Question

Consider a bond that pays $50 semi-annual interest and has a remaining life of 10 years. The bond currently sells for $987 and has a yield to maturity of 10.20%.

a. What is the bond’s duration?

b. What is its modified duration?

c. What is its convexity?

d. What percentage change in bond price would be expected for one percent decrease in interest rate?

Answer 1

$50 Semi annual Interest means, $100 annual interest or yield of bond is 10% (on $1000 Bond value)

Duration = time weighted PV / PV of cash flows

Convexity = [PV*(t^2+t) / (1+i)^2] / PV

Modified Duration is Duration / (1+y/k) Where y is the Yield to Maturity and k is the number of periods in a year.

As shown in the calculations below,

Duration = 6.52

Convexity = 209.9

Modified Duration = 6.20

D) For 1% decrease in the interest rate, bond price would increase by 1%*4.45 = 6.52% increase in Bond price.

				1)	2)			3)
			i=10.2%/2	PV of cash flow	Time weighted value			change of change
t	Period	Cash flow	(1+i)^t	=cash flow / (1+i)^t	=PV*period	(1+i)^2	t^2+t	=PV*(t^2+t) / (1+i)^2
1	0.5	50	1.051	47.5737393	23.78686965	1.1046	2	86.13741848
2	1	50	1.104601	45.26521341	45.26521341	1.1046	6	245.8727454
3	1.5	50	1.160936	43.06870924	64.60306386	1.1046	12	467.8834356
4	2	50	1.220143	40.9787909	81.95758181	1.1046	20	741.9654863
5	2.5	50	1.282371	38.9902863	97.47571576	1.1046	30	1058.942178
6	3	50	1.347772	37.09827431	111.2948229	1.1046	42	1410.579495
7	3.5	50	1.416508	35.29807261	123.5432541	1.1046	56	1789.507764
8	4	50	1.48875	33.58522608	134.3409043	1.1046	72	2189.149093
9	4.5	50	1.564676	31.95549579	143.7997311	1.1046	90	2603.650206
10	5	50	1.644475	30.40484852	152.0242426	1.1046	110	3027.820305
11	5.5	50	1.728343	28.92944674	159.111957	1.1046	132	3457.073612
12	6	50	1.816488	27.52563914	165.1538348	1.1046	156	3887.376262
13	6.5	50	1.909129	26.18995161	170.2346855	1.1046	182	4315.197246
14	7	50	2.006495	24.9190786	174.4335502	1.1046	210	4737.463126
15	7.5	50	2.108826	23.70987498	177.8240623	1.1046	240	5151.516244
16	8	50	2.216376	22.55934822	180.4747857	1.1046	272	5555.07619
17	8.5	50	2.329411	21.46465101	182.4495336	1.1046	306	5946.204295
18	9	50	2.448211	20.42307423	183.8076681	1.1046	342	6323.270924
19	9.5	50	2.57307	19.43204018	184.6043817	1.1046	380	6684.925388
20	10	1050	2.704297	388.2710217	3882.710217	1.1046	420	147631.4335
				987.6427829	6438.896075			207311.0449
	Duration		6.52	=2) / 1)
	Convexity		209.90	=3) / 1)
	Modified Duration		6.2031	= D / (1+y/k)	Where y is yield to maturity and k is the number of periods in a year