###### The Standard Normal Distribution: Calc ...

The standard normal distribution refers to a normal distribution that has been standardized... **Read More**

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 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:

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

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.

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.

QuestionAssume 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.*