Question

Problem A. Construct a Replicating Portfolio (RP) to replicate a 1.5-year Bond-0 that pays A1 percent of coupon per year. The available bonds for replication are: a one year zero coupon Bond-1, a 1.5-year Bond-2 that pays A2 percent coupon per year, and a 1-year Bond-3 which pays A3 percent coupon per year. All the bonds (Bond-0, Bond-1, Bond-2, and Bond-3) have the same face value of $100 and pay their annual coupons two times a year. Compute an arbitrage trading strategy to generate profits, if any, when the current market prices of the four bonds, respectively, are A4, A5, A6 and A7.
1. What is the dollar face value of Bond-1 in the RP?
2. What is the dollar face value of Bond-2 in the RP?
3. What is the dollar face value of Bond-3 in the RP?
4. What is the cost of the RP?
5. What is the arbitrage fair price of Bond-0?
6. Write 1 if Bond-0 is to be held long and 0 if Bond-0 is to be held short in arbitrage trading strategy.
7. Write 1 if Bond-1 is to be held long and 0 if Bond-1 is to be held short in an arbitrage trading strategy.
8. Write 1 if Bond-2 is to be held long and 0 if Bond-2 is to be held short in an arbitrage trading strategy.
9. Write 1 if Bond-3 is to be held long and 0 if Bond-3 is to be held short in an arbitrage trading strategy.

A1 = 9
A2 = 7
A3 =11
A4 = 103
A5 = 96.5
A6 = 100.75
A7 = 105.6

Please show work.

Answer 1

Let's summarize the payoffs from the given bonds in a table with respect to time. ALso let's say we need P, Q & R numbers of Bond 1, 2 & 3 to replicate Bond 0.

Please see the table below:

Payoff at time	0.5 year	1 year	1.5 Year	Price	Numbers in replicating portfolio
Bond 0	A1/2	A1/2	A1/2+FV	A4
Bond 1		FV		A5	P
Bond 2	A2/2	A2/2	A2/2+FV	A6	Q
Bond 3	A3/2	A3/2+FV		A7	R

Let's put figures to these tables now:

A1 / 2 = 9 / 2 = 4.5; A2 / 2 = 7 / 2 = 3.5; A3 / 2 =11 / 2 = 5.5; A4 = 103; A5 = 96.5; A6 = 100.75; A7 = 105.6

Please note that all the coupon rates are in %age and need to be applied on FV of $ 100. So the resultant amount will be value of coupon without %age. Say for example, A2 = 7% = 7% x 100 = 7

So, the table with figures are as shown below:

Payoff at time	0.5 year	1 year	1.5 Year	Price	Numbers in replicating portfolio
Bond 0	4.5	4.5	104.5	103
Bond 1		100		96.5	P
Bond 2	3.5	3.5	103.5	100.75	Q
Bond 3	5.5	105.5		105.6	R

By the concept of replicating portfolio, payoff of bond is 0 will be replicated by a combination of P no. of Bond 1, Q no. of Bond 2 and R no. of Bond 3.

Hence, equating the payoff at t= 0.5 years; we get equation (1) : 3.5Q + 5.5R = 4.5

Hence, equating the payoff at t= 1 years; we get equation (2) : 100P + 3.5Q + 105.5R = 4.5

Hence, equating the payoff at t= 1.5 years; we get equation (3) : 103.5Q = 104.5

From equation (3): Q = 104.5 / 103.5 = 1.009661836

Putting this in equation (1), we get R = (4.5 - 3.5Q) / 5.5 = 0.175669741

From equation (2); we get P = (4.5 - 3.5Q - 105.5R) / 100 = -0.17566974

We are now ready to answer the questions:

Please round off the figures as per your requirement.

1. What is the dollar face value of Bond-1 in the RP?

P x 100 = 0.17566974 x 100 = $ 17.566974

2. What is the dollar face value of Bond-2 in the RP?

Q x 100 = 1.009661836 x 100 = $ 100.9661836

3. What is the dollar face value of Bond-3 in the RP?

R x 100 =  0.175669741 x 100 = $ 17.5669741

4. What is the cost of the RP?

= Cost of P numbers of Bond 1 + Cost of Q numbers of Bond 2 + Cost of R number of Bond 3 = -0.17566974 x 96.5 + 1.009661836 x 100.75 + 0.17566974 x 105.6 = $ 103.3220246

5. What is the arbitrage fair price of Bond-0?

Arbitrage fair price of bond should be cost of the replicating portfolio = $ 103.3220246(as calculated above)

But since the actual cost of bond 0 = 103 < 103.0147429; the arbitrage strategy should be buy the bond and short the replicating portfolio. Since replicating portfolio is:

• Short 0.17566974 no. of bond 1
• Buy 1.009661836 no. of bond 2
• Buy 0.17566974 no. of Bond 3

My arbitrage portfolio will be:

• Buy 1 no. of Bond 0
• Short the replicating portfolio i.e
  • Buy 0.17566974 no. of bond 1
  • Short 1.009661836 no. of bond 2
  • Short 0.17566974 no. of Bond 3

And the arbitrage profit will be proceeds from sell of replicating portfolio - cost of bond 0 = $ 103.3220246
- 103 = $ 0.322024594 without any future liabilities because the cash flows from bond 0 will be matched by the cash flows from the short position in the replicating portfolio.

6. Write 1 if Bond-0 is to be held long and 0 if Bond-0 is to be held short in arbitrage trading strategy.

We have buy or long position in Bond 0. Hence write 1.

7. Write 1 if Bond-1 is to be held long and 0 if Bond-1 is to be held short in an arbitrage trading strategy.

We have buy or long position in Bond 1. Hence write 1.

8. Write 1 if Bond-2 is to be held long and 0 if Bond-2 is to be held short in an arbitrage trading strategy.

We have short position in Bond 2. Hence write 0.

9. Write 1 if Bond-3 is to be held long and 0 if Bond-3 is to be held short in an arbitrage trading strategy.

We have short position in Bond 3. Hence write 0.