Question

You owe a mobster in Chicago money in 5 years. You want to earn a little bit on money you set aside to pay this liability. However, you do not know if interest rates will rise or fall. You have two bonds to choose from with semi-annual coupon payments:

	Bond A	Bond B
Time to maturity (years)	5	6
Annual yield to maturity	4.00%	4.00%
Annual coupon payment	40.00	65.94
Current price	-1000	-1000
Face value	1000	826.02

So we start with two bonds of equal price ($1000) and annual yield to maturity (4.00%).

1. What is the Macaulay duration (in years) for Bond A?

2. What is the Macaulay duration (in years) for Bond B?

3. Which bond should you choose to immunize your bond holdings from interest rate fluctuations?

4. Assume annual yield to maturity drops from 4.00% to 3.00%. What is the total ending wealth of Bond B?

5. Assume annual yield to maturity drops from 4.00% to 3.00%. What is the total ending wealth of Bond A?

Answer 1

1. For Bond A

Year	Cash flow	PV @ 4%	Present Value of cash flows	Proportion	Proportion of bond value * time
1	40	0.9615	38.46	0.038	0.038
2	40	0.9246	36.984	0.037	0.074
3	40	0.889	35.57	0.036	0.107
4	40	0.8548	34.202	0.034	0.137
5	1040	0.8219	854.79	0.855	4.274
			1000.00	1.000	4.630

(In year 5, redemption at face value of 1000 has been added)

Therefore, Macaulay's duration of Bond A is 4.630 years

2. For Bond B

Year	Cash flow	PV @ 4%	Present Value of cash flows	Proportion	Proportion of bond value * time
1	65.94	0.9615	63.401	0.0634	0.0634
2	65.94	0.9246	60.968	0.0610	0.1219
3	65.94	0.889	58.621	0.0586	0.1759
4	65.94	0.8548	56.366	0.0564	0.2255
5	65.94	0.8219	54.196	0.0542	0.2710
6	891.96	0.7903	706.448	0.7064	4.2387
			1000.000	1.0000	5.0963

(In year 5, redemption at face value of 826.02 has been added)

Therefore, Macaulay's duration is 5.0963 years
[The sum of present value of cash flows has been rounded off to 1000]

3. Higher the duration, the more the bond's price will drop as interest rates rise. Therefore, between bond A and bond B, one should bond A which has a lower duration as compared to the bond B.

4. YTM reduces to 3% for Bond B

Year	Cash flow	PV @ 3%	Present Value of cash flows	Proportion	Proportion of bond value * time
1	65.94	0.9709	64.021	0.0640	0.0640
2	65.94	0.9426	62.155	0.0622	0.1243
3	65.94	0.9151	60.342	0.0603	0.1810
4	65.94	0.8885	58.588	0.0586	0.2344
5	65.94	0.8626	56.880	0.0569	0.2844
6	891.96	0.8375	748.549	0.7485	4.4913
			1050.534	1.0505	5.3794

5. YTM reduces to 3% for Bond A

Year	Cash flow	PV @ 3%	Present Value of cash flows	Proportion	Proportion of bond value * time
1	40	0.9709	38.836	0.039	0.039
2	40	0.9426	37.704	0.038	0.075
3	40	0.9151	36.614	0.037	0.110
4	40	0.8885	35.55	0.036	0.142
5	1040	0.8626	897.11	0.897	4.486
			1045.82	1.046	4.852