Question

A bond has just been issued. The bond has an annual coupon rate of 9% and coupons are paid annually. The bond has a face value of $1,000 and will mature in 10 years. The bond’s yield to maturity is 12%.

e. Calculate the bond’s duration at a yield to maturity of 10.5%.

f. Use the bond’s duration to calculate the approximate bond price change as the yield to maturity changes from 12% to 10.5%.

g. Use the bond’s modified duration to calculate the approximate bond price change as the yield to maturity changes from 12% to 10.5%.

Answer 1

First calculate the price of the bond

The current price of the bond can be calculated in following manner

Bond price P0 = C* [1- 1/ (1+YTM) ^n] /YMT + M / (1+YTM) ^n

Where,

M = value at maturity, or par value = $ 1000

P0 = the current market price of bond =?

C = coupon payment =9% of $1,000 = $90

n = time to maturity or number of payments = 10 (years)

YTM = Current market interest rate, or yield to maturity =12%

Now we have,

P0 = $90 * [1 – 1 / (1+12%) ^10] /12% + $1000 / (1+12%) ^10

Or P0 = $508.52 + $321.97

= $830.49

e. Calculate the bond’s duration at a yield to maturity of 10.5%.

Year (t)	Cash Flow from coupon payments	Cash Flow from maturity amount	Total Cash Flow from coupon payments and maturity amount (CF)	Present value (PV) discounted at 10.5%	PV *t
1	$90.0		$90.0	$81.45	$81.45
2	$90.0		$90.0	$73.71	$147.42
3	$90.0		$90.0	$66.70	$200.11
4	$90.0		$90.0	$60.37	$241.46
5	$90.0		$90.0	$54.63	$273.15
6	$90.0		$90.0	$49.44	$296.63
7	$90.0		$90.0	$44.74	$313.19
8	$90.0		$90.0	$40.49	$323.92
9	$90.0		$90.0	$36.64	$329.78
10	$90.0	$1,000.0	$1,090.0	$401.61	$4,016.09
sum				$909.78	$6,223.20
				Bond's Price↑
			Macaulay Duration = sum of (PV*t)/sum of PVs =	$6223.20/$909.78	6.84	Years

f. Use the bond’s duration to calculate the approximate bond price change as the yield to maturity changes from 12% to 10.5%.

Predicted price change = – Duration * (change in y)/ (1+y)* P0

= - 6.84 *(-1.50%)/ (1+12%) * $830.49

= $76.08

Therefore, predicted new price = $76.08 + $830.49 = $906.57

g. Use the bond’s modified duration to calculate the approximate bond price change as the yield to maturity changes from 12% to 10.5%.

Modified Duration = Macaulay Duration / (1 + YTM/n)

Where,

Macaulay Duration = 6.84 years

Yield to maturity, YTM = 10.5% per year

Number of discounting periods in year, n = 1 (for annual coupon payments)

Therefore,

Modified Duration = 6.84/ (1+ 10.5%)

= 6.19 years

New price of bond can be calculated with following formula

% Change in bond’s Price = -1 * Modified Duration * Change in interest rate

= -1 *6.19 * (-1.50%) = 9.29%

New price of bond = current price * % Change in bond’s Price

Where, Current price of the bond = $830.49

New price of bond = $830.49 * (1+ 9.29%) = $907.61

Therefore new price of bond is approx. $907.61