Question

This is a zero rates bootstraping question. Suppose there are three coupon bearing treasury bonds. The face value of all three bonds are $100. The first coupon payments for all three bonds will take place in 6 months. The first bond provides coupon at rate of 2% per annum semiannually and will mature in 1.5 years. The current price of the first bond is 98.8102. The second bond pays coupon at the rate of 2% per annum semiannually and will mature in 1 year. The current price for the second bond is 99.4954. The third bond pays coupon at the rate of 1 % per annum annually and will mature in 1.5 years. The price of the third bond is 97.8349. What are 0.5-year zero rate, 1-year zero rate, and 1.5-year zero rate, assuming the zeros rates are measured with continuously compounding? (Hint: using simultaneous equations to find the zero rates. You should have 3 equations with 3 unknown. This simultaneous equations system can be solved analytically.)

Answer 1

Let the interest rate for 0.5 year. 1 year and 1.5 years be r1, r2 and r3 respectively

For 1st bond, 3 coupon amounts of $ 1 are paid at the end of 6,12 and 18 months and $100 at the end of 18 months

So, 1*exp(-r1/2) +1*exp(-r2)+101*exp(-r3*1.5) = 98.8102 ...................... (1)

For 2nd bond, 2 coupon amounts of $1 are paid at the end of 6 and12 months and $100 at the end of 12 months

So, 1*exp(-r1/2) +101*exp(-r2) = 99.4954 ...................... (2)

For 3rd bond, 3 coupon amounts of $0.5 are paid at the end of 6,12 and 18 months and $100 at the end of 18 months

So, 0.5*exp(-r1/2) +0.5*exp(-r2)+100.5*exp(-r3*1.5) = 97.8349 ...................... (3)

2* equation(3) -equation (1) gives

201*exp(-r3*1.5) - 101*exp(-r3*1.5) = 96.8596

=> exp(-r3*1.5) = 96.8596/100 = 0.968596

=> -r3*1.5 = ln(0.968596) = -0.03191

r3 = 0.02127 or 2.127%

equation(2) -equation (1) gives

100*exp(-r2) - 101*exp(-r3*1.5) = 0.6852

=>100*exp(-r2) -101*0.968596 =0.6852

=>  exp(-r2) = 0.985134

-r2 = ln(0.985134) = -0.01498

r2 = 1.498%

Equation (2) gives

1*exp(-r1/2) +101*exp(-r2) = 99.4954

=> exp(-r1/2) + 101*0.985134 = 99.4954

=> exp(-r1/2) = -0.00313

r1/2 = ln(-0.00313) which is undefined

So, the zero rate for 0.5 years is undefined

zero rate for 1 year is 1.498% p.a. continuously compounded

zero rate for 1.5 year is 2.127% p.a.continuously compounded