Question

The (zero coupon) U.S. Treasury strip maturing in one year and having a face value of $1000 is selling at an annualized yield to maturity of 2.40 percent, which is equivalent to a price of 97.65625 percent of par (face) value. The (zero coupon) U.S. Treasury strip (with a face value of $1000) maturing in two years is selling at an annualized yield to maturity of 2.8 percent, which is equivalent to a price of 95.083 percent of par value. A coupon-paying U.S. Treasury bond having a coupon rate of 2.5 percent, a face value of $1000 and three years to maturity is selling at 97.5247 percent of par (face) value. Assuming that interest payments for coupon-paying bonds are paid annually (once per year) and that bond yields are quoted as annualized interest rates (don't worry about semiannual compounding),

a. determine the price and annualized yield for a U.S. Treasury strip having a face value of $1000 and 3 years to maturity (5 points),

b. determine the forward rate of interest for the 2-year period that begins at the end of 1 year (5 points),

c. determine the price for a U.S. Treasury bond (making annual coupon payments) having a face value of $1000, a 5.5 percent coupon rate and 3 years to maturity.

Answer 1

Part a

Since the the price of 3 year treasury is 97.5247% of Face value which 1000 hence the price of this srip is = 1000 ×.975247= 975.247$

Annualised yield for a coupon bearing bond is given by

Y =(I +(F- P)/n)/((F+P)/2)

Where Y = annualised yield

I = coupon amount = 1000×2.5%=25

F= face value = 1000$

P= price = 975.247$

n = years to maturity = 3 years

Therefore

Y = (25 +(1000 -975.247)/3)/((1000 +975.247)/2) = 3.37%

Part b

We need 2 year running forward rate starting from year the end of year 1 i.e. we need f(1,2) which is given by

Factor of lower spot rates × factors of forward rates = factors of higher spot rates

(1+r01)^1 ×(1+f(1,2))^2 = (1+ r03)^3

Where

r01 = 1 yesr spot rate = yield of 1 year Zero coupon bond which is given in the question = 2.4%

But we don't have r03 niether we have 3 year zero coupon bond so we have to extract 3 year spot rate from coupon bearing bond using bootstrapping technique which says that discount N th cashflows at Nth unknown spot rate and all previous cashflows at (N-1)th known spot rates i.e. we have following relation

Price = I/(1+r01) +I/(1+r02)^2 + (I + F)/ (1+ r03)^3

We know r02 i.e. 2 year spot rate = yield of 2 year ZCB = 2.8%

Hence

975.247 = 25/1.024 +25/(1.028)^2 + 1025/(1+r03)^3

Solving for r03 we get r03 = 3.4%

Now lets calculate the forward rate f(1,2)

(1.024)×(1+f(1,2))^2 =1.034^3

Solving for f(1,2) we get f(1,2) = 3.90%this is our required forward rate.

Part c

Price of bond = present value of its future cashflows

Since we are not given the yield of this bond nor of any similar bond so we have to use spot rates to calculate the present values

Therefore the price = 55/1.024 +55/1.028^2 +1055/1.034^3

Price = 1060.06

Note that by using this price we can now calculate the yield for this bond