### 1. Break Down the Problem
To determine if there's an arbitrage opportunity, we can compare the expected returns of the portfolios with their respective beta values using the Capital Asset Pricing Model (CAPM). The CAPM formula is given by:

$$
E(R) = R_f + \beta(E(R_m) - R_f)
$$

where:
- $E(R)$ = expected return of the asset
- $R_f$ = risk-free rate
- $\beta$ = beta of the asset
- $E(R_m)$ = expected return of the market

We'll need to assess the expected return for portfolio E using the expected returns and betas of portfolios A and B to identify any inconsistencies.

### 2. Relevant Concepts
1. **CAPM Formula:** Understanding the relationship between expected returns and risk (beta).
2. **Portfolio Comparison:** Analyze the expected returns against the beta values of each portfolio.

### 3. Analysis and Detail
1. **Identify Risk-Free Rate (Assuming):**
Since we don't have a specified risk-free rate, we can estimate it using the information from the portfolios. Portfolios A and B can help us define the market return.

2. **Calculate Market Return (E(R_m)):**
Using portfolios A (8%, beta 1.5) and B (5%, beta 0):
- For portfolio B (with $\beta = 0$), we assume the return represents the risk-free rate $R_f = 5\%$.
- Now using portfolio A:
$$
E(R_A) = R_f + \beta(E(R_m) - R_f)
$$
$$
8\% = 5\% + 1.5(E(R_m) - 5\%)
$$
$$
8\% - 5\% = 1.5(E(R_m) - 5\%)
$$
$$
3\% = 1.5(E(R_m) - 5\%)
$$
$$
\frac{3\%}{1.5} = E(R_m) - 5\%
$$
$$
2\% = E(R_m) - 5\%
$$
$$
E(R_m) = 7\%
$$

3. **Verify Portfolio E:**
Now using the market return to determine if portfolio E is appropriately valued:
$$
E(R_E) = R_f + \beta(E(R_m) - R_f)
$$
Substituting $R_f = 5\%$, $\beta = 0.5$, and $E(R_m) = 7\%$:
$$
E(R_E) = 5\% + 0.5(7\% - 5\%)
$$
$$
= 5\% + 0.5(2\%)
$$
$$
= 5\% + 1\% = 6\%
$$

### 4. Verify and Summarize
The expected return from the calculated value of portfolio E is 6%, while its stated expected return is 7%. Since the expected return (7%) exceeds the calculated expected return (6%), portfolio E is offering an overestimated return compared to its risk (beta), indicating a deviation from CAPM expectations.

### Final Answer
Yes, there exists an arbitrage opportunity because portfolio E is priced too high for its risk, as its expected return (7%) exceeds the calculated expected return (6%).

Question

### 1. Break Down the Problem
To determine if there's an arbitrage opportunity, we can compare the expected returns of the portfolios with their respective beta values using the Capital Asset Pricing Model (CAPM). The CAPM formula is given by:

$$
E(R) = R_f + \beta(E(R_m) - R_f)
$$

where:
- $E(R)$ = expected return of the asset
- $R_f$ = risk-free rate
- $\beta$ = beta of the asset
- $E(R_m)$ = expected return of the market

We'll need to assess the expected return for portfolio E using the expected returns and betas of portfolios A and B to identify any inconsistencies.

### 2. Relevant Concepts
1. **CAPM Formula:** Understanding the relationship between expected returns and risk (beta).
2. **Portfolio Comparison:** Analyze the expected returns against the beta values of each portfolio.

### 3. Analysis and Detail
1. **Identify Risk-Free Rate (Assuming):**
   Since we don't have a specified risk-free rate, we can estimate it using the information from the portfolios. Portfolios A and B can help us define the market return.

2. **Calculate Market Return (E(R_m)):**
   Using portfolios A (8%, beta 1.5) and B (5%, beta 0):
   - For portfolio B (with $\beta = 0$), we assume the return represents the risk-free rate $R_f = 5\%$.
   - Now using portfolio A:
   $$
   E(R_A) = R_f + \beta(E(R_m) - R_f)
   $$
   $$
   8\% = 5\% + 1.5(E(R_m) - 5\%)
   $$
   $$
   8\% - 5\% = 1.5(E(R_m) - 5\%)
   $$
   $$
   3\% = 1.5(E(R_m) - 5\%)
   $$
   $$
   \frac{3\%}{1.5} = E(R_m) - 5\%
   $$
   $$
   2\% = E(R_m) - 5\%
   $$
   $$
   E(R_m) = 7\%
   $$

3. **Verify Portfolio E:**
   Now using the market return to determine if portfolio E is appropriately valued:
   $$
   E(R_E) = R_f + \beta(E(R_m) - R_f)
   $$
   Substituting $R_f = 5\%$, $\beta = 0.5$, and $E(R_m) = 7\%$:
   $$
   E(R_E) = 5\% + 0.5(7\% - 5\%)
   $$
   $$
   = 5\% + 0.5(2\%)
   $$
   $$
   = 5\% + 1\% = 6\%
   $$

### 4. Verify and Summarize
The expected return from the calculated value of portfolio E is 6%, while its stated expected return is 7%. Since the expected return (7%) exceeds the calculated expected return (6%), portfolio E is offering an overestimated return compared to its risk (beta), indicating a deviation from CAPM expectations.

### Final Answer
Yes, there exists an arbitrage opportunity because portfolio E is priced too high for its risk, as its expected return (7%) exceeds the calculated expected return (6%).

Knowee AI · Accepted Answer