Portfolio theoryTable 7.1Risk and expected returns for various portfolios of securities A and B with
differing assumptions as to the correlation coefficientStandard deviation for
Proportion Expected various correlation coefficients (R)
of security A returns of
in portfolio portfolio (rP) −1.0 −0.5 0.0 +0.5 +1.0
% % %%%%%0 14 12.0 12.0 12.0 12.0 12.0
25 13 6.5 8.0 9.3 10.5 11.5
50 12 1.0 5.6 7.8 9.5 11.0
75 11 4.5 6.5 8.1 9.4 10.5
100 10 10.0 10.0 10.0 10.0 10.0To illustrate where the figures in Table 7.1 come from, let us take the case where the
proportions of the securities in the portfolio are 25 per cent security A and 75 per cent
security B, and the correlation coefficient between the returns of the two securities is
+0.5 (i.e. the figures in row 2, columns 2 and 6 of the table).
Remember that security A has an expected return of 10 per cent with a standard
deviation of 10 per cent, and that security B has an expected return of 14 per cent with
a standard deviation of 12 per cent.
The expected return on the portfolio is given by:rP=αAαÂ+αBαBˆ
=(0.25 ×10%) +(0.75 ×14%)
=13%The standard deviation of the portfolio is given by:σP==
=10.476%, say 10.5%Other portfolios of A and B
The expected return/risk profile of portfolios of various proportions of A and B,
assuming a correlation coefficient of +0.5, is shown in Figure 7.3. This is a plot of col-
umn 2 values against those from column 6 (Table 7.1). Each of the points along this line
represents the combination of return and risk for a portfolio containing particular pro-
portions of A and B. We can see that the risk is at a minimum where the portfolio con-
tains between 50 and 75 per cent of A.
Figure 7.4 shows the profiles of expected return and risk following the five assumed
correlation coefficients (−1.0, −0.5, 0.0, +0.5 and +1.0), rather than just the one (+0.5)
shown in Figure 7.3. Here we can see that the greatest risk reduction possibilities occur(. 22 22 %) ( .×+×+×××+×× %) (. %.. %)0 25 10 0 75 12 2 0 25 10 0 5 0 75 12
ασAA^22 ++ασBB^22 2 ασ ασAAR BB