Question

stock boom and stock bust are perfectly negatively correlated and their standard deviation are 24.495% and 9.798% respectively. what is the approximate weight of stock boom to form a portfolio that has no volatility?

a. 15%

b. 20%

c. 28%

d. 49%

e. it is not possible to form a risk-less portfolio with stocks boom and bust

Answer 1

correlation between stocks , c = -1

Standard deviation of stock boom , S1 = 24.495% = 0.24495

Standard deviation of stock bust , S2 = 9.798% = 0.09798

let weight of stock boom in portfolio = w

weight of stock boom in portfolio = 1- w

volatility of portfolio = variance of portfolio, V = 0

V = (w*S1)² + ((1-w)*S2)² + 2*w*(1-w)*S1*S2*c

V = (w*24.495)² + ((1-w)*9.798)² + 2*w*(1-w)*24.495*9.798*-1

V =( 600.005025*w²) + ((1-w)²*96.000804) - (480.00402w*(1-w))

V =  600.005025*w² + ((1+w²-2w)*96.000804) - 480.00402w + 480.00402w²

V = 600.005025*w² + 96.000804 + 96.000804w² - 192.001608w - 480.00402w + 480.00402w²

V = 1176.009849w² - 672.005628w + 96.000804

V = 0

1176.009849w² - 672.005628w + 96.000804 = 0

we will substitute the given options in the question for w in the above equation and confirm the answer

when we substitue, w = 28% = 0.28 in the above equation

1176.009849(0.28)² - 672.005628*0.28 + 96.000804 = 0.038 or 0 ( after rounding off)

hence correct option is c) 28%