Question

4.23. The cash prices of 6-month and 1-year Treasury bills are 94.0 and 89.0. A 1.5-year Treasury bond that will pay coupons of $4 every 6 months currently sells for $94.84. A 2-year Treasury bond that will pay coupons of $5 every 6 months currently sells for $97.12. Calculate the 6-month, 1-year, 1.5-year, and 2-year Treasury zero rates 4.24. "An interest rate swap where 6-month LIBOR is exchanged for a fixed rate of 5% on a

Answer 1

Solution

Spot Rate=((Face Value of the bond/Current Bond Price)^(1/Years To Maturity)−1) *100

Hence, the spot rate of the 6-month old bond, assuming face value of $100 is (((100/94)^(1/0.5))-1)*100 = 13.17%

Similarly, the spot rate for the one-year bond is ((100/89)^(1/1) -1)*100 = 12.36%

The next step is to figure out the spot rate for the 1.5 year bond, using these two spot rate. The 1.5 year bond has three payments. We know the spot rate for the first two payments. we discount the cash flows using their respective cash flows.

We can calculate the 1.5 year zero rate or spot rate as follows

(4/(1+ 0.1317/2)) + (4/(1+ 0.1236/2)^2) + (104/(1 + S1.5/2)^3) = $94.84

Where, S1.5 is the spot rate of the 1.5-year bond and $94.84 is the price of the bond as mentioned in the question.

Hence, $94.84 - $3.753 - $3.548 = 104/(1 + S1.5/2)^3

$87.54 = $104/(1 + S1.5/2)^3

(1 + S1.5/2)^3 = $104/$87.54 =1.188

1 + S1.5/2 = 1.188^ (1/3) =1.0591

S1.5/2 = 0.0591

Spot rate for the 1.5 year Treasury note = 11.82%

Finally, we can calculate the 2-year spot rate

(5/(1+ 0.1317/2)) + (5/(1+ 0.1236/2)^2) + (5/(1 + 0.1182/2)^3) +(105/(1 + S2/2)^4) = $97.12 , where S2 is the 2-year spot rate

$4.691 + $4.435 + $4.209 +(105/(1 + S2/2)^4) = $97.12

(105/(1 + S2/2)^4) = $97.12 - ($4.691 + $4.435 + $4.209)

($105/(1 + S2/2)^4) = $83.785

(1 + S2/2)^4) = $105/$83.785

(1 + S2/2)^4) = 1.2532

1 + S2/2 = (1.2532)^(1/4) = 1.05805

S2/2= 1.05805

S2 = 0.11609 or 11.61%