Question

Calculate the duration for a bond with an annual coupon of 8%, nominal value of 100 euros, time until maturity 5 years and yield at maturity equal to 10%.
a) What is the modified duration of the bond?
b) What do we call yield to maturity at the end of a bond and what does it represent? What is the relationship between the current and the nominal value of the bond when the bond is traded for, under, or even?
c) If the yield on maturity increases sharply from 10% to 10.30%, what will be the new bond price?

Answer 1

a)

Duration = sum of (weights * years) / sum of years where weights are proportion of present value of cash flows to the total value of present value of cash flows.

Years	Cashflow	Discounting Factor @ 10%(1/ (1+r)^n)	Present Value of cash flows ( C= A*B)	Weight	weight * year
1	8	0.909090909	7.27	0.078693	0.078693477
2	8	0.826446281	6.61	0.071540	0.143079048
3	8	0.751314801	6.01	0.065036	0.195107793
4	8	0.683013455	5.46	0.059124	0.236494295
5	108	0.620921323	67.06	0.725607	3.628037474
Total	92.42	1.000000	4.281412086

Duration = 4.2814 years

Modified duration = Duration / ( 1 + ytm factor) = 4.2814 / ( 1.1) = 3.89%

b)

the yield to maturity moves closer to coupon rate as an when the maturity is nearing by. So here ytm of 10 % will be close to 8% when the bond reaches maturity.

When the bond is traded under, current value is lower than the nominal value and the ytm shall be higher than the coupon rate.

When the bond is traded over, current value is higher than the nominal value and the ytm shall be lower than the coupon rate.

c)

Price of the bond = coupon * [ ( 1 - ( 1+r)^-n ] / r + pricipal * 1/ (1+r)^n

= 8 * [ ( 1- 1.1^-5 )/ 0.1 ] + 100 * 1/ 1.1^5

Years	Cashflow	Discounting Factor @ 10.3%(1/ (1+r)^n)	Present Value of cash flows ( C= A*B)
1	8	0.906618314	7.25
2	8	0.821956767	6.58
3	8	0.745201058	5.96
4	8	0.675612926	5.40
5	108	0.612523052	66.15
Total	91.35

New bond price = $91.35