Chapter 7 • Portfolio theory and its relevance to real investment decisions
* R=Readers who are not too confident with these statistical measures should refer to Bancroft and
O’Sullivan (2000) or one of the other texts that cover the ground.Covariance (A, B)
σAσBIn Figure 7.2, all risk-averse investors would prefer investment B to investment C
as it has higher expected value for the same level of risk. All risk-averse investors
would prefer investment A to investment C as it has lower risk for the same level of
expected return. Thus both A and B are said to ‘dominate’ C. The choice between A
and B, however, depends upon the attitude to risk of the investor making the decision
(that is, on the shape of the particular investor’s utility curve).
The positions of A and B in Figure 7.2 are dictated by their expected values and
standard deviations. B is expected to yield higher returns than A (14 per cent as
against 10 per cent), but it is also expected to be more risky (standard deviation of 12
per cent as against 10 per cent). Assuming that all investors have similar expectations
of these and other securities, then they would all see these securities as occupying the
same position on the graph.Expected return and risk of a portfolio
Suppose that we were to form a portfolio containing a proportion (α) of each of A and
B. The expected return from that portfolio would be:rP=αArA+αBrBwhere rP,rAand rBare the expected returns from the portfolio, security A and security
B, respectively: in other words, the returns of the portfolio are simply the weighted
average of the returns of the constituent securities. The standard deviation of the port-
folio would be:σP=where σP, σAand σBare, respectively, the standard deviation of the expected returns
of the portfolio, security A and security B. R is the coefficient of correlationbetween
the expected returns of the two securities.* If Rwere +1 it would mean that the
expected returns would be perfectly correlated one with the other: an increase of xper
cent in the returns of A would always imply a yper cent increase in the returns of B,
and vice versa. If Rwere −1 it would imply as close a relationship but a negative one:
an xper cent increase in the returns of A would always mean a yper cent decrease in
the returns from B. An Rof between −1 and +1 means less direct relationships, with a
value of zero meaning that there is no relationship at all between the returns of the two
securities. (Note that there would be a separate Rfor any two variables. The particu-
lar value of Rfor A and B would be peculiar to A and B and not the same, except by
coincidence, for any other two securities.)
Table 7.1 shows the expected returns and standard deviations for a selection of
possible portfolios of A and B for a range of hypothetical correlation coefficients (R).α σAA^22 ++ασBB^22 2 αα σσABR AB‘