Portfolio Returns

A portfolio is basically a collection of investments held by a company, mutual fund, or even an individual investor, consisting of assets such as stocks, bonds, or cash equivalents. Financial professionals usually manage a portfolio.

Portfolio Expected Return

Portfolio expected return is the sum of each of the individual asset’s expected returns multiplied by its associated weight. Therefore:

E(R_p) = ΣW_i R_i where i = 1,2,3 … n

Where W_i represents the weight attached to the asset, i, and R_i is the asset’s return.

The weight attached to an asset = market value of an asset/market value of a portfolio

Example

Assume that we have a simple portfolio of two mutual funds, one invested in bonds and the other invested in stocks. Let’s further assume that we expect a stock return of 8% and a bond return of 6% and our allocation is equal in both funds. Then:

$$ \begin{align*} E(R_p)& = 0.5 * 0.08 + 0.5 * 0.06 \\ & = 0.07 \text{ or } 7\% \\ \end{align*} $$

Portfolio Variance

The variance of a portfolio’s return is always a function of the individual assets as well as the covariance between each of them. If we have two assets, A and B,

$$ \text{Portfolio variance} = { W }_{ A }^{ 2 }\ast { \sigma }^{ 2 }\left( { R }_{ A } \right) +{ W }_{ B }^{ 2 }\ast { \sigma }^{ 2 }\left( { R }_{ B } \right) +2\ast \left( { W }_{ A } \right) \ast \left( { W }_{ B } \right) \ast Cov\left( { R }_{ A },{ R }_{ B } \right) $$

Portfolio variance is a measure of risk. More variance translates to more risk. Investors usually reduce the portfolio variance by choosing assets that have low or negative covariance, e.g. stocks and bonds.

Portfolio Standard Deviation

This is simply the square root of the portfolio variance. Therefore:

$$ S.D={ \left\{ { W }_{ A }^{ 2 }\ast { \sigma }^{ 2 }\left( { R }_{ A } \right) +{ W }_{ B }^{ 2 }\ast { \sigma }^{ 2 }\left( { R }_{ B } \right) +2\ast \left( { W }_{ A } \right) \ast \left( { W }_{ B } \right) \ast Cov\left( { R }_{ A },{ R }_{ B } \right) \right\} }^{ \frac { 1 }{ 2 } } $$

It is also a measure of the riskiness of a portfolio.

Question

Assume that we have invested equally in two different companies; ABC and XYZ. We anticipate a 15% chance that next year’s stock returns for ABC Corp will be 6%, a 60% probability that they will be 8% and a 25% probability that they will be 10%. In addition, we already know that the expected value of returns is 8.2%, and the standard deviation is 1.249%.

We also anticipate that the same probabilities and states are associated with a 4% return for XYZ Corp, a 5% return, and a 5.5% return. The expected value of returns is then 4.975 and the standard deviation is 0.46%.

Calculate the portfolio standard deviation:

A. 0.0000561

B. 0.0000

C. 0.00851

The correct answer is C.

Working

$$ \text{Portfolio variance} = { W }_{ A }^{ 2 }\ast { \sigma }^{ 2 }\left( { R }_{ A } \right) +{ W }_{ B }^{ 2 }\ast { \sigma }^{ 2 }\left( { R }_{ B } \right) +2\ast \left( { W }_{ A } \right) \ast \left( { W }_{ B } \right) \ast Cov\left( { R }_{ A },{ R }_{ B } \right) $$

First, we must calculate the covariance between the two stocks:

$$ \begin{align*} \text{Covariance}, \text{cov}(\text R_{ \text{ABC}},\text R_{ \text{XYZ}}) & = 0.15(0.06 – 0.082)(0.04 – 0.04975) \\ & + 0.6(0.08 – 0.082)(0.05 – 0.04975) \\ & + 0.25(0.10 – 0.082)(0.055 – 0.04975) \\ & = 0.0000561 \\ \end{align*} $$

Since we already have the weight and the standard deviation of each asset, we can proceed and calculate the portfolio variance:

$$ \begin{align*} & = 0.5^2* 0.01249^2+ 0.5^2* 0.0046^2+ 2 * 0.5 * 0.5 * 0.0000561 \\ & = 0.00007234 \\ \end{align*} $$

Therefore, the standard deviation is \(0.00007234^{\frac {1}{2}} = 0.00851\)

Reading 8 LOS 8l

Calculate and interpret the expected value, variance, and standard deviation of a random variable and returns on a portfolio.