Question

Consider the following bonds currently traded in the market. Using this information find the no-arbitrage price of a 5-Year bond with a coupon of 5%. Suppose this bond is currently selling for $102 in the market. Is there an arbitrage opportunity? Explain how you would execute this arbitrage (All coupons are annual payment, including the bond you are asked to price)

	Annual Coupon	Maturity in Years	Price
Bond 1	8%	1	102.800
Bond 2	9%	2	107.250
Bond 3	11%	3	116.400
Bond 4	6%	4	104.410
Bond 5	7%	5	108.030
Bond 6	8%	6	113.950
Bond 7	10%	7	127.020

Answer 1

First we need to calculate the arbitrage free value of the 5 year, 5% coupon bond calculated as:

Step 1: Calculate spot rates of year 1, 2, 3, 4, 5 using a financial calculator.

Spot Rate of Year 1: PV= -102.8, PMT= 100*8%= 8, N= 1, FV= 100, Compute I/Y i.e. 5.058366

Spot Rate of Year 2: PV= -107.25, PMT= 100*9%= 9, N= 2, FV= 100, Compute I/Y i.e. 5.095630

Spot Rate of Year 3: PV= -116.4, PMT= 100*11%= 11, N= 3, FV= 100, Compute I/Y i.e. 4.980033

Spot Rate of Year 4: PV= -104.41, PMT= 100*6%= 6, N= 4, FV= 100, Compute I/Y i.e. 4.763163

Spot Rate of Year 5: PV= -108.03, 100*7%= 7, N= 5, FV= 100, Compute I/Y i.e. 5.138182

Therefore,

Arbitrage free value of the bond=

5/1.05058366 + 5/(1.05095630)² + 5/(1.04980033)³ + 5/(1.04763163)⁴ +105/(1.05138182)⁵ = 99.489678

Since the bond is overvalued i.e. selling at a higher price, we can arbitrage by short selling the bond.

The bond will be borrowed and sold for 102. Later when the bond is correctly priced, it will be purchased for 99.489678 and returned to the lender.