Question

4.[10 points] There are four bonds, A, B, C, D, with the following cash flows (dates in years, entries in $): Date 0 1 2 3 -134 150 10 30 -0.8 1 0 0 C -0.7 0 1 0 D 0 0 -0.5 1 (a) (4pts) Construct an arbitrage in this market. (b) (1pt) To eliminate the arbitrage opportunity, what should be the price for A? (c) (5pts) To eliminate the arbitrage opportunity, what should be the price for D?2

Answer 1

Assumptions - At t=0, the negative flows mean that they are currently trading prices of the bond.

A.

Based on the given cash flows of the bond we can see that for Bond A, we can calculate the yield to maturity.

134 = CF₁ / (1+YTM) + CF2 / (1+YTM)² + CF3 / (1+YTM)^₃

134 = 150 / (1+YTM) + 10/ (1+YTM)² + 30 / (1+YTM)_³

YTM = 30.74%

For the Bond B,

0.7 = 1 / (1+YTM)

YTM = 19.52%

For the Bond C,

0.8 = 1 / (1+YTM)^2

YTM = 25%

Hence, the different instruments are yielding different yields,we have arbitrage opportunioties in the market. We can sell the bonds with the lower YTM and buy the bonds with the higher YTM. Buy bond A and sell bond B and C.

B. Let's assume that bond B is priced correctly and the bond should be yielding 19.52%.

Hence,

price = 150 / (1+19.52%) + 10/ (1+19.52%)² + 30 / (1+19.52%)_³

Bond A price = 150.07

C.

For bond A three year spot rate is 30.74%, for bond B two year spot rate is 19.52% and for bodn C 2 year spot rate is 25%.

Hence, (1+ S3)^3 = (1+S2)^2*( 1+f(2,1))

s3 = three year spot rate

s2= 2 year spot rate

f(2,1) = one year forward rate after two years

Hence, the yield f(2,1) = 56.44%.

(1+ 30.74%)^3 = (1+19.52%)^2* (1+f(2,1))

F(2,1) = 56.44%

price bond D = 1/ (1+56.44%) = 0.64