Question 9.3: Let us assume that standard deviations of assets L and H, co......

If the coefficient of correlation between their returns is 0.6 and the two assets are combined in the ratio of 3:1, the expected return of the portfolio is determined as follows:
E(r_{portfolio}) = w_LE(r_L) + w_VE(r_H)
E(r_{portfolio}) = (0.75 × 12%) + (0.25  × 16%) = 9.0% + 4.0% = 13 per cent
The variance of the portfolio is given by:
\sigma^ 2_p = (w_1\, \sigma_1)^2 + (w_2 \,\sigma_2)62 + 2 \,w_1\, w_2\, (\rho_{12} \, \sigma_1\, \sigma_2)
\sigma^ 2_p = (0.75 × 16)2 + (0.25 × 20)2 + 2 (0.75) (0.25) [(0.6) (16 × 20)]
= 144 + 25 + (0.375)(192) = 144 + 25 + 72 = 241

Thus,   \sigma_p = 15.52 per cent
The above discussion shows that the portfolio risk depends on three factors: (a) Variance (or standard deviation) of each asset in the portfolio; (b) Relative importance or weight of each asset in the portfolio; (c) Interplay between returns on two assets or interactive risk of an asset relative to other, measured by the covariance of returns. Among these only weights can be controlled by the investor/ portfolio manager. Thus, the primary task of a portfolio manager is to decide the proportion of each security in the portfolio.
\,  The portfolio’s expected rate of return and standard deviation (risk), for various combinations of assets L and H, with different degrees of correlation between their returns, are summarised in Table 9.1.
A perusal of the Table 9.1 leads to the following notable inferences:
(i) Two assets/ securities can be combined in such a way that the portfolio risk is less than the risk of individual assets comprising the portfolio. For example, portfolio standard deviation is 15.20 per cent when correlation coefficient (ρ) is 0.5 and L and H are combined in the ratio of 80:20. This is lower than the standard deviation of L (16 per cent) and H (20 per cent).
(ii) For given weights, portfolio standard deviation declines as correlation coefficient moves from + 1.0 to – 1.0. For example, when L and H are combined in the ratio of 80:20, the range of portfolio standard deviation is 16.80 per cent for perfect positive correlation (ρ = + 1.0) to 8.80 per cent for perfect negative correlation (ρ = –1.0).
(iii) When returns have less than perfect positive correlation, some combinations are more efficient than others; they do not involve risk-return trade-off. For correlation coefficient 0.5, increase in the weight of H from 0 per cent to 30 per cent raises the expected return from 12 per cent to 13.2 per cent, but standard deviation (risk) declines from 16 per cent to 15.12 per cent.
(iv) For given correlation coefficient, there is a minimum \>variance or minimum \>risk\> portfolio. The minimum variance portfolio has a standard deviation smaller than that of either of the individual component assets (securities). The optimal weights (w^*) that produce the minimum variance may be obtained from Equation (9.5) and Equation (9.6):
\qquad\,w^*_1 = [\sigma_ 2 ^2 – (\rho_{12}\, \sigma_ 1 \,\sigma_ 2) ]/[\sigma^2_ 1+ \sigma_ 1^2 – 2(\rho_{12}\, \rho_{1}\,\sigma_2)]        (9.5)
\qquad\,w^*_2= 1 – w^*_1     (9.6)
where  w^*_1  = Optimal weight of asset 1
\qquad\,w^*_2  = Optimal weight of asset 2
\qquad\sigma_ 1 ^2  = Variance of asset 1
\qquad\sigma_ 2 ^2  = Variance of asset 2
\qquad\rho_{12}\, \sigma_ 1 \,\sigma_ 2  = Covariance of returns
\qquad\rho_{12} = Coefficient of correlation between the returns of two assets