Question

Assume the following characteristics for a particular bond: A face value of $1,000; annual coupon payments of $60 (the first payment due in 1 year); an internal yieldto-maturity of 7% (compounded annually); and a three year term. (a) Compute the Macaulay duration of the bond. (b) Given your answer above, compute the approximate change in the bond’s value if the yield fell to 6.5%. (c) Now compute the actually change in the bond’s value. Comment on the difference.

Answer 1

a

K = N

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

k=1

K =3

Bond Price =∑ [(6*1000/100)/(1 + 7/100)^k]     +   1000/(1 + 7/100)^3

k=1

Bond Price = 973.76

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($973.76)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	60.00	1.07	56.07	56.07
2	60.00	1.14	52.41	104.81
3	1,060.00	1.23	865.28	2,595.83
			Total	2,756.71

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

=2756.71/(973.76*1)

=2.831

b

Modified duration = Macaulay duration/(1+YTM)

=2.83/(1+0.07)

=2.645794

Using only modified duration

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

=-2.65*-0.005*973.76

=12.88

c

Actual bond price change

K = N

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

k=1

K =3

Bond Price =∑ [(6*1000/100)/(1 + 6.5/100)^k]     +   1000/(1 + 6.5/100)^3

k=1

Bond Price = 986.76

change in price =(New price-Old price)

change in price = (986.76-973.76)=13

d

Difference is due to convexity of yield curve whereas duration is a straight line approximation