# Practice Problem Answers for Lecture 4

1. Consider the following situation. Today you are looking at a Treasury bond with a coupon of 10% paid semiannually and a face value of \$100 which is one year from maturity and is currently trading at \$99.10 after this period's coupon has already been paid. You have also obtained prices of Treasury STRIPs -- one that matures in 6 months and is currently trading at \$95.24 with a face of \$100 and another one maturing in a year with a price of \$89.85 with a face of \$100.

(A) What is the implied forward rate for borrowing and lending 6 months from today?

Given the two STRIP prices we can compute the discount factors as D(0,1/2) = 95.24/100 = 0.9524 and D(0,1) = 89.85/100 = 0.8985. The forward rate f(1/2,1) is then given by the following formula: f(1/2,1) = D(0,1/2)/D(1/2,1) - 1 = 5.9989% or roughly 6%.

(B) What is the bond-equivalent yield of the Treasury bond?

By definition the bond-equivalent yield is the yield which equates the present value of coupons and face to the current price, i.e.

99.10 = (5/(1+y/2)) + (105/(1+y/2)^2)

where a^b means a raised to the power b. Instead of solving for y directly it pays to substitute x = 1/(1+y/2) in the above equation which then becomes:

99.10 = 5x + 105x^2

which is just a quadratic equation for x. Solving for the positive root we obtain

x = (-5 + sqrt(25+4*105*99.10))/(2*105) = 0.9479808

Now it is easier to get y which is just equal to 2*(1/x - 1) = 10.9747%.

(C) What is the Macaulay duration of the Treasury bond with respect to its own yield?

Before we proceed it pays to compute x^2 from (B) above since it will come in handy for computing the duration: x^2 = 0.8986676. Now the duration is simply:

duration = (5*0.5*0.9479808 + 105*1*0.8986676) / (5*0.9479808 + 105*0.8986676) = 0.976085.

(D) What is the Macaulay duration of the Treasury bond with respect to the discount factors in the economy?

Here we are going to be using the discount factors implied by the STRIP prices in (A) above instead the discount factors computed using the bond's own yield:

duration = (5*0.5*0.9524 + 105*1*0.8985) / (5*0.9524 + 105*0.8985) = 0.975975.

(E) What is the effective duration of the Treasury bond with respect to the discount factors in the economy?

First we need to compute the sensitivities of the two cash flows as follows:

sens(1/2) = (1-exp(-0.125*0.5))/0.125 = 0.484695497
sens(1) = (1-exp(-0.125*1))/0.125 = 0.940024779

Then the sensitivity of the duration is simply equal to:

sens(effective duration) = (5*0.484695497*0.9524 + 105*0.940024779*0.8985) / (5*0.9524 + 105*0.8985) = 0.918146

Note that in the above formula we have used the economy-wide discount factors. Finally, using the formula from the slides inverting the sensitivity we obtain the effective duration as:

effective duration = -log(1-0.125*0.918146)/0.125 = 0.975247.

A note on notation -- be careful what logarithms you are using on a computer/calculator or in Excel. The logarithm above should be the natural logarithm, i.e. the one which has a base equal to Euler's number e = 2.718281828... In Excel and on most calculators the symbol for that logarithm is LN. The symbol LOG usually refers to the logarithm with base 10.

2. Consider a fixed income security which pays one half of its value in 20 years, one quarter in 10 years and one quarter in 5 years.

(A) What is the Macaulay duration of this security?

duration = (20*(1/2)+10*(1/4)+5*(1/4)) / ((1/2)+(1/4)+(1/4)) = 55/4 = 13.75.

(B) What is the effective duration of this security?

Again, as in (E) above we need to compute the sensitivities for the three maturities:

sens(20) = (1-exp(-0.125*20))/0.125 = 7.343320011
sens(10) = (1-exp(-0.125*10))/0.125 = 5.707961625
sens(5) = (1-exp(-0.125*5))/0.125 = 3.717908572

sens(effective duration) = ((1/2)*7.343320011 + (1/4)*5.707961625 + (1/4)*3.717908572) / ((1/2)+(1/4)+(1/4))  = 6.028127555

Finally, inverting the sensitivity we obtain:

effective duration = -log(1-0.125*6.028127555)/0.125 = 11.20366376.