Question

Assume all the following bonds are risk-free and with a par value $100:

Bond A: one-year zero with 2% YTM

Bond B: two-year zero with 3% YTM

Bond C: Three-year zero with 4% YTM

Bond D: three-year 5% coupon-paying bond with YTM 4%.

Question: Is there any violation of the No-Arbitrage Principle? If there is, can you develop a strategy to exploit this opportunity?

Answer 1

Bond A: one year zero with 2% YTM

Bond B: two year zero with 3% YTM

Bond C: three year zero with 4% YTM

Bond D: three year 5% coupon paying bond with 4% YTM

Price of Bond = C/(1+YTM) + C/(1+YTM)² + C/(1+YTM)³ + ….. C/(1+YTM)^t + F/(1+YTM)^t

Where, C is Coupon payment

YTM is yield of bond

F is par value

t is time to maturity

Price of Bond A = 100/(1+2%) =$98.04

Price of Bond B = 100/(1+3%)² =$94.26

Price of Bond C = 100/(1+4%)³ =$88.90

Price of Bond D = 5/(1+4%) + 5/(1+4%)² + 5/(1+4%)³ + 100/(1+4%)³ = $102.78

Price of Bond D as per zero bond yields:

Price of Bond = C/(1+r1) + C/(1+r2)² + C/(1+r3)³ + ….. C/(1+rt)^t + F/(1+r)^t

Where, C is Coupon payment

rt is zero coupon yield

F is par value

t is time to maturity

C = 5%

t= 3

Price of Bond D = 5/(1+2%) + 5/(1+3%)² + 5/(1+4%)³ + 100/(1+4%)³ = $102.96

Price of Bond D (as calculated based on zero coupon yield)= $102.96

So, Price of Bond D = $102.78 hence Bond is under-priced.

Arbitrage position:

The bond is underpriced relative to the replicating portfolio. To take advantage of the arbitrage opportunity Buy the coupon bond and Sell the replicating portfolio.

	Cash flow at t =0	Cash flow at t =0.5	Cash flow at t =1	Cash flow at t =1.5
Buy 100 unit of Bond D	-10278.00	5	5	105
Sell 5 unit of Bond A	5*98.04= 490.20	-5	0	0
Sell 5 unit of Bond B	5*94.26= 471.30	0	-5	0
Sell 105 unit of Bond C	105*88.90= 9334.50	0	0	-105
Total Cash flow	18.00	0	0	0