Question

Consider a portfolio made up of two stocks, S₁ an S₂. The variance of the 1-year returns on the two stocks are denoted s₁² and s₂². 

 

[a] Assume we construct a portfolio consisting of long positions in the two stocks. Explain why the variance of the 1-year return on the portfolio can’t exceed the larger of s₁² and s₂²_.

 

Suppose we invest $150 in a stock S₁:

$100 of our own capital

$50 of borrowed 1-year money (at an interest rate of 3%)

 

Let R₁ denote the 1-year return on S₁. Let E[R₁] = m. 

 

[b] In terms of s₁² what’s the expected value and variance of the 1-year return on our $100 investment?

Answer 1

 The variance of the 1-year returns on the two stocks are denoted s₁² and s₂². 

 

(a) When we construct a portfolio consisting of long positions in the two stocks. variance of the 1-year return on the portfolio can't exceed because the portfolio variance is the maximum variance occur on a investment.

Investment I=$150

$R_1$ return after 1 year on $s_1$

(b)  expected value $= E[R_1]-\sqrt{s_1^2}$

and variance of the 1-year return $= s_1^2-\dfrac{3\times 50}{100}=s_1^2-1.5$