Question

Problem 3: You have a portfolio of 5 stocks with equal shares invested in each stock. Variances of individual stock are the same and equal to ơ2-16. Covariances between stocks 1, 2, and 5 are equal to 2. Stocks 3 and 4 are uncorrelated with each other and are uncorrelated with stocks 1,2 and 5. a) Find the variance of this portfolio using "box" method. Further assume that stocks 1, 2, and 5 have beta's of1.1 each and stocks 3 and 4 have zero betas b) What is the beta of your portfolio.

Answer 1

a)

Given:

• It is given that each stock has an equal share in the portfolio. This means the weight ($w$) of all stocks is equal to (1/5) or 0.2.
• Variance ($\sigma ^{2}$) of all stocks is equal to 16.
• Covariance ($\sigma _{xy}$ ) between stocks 1, 2 and 5 is equal to 2 whereas covariance of stocks 3 and 4 is zero with all the other stocks and among themselves.

Now, the covariance matrix or the box matrix for a 5 stock portfolio looks like this:

$\begin{bmatrix} &1 & 2 & 3 & 4 & 5\\ 1 &w_{1}^{2}\sigma _{1}^{2} &w_{2}w_{1}\sigma _{21} & w_{3}w_{1}\sigma _{31} &w_{4}w_{1}\sigma _{41} &w_{5}w_{1}\sigma _{51} \\ 2 & w_{1}w_{2}\sigma _{12} &w_{2}^{2}\sigma _{2}^{2} &w_{3}w_{2}\sigma _{32} & w_{4}w_{2}\sigma _{42} &w_{5}w_{2}\sigma _{52} \\ 3 &w_{1}w_{3}\sigma _{13} &w_{2}w_{3}\sigma _{23} & w_{3}^{2}\sigma _{3}^{2} & w_{4}w_{3}\sigma _{43} &w_{5}w_{3}\sigma _{53} \\ 4& w_{1}w_{4}\sigma _{14} &w_{2}w_{4}\sigma _{24} & w_{3}w_{4}\sigma _{34} & w_{4}^{2}\sigma _{4}^{2} &w_{5}w_{4}\sigma _{54} \\ 5 & w_{1}w_{5}\sigma _{15} &w_{2}w_{5}\sigma _{25} & w_{3}w_{5}\sigma _{35} & w_{4}w_{5}\sigma _{45} & w_{5}^{2}\sigma _{5}^{2} \end{bmatrix}$

where the first row and column represents the stock number (1 through 5). It looks complicated but is quite easy to create. Every cell corresponds to a stock pair. Each cell is filled with a term that contains the product of weights of each stock in the stock pair and either:

• the variance, if the stock is paired with itself (i.e. on the diagonal)
• or, the covariance between the stock pair

Now, the variance of the portfolio is calculated by adding all these terms up.

When we substitute the values, the matrix simplifies as shown:

$\begin{bmatrix} &1 & 2 & 3 & 4 & 5\\ 1& 0.64 & 0.8 & 0 & 0 &0.8 \\ 2&0.8 & 0.64 & 0 &0 &0.8 \\ 3& 0 &0 & 0.64 &0 &0 \\ 4 & 0 & 0 & 0 & 0.64 &0 \\ 5 &0.8 & 0.8 & 0 & 0 &0.64 \end{bmatrix}}$

So, all cells take up one of 3 values 0.64, 0.8 or 0.

the variance of the portfolio is simply the sum of all these values. There are 5 terms of 0.64 and 6 terms of 0.8. Rest are zero.

So, variance = 5(0.64) + 6(0.8)

Variance = 8

b)

Portfolio beta is simply the weighted average of individual stock betas.

It is given that betas of stocks 3 and 4 are zero, so they can be ignored totally.

For stocks 1,2 and 5, beta is equal to 1.1 and their weights are 0.2 each. So, their weighted average is calculated as:

$\beta _{portfolio}=3(0.2*1.1) = 0.66$