## 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. 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(Rp) = ?Wi Ri where i = 1,2,3 … n

Where Wi represents the weight attached to asset I and Ri 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(Rp) = 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 = W2A * σ2(RA) + W2B * σ2(RB) + 2 * ( WA)  * (WB) * Cov(RA,RB)

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 = {W2A * σ2(RA) + W2B * σ2(RB) + 2 * ( WA) * (WB) * Cov(RA,RB)}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

Working

Portfolio variance = W2A * σ2(RA) + W2B * σ2(RB) + 2 * ( WA)  * (WB) * Cov(RA,RB)

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

Covariance, cov(RABC,RXYZ) = 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:

= 0.52 * 0.012492 + 0.52 * 0.00462 + 2 * 0.5 * 0.5 * 0.0000561

= 0.00007234

Therefore, the standard deviation is 0.000072341/2 = 0.00851

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

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