Risk Minimisation and the Two-Asset Portfolio
When investment returns exhibit perfect positive correlation a portfolio's risk is at a maximum, defined by the weighted average of its constituents. As the correlation coefficient falls there is a proportionate reduction in portfolio risk relative to this weighted average. So, if we diversify investments; risk is minimised when the correlation coefficient is minus one.
To illustrate this general proposition, Figure 3.1 roughly sketches the various two-asset portfolios that are possible if corporate management combine two investments, A and B, in various proportions for different correlation coefficients.
Specifically, the diagonal line A (+1) B; the curve A (E) B and the "dog-leg" A (-1) B are the focus of all possible risk-return combinations when our correlation coefficients equal plus one, zero and minus one, respectively.
Thus, if project returns are perfectly, positively correlated we can construct a portfolio with any risk-return profile that lies along the horizontal line, A (+1) B, by varying the proportion of funds placed in each project. Investing 100 percent in A produces a minimum return but minimises risk. If management put all their funds in B, the reverse holds. Between the two extremes, having decided to place say two-thirds of funds in Project A, and the balance in Project B, we find that the portfolio lies one third along A (+1) B at point +1.
Figure 3.1: The Two Asset Risk-Return Profile and the Correlation Coefficient
Similarly, if the two returns exhibit perfect negative correlation, we could construct any portfolio that lies along the line A (-1) B. However, because the correlation coefficient equals minus one, the line is no longer straight but a dog-leg that also touches the vertical axis where a(P) equals zero. As a consequence, our choice now differs on two counts.
- It is possible to construct a risk-free portfolio.
- No rational, risk averse investor would be interested in those portfolios which offer a lower expected return for the same risk.
As you can observe from Figure 3.1, the investment proportions lying along the line -1 to B offer higher returns for a given level of risk relative to those lying between -1 and A. Using the terminology of Markowitz based on mean-variance criteria; the first portfolio set is efficient and acceptable whilst the second is inefficient and irrelevant. The line -1 to B, therefore, defines the efficiency frontier for a two-asset portfolio.
Where the two lines meet on the vertical axis (point -1 on our diagram) the portfolio standard deviation is zero. As the horizontal line (-1, 0, +1) indicates, this riskless portfolio also conforms to our decision to place two-thirds of funds in Project A and one third in Project B.
Finally, in most cases where the correlation coefficient lies somewhere between its extreme value, every possible two-asset combination always lies along a curve. Figure 3.1 illustrates the risk-return trade-off assuming that the portfolio correlation coefficient is zero. Once again, because the data set is not perfect positive (less than +1) it turns back on itself. So, only a proportion of portfolios are efficient; namely those lying along the E-B frontier. The remainder, E-A, is of no interest whatsoever. You should also note that whilst risk is not eliminated entirely, it could still be minimised by constructing the appropriate portfolio, namely point E on our curve.
The Minimum Variance of a Two-Asset Portfolio
Investors trade financial securities to earn a return in the form of dividends and capital gains. Companies invest in projects to generate net cash inflows on behalf of their shareholders. Returns might be higher or lower than anticipated and their variability is the cause of investment risk. Investors and companies can reduce risk by diversifying their portfolio of investments. The preceding analysis explains why risk minimisation represents an objective standard against which investors and management compare their variance of returns as they move from one portfolio to another.
To prove this proposition, you will have observed from Figure 3.1 that the decision to place two-thirds of our funds in Project A and one-third in Project B falls between E and A when COR(A,B) = 0. This is defined by point 0 along the horizontal line (-1, 0, +1).
Because portfolio risk is minimised at point E, with a higher return above and to the left in our diagram, the decision is clearly suboptimal. At one extreme, speculative investors or companies would place all their money in Project B at point B hoping to maximise their return (completely oblivious to risk). At the other, the most risk-averse among them would seek out the proportionate investment in A and B which corresponds to E. Between the two, a higher expected return could also be achieved for any degree of risk given by the curve E-A. Thus, all investors would move up to the efficiency frontier E-B and depending upon their risk attitudes choose an appropriate combination of investments above and to the right of E.
However, without a graph, let alone data to fall back on, this raises another fundamental question.
How do investors and companies mathematically model an optimum portfolio with minimum variance from first principles?
According to Markowitz (op. cit) the mathematical derivation of a two-asset portfolio with minimum risk is quite straightforward.
Where a proportion of funds x is invested in Project A and (1-x) in Project B, the portfolio variance can be defined by the familiar equation:
The value of x, for which Equation (7) is at a minimum, is given by differentiating VAR(P) with respect to x and setting AVAR(P) / A x = 0, such that:
Since all the variables in the equation for minimum variance are now known, the risk-return trade-off can be solved. Moreover, if the correlation coefficient equals minus one, risky investments can be combined into a riskless portfolio by solving the following equation when the standard deviation is zero.
Because this is a quadratic in one unknown (x) it also follows that to eliminate portfolio risk when COR(A,B) = -1, the proportion of funds (x) invested in Project A should be:
Algebraically, mathematically and statistically, we have covered a lot of ground since Chapter Two. So, the previous section, like those before it, is illustrated by the numerical application of data to theory in the bookroom companion text.
Portfolio Theory and Financial Analyses; Exercises (PTFAE): Chapter Three, 2010.
You might find it useful at this point in our analysis to cross-reference the appropriate Exercises (3.1 and 3.2) before we continue?