Question

Bond J has a coupon rate of 3 percent. Bond K has a coupon rate of 9 percent. Both bonds have 14 years to maturity, make semiannual payments, and have a YTM of 6 percent. If interest rates suddenly rise by 2 percent, what is the percentage price change of these bonds? What if rates suddenly fall by 2 percent instead? What does this problem tell you about the interest rate risk of lower-coupon bonds?" PLEASE show how to solve using a calculator not excel

Answer 1

In the given problem, in order to calculate the % change in price, we are required to ascertain the Duration of Bond J & Bond K.

Duration of the bond measures the sensitivity of the Bond Price for % change in YTM, i.e. how much % the bond price will change for every 1% change in YTM.

DURATION of a BOND = WEIGHTED AVERAGE TIME OF BOND PAYMENTS

I.e. D = (w * t)

Where, weight (w) = Present Value of each Cash Flow / Bond Price

& Time (t) = Bond’s maturity life

Calculating the Duration and current price of Bond J at YTM @ 6%p.a.

The price of a bond is sum total of present values of all coupon payments and present value of maturity value. Further, the cash flows value and the discount rate should always be consistent.

Since, cash flows are received semiannually, time interval and required rate of return is also compounded semiannually,

Thus,

Coupon rate will be 3%/2 = 1.50%

YTM will be 6% / 2 = 3%

And; years to maturity will be 14*2 = 28

Hence, Duration and Bond Price of the bond J will be:

Time	Cash Flow @ coupon rate 1.50%	Discount factor @ 4% = 1/ (1+YTM%)^n	Bond Price (PV of all coupon payments + PV of maturity value)	Weight =(PV of cash flows)/ Bond Price	Duration =(weight * time)
1	$ 30	0.9709	14.56	0.020	0.020
2	$ 30	0.9426	14.14	0.020	0.039
3	$ 30	0.9151	13.73	0.019	0.057
4	$ 30	0.8885	13.33	0.019	0.074
5	$ 30	0.8626	12.94	0.018	0.090
6	$ 30	0.8375	12.56	0.017	0.105
7	$ 30	0.8131	12.20	0.017	0.119
8	$ 30	0.7894	11.84	0.016	0.132
9	$ 30	0.7664	11.50	0.016	0.144
10	$ 30	0.7441	11.16	0.016	0.155
11	$ 30	0.7224	10.84	0.015	0.166
12	$ 30	0.7014	10.52	0.015	0.176
13	$ 30	0.6810	10.21	0.014	0.185
14	$ 30	0.6611	9.92	0.014	0.193
15	$ 30	0.6419	9.63	0.013	0.201
16	$ 30	0.6232	9.35	0.013	0.208
17	$ 30	0.6050	9.08	0.013	0.215
18	$ 30	0.5874	8.81	0.012	0.221
19	$ 30	0.5703	8.55	0.012	0.226
20	$ 30	0.5537	8.31	0.012	0.231
21	$ 30	0.5375	8.06	0.011	0.236
22	$ 30	0.5219	7.83	0.011	0.240
23	$ 30	0.5067	7.60	0.011	0.243
24	$ 30	0.4919	7.38	0.010	0.246
25	$ 30	0.4776	7.16	0.010	0.249
26	$ 30	0.4637	6.96	0.010	0.252
27	$ 30	0.4502	6.75	0.009	0.254
28	$1030	0.4371	443.63	0.617	17.287
			$ 718.54	1.00	21.96

The Bond Price of Bond J is $ 718.54 and duration is 21.96 years

To calculate the % change in price, we need to measure the volatility of the bond J by calculating Modified Duration

Modify Duration (MD) = D / (1 + ytm)

                                           = 21.96 / (1+3%)

                                             = 21.32

% change in price of Bond J,

1. If YTM increases by 2%

Change in price / price = - MD * % CHANGE IN YTM

Change in price / $ 718.54 = - 21.32 * 2%

Change in price of Bond J = $ 718.54 * -21.32 * 2%

                                               = - 306.39

i.e. New Bond Price of Bond J will be 718.54 – 306.39 = $412.15

2. If YTM decreases by 2%

Change in price / price = - MD * % CHANGE IN YTM

Change in price / $ 718.54 = - 21.32 *- 2%

Change in price of Bond J = $ 718.54 * -21.32 * 2%

                                               = 306.39

i.e. New Bond Price of Bond J will be 718.54 + 306.39 = $ 1024.93

Calculating the Duration and current price of Bond K at YTM @ 6%p.a.

Since, cash flows are received semiannually, time interval and required rate of return is also compounded semiannually,

Thus,

Coupon rate will be 9%/2 = 4.50%

YTM will be 6% / 2 = 3%

And; years to maturity will be 14*2 = 28

Hence, Duration and Bond Price of the bond J will be:

Time	Cash Flow @ coupon rate %	Discount factor @ 3% = 1/ (1+YTM%)^n	Bond Price (PV of all coupon payments + PV of maturity value)	Weight =(PV of cash flows)/ Bond Price	Duration =(weight * time)
1	$ 45	0.9709	43.69	0.034	0.034
2	$ 45	0.9426	42.42	0.033	0.066
3	$ 45	0.9151	41.18	0.032	0.096
4	$ 45	0.8885	39.98	0.031	0.125
5	$ 45	0.8626	38.82	0.030	0.151
6	$ 45	0.8375	37.69	0.029	0.176
7	$ 45	0.8131	36.59	0.029	0.200
8	$ 45	0.7894	35.52	0.028	0.222
9	$ 45	0.7664	34.49	0.027	0.242
10	$ 45	0.7441	33.48	0.026	0.261
11	$ 45	0.7224	32.51	0.025	0.279
12	$ 45	0.7014	31.56	0.025	0.296
13	$ 45	0.6810	30.64	0.024	0.311
14	$ 45	0.6611	29.75	0.023	0.325
15	$ 45	0.6419	28.88	0.023	0.338
16	$ 45	0.6232	28.04	0.022	0.350
17	$ 45	0.6050	27.23	0.021	0.361
18	$ 45	0.5874	26.43	0.021	0.371
19	$ 45	0.5703	25.66	0.020	0.380
20	$ 45	0.5537	24.92	0.019	0.389
21	$ 45	0.5375	24.19	0.019	0.396
22	$ 45	0.5219	23.49	0.018	0.403
23	$ 45	0.5067	22.80	0.018	0.409
24	$ 45	0.4919	22.14	0.017	0.415
25	$ 45	0.4776	21.49	0.017	0.419
26	$ 45	0.4637	20.87	0.016	0.423
27	$ 45	0.4502	20.26	0.016	0.427
28	$1045	0.4371	456.75	0.356	9.980
			1281.46	1.00	17.85

Now