Determining the formula for the instantaneous forward rates

Question

1. Extract the term structure of interest rates out to 3 years given the following bond data:

Maturity (yrs)       Coupon rate (%)          Yield to maturity (%)
    0. 5                      1. 0                             1. 5
    1. 0                      1. 5                             1. 75
    1. 5                      2. 0                              2. 0
    2. 0                      2. 5                              2. 25
    2. 5                      3. 0                              2. 50
    3. 0                      3. 5                              2. 75

2. The time t= 0 continuously compounded term structure of interest rates is given by r(0,T)=0. 05 - 0. 005e^-0.10T. Find the price of a Treasury bond with exactly 3 years to maturity and a coupon rate of 4. 0%.

3. Find a formula for the instantaneous forward rates f(0;T) associated with the continuously compounded term structure of interest rates

r(0,T)=0. 05 - 0. 005e^-0.10T.

4. At time t an investor shorts a $1 face value zero coupon bond that matures at time T >t and uses the entire proceeds to purchase a zero coupon bond that matures at time S > T.

(a) In what quantity is the zero coupon bond that matures at time S purchased? Your answer should be expressed in terms of the time t prices P(t,T) and P(t,S).

(b) Explain why these transactions are equivalent to agreeing to lend over the future period [T, S] at a rate that is determined at time t.

(c) What is the continuously compounded forward rate f(t;T;S) associated with this loan?

5. A bond is said to be at par if its flat price is equal to its face value. The Par Bond Yield Curve out to N years is a plot of their yields to maturity {y₀⁽ⁱ⁾}_i-1^2N on the vertical axis versus their respective maturities {i/2yrs}^2N i-1 on the horizontal axis. Given the following term structure of interest rates

tj (yrs)      r₀^(j) (%)
0.5               2.0
1. 0              2. 5
1. 5              3. 0
2. 0              3. 5
2. 5              4. 0
3. 0              4. 5

find the par bond yields to maturity {y₀⁽ⁱ⁾}_i=1⁶