Question

Find i.) the yield to maturity and ii.) the Macauly duration of a $1,000 par value 10-year bond with coupons at 8% convertible semiannually has a market value of $910. The bond is redeemable at par.

Answer 1

(i) We have following formula for calculation of bond’s yield to maturity (YTM)

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

Where,

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

P0 = the current market price of bond = $910

C = coupon payment = 8% of $1000 = $80 but semiannual coupon, therefore C = $80/2 = $40

n = number of payments (time remaining to maturity) = 10 years; therefore number of payments n = 10 *2 = 20

YTM = interest rate, or yield to maturity =?

Now we have,

$910 = $40 * [1 – 1 / (1+YTM) ^20] /YTM+ 1000 / (1+YTM) ^20

By trial and error method we can calculate the value of YTM = 4.7042% semiannual

Or annual YMT = 2 * 4.7042% = 9.41% per year

[Or you can use excel function for YTM calculation in following manner

“= Rate(N,PMT,PV,FV)”

“Rate(20,-40,910,-1000)” = 4.7042%]

The YTM is 9.41%

(ii) Macaulay Duration Calculation:

Year (t)	Payments (n)	Cash Flow from coupon payments (8%/2 of $1000)	Cash Flow from maturity amount	Total Cash Flow from coupon payments and maturity amount (CF)	Present value (PV) discounted at 9.41%/2 =4.7042% semiannual yield to maturity [PV = CF/(1+4.7042%)^n]	PV *t
0.5	1.0	$40.0		$40.0	$38.20	$19.10
1.0	2.0	$40.0		$40.0	$36.49	$36.49
1.5	3.0	$40.0		$40.0	$34.85	$52.27
2.0	4.0	$40.0		$40.0	$33.28	$66.56
2.5	5.0	$40.0		$40.0	$31.79	$79.47
3.0	6.0	$40.0		$40.0	$30.36	$91.07
3.5	7.0	$40.0		$40.0	$28.99	$101.48
4.0	8.0	$40.0		$40.0	$27.69	$110.77
4.5	9.0	$40.0		$40.0	$26.45	$119.01
5.0	10.0	$40.0		$40.0	$25.26	$126.30
5.5	11.0	$40.0		$40.0	$24.12	$132.68
6.0	12.0	$40.0		$40.0	$23.04	$138.24
6.5	13.0	$40.0		$40.0	$22.01	$143.03
7.0	14.0	$40.0		$40.0	$21.02	$147.12
7.5	15.0	$40.0		$40.0	$20.07	$150.54
8.0	16.0	$40.0		$40.0	$19.17	$153.36
8.5	17.0	$40.0		$40.0	$18.31	$155.63
9.0	18.0	$40.0		$40.0	$17.49	$157.38
9.5	19.0	$40.0		$40.0	$16.70	$158.66
10.0	20.0	$40.0	$1,000.0	$1,040.0	$414.72	$4,147.16
sum					$910.00	$6,286.33
					Bond's Price↑
				Macaulay duration = sum of (PV*t)/sum of PVs =	$6286.35/ $910	6.91

Formulas used in excel: