Question

Find the duration of a 5% coupon bond making annual coupon payments if it has three years until maturity and a yield to maturity of 6.3%. What is the duration if the yield to maturity is 10.3%?

Answer 1

K = N

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

k=1

K =3

Bond Price =∑ [(5*1000/100)/(1 + 6.3/100)^k]     +   1000/(1 + 6.3/100)^3

k=1

Bond Price = 965.44

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($965.44)	=(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	50.00	1.06	47.04	47.04	83.25
2	50.00	1.13	44.25	88.50	234.96
3	1,050.00	1.20	874.16	2,622.47	9,283.34
			Total	2,758.01	9,601.55

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

=2758.01/(965.44*1)

=2.856735

K = N

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

k=1

K =3

Bond Price =∑ [(5*1000/100)/(1 + 10.3/100)^k]     +   1000/(1 + 10.3/100)^3

k=1

Bond Price = 868.89

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($868.89)	=(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	50.00	1.10	45.33	45.33	74.52
2	50.00	1.22	41.10	82.20	202.68
3	1,050.00	1.34	782.46	2,347.38	7,717.79
			Total	2,474.91	7,994.99

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

=2474.91/(868.89*1)

=2.848358