Question

Consider three traded bonds A, B, and C with the following data: Bond A has face value $1,000, zero coupon, maturity in 1 year and YTM of 5%; bond B has face value of $1,000, 5% coupon rate, maturity in 2 years and YTM of 5.85%; bond C has face value of $1,000, 10% coupon rate, maturity in 2 years and YTM of 6%. Coupon payments are made at the end of years 1 and 2.

(i) What are the current prices of these bonds?

(ii) There is an arbitrage opportunity at the prices you found above; explain why and construct an arbitrage portfolio of bonds (assuming that bonds can be sold short at prices you found in (i)).

Answer 1

I) a) Bond A price = Par value * (1 / (1+i)^n)

Here,

Bond A is zero coupon bond & hence coupon interest is nil.

i = 5% or 0.05, n = 1 year

Bond A price = $1000 * (1 / (1+0.05)^1)

Bond A price = $1000 * 0.9524

Bond A price = $952.40

b) Bond B price = Coupon * ((1 - (1/(1+i)^n)) / i) + Par value * (1 / (1+i)^n)

Here,

Coupon = Par value * Coupon rate = $1000 * 5%

Coupon = $50

i (yield) = 5.85% or 0.0585

n (years) = 2

Now,

Bond B price =$50*((1 - (1/(1+0.0585)^2))/0.0585) + $1000 * (1 / (1+0.0585)^2)

Bond B Price = $50 *( (1 - 0.8925) / 0.0585) + $1000 * 0.8925

Bond B price = ($50 * 1.8376) + $892.50

Bond B price = $984.38

c) Bond C price = Coupon * ((1 - (1/(1+i)^n)) / i) + Par value * (1 / (1+i)^n)

Here,

Coupon = Par value * Coupon rate = $1000 * 10%

Coupon = $100

i (yield) = 6% or 0.06

n (years) = 2

Now,

Bond C price = $100 * ((1 - (1/(1+0.06)^2)) / 0.06) + $1000 * (1 / (1+0.06)^2)

Bond C price = $100 * ((1 - 0.8900) / 0.06) + $1000 * 0.8900

Bond C price = ($100 * 1.8333) + $890

Bond C price = $1073.33

II) Arbitrage opportunity : If bonds can be sold short at price calculated above then this is a indication of arbitrage opportunity. As bonds can be purchased at lower price in one market & sold at higher price in another market which leads to arbitrage opportunity.