# Solutions for 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.

SOLUTION:

It is easiest to compute D(0,1) first -- it is just equal to the ratio of the current price of bond C to its last cashflow at maturity, i.e. D(0,1) = 113.63/125 = 0.90904. In order to figure out D(0,2) we need to construct a portfolio of bonds that has only cashflows in the current year and in year 2 -- no cashflows in year 1. This can be accomplished by buying 1 bond B and selling short 0.2 bonds C resulting in a cashflow of 94.694 (= 117.42 - 0.2 x 113.63) in the current year and 125 in year 2. This yields a value of D(0,2) = 94.694/125 = 0.757552. Finally, obtaining D(0,3) involves constructing a portfolio of bonds that has no intermediate cashflows, i.e. no cashflows in years 1 and 2. This can be done by buying 1 Bond A and selling short 0.2 of Bond B and 0.16 of Bond C. This portfolio produces the following cashflows: 72.845 in the current year, 0 in years 1 and 2, and 125 in year 3. Hence, D(0,3) = 72.845/125 = 0.58276.

(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)).

SOLUTION:

This is straightforward to do given our results from (A) above and the formulas on page 14 of Lecure 2.

z(0,1) = D(0,1)^(-1) - 1 = 0.100062

z(0,2) = D(0,2)^(-1/2) - 1 = 0.148931

z(0,3) = D(0,3)^(-1/3) - 1 = 0.197208

where a^b means a raised to the power b.

(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)).

SOLUTION:

Again this is going to be easy given that the hard work is done in (A) above. Just use the formulas from page 17 from Lecture 2.

f(0,1) = (D(0,0)/D(0,1)) - 1 = (1/D(0,1)) - 1 = 0.100062

f(0,2) = (D(0,1)/D(0,2)) - 1 = 0.19997

f(0,3) = (D(0,2)/D(0,3)) - 1 = 0.299935

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

SOLUTION:

The following utilizes the formulas from the last slide in Lecture 2:

y(1) = (1-D(0,1))/D(0,1) = 0.100062

y(2) = (1-D(0,2))/(D(0,1)+D(0,2)) = 0.145475

y(3) = (1-D(0,3))/(D(0,1)+D(0,2)+D(0,3)) = 0.185473

## Question 2

### Arb Exercise Solution

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?

The implied forward rate f(0,2) is 0.19997, which is greater than 15%. Borrowing money is too cheap, so we should borrow money.

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

At the scale of \$1, the arb is:

 time (years from now) 0 1 2 borrow forward 0 1 -1.15 sell one-year zero 0.90904 -1 0 buy two-year zero -0.871185 0 1.15 total net cash flow 0.037855 0 0

Since we use zero-coupon bonds, it does not matter which year is used as starting point. Here we start from year 1. We sell one-year zero-coupon bond, and net out year 1 cash flow by borrowing forward \$1, then we net out year 2 cash flow by buying a fraction of a two-year zero-coupon bond (0.757552*1.15 = 0.87118). The net profit is 0.037855.

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

Assume one cannot buy fractional coupon bonds, the arb is as follows.

 time (years from now) 0 1 2 borrow forward 0 108.7 -125 borrow 7.91 -8.7 0 sell one-year zero 113.63 -125 0 buy two-year zero -117.42 25 125 total net cash flow 4.12 0 0

Basically, we start from the last year (the second year in this case), and work back to the present time (t = 0). We first net out year 2 cash flow by borrowing forward \$108.7(\$125/1.15), then we net out year 1 cash flow by borrowing spot \$7.91(\$8.7/1.1), which gives us a net porfit of \$4.12 at time 0.

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.

The discount factors are 1, 1/1.02=0.9804, 1/1.03^2=0.9426, 1/1.04^3=0.8890, 1/1.05^4=0.8227.

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.

The forward rates are 1/0.9804-1=2.00%, 0.9804/0.9426-1=4.010%, 0.9426/0.8890-1=6.029%, and 0.8890/0.8227-1 =8.058%. The initial forward rate (from 0 to 1) is the spot rate.

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. Compare with the original data.

The approximate discount bond yields are 2.000=2.000%, (2.000+4.010)/2=3.005%, (2.000+4.010+6.029)/3=4.013%, and (2.000+4.010+6.029+8.058)/4=5.024%. These are only slightly different (due to compounding) from the original values.

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

The coupon bond yields are (1-0.9804)/0.9804=2.000%, (1-0.9426)/(0.9804+0.9426)=2.985%, (1-0.8890)/(0.9804+0.9426+ 0.8890)=3.948%, and (1-0.8227)/(0.9804+0.9426+ 0.8890+0.8227)=4.878%.