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


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.


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.


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

Reading 9 LOS 9l

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

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