Question

The cash prices of six-month and one-year Treasury bills are 98.0 and 95.0, respectively. A 1.5-year bond that will pay coupons of $3 every six months currently sells for $96. A two-year bond that will pay coupons of $7 every six months currently sells for $101. Assume the principal of the bond is $100. Calculate the (a) six-month, (b) one-year, (c) 1.5-year, and (d) two-year zero rates.

Please give me the process, thank you!

Answer 1

The 6-month Treasury bill provides a return of 2/98 = 2.041% in six months. This is 2*2.041 = 4.082% per annum with semiannual compounding or 2ln(1.02041) = 4.041% per annum with continuous compounding.

The 12-month rate is 5/95 = 5.263% with annual compounding or ln(1.05263) = 5.130% with continuous compounding.

For the 1.5 years bond we must have the following:

3e^-0.04041*0.5 + 3e^-0.05130*1 + 103e^-1.5R = 96

(where R is the 1.5 years zero rate)

2.94 + 2.85 + 103e^-1.5R = 96

e^-1.5R = 0.8752

R = ln(0.8752)/-1.5 (taking inverse 'ln' both side)

R = 8.89%

For the 2 years bond we must have the following:

7e^-0.04041*0.5 + 7e^-0.05130*1 + 7e^-0.0889*1.5 + 107e^-2R = 101

(where R is the 2 years zero rate)

6.86 + 6.65 + 6.13 + 107e^-2R = 101

e^-2R = 0.7604

R = ln(0.7604)/-2 (taking inverse 'ln' both side)

R = 13.70%

So, 2.020%, 5.130%, 8.89% and 13.70% are the respective answers.