Both Bond A and Bond B have 7.8 percent coupons and are priced at par value. Bond A has 8 years to maturity, while Bond B has 18 years to maturity. If interest rates suddenly rise by 1 percentage points, what is the difference in percentage changes in prices of Bond A and Bond B? (i.e., Bond A - Bond B). The bonds pay coupons twice a year.
YTM = [Annual interest +(Face value-Current price)/n]/(Face value + 2*Current Price)/3 | |||||
Bond A | Bond B | ||||
Maturity Years =n | 8 | 18 | |||
Face value -suppose | 100 | 100 | |||
Market value | 100 | 100 | |||
Annual Interest amount | 7.8 | 7.8 | |||
Yield to maturity | 7.80% | 7.80% | |||
Bond A =[7.8+0/8](100+200)/3=7.8/100=7.8% | |||||
Bond B=[7.8+0/18]/(100+200)/3=7.8/100=7.8% |
Caculation of Modified Duration of Bond A | |||||
Period | Cash Flow | Discount factor @3.9% semi annually | PV cash floW | Time period*Cash flow | PV of time adjusted cash flow |
1 | 3.9 | 0.962 | 3.7518 | 3.9 | 3.7518 |
2 | 3.9 | 0.926 | 3.6114 | 7.8 | 7.2228 |
3 | 3.9 | 0.892 | 3.4788 | 11.7 | 10.4364 |
4 | 3.9 | 0.858 | 3.3462 | 15.6 | 13.3848 |
5 | 3.9 | 0.826 | 3.2214 | 19.5 | 16.107 |
6 | 3.9 | 0.795 | 3.1005 | 23.4 | 18.603 |
7 | 3.9 | 0.765 | 2.9835 | 27.3 | 20.8845 |
8 | 3.9 | 0.736 | 2.8704 | 31.2 | 22.9632 |
9 | 3.9 | 0.709 | 2.7651 | 35.1 | 24.8859 |
10 | 3.9 | 0.682 | 2.6598 | 39 | 26.598 |
11 | 3.9 | 0.656 | 2.5584 | 42.9 | 28.1424 |
12 | 3.9 | 0.632 | 2.4648 | 46.8 | 29.5776 |
13 | 3.9 | 0.608 | 2.3712 | 50.7 | 30.8256 |
14 | 3.9 | 0.585 | 2.2815 | 54.6 | 31.941 |
15 | 3.9 | 0.563 | 2.1957 | 58.5 | 32.9355 |
16 | 103.9 | 0.542 | 56.3138 | 1662.4 | 901.0208 |
Total= | 100.0 | 1219.2803 |
Macaulays duration= sum of PV of time adjusted cash flow/ Sum of PV of cash flows= | 12.19593736 |
Modified duration =Macaulay's duration/(1+YTN/n) | 11.73814952 |
n= no of coupon payment per year |
% change in Price = - modified Duration*% interest change |
Interest rate rise 1% |
So change in Price in Bond A=-11.73*1%=-11.73% |