Question

There is a 9%, 23 year note bond which has a ytm of 9%. The ytm alters by one percent down. By how much does the price alter? If the ytm drops by 2%, by how much does the price change?

What is the exact percentage change of the bond in the 2 cases?

Please show all work and the formula(s) used.

Answer 1

Given a 9%, 23 year note bond which has a YTM of 9%

Modified duration measures percentage change in bond price when the YTM changes by 1 percentage point, i.e., if the YTM increases (decreases) by k% then the bond price will decrease (increase) by [modified duration x k]%

The formula for modified duration is:

We know that duration is the weighted average of the times when bond's coupons are received.

Duration can be calculated using the below formula:

where PV is the sum of present value of all future cash flows (C₁, C₂,..., C_n) = price of bond

Suppose face valueof the bond = 100

Coupon rate = 9%

Hence, yearly coupon payments = 9% * 100 = 9

C₁ = C₂ =..,= C_n-1 = 9 and Cn = 109

YTM = 9%

PV(C₁) = 9/(1+9%)¹

PV(C₂) = 9/(1+9%)²

^.

^.

PV(C₂₂) = 109/(1+9%)22

PV(C₂₃) = 109/(1+9%)²³

Now Duration is calculated by:

Duration = 10.44243 years

Below table shows the present value of all the cash flows and the weighted average of these cash flows. The sum of present value of all the cash flows gives the present value of the bond, while the sum of weighted average of these cash flows gives us the duration of the bond which is 10.44243 years.

Period	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
Cashflows	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	9	109
Present value of cash flows	8.256881	7.57512	6.949651	6.375827	5.849382	5.366406	4.923308	4.516797	4.14385	3.801697	3.487796	3.199813	2.935608	2.693218	2.470842	2.266828	2.079659	1.907944	1.750407	1.605878	1.473283	1.351635	15.01817
weighted average	0.082569	0.151502	0.20849	0.255033	0.292469	0.321984	0.344632	0.361344	0.372947	0.38017	0.383658	0.383978	0.381629	0.377051	0.370626	0.362692	0.353542	0.34343	0.332577	0.321176	0.309389	0.29736	3.454179

Now, modified duration = Duration/(1+YTM) = 10.44243/(1+9%) = 9.580207 years

Modified duration measures percentage change in bond price when the YTM changes by 1 percentage point, i.e., if the YTM increases (decreases) by k% then the bond price will decrease (increase) by [modified duration x k]%

We know that there is inverse relationship between the price of bond and YTM. So, when the YTM decreases by 1%, the bond price will increase by 9.580207%

And when the YTM drops by 2%, bond price will increase by 2*9.580207% = 19.16041%