Question

What is the modified duration of an 5% bond with 15 years to maturity that is trading at a yield of 8%? Assume that coupon is paid semi-annually. (Keep your answer to 2 decimal places, xx.12.)

Answer 1

The coupon is paid semi-annually

semi-annual coupon = 2.5%*1000 = 25

Total number of periods = 15*2 = 30

Semi-annual YTM = 8%/4 = 4%

The following table shows the future cash flows and the present value calculation of these cashflows in Excel:

Period	Cashflow	Present value of Cashflow
1	25	24.03846154
2	25	23.11390533
3	25	22.22490897
4	25	21.37010478
5	25	20.54817767
6	25	19.75786314
7	25	18.99794533
8	25	18.26725513
9	25	17.56466839
10	25	16.88910422
11	25	16.23952329
12	25	15.61492624
13	25	15.01435215
14	25	14.43687707
15	25	13.88161257
16	25	13.34770439
17	25	12.83433115
18	25	12.34070303
19	25	11.8660606
20	25	11.40967366
21	25	10.97084005
22	25	10.54888467
23	25	10.14315833
24	25	9.753036859
25	25	9.377920056
26	25	9.017230823
27	25	8.670414253
28	25	8.336936782
29	25	8.016285367
30	1025	316.0266347
	PV of Bond	740.6195005

In the table above, Cashflows are C₁=C₂ =.........=C₂₉ = 25, C₃₀ = 1025

Present value of C₁ = PV(C₁) = C₁/(1+4%)¹

Similarly, we can calculate the present values of all the future cash flows and the sum of the present values of all the future cash flows gives the value of the bond.

So, Present value of the bond = PV_Bond = PV(C₁) + PV(C₂) +.....+PV(C₃₀) = 740.6195

Alternatively, PV of Bond can also be calculated using the Excel function

=PV(4%,30,-25,-1000) = 740.6915

Now, we need to calculate the weight of all the future cash flows:

The formula to calculate the modified duration is:

Modified Duration = 19.0957687480666 (in semi-annual periods)

Modified duration in years will be 19.0957687480666/2 = 9.54788437403329 (in years)

Answer -> 9.55 years