# Practice Problems for Lecture 2

## Question 1

In later problems we will work with semi-annual yields quoted annually (like bond-equivalent yields used in practice) or continuous yields (also used in practice), but for these problems we will keep it simple and use annual interest rates and cash flows.

Assume you observe the following three coupon bond prices and remaining cashflows.

Bond A is currently trading at a price of 114.51, has a face value of 100 and 25% coupon and three years to maturity.

Bond B is currently trading at a price of 117.42, has a face value of 100 and 25% coupon and two years to maturity.

Finally, Bond C is currently trading at a price of 113.63, has a face value of 100 and 25% coupon and 1 year to maturity.

(A) First, find the zero-coupon discount factors for one, two and three years out (i.e. find D(0,1), D(0,2) and D(0,3)).

Hint: D(0,1) is easy to find just from the information about Bond C. For D(0,2) you need to construct a portfolio of Bond C and Bond B that has a zero payoff 1 year from now. Finally, to get D(0,3) you need to construct a portfolio of all three bonds that has zero payoffs in both 1 and 2 years from now.

(B) Next, compute the zero-coupon yields for one, two and three years out (i.e. z(0,1), z(0,2) and z(0,3)).

(C) Compute the forward rates implied by the prices of these bonds for one, two and three years out (i.e. f(0,1), f(0,2) and f(0,3)).

(D) Finally, compute the par coupon bond yields for one, two and three years out (see the last two slides in Lecture 2).

## Question 2

This problem uses the same information as in the previous problem, but only considers cash flows out to a two-year horizon. Assume that the one-year spot rate is the same as the forward rate f(0,1)(10%). Furthermore, you believe/know that in year 1 you can borrow at a rate of 15%.

A. Is there any arbitrage opportunity?

B. If there is an arbitrage opportunity, try to construct a strategy using one-year and two-year zero-coupon bonds.

C. (optional) Try to construct a strategy using spot borrowing, Bond B and Bond C from the previous problem.

Note: Please refer to the previous problem for forward rates, discount factors, and coupon bond prices.

## Question 3

Suppose the yield on 1-year LIBOR is 2\%, on 2-year LIBOR is 3\%, on 3-year libor is 4\%, and on 4-year LIBOR is 5\%. (These are rates for discount bonds, with borrowing/lending today for repayment at a single future point in time.)

A. Use the LIBOR yields to compute the discount factors 0,1,2,3, and 4 years out.

B. Use the discount factors to compute the spot rate and the forward rate for borrowing or lending from year 1 to year 2, year 2 to year 3, and year 3 to year 4.

C. Use the forward rates to compute the discount bond yields using the approximation that the discount bond yield is approximately the average of the corresponding spot/forward rates over the subperiods.

D. Use the discount factors to compute the par coupon bond yield at each maturity 1,2,3, and 4.