# 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(Rp) = ΣWi Ri where i = 1,2,3 … n

Where Wi represents the weight attached to the asset, i, and Ri 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

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 $$0.00007234^{\frac {1}{2}} = 0.00851$$

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

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