Question

A six-year bond, with a Face Value of $1000 has yield rate of 5% compounded continuously, and coupon rate of 6% compounded semi-annually, paid every half-year. You are required to: a) compute the price of bond b) compute the the duration of bond c) compute the convexity of bond. d) Use duration to estimate the effect of a 1% increase in the yield on the price of bond. e) Use convexity to estimate the eect of a 1% increase in the yield on the the price of bond. f) How accurate is the estimated price of the bond based on your answers in (d) and (e). Hint: You will need to calculate the price of the bond given a 1% increase of the yield and compare your answers with parts (d) and (e).

Answer 1

EAR =[ e^(Annual percentage rate) -1]*100

5=(e^(APR%/100)-1)*100

APR% = 4.88

EAR = [(1 +stated rate/no. of compounding periods) ^no. of compounding periods - 1]* 100

4.88 = ((1+Stated rate%/(2*100))^2-1)*100

Stated rate% = 4.82

a

K = Nx2

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

k=1

K =6x2

Bond Price =∑ [(6*1000/200)/(1 + 4.82/200)^k]     +   1000/(1 + 4.82/200)^6x2

k=1

Bond Price = 1060.85

b

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($1.060.85)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	30.00	1.02	29.29	29.29
2	30.00	1.05	28.60	57.21
3	30.00	1.07	27.93	83.79
4	30.00	1.10	27.27	109.10
5	30.00	1.13	26.63	133.16
6	30.00	1.15	26.01	156.03
7	30.00	1.18	25.39	177.76
8	30.00	1.21	24.80	198.37
9	30.00	1.24	24.21	217.91
10	30.00	1.27	23.64	236.43
11	30.00	1.30	23.09	253.95
12	1.030.00	1.33	773.98	9.287.74
			Total	10.940.74

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

=10940.74/(1060.85*2)

=5.156593

Modified duration = Macaulay duration/(1+YTM)

=5.16/(1+0.0482)

=5.035244

c

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($1.060.85)	=(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	30.00	1.02	29.29	29.29	55.86
2	30.00	1.05	28.60	57.21	163.65
3	30.00	1.07	27.93	83.79	319.59
4	30.00	1.10	27.27	109.10	520.11
5	30.00	1.13	26.63	133.16	761.81
6	30.00	1.15	26.01	156.03	1.041.43
7	30.00	1.18	25.39	177.76	1.355.90
8	30.00	1.21	24.80	198.37	1.702.28
9	30.00	1.24	24.21	217.91	2.077.77
10	30.00	1.27	23.64	236.43	2.479.74
11	30.00	1.30	23.09	253.95	2.905.66
12	1.030.00	1.33	773.98	9.287.74	115.124.75
			Total	10.940.74	128.508.54

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

=128508.54/(1060.85*2^2)

=30.284

d

Using only modified duration

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

=-5.04*0.01*1060.85

=-53.42

%age change in bond price=Mod.duration prediction/bond price

=-53.42/1060.85

=-5.04%

New bond price = bond price+Modified duration prediction

=1060.85-53.42

=1007.43

Actual bond price change

K = Nx2

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

k=1

K =6x2

Bond Price =∑ [(6*1000/200)/(1 + 5.82/200)^k]     +   1000/(1 + 5.82/200)^6x2

k=1

Bond Price = 1009.01

%age change in price =(New price-Old price)*100/old price

%age change in price = (1009.01-1060.85)*100/1060.85

= -4.89%

Difference in price predicted and actual

=predicted price-actual price

=1007.43-1009.01

=-1.58

%age difference = difference/actual-1

=-1.58/1009.01

=-0.1562%

e

Using convexity adjustment to modified duration

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

0.5*30.28*0.01^2*1060.85

=1.61

%age change in bond price=(Mod.duration pred.+convex. Adj.)/bond price

=(-53.42+1.61)/1060.85

=-4.88%

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

=1060.85-53.42+1.61

=1009.04

Difference in price predicted and actual

=predicted price-actual price

=1009.04-1009.01

=0.03

%age difference = difference/actual-1

=0.03/1009.01

=0.003%