The problem provides the index models for Amazon and Meta stocks: $R_{Amazon} = 0.032 + 0.9R_M + e_{Amazon}$ $R_{Meta} = 0.075 + 1.2R_M + e_{Meta}$ Also given are: $\sigma_M = 0.25$ $\sigma_{eAmazon} = \sqrt{0.28}$ $\sigma_{eMeta} = 0.10$ We need to find: 1. Covariance between the returns on Amazon and Meta.

1. Problem Description

The problem provides the index models for Amazon and Meta stocks:

$R_{Amazon} = 0.032 + 0.9R_M + e_{Amazon}$

$R_{Meta} = 0.075 + 1.2R_M + e_{Meta}$

Also given are:

$\sigma_M = 0.25$

$\sigma_{eAmazon} = \sqrt{0.28}$

$\sigma_{eMeta} = 0.10$

We need to find:

1. Covariance between the returns on Amazon and Meta.

2. Standard deviation of Amazon's returns.

3. Standard deviation of Meta's returns.

4. Correlation coefficient between the returns on Amazon and Meta.

2. Solution Steps

1. Covariance between Amazon and Meta:

$Cov(R_{Amazon}, R_{Meta}) = Cov(0.032 + 0.9R_M + e_{Amazon}, 0.075 + 1.2R_M + e_{Meta})$

$Cov(R_{Amazon}, R_{Meta}) = Cov(0.9R_M, 1.2R_M) + Cov(0.9R_M, e_{Meta}) + Cov(e_{Amazon}, 1.2R_M) + Cov(e_{Amazon}, e_{Meta})$

Since $e_{Amazon}$ and $e_{Meta}$ are independent of $R_M$, $Cov(R_M, e_{Amazon}) = Cov(R_M, e_{Meta}) = Cov(e_{Amazon}, e_{Meta}) = 0$.

$Cov(R_{Amazon}, R_{Meta}) = (0.9)(1.2)Cov(R_M, R_M) = (0.9)(1.2)Var(R_M) = (0.9)(1.2)\sigma_M^2$

$Cov(R_{Amazon}, R_{Meta}) = (0.9)(1.2)(0.25)^2 = (1.08)(0.0625) = 0.0675$

2. Standard deviation of Amazon's returns:

$Var(R_{Amazon}) = Var(0.032 + 0.9R_M + e_{Amazon})$

$Var(R_{Amazon}) = Var(0.9R_M) + Var(e_{Amazon}) + 2Cov(0.9R_M, e_{Amazon})$

Since $R_M$ and $e_{Amazon}$ are independent, $Cov(R_M, e_{Amazon}) = 0$

$Var(R_{Amazon}) = (0.9)^2Var(R_M) + Var(e_{Amazon}) = (0.9)^2\sigma_M^2 + \sigma_{eAmazon}^2$

$Var(R_{Amazon}) = (0.81)(0.25)^2 + 0.28$

$Var(R_{Amazon}) = (0.81)(0.0625) + 0.28 = 0.050625 + 0.28 = 0.330625$

$\sigma_{Amazon} = \sqrt{0.330625} \approx 0.5750$

3. Standard deviation of Meta's returns:

$Var(R_{Meta}) = Var(0.075 + 1.2R_M + e_{Meta})$

$Var(R_{Meta}) = Var(1.2R_M) + Var(e_{Meta}) + 2Cov(1.2R_M, e_{Meta})$

Since $R_M$ and $e_{Meta}$ are independent, $Cov(R_M, e_{Meta}) = 0$

$Var(R_{Meta}) = (1.2)^2Var(R_M) + Var(e_{Meta}) = (1.2)^2\sigma_M^2 + \sigma_{eMeta}^2$

$Var(R_{Meta}) = (1.44)(0.25)^2 + (0.1)^2 = (1.44)(0.0625) + 0.01 = 0.09 + 0.01 = 0.10$

$\sigma_{Meta} = \sqrt{0.10} \approx 0.3162$

4. Correlation coefficient between Amazon and Meta:

$\rho_{Amazon, Meta} = \frac{Cov(R_{Amazon}, R_{Meta})}{\sigma_{Amazon}\sigma_{Meta}}$

$\rho_{Amazon, Meta} = \frac{0.0675}{(0.5750)(0.3162)} = \frac{0.0675}{0.181815} \approx 0.3713$

3. Final Answer

1. Covariance between Amazon and Meta stocks: 0.0675

2. Standard deviation of Amazon's returns: 0.5750

3. Standard deviation of Meta's returns: 0.3162

4. Correlation coefficient between Amazon and Meta stocks: 0.3713