Question

Hi, I'm having some trouble with this question, especially part B.

Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%.

a) In light of the apparent inferiority of gold with respect to average return and volatility, would anyone hold gold in his portfolio? (4 points)

b) Assume that the correlation between Stocks and Gold is -0.5. Find the weights wS and wG of the efficient risky portfolio which is invested in Stocks and Gold and which has an expected return of 15%. (4 points)

Answer 1

a.)

In light of the apparent inferiority of gold with respect to average return and volatility, most investors may not like holding gold. Still, Investors often/can hold gold to have a diversified portfolio. Especially during the volatile stock market as nowadays due to Coronavirus / COVID-19, stocks are tumbling. So, investors are increasing their holding of gold. Note that given data is for the expected case, actual return and volatility might differ.

b.) rS = 18%, rG =10%, σs = 22%, σg = 30%, ρ = correlation = -0.5

ws=?

wg =?

Expected return = rp = 15%

rp = rs Ws + rg Wg =

=

σ2p = ws2σs2 + wg2σg2 + 2 wgwsσsσg ρ

or, σ2p = ws2 222+ wg2302 + 2 wgws 22*30* (-0.5)

= 484* ws2 + 900* wg2 - 660wgws

= 484* ws2 + 900* (1.5 - 1.8Ws)2 - 660 (1.5 - 1.8Ws)*ws

σ2p = 2025 -5860Ws + 4588Ws2

for efficient risky portfolio, σ2p should be minimum.

taking diffrentiation, d( σ2p ) = d(2025 -5860Ws + 4588Ws2) = -5860 + 2*4588*Ws

at minmuim σ2p ,   d( σ2p ) = 0

so, -5860 + 2*4588*Ws = 0,

so, ws = 5860/9176 = = 0.64

and Wg = 1.5 - 1.8*0.64 = 0.35