BUSF_A01.qxd

Chapter 7 • Portfolio theory and its relevance to real investment decisions

with negatively correlated securities. In fact there is a combination* of A and B that has

no risk when the correlation coefficient is assumed to be −1. In reality such perfectly

negativelycorrelated securities would be impossible to find. Their existence would

imply that it would be possible to form portfolios possessing no risk at all, which is

not likely to be the case. It is probably equally unlikely that securities whose expected

returns are perfectly positivelycorrelated exist in the real world. If they did, forming

Figure 7.3
The risk /return
profiles of portfolios
of securities A and
B, assuming a 0.5
positive correlation
coefficient

By combining securities A and B into a portfolio, risk is reduced because the two securities

have a correlation coefficient of less than +1.0.

* This is discoverable by finding the minimum value for σPby means of differential calculus. Writing

αBas 1 −αA, we get:

σ^2 P=α^2 Aσ^2 A+(1 −αA)^2 σ^2 B+ 2 αA(1 −αA)RσAσB

Differentiating with respect to αAgives:

= 2 αAσ^2 A+ 2 αAσ^2 B− 2 σ^2 B+ 2 RσAσB− 4 αARσAσB

Setting this equal to zero and putting in the actual values for σAand σBand assuming R =−1, we get:

(2αA×100) +(2αA×144) −(2 ×144) −(2 × 10 ×12) +(4αA×120)= 0

200 αA+ 288 αA+ 480 αA= 288 + 240

αA=

• that is, about 55 per cent security A and 45 per cent security B

528

968

dσ^2 P

dαA