Question

A zero-coupon bond has a principal of $100 and matures in four years. The market price for the bond is $72. Calculate the yield to maturity, duration and convexity of the bond.

(Please provide a well detailed answer with the equations that are being used. Thank you!)

Answer 1

K = N

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

k=1

K =4

72 =∑ [(0*100/100)/(1 + YTM/100)^k]     +   100/(1 + YTM/100)^4

k=1

YTM% = 8.56

Duration

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($72.00)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	-	1.09	-	-
2	-	1.18	-	-
3	-	1.28	-	-
4	100.00	1.39	72.00	287.99
			Total	287.99

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

=287.99/(72*1)

=4

Modified duration = Macaulay duration/(1+YTM)

=4/(1+0.0856)

=3.68

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($72.00)	=(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	-	1.09	-	-	-
2	-	1.18	-	-	-
3	-	1.28	-	-	-
4	100.00	1.39	72.00	287.99	1,221.83
			Total	287.99	1,221.83

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

=1221.83/(72*1^2)

=16.97