Question

2(A) A 10-year bond has a face value of EUR1000 pays a 6% annual coupon rate. The required market yield is 6.5%. What is its convexity?

2(B) A 10-year bond has a face value of EUR1000 pays a 6% annual coupon rate and is traded at 102%. The market yield is 5.73%. What are its duration and convexity? If the required yield changes by +200 basis points, compare the actual bond price change with using duration and convexity rule to estimate the bond price change?

Answer 1

2A

K = N

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

k=1

K =10

Bond Price =∑ [(6*1000/100)/(1 + 6.5/100)^k]     +   1000/(1 + 6.5/100)^10

k=1

Bond Price = 964.06

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($964.06)	=(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	60.00	1.07	56.34	56.34	99.34
2	60.00	1.13	52.90	105.80	279.84
3	60.00	1.21	49.67	149.01	525.51
4	60.00	1.29	46.64	186.56	822.40
5	60.00	1.37	43.79	218.96	1,158.31
6	60.00	1.46	41.12	246.72	1,522.66
7	60.00	1.55	38.61	270.27	1,906.31
8	60.00	1.65	36.25	290.03	2,301.38
9	60.00	1.76	34.04	306.37	2,701.15
10	1,060.00	1.88	564.69	5,646.90	54,765.02
			Total	7,476.96	66,081.92

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

=66081.92/(964.06*1^2)

=68.55

2B

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($1,020.13)	=(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	60.00	1.06	56.75	56.75	101.53
2	60.00	1.12	53.67	107.35	288.08
3	60.00	1.18	50.76	152.29	544.93
4	60.00	1.25	48.01	192.05	859.00
5	60.00	1.32	45.41	227.05	1,218.67
6	60.00	1.40	42.95	257.70	1,613.67
7	60.00	1.48	40.62	284.36	2,034.96
8	60.00	1.56	38.42	307.37	2,474.58
9	60.00	1.65	36.34	327.05	2,925.59
10	1,060.00	1.75	607.19	6,071.89	59,747.50
			Total	7,983.84	71,808.50

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

=7983.84/(1020.13*1)

=7.83

Modified duration = Macaulay duration/(1+YTM)

=7.83/(1+0.0573)

=7.4

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

=71808.5/(1020.13*1^2)

=70.39

Actual bond price change

K = N

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

k=1

K =10

Bond Price =∑ [(6*1000/100)/(1 + 7.73/100)^k]     +   1000/(1 + 7.73/100)^10

k=1

Bond Price = 882.49

Using convexity adjustment to modified duration

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

0.5*70.39*0.02^2*1020.13

=14.36

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

=(-151.02+14.36)/1020.13

=-13.4%

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

=1020.13-151.02+14.36

=883.47

Difference in price predicted and actual

=predicted price-actual price

=883.47-882.49

=0.978

%age difference = difference/actual-1

=0.98/882.49

=0.1109%