Question

The portfolio of an investment firm comprises of two risky assets, 𝑆 and 𝑇, whose returns are denoted by random variables 𝑅_𝑠 and 𝑅_𝑇 respectively. The mean, the variance and the covariance of the returns are
𝐸(𝑅_𝑠 ) = 0.08, 𝑉𝑎𝑟(𝑅_𝑠 ) = 0.07, 
𝐸(𝑅_𝑇 ) = 0.05, 𝑉𝑎𝑟(𝑅_𝑇 ) = 0.05, 𝐶𝑜𝑣(𝑅𝑠 , 𝑅_𝑇 ) = 0.04. 
Let 𝑤 be the proportion of assets allotted to 𝑆 so that the return from the portfolio is 𝑅 = 𝑤𝑅_𝑠 + (1 − 𝑤)𝑅_𝑇 . The value of 𝑤 which minimizes 𝑉𝑎𝑟(𝑅) is _____(round off to 2 decimal places)

Answer 1

Correct Answer: 0.24

Given:
\(E(R_S)=0.08,\; \mathrm{Var}(R_S)=0.07,\; E(R_T)=0.05,\; \mathrm{Var}(R_T)=0.05,\; \mathrm{Cov}(R_S,R_T)=0.04.\)
Portfolio return: \(R = wR_S+(1-w)R_T\). We want the value of \(w\) that minimizes \(\mathrm{Var}(R)\). 
Step 1 — Write \(\mathrm{Var}(R)\) 
\[ \mathrm{Var}(R)=w^2\mathrm{Var}(R_S)+(1-w)^2\mathrm{Var}(R_T)+2w(1-w)\,\mathrm{Cov}(R_S,R_T). \] Step 2 — Differentiate w.r.t. \(w\) and set to zero
Differentiate: \[ \frac{d}{dw}\mathrm{Var}(R) =2w\mathrm{Var}(R_S)-2(1-w)\mathrm{Var}(R_T)+2(1-2w)\mathrm{Cov}(R_S,R_T). \] Set equal to zero and simplify (divide by 2): \[ w\mathrm{Var}(R_S)-(1-w)\mathrm{Var}(R_T)+(1-2w)\mathrm{Cov}(R_S,R_T)=0. \] Collect terms in \(w\): \[ w\big(\mathrm{Var}(R_S)+\mathrm{Var}(R_T)-2\mathrm{Cov}(R_S,R_T)\big) = \mathrm{Var}(R_T)-\mathrm{Cov}(R_S,R_T). \] So \[ \boxed{\,w^\ast=\dfrac{\mathrm{Var}(R_T)-\mathrm{Cov}(R_S,R_T)}{\mathrm{Var}(R_S)+\mathrm{Var}(R_T)-2\mathrm{Cov}(R_S,R_T)}\,} \] Step 3 — Substitute numbers
\[ \text{Numerator}=0.05-0.04=0.01, \qquad \text{Denominator}=0.07+0.05-2(0.04)=0.12-0.08=0.04. \] Therefore \[ w^\ast=\frac{0.01}{0.04}=0.25. \] Final answer (rounded to 2 d.p.):
\[ \boxed{w=0.25} \]