Question

1.Suppose that you are tasked with managing a liability of $5,000 worth of 6% 4-year, annual coupon bonds when the interest rate is 4.5%. You want to minimize the interest rate risk by immunizing this position through value and duration matching. If you have 2-year and 10-year zero-coupon bonds available to create the hedge, how many dollars should you invest in each bond? Explain at least three reasons this hedge will not be perfect one year after you set it up. 


Answer 1

Formula sheet

A	B	C	D	E	F	G	H	I	J	K	L	M
2
3		During Immunization of the Bonds:
4		1) Duration of the portfolio should math the duration of the obligation
5		2) Value of portfolio today should be equal to the present value of the obligation
6
7		Calculation of Duration of Obligation:
8
9		Macaulay Duration is the weightage average of the time to present value of cash flows.
10		Formula for Macaulay duration is as follows:
11
12
13
14
15
16
17		Where, Ct is cash flow at time t, PV(Ct) is the present value of cash flow at time t and T is the total time horizon.
18		Face value	5000
19		Coupon rate	0.06
20		Maturity	4	years
21		Annual Coupon	=D18*D19
22		YTM	0.045
23
24		Cash flow to investor will be as follows:
25		Year (t)	0	1	2	3	4
26		Payment (Ct)		=$D21	=$D21	=$D21	=$D21+D18
27		Yield to maturity (i)	=D22
28		Present value factor (P/F,i,n) for each year		=1/((1+$D27)^E25)	=1/((1+$D27)^F25)	=1/((1+$D27)^G25)	=1/((1+$D27)^H25)
29		PV (Ct) = (Ct)*(P/F,i,n)		=E26*E28	=F26*F28	=G26*G28	=H26*H28
30		? PV (Ct)	=SUM(E29:L29)
31		Fraction of total Value [PV(Ct)/? PV (Ct)]		=E29/$D30	=F29/$D30	=G29/$D30	=H29/$D30
32		Year* Fraction of Total Value                                       [t *PV(Ct)/? PV (Ct)]		=E25*E31	=F25*F31	=G25*G31	=H25*H31
33		Macaulay Duration	=SUM(E32:H32)	=SUM(E32:H32)
34
35		Hence,
36		Duration of the Obligation	=D33	Years
37		Present Value of the Obligation	=D30
38
39		Portfolio consist of 2 Year and 10 year zero coupon Bond.
40		Since for zero coupon bonds the duration is equal to the maturity of the zero coupon bond,
41		therefore,
42		Duration of 2 Year Zero Coupon Bond (D1)	2
43		Duration of 10 Year Zero Coupon Bond (D2)	10
44
45		Assuming,
46		Amount invested in 2 -Year zero coupon bond	V1
47		Amount invested in 10 -Year zero coupon bond	V2
48
49		Then present value of the portfolio should be equal to the present value of the obligation i.e.
50		V1+V2 = PV of obligation
51		V1+V2 = $5,269.06	------------(1)
52
53		Duration of the portfolio will be the weighted average of duration i.e.
54		Duration of the portfolio	=(V1/(V1+V2))*D1+(V2/(V1+V2))*D2
55			=(V1/(V1+V2))*2+(V2/(V1+V2))*10
56
57		Since the Duration of the portfolio should be equal to the duration of the obligation,
58		therefore,
59		(V1/(V1+V2))*2+(V2/(V1+V2))*10 = Duration of obligation
60		(V1/(V1+V2))*2+(V2/(V1+V2))*10 = 3.68
61		(V1)*2+(V2)*10 = 3.68*(V1+V2)
62		(V1)*2+(V2)*10 = 3.68*$5269.06
63		(V1)*2+(V2)*10 = $19,402.69
64		(V1)+(V2)*5= $9,701.34	------------(2)
65
66		Using Equations 1 and 2,
67		V1	=((D36*D37/2)-D37)/4
68		V2	=D37-D67
69
70		Hence,
71		Amount to be invested in 2-Year Bond	=D67
72		Amount to be invested in 10-Year Bond	=D68
73
74		Three reason the this hedge will not be perfect after one year are:
75		1) Since duration depends on yield, if the yield changes next year then the duration will change and hence existing bond immunization will not be work
76		2) Due to change in time to maturity the duration of the obligation will change
77		3) Change in interest rate may affect the yields of different bonds to a different extent
78