Question

A bond with a face value of $1,000 and matures after 15 years is currently selling for $980. The semiannual coupon on the bond is $80 (paid in two payments of $40 each).

Part a. Calculate the duration of the bond.

Part b. Calculate the change in the price of the bond if interest rates increase by 0.25%

Answer 1

YTM:

YTM :
YTM is the rate at which PV of Cash inflows are equal to Bond price when the bond is held till maturity.

YTM = [ Rate at which least +ve NPV + [ NPV at that Rate / Change in NPV due to 0.5% inc in disc rate ] * 0.5% ] *2

Period	Cash Flow	PVF/[email protected]	PV of Cash Flows	PVF/[email protected]	PV of Cash Flows
1-30	$                  40.00	17.2920	$                        691.68	16.2889	$                     651.56
30	$             1,000.00	0.3083	$                        308.32	0.2670	$                     267.00
PV of Cash Inflows	$                    1,000.00		$                     918.56
PV of Cash Oiutflows	$                        980.00		$                     980.00
NPV	$                          20.00		$                     -61.44

YTM = [ Rate at which least +ve NPV + [ NPV at that Rate / Change in NPV due to 0.5% inc in disc rate ] * 0.5% ] *2

= [ [ 4% + [ 20 / 81.44 ] * 0.5% ] * 2 ]

= [ [ 4% + [ 20 / 81.44 ] * 0.5% ] * 2 ]

= [ 4 % + [ 0.2455 * 0.5% ] ] * 2

= [ 4 % + 0.12% ] * 2

= 4.12% * 2

= 8.25%

Duration:
Duaration = Sum [ Weight * Periods ]

Period	CF	PVF @4.12%	Disc CF	Weight	Wt * period
1	$   40.00	0.9604	$   38.42	0.0392	0.0392
2	$   40.00	0.9235	$   36.94	0.0377	0.0754
3	$   40.00	0.8880	$   35.52	0.0362	0.1087
4	$   40.00	0.8538	$   34.15	0.0348	0.1394
5	$   40.00	0.8210	$   32.84	0.0335	0.1675
6	$   40.00	0.7894	$   31.58	0.0322	0.1933
7	$   40.00	0.7590	$   30.36	0.0310	0.2169
8	$   40.00	0.7298	$   29.19	0.0298	0.2383
9	$   40.00	0.7018	$   28.07	0.0286	0.2578
10	$   40.00	0.6748	$   26.99	0.0275	0.2754
11	$   40.00	0.6488	$   25.95	0.0265	0.2913
12	$   40.00	0.6239	$   24.96	0.0255	0.3056
13	$   40.00	0.5999	$   24.00	0.0245	0.3183
14	$   40.00	0.5768	$   23.07	0.0235	0.3296
15	$   40.00	0.5546	$   22.18	0.0226	0.3396
16	$   40.00	0.5333	$   21.33	0.0218	0.3483
17	$   40.00	0.5128	$   20.51	0.0209	0.3558
18	$   40.00	0.4931	$   19.72	0.0201	0.3622
19	$   40.00	0.4741	$   18.96	0.0194	0.3677
20	$   40.00	0.4559	$   18.23	0.0186	0.3721
21	$   40.00	0.4383	$   17.53	0.0179	0.3757
22	$   40.00	0.4215	$   16.86	0.0172	0.3785
23	$   40.00	0.4053	$   16.21	0.0165	0.3804
24	$   40.00	0.3897	$   15.59	0.0159	0.3817
25	$   40.00	0.3747	$   14.99	0.0153	0.3823
26	$   40.00	0.3603	$   14.41	0.0147	0.3823
27	$   40.00	0.3464	$   13.86	0.0141	0.3818
28	$   40.00	0.3331	$   13.32	0.0136	0.3807
29	$   40.00	0.3203	$   12.81	0.0131	0.3791
30	$   40.00	0.3080	$   12.32	0.0126	0.3771
30	$ 1000.00	0.3080	$ 307.96	0.3142	9.4274
Duration in Periods	18.3296
Period per year	2.0000
Duration in Years	9.1648

Modified duaration :
Modified duration = Duration / [ 1 + YTM ]
It specifies% change in Price in opposite direction due to 1% change in YTM.

= 9.1648 / [ 1 + 0.0825 ]

= 9.1648 / 1.0825

= 8.47

1% change in Disc Rate will leads to 8.47% change in Price .

0.25% change in disc rate leads to 8.47% * 0.25 / 1

= 2.12%

Pls do rate, if the answer is correct and comment, if any further assistance is required.