Correlation and diversification

We have seen that the diversification effect increases with the number of assets because their covariance terms can be low or negative. We will now analyze in more detail when they are low or negative. To make covariances more comparable they can be standardized into a correlation coefficient by dividing them by the product of the standard deviations of the variables:

correlation coefficient ρij = σ_ij σ_i× σj

(3.6)

59 3.1 Risk and return

The correlation coefficient has the advantage of being independent of the units of mea-surements and is bounded by minus 1 and plus 1: −1 ≤ ρ ≤ 1. In the extreme cases we speak of perfectly negatively (−1) or perfectly positively (+1) correlated variables, while variables with a correlation coefficient of zero are statistically independent (or orthogonal). Investments are generally characterized by moderately positive correlations.

We can make the relation between correlation and diversification explicit by writ-ing covariance in terms of correlation: σij= ρ_ijσ_iσ_j and substituting this, properly subscripted, into equation (3.3) for portfolio variance:

σ_p² = x₁²σ₁²+ x₂²σ₂²+ 2x₁x₂ρ_1,2σ₁σ₂ (3.7) As equation (3.7) shows, the diversification effect depends entirely on the correlation coefficient in the third term. If ρ = 1, the third term is included in portfolio variance with its full weight and there is no diversification effect. As ρ gets smaller, the third term gets smaller and the diversification effect increases. If ρ = 0, the third term will disappear altogether and negative values of ρ will make the third term contribute negatively to portfolio variance. The diversification effect reaches its maximum when ρ reaches its minimum at −1.

We will illustrate the diversification effect with a numerical example of four stocks in five different scenarios. Table 3.3 gives their returns, expected returns and standard deviations.

Table 3.3 Stock returns in scenarios

Scenario Prob. r₁ r₂ r₃ r₄

1 0.2 0.125 0.125 0.225 0.035

2 0.2 0.1 0.075 0.275 0.2

3 0.2 0.15 0.175 0.175 0.225

4 0.2 0.2 0.275 0.075 0.2

5 0.2 0.175 0.225 0.125 0.215

E[r] 0.15 0.175 0.175 0.175

σ(r) 0.0354 0.0707 0.0707 0.0706

The expected returns and standard deviations are calculated as in (3.1) and the square root of (3.2). Notice that the stocks 2, 3 and 4 all have the same expected return and standard deviation. So their risk-return characteristics are identical, they differ only with respect to their correlation with stock 1. The relevant covariances and correlations, calculated as in (3.4) and (3.6), are:

σ_1,2= 0.0025 ρ_1,2= 1 σ_1,3= −0.0025 ρ_1,3= −1 σ_1,4= 0.0009 ρ_1,4= 0.36

The correlations between stocks 1 and 2, and 1 and 3 are extreme cases with per-fectly positive and negative correlations. The combination of stocks 1 and 4 represents the normal cases with a moderately positive correlation coefficient.

We now combine stock 1 with, in turn, each of the three other stocks, so stock 1 and 2, 1 and 3, and 1 and 4. For each pair of stocks we make five different portfolios by combining them in different proportions. The composition of the portfolios, i.e. the weights of stock 1, x₁,and the other stock, 1 − x₁, as well as their expected returns and standard deviations are given in Table 3.4. For example, the expected return of the portfolio with equal parts of stock 1 and 2 is 0.5 × 0.15 + 0.5 × 0.175 = 0.163 and its variance is 0.5²× 0.0354²+ 0.5² × 0.0707²+ 2 × 0.5 × 0.5 × 0.0025 = 0.0028129 so that its standard deviation is√

0.0028129 = 0.053. Notice that the returns in the columns 3, 5 and 7 are the same.

Since stocks 2, 3 and 4 have the same expected returns, it is irrelevant for portfolio return which one is included. Also notice that all the weights are ≥ 0, so the possibility of short sellingis not included in this example.²

Table 3.4 Portfolios of stock 1, and 2, 3 and 4

Stock 1, 2 Stock 1, 3 Stock 1, 4

x1 1-x1 E[rp] σp E[rp] σp E[rp] σp

1 0 0.15 0.035 0.15 0.035 0.15 0.035

0.75 0.25 0.156 0.044 0.156 0.001 0.156 0.037

0.50 0.50 0.163 0.053 0.163 0.018 0.163 0.045

0.25 0.75 0.169 0.062 0.169 0.044 0.169 0.057

0 1 0.175 0.071 0.175 0.071 0.175 0.071

The return and risk data in Table 3.4 are plotted in Figure 3.5 in two different ways.

On the left-hand side, E[rp] and σp are plotted as functions of portfolio composition (proportion of stock 1). The return line E[r] is the same for all combinations of stocks:

it is a straight line interpolation between the high return (0.175) of stock 2, 3 or 4 and the low return (0.15) of stock 1. Portfolio risk, on the other hand, is seen to depend on correlation. When the correlation coefficient is 1, there is no diversification effect and portfolio risk is a straight line interpolation between the standard deviations of the two stocks (of 0.071 and 0.035). This is depicted as the straight line connecting the boxes (). But as the correlation coefficient becomes lower, the diversification effect increases and the line becomes more curved downwards. In the normal case of moder-ately but positively correlated stocks, the diversification effect is present but not strong.

For ρ = 0.38 this is depicted as the slightly curved line connecting the diamonds (✸^).

The diversification effect is maximal when perfectly negatively correlated stocks are combined. This is depicted by the line connecting the dots (◦), which almost reaches the x-axis.

On the right-hand side of Figure 3.5, the portfolios are plotted in a risk-return space, which visualizes the risk-return trade-off. The diversification effect is expressed in the curvature to the left, that reduces risk along the x-axis. Again, this effect is seen to increase

...

2 Recall from Chapter 2 that you short a share by borrowing it from your stock broker and selling it. After an agreed period you buy it back in the market, hopefully at a lower price than you sold it for: short selling is a strategy to profit from price decreases.

61 3.2 Selecting and pricing portfolios

0.0 0.5 1.0

0.00 0.05 0.10 0.15 0.20

E[r]

risk

prop.stock 1 ^{0.00 0.02 0.04 0.06} 0.10

0.15 0.20

risk E[r]

Figure 3.5 Portfolios’ risk and return when correlation = 1 (boxes), 0.36 (diamonds) and −1 (dots)

from zero on the right (perfectly positively correlated stocks) to its maximum on the left (perfectly negatively correlated stocks). If ρ = −1, risk can even be completely elimi-nated by choosing the portfolio weights in inverse proportion to the standard deviations:

x₁= σ₃/(σ₁+ σ₃)= 0.66 (not depicted).³

Obviously, stocks that correlate negatively with other stocks would be very attractive to investors. Unfortunately, such stocks are not available in the market, although stocks do correlate negatively over short periods of time. The returns of most real-life assets depend on the economy as a whole, so most assets are positively correlated. However, correlation is already less than perfect if assets that go up and down together reach their peak returns in different scenarios or periods. This is very common in practice, and when it occurs, diversification lowers the risk of a portfolio. Thus, the behaviour of real-life investments resembles combinations of stock 1 and 4, which correlate positively and moderately.