Appendix 5.1: Statistical Review

Asset returns are not perfectly anticipated. Therefore, they are modeled as draws from an underlying probability distribution (for example, the Normal density with mean return μ and variance σ²). The probability distribution describes the riskiness of returns. For example, if returns are normally distributed, then observed returns will have a relatively high likelihood of being close to the average, or central tendency of the distribution and, consequently, would have a low likelihood of taking on extreme values in the tails of the distribution. The following is a picture of the standard normal density for the random variable . The likelihood is higher for values of z closer to the mean (zero) than for extreme values.

We can therefore think of a random draw as having an expected value that corresponds to some notion of central tendency. Thus, the expected value of a random variable is some kind of average of all its possible outcomes. Figure 5.1 is such an example—the expected values of the random payoffs are weighted averages of their possible outcomes. In repeated draws, we would expect an outcome to be represented by:

Here, f_i is the relative frequency (probability) of observing return (or payoff) r_i, and there are N possible outcomes. If all N returns are equally likely as they would be in a simple random sample, then the return is expected to be:

Variances are also expected values, but for squared deviations of the possible returns from what we would expect them to be:

Like an expected value, the expectation is simply the sum of the squared terms divided by N. Before we evaluate the expectation (E), let's first expand the square to get:

Now, recall that expectations are for random variables, so if c is constant and r is a random variable, then . Therefore,

If we denote , then equivalently, we have

This is a standard formula to estimate variances. Now take two random variables, r₁ and r₂, which are the random returns on two assets. What is the covariance? It is the expectation of the variation across these two returns over the sample.

Let's denote . Therefore:

Now, passing the expectations operator through and evaluating the expected values of the random variables yields:

Since , then this simplifies to the following:

Correlations are closely related—they are standardized covariances. That is,

Rho (ρ) is the correlation coefficient and σ_i are standard deviations (square roots of the variances). The denominator is always positive but the numerator can take positive or negative values. Therefore, ρ may be of either sign. It is, however, bounded by one in absolute value since by the Cauchy-Schwartz inequality.

Extrapolation

Assume a security's returns are independently distributed over time; that is, the return today does not depend on the return in any other time. Over a period of time T, we observe total return

Taking logarithms produces the sum:

For simplicity, let . Then, the return for the entire period T is the sum:

Since returns are assumed to be independent, the variance of the return over T periods is the sum of the individual variances (since independence requires the cross-product terms to be zero):

This suggests that volatility over any time span T be a simple constant extrapolation of the single period volatility:

Diversification on the Margin

Suppose a portfolio consists of a single risky asset with variance σ₁². Does adding another risky asset always reduce risk as long as the correlation between the two assets is less than one? To answer this question, let's find out under which conditions the risk on the two-asset portfolio is less than the risk on the single risky asset. We express this relationship as an inequality:

The left side of the inequality is the risk on the two-asset portfolio. With no loss of generality, assume that the second asset's risk is linear in the first asset's risk, so that:

The constant α can be any number. Making this substitution into the first equation from before and collecting a few terms yields the following:

Now, if we isolate the correlation coefficient ρ, we can show that the risk on the expanded portfolio will depend on the relative values between the two assets’ correlation, their weights, and their individual risks. Solving for ρ:

Therefore, whether diversification of risk occurs will depend on the strength of their correlation; in general, smaller correlations are required. If we were to examine this problem in a spreadsheet, we would quickly find that the correlation alone is not the arbiter of diversification. Rather, it is the risk on the second asset relative to the first asset, captured by α, as well as their relative weights. Diversification is not guaranteed simply because the asset being added to the portfolio has a low correlation to the existing assets. Rather, diversification depends on how risky the second asset is as well as how heavily weighted the bigger portfolio is to this asset. Therefore, as α gets larger, ρ will have to be smaller to compensate for the additional assets’ higher risk.