In an efficient capital market where the random returns from two investments are normally distributed (symmetrical) we have explained how rational (risk averse) investors and companies who seek an optimal portfolio can maximise their utility preferences by efficient diversification. Any combination of investments produces a trade-off between the two statistical parameters that define a normal distribution; their expected return and standard deviation (risk) associated with the covariability of individual returns. According to Markowitz (1952) this is best measured by the correlation coefficient such that:

Efficient diversified portfolios are those which maximise return for a given level of risk, or minimise risk for a given level of return for different correlation coefficients.

The purpose of this chapter is to prove that when the correlation coefficient is at a minimum and portfolio risk is minimised we can derive an optimum portfolio of investments that maximises there overall expected return.

The Mathematics of Portfolio Risk

You recall from Chapter Two (both the Theory and Exercises texts) that substituting the relative linear correlation coefficient for the absolute covariance term into a two-asset portfolio's standard deviation simplifies the wealth maximisation analysis of the risk-return trade-off between the covariability of returns. Whenever the coefficient falls below one, there will be a proportionate reduction in portfolio risk, relative to return, by diversifying investment.

For example, given the familiar equations for the return, variance, correlation coefficient and standard deviation of a two-asset portfolio:

Harry Markowitz (op. cit.) proved mathematically that:

However, he also illustrated that if the returns from two investments exhibit perfect positive, zero, or perfect negative correlation, then portfolio risk measured by the standard deviation using Equation (8) can be simplified further.

To understand why, let us return to the original term for the portfolio variance:

Because the correlation coefficient is given by:

We can rearrange its terms, just as we did in Chapter Two, to redefine the covariance:

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

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

Armed with this information, we can now confirm that:

If the returns from two investments exhibit perfect, positive correlation, portfolio risk is simply the weighted average of its constituent's risks and at a maximum.

If the correlation coefficient for two investments is positive and COR(A,B) also equals plus one, then the correlation term can disappear from the portfolio risk equations without affecting their values. The portfolio variance can be rewritten as follows:

Simplifying, this is equivalent to:

And because this is a perfect square, our probabilistic estimate for the risk of a two-asset portfolio measured by the standard deviation given by Equation (8) is equivalent to:

To summarise:

Whenever COR(A, B) = +1 (perfect positive) the portfolio variance VAR(P) and its square root, the standard deviation c(P), simplify to the weighted average of the respective statistics, based on the probabilistic returns for the individual investments.

But this is not all. The substitution of Equation (6) into the expression for portfolio variance has two further convenient properties. Given:

(6) COV(A,B) = COR(A,B) aA aB

If the relationship between two investments is independent and exhibits zero correlation, the portfolio variance given by Equation (7) simplifies to:

(12) VAR(P) = x2 VAR(A) + (1-x)2 VAR(B)

And its corresponding standard deviation also simplifies:

(13) a(P) = V [x2 VAR(A) + (1-x)2 VAR(B)]

Similarly, with perfect inverse correlation we can deconstruct our basic equations to simplify the algebra.

Activity 1

When the correlation coefficient for two investments is perfect positive and equals one, the correlation term disappears from equations for portfolio risk without affecting their values. The portfolio variance VAR(P)) and its square root, the standard deviation a(P), simplify to the weighted average of the respective statistics.

Can you manipulate the previous equations to prove that if COR(A,B) equals minus one (perfect negative) they still correspond to a weighted average, like their perfect positive counterpart, but with one fundamental difference? Whenever COR(A, B) = +1 (perfect positive) the portfolio variance VAR(P) and its square root, the standard deviation a(P), simplify to the weighted average of the respective statistics, based on the probabilistic returns for the individual investments.

Let us begin again with the familiar equation for portfolio variance.

(7) VAR(P) = x2 VAR(A) + (1-x)2 VAR(B) + 2x (1-x) COR(A,B) a A a B

If the correlation coefficient for two investments is negative and COR(A,B) also equals minus one, then the coefficient can disappear from the equation's third right hand term without affecting its value. It can be rewritten as follows with only a change of sign (positive to negative):

And because this is a perfect square, our probabilistic estimate for the risk of a two-asset portfolio measured by the standard deviation is equivalent to:

The only difference between the formulae for the risk of a two-asset portfolio where the correlation coefficient is at either limit (+1 or -1) is simply a matter of sign (positive or negative) in the right hand term for a (P).

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