Question

There are two bonds in the market: Bond A is a coupon bond with a nominal value of $100, maturing in one year, with coupon of $5 paid every six months. Bond B is a six-month pure-discount bond which pays $100. Suppose that the annual interest rate is 5% compounded monthly.

(a)What is the non-arbitrage price of the bonds?

(b)Explain how to replicate a pure-discount bond maturing in one year, by using a combination of the bonds in the market.

Answer 1

(a) Non-arbitrage price of bonds

Pa: Price of bond A

Face value = 100

Coupon = 5 (semi-annual)

interest rate = 5% compounded monthly = 5%/12 per month

time = 1 year = 12-months

Pa = 5/(1+5%/12)^6 + (100+5)/(1+5%/12)^12 = $104.766

Pb: Price of bond B

zero-coupon

time = 6-months

Pb = 100/(1+5%/12)^6 = $97.536

(b)

Da: Duration of bond A = (0.5*5/(1+5%/12)^6 + 1*(100+5)/(1+5%/12)^12)/104.766 = 0.977 years

Db: Duration of bond B = 0.5 years (zero-coupon bond)

Pc: price of 1 year pure discount bond

Pc = 100/(1+5%/12)^12 = $95.133

Dc: duration of bond C = 1 year

Replicating portfolio

N1: no. of bonds A

N2: no. of bonds B

95.133 = N1*104.766 + N2*97.536.............equ 1 (matching price of the portfolio)

1*95.133 = (0.977*104.766*N1 + 0.5*97.536*N2).............equ 2 (matching duration of the portfolio)

Solving these two equations we get,

N1 = 3.098 & N2 = -2.352

Therefore the replicating portfolio should consist of 3.098 units of bond A & -2.352 units of bond B (short sell)