Question

 

IMMUNIZATION USING INDIVIDUAL BONDS
Yield to maturity (Expected/Current)	9%
Number of Years to Future Liability	11
Future Liability	$       3,600.00
Amount Invested to Cover Future Liability
	Bond 1	Bond 2	Bond 3
Coupon rate	8.00%	12.000%	6.00%
Maturity	12	18	30
Face value	1,000	1,000	1,000

a. Compute the amount to be invested to meet the future liability noted in the data. This future liability is due in 7 years.

b. Find a combination of Bond 1 and Bond 2 having a target duration of 7.

c.Find a combination of Bond 1 and Bond 3 having a target duration of 7.

d. Perform an analysis using a data table and an accompanying graph to determine which

portfolio would be preferred to attempt to immunize this obligation.

i. Construct a data table by varying the yield to maturity that shows the value of

each portfolio at the end of 7 years.

ii. Based on your data table, construct a graph that demonstrates the performance

of these two portfolios.

Answer 1

Soln : a) Amount to be invested today to meet future iability of $3600 after 7 years, let be A

A = 3600/1.09⁷ = $1969.32

Duration of the futire liabiliy = 7 years

b) Need to calculate the duration of each of the bonds, by using the formula

Duration =

Period	1	2	3	4	5	6	7
Bond 1	80	80	80	80	80	80	80
PV	73.39	67.33	61.77	56.67	51.99	47.70	43.76
NPV	928.39
period*PV (PPV)	73.39	134.67	185.32	226.70	259.97	286.21	306.34
	7411.99
Duration of bond1 =sum of PPV/NPV	7.98

8	9	10	11	12
80	80	80	80	1080
40.15	36.83	33.79	31.00	383.98
321.19	331.51	337.93	341.03	4607.73

We will get the duration of bond 1 = 7.98

Similarly for bond 2 , duration = 9.07 years

And for bond 3, duration = 11.88 years

b) Let W be the weight when combination of bond 1 & 2 is used

W*7.98 +(1-W)*9.07 = 7 on solving , we get W = 1.9 of bond 1 to be bought and 0.9 of the bond 2 is to be sold for immunization.

c) Let w be the weight of bond 1 and 1-w be the wt. of bond 3

We can say that for immunization, w*7.98 +(1-w)*11.88 = 7

on solving w = 1.25 and 1-w = -0.25

i.e. need to buy bond 1 of value = 1.25*1969.32 and sell bond 3 of value = 0.25*1969.32