Question

Consider a bond with a coupon of 5.6 percent, ten years to maturity, and a current price of $1,057.70. Suppose the yield on the bond suddenly increases by 2 percent.

a. Use duration to estimate the new price of the bond. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. Calculate the new bond price using the usual bond pricing formula. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Answer 1

Firstly we calculate the YTM. For this use the excel function ..... " RATE"

Syntax .............. = Rate(nper, pmt, -pv, fv, type)

nper = number of periods = 10

pmt = coupon payment = 1000 * 0.056 = 56

pv = current price of the bond = 1057.70

fv = future value or face value of the bond = 1000

=Rate(10,56,-1057.70,1000,0)

= 4.8579%

Now compute the Macaulay duration and then estimate the modified duration.

t	CF	DF	PV	W	W*t
1	56	0.953672	53.40561	0.050492	0.050492
2	56	0.909489	50.93141	0.048153	0.096306
3	56	0.867354	48.57184	0.045922	0.137766
4	56	0.827171	46.32158	0.043795	0.175179
5	56	0.78885	44.17558	0.041766	0.208828
6	56	0.752303	42.12899	0.039831	0.238985
7	56	0.71745	40.17722	0.037985	0.265898
8	56	0.684212	38.31588	0.036226	0.289805
9	56	0.652514	36.54076	0.034547	0.310926
10	1056	0.622284	657.1316	0.621284	6.212836
			Macaulay Duration		7.987022
			Modified Duration		7.948409

Modified duration = Macaulay duration / ( 1 + ytm/n) = 7.9870 / (1 + 0.048579 / 10) = 7.9484

Question - 1

Bond duration of 7.8494 indicates that for every 1% change in YTM we have 7.9484 % change in bond price. In the given case YTM increase by 2% ........... so bond price decreases by 2 * 7.9484 = 15.8968 %.

New Bond price = current bond price * ( 1 - % decrease )

= 1057.70 * ( 1 - 0.158968 ) = 889.56

Question - 2

Using the bond formula

Price of bond = coupon * [ 1 - (1+r)^-n ] / r + Face value * ( 1 + r )^-n

r = new YTM = 0.048579 + 0.02 = 0.068579

= 56 * [ 1 - (1.068579)^-10 ] / 0.068579 + 1000 * (1.068579)^-10

= 56 * 7.06994947 + 1000 * 0.51514994

= 911.07