Question

Consider following two bonds:

Maturity Coupon YTM

Bond 1   4 years 6 5%

Bond 2 8 years 7% 5%

Coupon frequency and compounding frequency are assumed to be annual.

What should be the weight of Bond1 and Bond2 if your investment horizon is six years and you want to be hedged against small changes in interest rates?

Answer 1

Let the face value of bonds be 100

lets calculate the duration of the both the Bonds

Year (t)	Cash flows	Discounting factor = 1 / ( 1+r)^n	Present value	Present value of time weighted cashflow
1	6	0.952380952	5.714285714	5.714285714
2	6	0.907029478	5.442176871	10.88435374
3	6	0.863837599	5.183025591	15.54907677
4	106	0.822702475	87.20646233	348.8258493
Total	103.5459505	380.9735655

Duration of Bond 1 = Sum of present value of time weighted cashflows / sum of present values

= 380.97 / 103.55 = 3.68

Year (t)	Cash flows	Discounting factor = 1 / ( 1+r)^n	Present value	Present value of time weighted cashflow
1	7	0.952380952	6.666666667	6.666666667
2	7	0.907029478	6.349206349	12.6984127
3	7	0.863837599	6.04686319	18.14058957
4	7	0.822702475	5.758917324	23.03566929
5	7	0.783526166	5.484683165	27.42341583
6	7	0.746215397	5.223507776	31.34104666
7	7	0.71068133	4.974769311	34.82338518
8	107	0.676839362	72.42181174	579.3744939
Total	112.9264255	733.5036798

Duration of Bond 2 = 733.50 / 112.93 = 6.50

Investment amount in bond 1 & bond will be such the weighted average of bond is equal to the duration of the investment. So let x be the amount invested in Bond 1 and amount invested in Bond 1-x , then

3.68x + ( 1 -x)6.50 = 6

3.68x + 6.50 -6.5x = 6

2.82x =0.50

x = 0.50 / 2.82 = 0.1773

1- x = 1 -0.1773 = 0.8227

weight of Bond1 = 17.73%

weight of Bond 2 = 82.27%