \]\[ X = 34.188\% + 14.7\% = 48.888\% \]
08

Step 5-b: Conclusion for part (b)

The cutoff for the highest 15% of annual returns is approximately 48.89%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution in Finance
The normal distribution is a key concept in finance, particularly in the Capital Asset Pricing Model (CAPM). When we say returns on a portfolio are "normally distributed," we imply that if we plotted the returns on a graph, they'd form a bell-shaped curve.
- **Symmetry**: The curve is symmetric around its mean, showing that most return observations cluster around the average return, with fewer cases occurring as you move further from the mean. - **Empirical Rule**: Around 68% of portfolio returns fall within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations.
This knowledge helps investors understand the likelihood of various investment outcomes. Understanding the distribution of returns enables better assessment of risks and expectations.
Understanding Z-Score
The z-score is a statistical figure that describes a value's relation to the mean of a group of values. In finance, it helps gauge how unusually high or low a specific return is compared to an average return.
- **Calculation**: The z-score is calculated using the formula \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the individual data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.- **Interpretation**: A z-score of 0 indicates the score is identical to the mean. A positive z-score means the score is above the mean, while a negative score is below the mean.
Utilizing a z-score in investment analysis allows more precise predictions on returns, helping investors to strategize accordingly.
The Role of Standard Deviation
Standard deviation is a crucial measure that indicates the volatility or risk of a portfolio's returns. It quantifies the amount of dispersion or spread in a set of data points.
- **Large Standard Deviation**: Indicates that the returns are spread out over a larger range of values, meaning greater volatility and potentially higher risk. - **Small Standard Deviation**: Demonstrates that the returns are clustered tightly around the mean, suggesting lower volatility and risk.
By understanding the standard deviation, investors can get a better grip on how risky or stable a particular investment may be, guiding decisions on whether to undertake a certain risk.
Evaluating Portfolio Returns
Portfolio returns reflect the gain or loss made on the investments in the portfolio over a specific period. Understanding these can aid in making informed investment decisions.
- **Positive Return**: Indicates a gain in the portfolio's value. - **Negative Return**: Suggests a loss in the portfolio's value.
Analyzing portfolio returns within the context of their normal distribution allows investors to evaluate the performance and make educated forecasts. Tools like z-scores help, while standard deviation outlines the risk associated with these returns. In essence, understanding these metrics and their interrelation offers a comprehensive view of financial strategies.

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Most popular questions from this chapter

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