The returns per annum on two securities are denoted by R and R2, and are modelled...

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The returns per annum on two securities are denoted by R₁ and R2, and are modelled as follows: R₁ ~N (0.1, 0.2²), R₂ ~ N (0.05, 0.04²), Corr(R₁, R₂): = 0.5 where Corr (R₁, R2) denotes the correlation between R₁ and R₂. (i) Suppose that $1000 are invested in the minimum variance portfolio formed from the two securities. Calculate the composition of the minimum variance portfolio and state how much is invested in each security. Next, suppose that, at the beginning of the year, the balance on an investment account is $1000. The account is invested in securities 1 and 2 only, and short-selling is permitted. Suppose also that, at the end of the year, a random sum of $1000 × (1+Q) is withdrawn from the investment account. The random variable is modelled as follows: Q~ N(0.08, 0.12), Corr(Q, R₁) = 0.8, Corr(Q, R₂) If x is the portfolio weight for security 1, then the end-of-year account balance is: W₁ = 1000(1+xR₁ + (1 − x)R₂) — 1000(1+Q) (ii) Calculate the composition of the portfolio such that the end-of-year balance on the investment account has the least variance. = 0.2 (iii) Let m(x) = E[W₁] and v(x) = Var[W₁]. (a) Derive expressions for m(x) and v(x). (b) Let x* be the weight of security 1 invested in the portfolio which maximises the probability of a positive end-of-year balance on the investment account. Show that: m(x*)v' (x*) = 2m'(x*)v(x*) The returns per annum on two securities are denoted by R₁ and R2, and are modelled as follows: R₁ ~N (0.1, 0.2²), R₂ ~ N (0.05, 0.04²), Corr(R₁, R₂): = 0.5 where Corr (R₁, R2) denotes the correlation between R₁ and R₂. (i) Suppose that $1000 are invested in the minimum variance portfolio formed from the two securities. Calculate the composition of the minimum variance portfolio and state how much is invested in each security. Next, suppose that, at the beginning of the year, the balance on an investment account is $1000. The account is invested in securities 1 and 2 only, and short-selling is permitted. Suppose also that, at the end of the year, a random sum of $1000 × (1+Q) is withdrawn from the investment account. The random variable is modelled as follows: Q~ N(0.08, 0.12), Corr(Q, R₁) = 0.8, Corr(Q, R₂) If x is the portfolio weight for security 1, then the end-of-year account balance is: W₁ = 1000(1+xR₁ + (1 − x)R₂) — 1000(1+Q) (ii) Calculate the composition of the portfolio such that the end-of-year balance on the investment account has the least variance. = 0.2 (iii) Let m(x) = E[W₁] and v(x) = Var[W₁]. (a) Derive expressions for m(x) and v(x). (b) Let x* be the weight of security 1 invested in the portfolio which maximises the probability of a positive end-of-year balance on the investment account. Show that: m(x*)v' (x*) = 2m'(x*)v(x*)

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i The minimum variance portfolio weights can be calculated using the formula w1 2R2 R1R2 2R1 2R2 2R1R2 w2 1 w1 where R1 and R2 are the standard deviations of R1 and R2 respectively and R1R2 is the cov... View the full answer