Question

1. Prices of several bonds are given below:

   *Half

   Bond Principal($)	Time to maturity(years)	Annual coupon*($)	Bond price($)
   100	0.5	0	98.9
   100	1	0	97.5
   100	1.5	4	101.6
   100	2	4	101.9

   the stated coupon is assumed to be paid semiannually.
   (a) Use the bootstrap method to find the 0.5-year, 1-year, 1.5-year and 2-year zero rates per annum with continuous compounding.
   (b) What is the continuously compounded forward rate for the period between the 1-year point and the 2-year point?

Answer 1

A. Zero Rates Calculation:

0.5 Year Zero Rate: It can be calculated using 1st bond which has maturity of 0.5 years only. YTM on this bond will be Zero Rate for 0.5 Years.

YTM on 1st bond =

YTM = [(100 / 98.9) ^{(1 / 0.5)} - 1] * 100 = 2.24% p.a. continuously compounded

1 Year Zero Rate: It can be calculated using 2nd bond which has maturity of 1 years only. YTM on this bond will be Zero Rate for 1 Years.

YTM on 2nd bond =

YTM = [(100 / 97.5) ^{(1 / 1)} - 1] * 100 = 2.56% p.a. continuously compounded

1.5 Year Zero Rate: It can be calculated by seperating the cashflow according to their maturity. All cashflows prior to 1.5 years will be seperated out and their value is subtracted from present value to get present value of cashflow occuring in 1.5 years from now. On that we can apply out above logic to determine zero rate.

Year (From Now) Cashflow (Semiannual Coupon) Discount Rate Present Value

0.5 2 2.24% 1.978

1 2     2.56% 1.950

Total 3.928

Present Value of 3rd Bond = 101.6

Present Value of above coupons = 3.928

Present Value of Cashflow at 1.5 Years = 101.6 - 3.928 = 97.672

Future CashFlow = 100 (Face Value) + 2 (Coupon) = 102

By using the logic of YTM,

YTM on 3rd bond =

YTM = [(102 / 97.672) ^{(1 / 1.5)} - 1] * 100 = 2.93% p.a. continuously compounded

2 Year Zero Rate: It can be calculated by seperating the cashflow according to their maturity. All cashflows prior to 2 years will be seperated out and their value is subtracted from present value to get present value of cashflow occuring in 2 years from now. On that we can apply out above logic to determine zero rate.

Year (From Now) Cashflow (Semiannual Coupon) Discount Rate Present Value

0.5 2 2.24% 1.978

1 2     2.56% 1.950

1.5 2 2.93% 1.915

Total 5.843

Present Value of 3rd Bond = 101.9

Present Value of above coupons = 5.843

Present Value of Cashflow at 1.5 Years = 101.9 - 5.843 = 96.057

Future CashFlow = 100 (Face Value) + 2 (Coupon) = 102

By using the logic of YTM,

YTM on 4th bond =

YTM = [(102 / 96.057) ^{(1 / 2)} - 1] * 100 = 3.05% p.a. continuously compounded

Year Zero Rates (p.a. Continuously Compounded)

0.5 Year 2.24%

1 Year 2.56%

1.5 Year 2.93%

2 Year 3.05%

2) Forward Rate between 1 and 2 Year:

Forward Rate =

Forward Rate = 3.54% p.a. continuously compounded