Question

Suppose that a bond has the following terms:

•10-years-to-maturity

•$1000 face value

•Semi-annual coupons, with an annual coupon rate of 5%

Suppose that all discount rates are 7%.

1. Calculate the price of the bond.

2. Calculate the bond’s modified duration.

3. Calculate the bond’s convexity.

4. If discount rates increase to 10%, what is the new price of the bond. Do (i) the actual calculation and (ii) approximate the new bond price using the duration and convexity. How well does the duration and convexity approximation work?

Answer 1

1

K = Nx2

Bond Price =∑ [( Coupon)/(1 + YTM/2)^k]     +   Par value/(1 + YTM/2)^Nx2

k=1

K =10x2

Bond Price =∑ [(5*1000/200)/(1 + 7/200)^k]     +   1000/(1 + 7/200)^10x2

k=1

Bond Price = 857.88

2

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($857.88)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	25.00	1.04	24.15	24.15
2	25.00	1.07	23.34	46.68
3	25.00	1.11	22.55	67.65
4	25.00	1.15	21.79	87.14
5	25.00	1.19	21.05	105.25
6	25.00	1.23	20.34	122.03
7	25.00	1.27	19.65	137.55
8	25.00	1.32	18.99	151.88
9	25.00	1.36	18.34	165.09
10	25.00	1.41	17.72	177.23
11	25.00	1.46	17.12	188.36
12	25.00	1.51	16.54	198.53
13	25.00	1.56	15.99	207.81
14	25.00	1.62	15.44	216.22
15	25.00	1.68	14.92	223.83
16	25.00	1.73	14.42	230.68
17	25.00	1.79	13.93	236.81
18	25.00	1.86	13.46	242.26
19	25.00	1.92	13.00	247.07
20	1,025.00	1.99	515.13	10,302.60
			Total	13,378.83

Macaulay duration =(∑ Duration calc)/(bond price*number of coupon per year)

=13378.83/(857.88*2)

=7.797613

Modified duration = Macaulay duration/(1+YTM)

=7.8/(1+0.07)

=7.533925

3

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($857.88)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period	=duration calc*(1+period)/(1+YTM/N)^2
1	25.00	1.04	24.15	24.15	45.10
2	25.00	1.07	23.34	46.68	130.72
3	25.00	1.11	22.55	67.65	252.59
4	25.00	1.15	21.79	87.14	406.75
5	25.00	1.19	21.05	105.25	589.49
6	25.00	1.23	20.34	122.03	797.38
7	25.00	1.27	19.65	137.55	1,027.22
8	25.00	1.32	18.99	151.88	1,276.05
9	25.00	1.36	18.34	165.09	1,541.13
10	25.00	1.41	17.72	177.23	1,819.90
11	25.00	1.46	17.12	188.36	2,110.03
12	25.00	1.51	16.54	198.53	2,409.35
13	25.00	1.56	15.99	207.81	2,715.85
14	25.00	1.62	15.44	216.22	3,027.71
15	25.00	1.68	14.92	223.83	3,343.22
16	25.00	1.73	14.42	230.68	3,660.86
17	25.00	1.79	13.93	236.81	3,979.19
18	25.00	1.86	13.46	242.26	4,296.94
19	25.00	1.92	13.00	247.07	4,612.92
20	1,025.00	1.99	515.13	10,302.60	201,969.35
			Total	13,378.83	240,011.76

Convexity =(∑ convexity calc)/(bond price*number of coupon per year^2)

=240011.76/(857.88*2^2)

=69.94

4

i

Actual bond price change

K = Nx2

Bond Price =∑ [( Coupon)/(1 + YTM/2)^k]     +   Par value/(1 + YTM/2)^Nx2

k=1

K =10x2

Bond Price =∑ [(5*1000/200)/(1 + 10/200)^k]     +   1000/(1 + 10/200)^10x2

k=1

Bond Price = 688.44

ii

Using only modified duration

Mod.duration prediction = -Mod. Duration*Yield_Change*Bond_Price

=-7.53*0.03*857.88

=-193.9

Using convexity adjustment to modified duration

Convexity adjustment = 0.5*convexity*Yield_Change^2*Bond_Price

0.5*69.94*0.03^2*857.88

=27

We can check how well it works by calculating %age between actual and predicted values

Difference in price predicted and actual

=predicted price-actual price

=690.99-688.44

=2.545

%age difference = difference/actual-1

=2.55/688.44

=0.3697%

New bond price = bond price+Mod.duration pred.+convex. Adj.

=857.88-193.9+27

=690.99