 # T-bond (Pure exp. theory, Arbitrage-free, Yield, Duration...)

A two-year 4% Treasury Note has 12 months to expiry. The Note pays semiannual coupons (hence there are two coupon payments left). Assume that you know the following: Price of the 6-month T-Bill is 98.6527 The 6-month forward rate is F(6m; 12m) = 3:25% (a) According to the pure-expectations theory of the term structure of interest rates, how should the 6-month spot rate evolve? (b) What is the arbitrage-free price of the 4% T-Note? © What is the yield to maturity of the 4% T-Note? Hint: The solution involves finding roots of a quadratic equation. Only one of the roots has economic meaning. (d) Calculate duration and modifed duration of the 4% T-Note.

My answers:** a)** The interest rate curve will be decreasing.

**b)**The current market price for 6m is 98,6527; thus, the price of the 1y is 95,6798 (102/(1+0.0325)^2,

Therefore,the arbitrage free price is: 2(coupon) x 98,6527/100 + 2 x 95,6798/100 + 100 x 95,6798 = 99,5665

c) How should I construct a quadratic equation ? ie (2/(1+yt) + 2/(2/(1+yt)^2 + 100/(1+yt)^2=99,5665

d) Should I calculate the duration & MD for the whole 2 years using just the 4% or else… ? I appreciate !

Anyone ?

Hi Mil,

just the answers I could resolve quickly:

a) Decreasing term structure as S(0,6) = 4% > F(6,6) = 3,25%

b) Given the term structure above I would calculate the yield/discount rate as follows: Yield = (((1 + 0,4) * (1 + 0,325))^0,5) - 1 = 0,0362 ~ 3,62%

Arbitrage-free price of the bond given the cash flow stream of \$2 in 6 months and \$102 at maturity in 12 months (semiannual coupons): FV = 102; PMT = 2; I/Y = 3,62/2 = 1,81%; N = 2 <=> CPT PV = 102,30

c) You can solve this algebraically by solving the equation: 102,30 = 2/(1 + YTM) + 102 (1+ YTM)^2 The result is simply the discount rate calculated above of 3,62%. (Maybe I did not understand the question fully.)

d) MD is calculated using the YTM. You can use the BOND Worksheet on your BA II Plus calculator to calculate it.

Regards, Oscar