Question

Calculate the approximate modified duration of a 4-year, 5% coupon, semi-annual bond if yields change by 50bps. Assume the bond currently sells at 5% yield to maturity (YTM).
a) 1.79
b) 3.59
c) 7.17
d) 11.95
e) None of the above

Answer 1

First calculate Macaulay duration of the bond  in following manner -

Year (t)	Payments (n)	Cash Flow from coupon payments (5%/2 of $1000)	Cash Flow from maturity amount	Total Cash Flow from coupon payments and maturity amount (CF)	Present value (PV) discounted at 5%/2 =2.5% semiannual yield to maturity	PV *t
0.5	1.0	$25.0		$25.0	$24.39	$12.20
1.0	2.0	$25.0		$25.0	$23.80	$23.80
1.5	3.0	$25.0		$25.0	$23.21	$34.82
2.0	4.0	$25.0		$25.0	$22.65	$45.30
2.5	5.0	$25.0		$25.0	$22.10	$55.24
3.0	6.0	$25.0		$25.0	$21.56	$64.67
3.5	7.0	$25.0		$25.0	$21.03	$73.61
4.0	8.0	$25.0	$1,000.0	$1,025.0	$841.27	$3,365.06
sum					$1,000.00	$3,674.70
					Bond's Price↑
				Macaulay Duration = sum of (PV*t)/sum of PVs =	$3674.70/$1,000	3.67

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

Where,

Macaulay Duration = 3.67 years

Yield to maturity, YTM = 5% per year

Number of discounting periods in year, n = 2 (for semi-annual coupon payments)

Therefore,

Modified Duration = 3.67/ (1+ 5%/2)

= 3.59 years

Therefore correct answer is option: b) 3.59