Reading 5

The expected return of a portfolio composed of n assets with weights, \(w_i\), and expected returns, \(R_i\), can be determined using the following formula:

\[E(R_P) = \sum_{i=1}^{n} w_i E(R_i) = w_1 E(R_1) + w_2 E(R_2) + \cdots + w_n E(R_n)\]

The expected return and variance for a portfolio of assets can be determined using the properties of the individual assets in the portfolio. To do this, it is necessary to establish the portfolio weight for each asset. As indicated in the formula below, the weight, w, of Asset i is simply the market value currently invested in the asset divided by the current market value of the entire portfolio:

\[w_i = \dfrac{\text{market value of investment in Asset } i}{\text{market value of the portfolio}}\]

In many finance situations, we are interested in how two random variables move in relation to each other. For investment applications, one of the most frequently analyzed pairs of random variables is the returns of two assets. Investors and managers frequently ask questions such as, "What is the relationship between the return for Stock A and Stock B?" or, "What is the relationship between the performance of the S&P 500 and that of the automotive industry?"

Covariance is a measure of how two assets move together. It is the expected value of the product of the deviations of the two random variables from their respective expected values. A common symbol for the covariance between random variables X and Y is Cov(X, Y). Because we will be mostly concerned with the covariance of asset returns, the following formula has been written in terms of the covariance of the return of Asset i, \(R_i\), and the return of Asset j, \(R_j\):

\[\text{Cov}(R_i, R_j) = E\{[R_i - E(R_i)][R_j - E(R_j)]\}\]

The following are properties of covariance:

• The covariance of a random variable with itself is its variance; that is, Cov(R_A, R_A) = Var(R_A).
• Covariance may range from negative infinity to positive infinity.
• A positive covariance indicates that when one random variable is above its mean, the other random variable also tends to be above its mean.
• A negative covariance indicates that when one random variable is above its mean, the other random variable tends to be below its mean.

The sample covariance for a sample of returns data can be calculated as follows:

\[s_{X,Y} = \dfrac{\sum_{i=1}^{n}\left\{\left[R_{1,i} - \bar{R}_1\right]\left[R_{2,i} - \bar{R}_2\right]\right\}}{n - 1}\]

where:

• \(R_{1,i}\) = an observation of returns on Asset 1
• \(R_{2,i}\) = an observation of returns on Asset 2
• \(\bar{R}_1\) = mean return of Asset 1
• \(\bar{R}_2\) = mean return of Asset 2
• n = number of observations in the sample

A covariance matrix shows the covariances between returns on a group of assets.

Note that the diagonal terms from top left are the variances of each asset's returns—in other words, Cov(R_A, R_A) = Var(R_A).

The covariance between the returns on two assets does not depend on order—in other words, Cov(R_A, R_B) = Cov(R_B, R_A)—so in this covariance matrix, only three of the (off-diagonal) covariance terms are unique. In general for n assets, there are n variance terms (on the diagonal) and \(n(n-1)/2\) unique covariance terms.

With portfolio variance, to calculate the variance of portfolio returns, we use the asset weights, returns variances, and returns covariances:

\[\text{Var}(R_P) = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{Cov}(R_i, R_j)\]

The variance of a portfolio composed of two risky assets, A and B, can be expressed as follows:

\[\text{Var}(R_P) = w_A w_A \text{Cov}(R_A, R_A) + w_A w_B \text{Cov}(R_A, R_B) + w_B w_A \text{Cov}(R_B, R_A) + w_B w_B \text{Cov}(R_B, R_B)\]

We can write this more simply as:

\[\text{Var}(R_P) = w_A^2 \text{Var}(R_A) + w_B^2 \text{Var}(R_B) + 2 w_A w_B \text{Cov}(R_A, R_B)\]

or:

\[\sigma_P^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}_{AB}\]

For a three-asset portfolio, the portfolio variance is:

\[\sigma_P^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + w_C^2 \sigma_C^2 + 2 w_A w_B \text{Cov}_{AB} + 2 w_A w_C \text{Cov}_{AC} + 2 w_B w_C \text{Cov}_{BC}\]

Consider a portfolio with three assets: an index of domestic stocks (60%), an index of domestic bonds (30%), and an index of international equities (10%). A covariance matrix of the three assets is shown here.

Portfolio returns variance =

\[(0.6^2)400 + (0.3^2)70 + (0.1^2)450 + 2(0.6)(0.3)44 + 2(0.6)(0.1)180 + 2(0.3)(0.1)35 = 194.34\]

Portfolio returns standard deviation = \(\sqrt{194.34} = 13.94\%\)

Note that the units of variance and covariance are \(\%^2\) (i.e., 0.001). When we put these values in as whole numbers (in \(\%^2\)), the portfolio variance is in \(\%^2\), and the standard deviation is in whole percentages. We could also put variance and covariance in as decimals and get both the portfolio returns variance and standard deviation as decimals.

From the formula for portfolio returns variance, we can see that the lower the covariance terms, the lower the portfolio variance (and standard deviation). This is true for positive values of covariance, as well as negative values.

Recall that the correlation coefficient for two variables is:

\[\rho_{AB} = \dfrac{\text{Cov}_{AB}}{\sigma_A \sigma_B}, \text{ so that } (\text{Cov}_{AB}) = \rho_{AB} \times \sigma_A \sigma_B\]

This can be substituted for \(\text{Cov}_{AB}\) in our formula for portfolio returns variance. With this substitution, we can use a correlation matrix to calculate portfolio returns variance, rather than using covariances.

Note that the correlations of asset returns with themselves (the diagonal terms) are all 1.

Shortfall risk is the probability that a portfolio value or return will fall below a particular target value or return over a given period.

Roy's safety-first criterion states that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level. This minimum acceptable level is called the threshold level. Symbolically, Roy's safety-first criterion can be stated as follows:

\[\text{minimize } P(R_P < R_L)\]

where:

• \(R_P\) = portfolio return
• \(R_L\) = threshold level return

If portfolio returns are normally distributed, then Roy's safety-first criterion can be stated as follows:

\[\text{maximize the } \mathbf{safety\text{-}first\ ratio}, \text{ which equals } \dfrac{E(R_P) - R_L}{\sigma_P}\]

The reasoning behind the safety-first criterion is illustrated in Figure 5.4. Assume an investor is choosing between two portfolios: Portfolio A, with an expected return of 12% and standard deviation of returns of 18%, and Portfolio B, with an expected return of 10% and standard deviation of returns of 12%. The investor has stated that he wants to minimize the probability of losing money (negative returns). Assuming that returns are normally distributed, the portfolio with the larger safety-first ratio using 0% as the threshold return (\(R_L\)) will be the one with the lower probability of negative returns.

Panel B of Figure 5.4 relates the safety-first ratio to the standard normal distribution. Note that the safety-first ratio is the number of standard deviations below the mean. Thus, the portfolio with the larger safety-first ratio has the lower probability of returns below the threshold return, which is a return of 0% in our example. Using a z-table for negative values, we can find the probabilities in the left-hand tails as indicated. These probabilities (25% for Portfolio A and 20% for Portfolio B) are also the shortfall risk for a target return of 0%—that is, the probability of negative returns. Portfolio B has the higher safety-first ratio, which means it has the lower probability of negative returns.

In summary, when choosing among portfolios with normally distributed returns using Roy's safety-first criterion, there are two steps:

• Step 1:Calculate the safety-first ratio = \(\dfrac{E(R_P) - R_L}{\sigma_P}\).
• Step 2:Choose the portfolio that has the largest safety-first ratio.