Because the covariance is an absolute measure of the correspondence between the movements of two random variables, its interpretation is often difficult. Not all paired deviations need be negative for diversification to produce a degree of risk reduction. If we have small or large negative or positive values for individual pairs, the covariance may also assume small or large values either way. So, in our previous example, COV(A, B) = minus 8. But what does this mean exactly?

Fortunately, we need not answer this question? According to Markowitz, the statistic for the linear correlation coefficient can be substituted into the third covariance term of our equation for portfolio risk to simplify its interpretation. With regard to the mathematics, beginning with the variance for a two asset portfolio:

(2) VAR(P) = x2 VAR(A) + (1-x)2 VAR(B) + 2x (1-x) COV(A,B) Let us define the correlation coefficient.

Now rearrange terms to redefine the covariance.

Clearly, the portfolio variance can now be measured by the substitution of Equation (6) for the covariance term in Equation (2).

The standard deviation of the portfolio then equals the square root of Equation (7):

Activity 4

So far, so good; we have proved mathematically that the correlation coefficient can replace the covariance in the equations for portfolio risk.

But, given your knowledge of statistics, can you now explain why Markowitz thought this was a significant contribution to portfolio analysis?

Like the standard deviation, the correlation coefficient is a relative measure of variability with a convenient property. Unlike the covariance, which is an absolute measure, it has only limited values between +1 and -1. This arises because the coefficient is calculated by taking the covariance of returns and dividing by the product (multiplication) of the individual standard deviations that comprise the portfolio. Which is why, for two investments (A and B) we have defined:

The correlation coefficient therefore measures the extent to which two investments vary together as a proportion of their respective standard deviations. So, if two investments are perfectly and linearly related, they deviate by constant proportionality.

Of course, the interpretation of the correlation coefficient still conforms to the logic behind the covariance, but with the advantage of limited values.

- If returns are independent, i.e. no relationship exists between two variables; their correlation will be zero (although, as we shall discover later, risk can still be reduced by diversification).

- If returns are dependent:

1) A perfect, positive correlation of + 1 means that whatever affects one variable will equally affect the other. Diversified risk-reduction is not possible.

2) A perfect negative correlation of -1 means that an efficient portfolio can be constructed, with zero variance exhibiting minimum risk. One investment will produce a return above its expected return; the other will produce an equivalent return below its expected value and vice versa.

3) Between +1 and -1, the correlation coefficient is determined by the proximity of direct and inverse relationships between individual returns So, in terms of risk reduction, even a low positive correlation can be beneficial to investors, depending on the allocation of total funds at their disposal.

Providing the correlation coefficient between returns is less than +1, all investors (including management) can profitably diversify their portfolio of investments. Without compromising the overall return, relative portfolio risk measured by the standard deviation will be less than the weighted average standard deviation of the portfolio's constituents.

Review Activity

Using the statistics generated by Activity 3, confirm that the substitution of the correlation coefficient for the covariance into our revised equations for the portfolio variance and standard deviation does not change their values, or our original investment decision?

Let us begin with a summary of the previous mean-variance data for the two-asset portfolio:

The correlation coefficient is given by:

Substituting this value into our revised equations for the portfolio variance and standard deviation respectively, we can now confirm our initial calculations for Activity 3.

Thus, the company's original portfolio decision to place an equal proportions of funds in both investments, rather than either A or B exclusively, still applies. This is also confirmed by a summary of the following inter-relationships between the risk-return profiles of the portfolio and its constituents, which are identical to our previous Activity.

2.4. Summary and Conclusions

It should be clear from our previous analyses that the risk of a two-asset portfolio is a function of its covariability of returns. Risk is at a maximum when the correlation coefficient between two investments is +1 and at a minimum when the correlation coefficient equals -1. For the vast majority of cases where the correlation coefficient is between the two, it also follows that there will be a proportionate reduction in risk, relative to return. Overall portfolio risk will be less than the weighted average risks of its constituents. So, investors can still profit by diversification because: