Question

A) Assume that stock returns follow a 2-factor structure. The risk-free return is3%. Portfolio A has average return 8% and factor-betas 0.7 and 0.9 (for factor 1 and 2, respectively). Portfolio B has average return 10% and factor-betas 1.2 and 1.1 (for factor 1 and 2, respectively). What is the average return for portfolio C that has factor-betas 1 and 1 (for factor 1 and 2,respectively)?

B) A 5-year bond with face value $1,000 (paid at maturity) and coupon rate 5%(coupon paid in arrears annually) has yield-to-maturity 4.5%. What is the convexity of the bond?

Answer 1

1.

Let x be the factor premium for factor 1 and y for factor 2 respectively

From Portfolio A:
8%=3%+x*0.7+y*0.9

From Portfolio B:
10%=3%+x*1.2+y*1.1

Solving the equation simultaneously we get
x=2.58065%
y=3.54839%

Return on Portfolio C=3%+x*1+y*1=3%+2.58065%*1+3.54839%*1=9.12904%

2.

Price=5%*1000/4.5%*(1-1/1.045^5)+1000/1.045^5=1021.95

Convexity=1/(1021.95*1.045^2)*(5%*1000/1.045*(1^2+1)+5%*1000/1.045^2*(2^2+2)+5%*1000/1.045^3*(3^2+3)+5%*1000/1.045^4*(4^2+4)+5%*1000/1.045^5*(5^2+5)+1000/1.045^5*(5^2+5))=24.20