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. A portfolio is usually managed by financial professionals.

**Portfolio Expected Return**

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

E(R_{p}) = ?W_{i} R_{i} where i = 1,2,3 … n

Where W_{i }represents the weight attached to asset I and R_{i }is the asset’s return

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

Example

Assume 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:

E(R_{p}) = 0.5 * 0.08 + 0.5 * 0.06

= 0.07 or 7%

**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,

Portfolio variance = W^{2}_{A} * σ^{2}(R_{A}) + W^{2}_{B} * σ^{2}(R_{B}) + 2 * ( W_{A}) * (W_{B}) * Cov(R_{A},R_{B})

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. Thus:

S. D = {W^{2}_{A} * σ^{2}(R_{A}) + W^{2}_{B} * σ^{2}(R_{B}) + 2 * ( W_{A}) * (W_{B}) * Cov(R_{A},R_{B})}^{1/2}

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

**Question**

Assume we have equally invested in two different companies; ABC and XYZ. We anticipate that there is 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 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

Portfolio variance = W

^{2}_{A}* σ^{2}(R_{A}) + W^{2}_{B}* σ^{2}(R_{B}) + 2 * ( W_{A}) * (W_{B}) * Cov(R_{A},R_{B})First, we must calculate the covariance between the two stocks:

Covariance, cov(R

_{ABC,}R_{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

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

= 0.5

^{2}* 0.01249^{2}+ 0.5^{2}* 0.0046^{2}+ 2 * 0.5 * 0.5 * 0.0000561= 0.00007234

Therefore, the standard deviation is 0.00007234

^{1/2}= 0.00851

*Reading 9 LOS 9l*

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