Question

You find a bond with 23 years until maturity that has a coupon rate of 8.0 percent and a yield to maturity of 5.8 percent. Suppose the yield to maturity on the bond increases by 0.25 percent. a. What is the new price of the bond using duration and using the bond pricing formula? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Estimated price Actual price b. Now suppose the original yield to maturity is increased by 1 percent. What is the new price of the bond? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Estimated price Actual price

Answer 1

Duration of the bond = 12.46 years

Duration calculation:

Formula		CF/(1+YTM)^n	PV/BP	w*n
Time until payment (n)	Cash flow (CF)	PV of CF	Weight (w)	Weighted Time
1	80	75.61	0.06	0.06
2	80	71.47	0.06	0.11
3	80	67.55	0.05	0.16
4	80	63.85	0.05	0.20
5	80	60.35	0.05	0.24
6	80	57.04	0.04	0.27
7	80	53.91	0.04	0.30
8	80	50.96	0.04	0.32
9	80	48.16	0.04	0.34
10	80	45.52	0.04	0.36
11	80	43.03	0.03	0.37
12	80	40.67	0.03	0.38
13	80	38.44	0.03	0.39
14	80	36.33	0.03	0.40
15	80	34.34	0.03	0.40
16	80	32.46	0.03	0.41
17	80	30.68	0.02	0.41
18	80	29.00	0.02	0.41
19	80	27.41	0.02	0.41
20	80	25.90	0.02	0.41
21	80	24.48	0.02	0.40
22	80	23.14	0.02	0.40
23	1080	295.29	0.23	5.32
	Bond price (BP)	1275.60	Duration (in years)	12.46

(Note: Duration can also be calculated using the DURATION() function in Excel.)

Current price of the bond: FV (or par value) = 1,000; PMT (coupon payment) = 8%*1,000 = 80; N = 23; rate = 5.80%, solve for PV.

Current price = 1,275.60

a). Change in price (using duration) = -current price*duration*change in yield/(1+ present yield) where

current price = 1,275.60; duration = 12.46 years; change in yield = + 0.25%; present yield (YTM) = 5.80%

Change in price = -1,275*12.46*0.25%/(1+5.80%) = -37.56

New price = current price + change in price = 1,275.60 -37.56 = 1,238.04 (estimated price)

Actual price: FV = 1,000; PMT = 80; N = 23; rate = 5.80%+0.25% = 6.05%, solve for PV.

Actual price = 1,238.84

b). Change in price (using duration) = -current price*duration*change in yield/(1+ present yield) where

current price = 1,275.60; duration = 12.46 years; change in yield = + 1.00%; present yield (YTM) = 5.80%

Change in price = -1,275*12.46*1.00%/(1+5.80%) = -150.24

New price = current price + change in price = 1,275.60 -150.24 = 1,125.36 (estimated price)

Actual price: FV = 1,000; PMT = 80; N = 23; rate = 5.80%+1.00% = 6.80%, solve for PV.

Actual price = 1,137.61