Question

The following securities are available in the market:

Maturity	Coupon	Price
1	0	97.56
2	0	94.26
2	4%	102.5
3	0	80.67
3	3.5%	99.52
4	0	82.90
4	5.2%	101.85

a)Is this market arbitrage-free? If not, how many arbitrage opportunities are there? b) How can you exploit the arbitrage opportunities? c) What are the profits that you could realize?

annual discount rate

The exercise does not give discount rate.

Answer 1

Remember that a coupon paying bond can be expressed as a series of zero coupon bonds of varying maturities.

Let's assume the par value of the bonds to be $ 100

Let's take the example of a two year maturity 4% coupon bond having a price of 102.5

Cash flows from this bond: 4% x 100 = $ 4 at the end of year 1 and coupon + par value = 4 + 100 = 104 at the end of year 2.

This can be achieved if I have 0.04 no. of a zero coupon bond (ZCB) with maturity of 1 year and 1.04 no. of ZCB with maturity of 2 years.

Hence, the price of this coupon bond should be = 0.04 x Price of ZCB with maturity of 1 year + 1.04 x Price of ZCB with maturity of 2 year = 0.04 x 97.56 + 1.04 x 94.26 = $  101.93

However the quoted price is $ 102.5

Hence, there is an arbitrage opportunity. And the arbitrage can be exploited by

1. Shorting 1 no. of 2 year maturity coupon bond
2. Buying 0.04 nos. of 1 year ZCB
3. Buying 1.04 nos. of 2 year ZCB

And the arbitrage profit = 102.5 - 101.93 = $  0.57

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The above was used as an example to help you understand. I will do the same calculation for all the coupon bonds and spot arbitrage. I will follow the same procedure as shown above in each case, but i do it in short as shown in the table below.

a)Is this market arbitrage-free? If not, how many arbitrage opportunities are there?

The market is not arbitrage free. There are three arbitrage opportunities there.

b) How can you exploit the arbitrage opportunities? c) What are the profits that you could realize?

Both the questions have been answered in each of the three tables below:

Arbitrage opportunity 1: Case of 4% coupon bond maturing in 2 years:

ZCB	Maturity	Price	Nos.	Component Price
ZCB1	1	97.56	0.040	3.90
ZCB2	2	94.26	1.040	98.03
ZCB3	3	80.67
ZCB4	4	82.9
Total Price				101.93
Quoted price				102.5
Arbitrage opportunity	Short the coupon bond; Buy 0.04 nos. of ZCB1 and 1.04 nos. of ZCB 2
Arbitrage profit				0.57

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Arbitrage opportunity 2: Case of 3.5% coupon bond maturing in 3 years:

ZCB	Maturity	Price	Nos.	Component Price
ZCB1	1	97.56	0.035	3.41
ZCB2	2	94.26	0.035	3.30
ZCB3	3	80.67	1.035	83.49
ZCB4	4	82.9
Total Price				90.21
Quoted price				99.52
Arbitrage opportunity	Short the coupon bond; Buy 0.035 nos. of ZCB1 & ZCB2 each and 1.035 nos. of ZCB 3
Arbitrage profit				9.31

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Arbitrage opportunity 3: Case of 5.2% coupon bond maturing in 4 years:

ZCB	Maturity	Price	Nos.	Component Price
ZCB1	1	97.56	0.052	5.07
ZCB2	2	94.26	0.052	4.90
ZCB3	3	80.67	0.052	4.19
ZCB4	4	82.9	1.052	87.21
Total Price				101.38
Quoted price				101.85
Arbitrage opportunity	Short the coupon bond; Buy 0.052 nos. of ZCB1, ZCB2 & ZCB3 each and 1.052 nos. of ZCB 4
Arbitrage profit				0.47