Using the bond valuation formulas (7.1), (7.3), (7.6) we obtain the following yields and prices:
Maturity Zero-Coupon Bond Yield
Zero-Coupon Bond Price
One-Year Implied Forward Rate
Par Coupon
Cont. Comp. Zero Yield
1 0.04000 0.96154 0.04000 0.04000 0.03922
2 0.04500 0.91573 0.05003 0.04489 0.04402
3 0.04500 0.87630 0.04500 0.04492 0.04402
4 0.05000 0.82270 0.06515 0.04958 0.04879
5 0.05200 0.77611 0.06003 0.05144 0.05069
Question 7.2
The coupon bond pays a coupon of $60 each year plus the principal of $1,000 after five years. We have cash flows of [60, 60, 60, 60, 1,060]. To obtain the price of the coupon bond, we multiply each cash flow by the zero-coupon bond price of that year. This yields a bond price of $1,037.25280.
In order to be able to solve this problem, it is best to take equation (7.6) of the main text and solve progressively for all zero-coupon bond prices, starting with Year 1. This yields the series of zero-coupon bond prices from which we can proceed as usual to determine the yields.
a) We are looking for r0 (1, 3). We will use equation (7.3) of the main text, and the known one-year and three-year zero-coupon bond prices. We have to solve the following equation:
Now, we have the relevant implied forward zero-coupon prices and can find the coupon of the par two-year coupon bond issued at time 1 according to formula (7.6).
c = ( )( ) ( )
0
0 0
1 1,3 0.13472 0.0748511,2 1,3 0.93456 0.86528
PP P
−= =
+ +
Question 7.8
a) We have to take into account the interest we (or our counterparty) can earn on the FRA settlement if we settle the loan on initiation day and not on the actual repayment day. Therefore, we tail the FRA settlement by the prevailing market interest rate of 5 percent. The dollar settlement is:
( )1 annually FRA
annually
r rr−
+× notional principal = ( )0.05 0.06
1 0.05−
+ × $500,000.00 = −$4,761.905
b) If the FRA is settled on the date the loan is repaid (or settled in arrears), the settlement amount is determined by:
( )annually FRAr r− × notional principal = (0.05 − 0.06) × $500,000.00 = −$5,000
We have to pay at the settlement because the interest rate we could borrow at is 5 percent, but we have agreed via the FRA to a borrowing rate of 6 percent. Interest rates moved in an unfavorable direction.
a) We have to take into account the interest we (or our counterparty) can earn on the FRA settlement if we settle the loan on initiation day and not on the actual repayment day. Therefore, we tail the FRA settlement by the prevailing market interest rate of 7.5 percent. The dollar settlement is:
( )1 annually FRA
annually
r rr−
+× notional principal = ( )0.075 0.06
1 0.075−
+ × $500,000.00 = −$6,976.744
b) If the FRA is settled on the date the loan is repaid (or settled in arrears), the settlement amount is determined by:
( )annually FRAr r− × notional principal = (0.05 − 0.06) × $500,000.00 = −$5,000
We receive money at the settlement because our hedge pays off. The market interest rate has gone up, making borrowing more expensive. We are compensated for this loss through the insurance that the short position in the FRA provides.
Question 7.10
We can find the implied forward rates using the following formula:
( ) ( )( )0
0
0,1 ,
0,P t
r t t sP t s
+ = =⎡ ⎤⎣ ⎦ +
This yields the following rates on the synthetic FRAs:
r0 (90, 180) = 0.990090.97943
− 1 = 0.010884
r0 (90, 270) = 0.990090.96525
− 1 = 0.025734
r0 (90, 360) = 0.990090.95238
− 1 = 0.039596
Question 7.11
We can find the implied forward rate using the following formula:
The following table follows the textbook in looking at forward agreements from a borrower’s perspective (i.e., a borrower goes long an FRA to hedge his position, and a lender is thus short the FRA).
Transaction t = 0 t = 180 t = 360
Enter short FRA −10M +10M × 1.028403
= 10.28403M
Buy 9.7943M Zero Coupons maturing at −9.7943M +10M
Coupons maturing at time t = 360 ×0.95238 = 9.7943M
TOTAL 0 0 0
By entering in the above mentioned positions, we are perfectly hedged against the risk of the FRA. Please note that we are making use of the fact that interest rates are perfectly predictable.
Question 7.12
We can find the implied forward rate using the following formula:
( ) ( )( )0
0
0,1 ,
0,P t
r t t sP t s
+ = =⎡ ⎤⎣ ⎦ +
With the numbers of the exercise, this yields:
r0 (270, 360) = 0.965250.95238
− 1 = 0.0135135
The following table follows the textbook in looking at forward agreements from a borrower’s perspective (i.e., a borrower goes long on an FRA to hedge his position and a lender is thus short the FRA). Since we are the counterparty for a lender, we are in fact the borrower and thus long the forward rate agreement.
Zero Coupons maturing at time t = 360 ×0.95238 = −9.6525M
TOTAL 0 0 0
By entering in the above mentioned positions, we are perfectly hedged against the risk of the FRA. Please note that we are making use of the fact that interest rates are perfectly predictable.
Question 7.13
First, let us calculate the value of the three-year par coupon bond after we have held it for two years. After two years, the bond still entitles us to receive one coupon and the repayment of the principal. We have to discount those payments, which we receive at t = 3, with the implied forward rate from Year 2 to 3. This determines the value of the three-year par coupon bond after two years. We have:
B2 = 106.954851.0800705
= 99.0258
Furthermore, after one year, we received a coupon of 6.95485 dollars, which we reinvested at the prevailing interest rate, the implied forward rate from Year 1 to Year 2, 1.0700237, and we receive a coupon of 6.95485 at the end of Year 2. The value of the coupons at the end of Year 2 is:
Sum of coupon values = 6.95485 × 1.0700237 + 6.95485 = 14.3967045
Therefore, our two-year gross return is:
2 − year return = 14.3967045 99.0258100
+ − 1 = 1.134225
Finally, we annualize this return by taking its square root. This yields an annualized gross return of 1.065, which was to be shown.
Question 7.14
We would like to guarantee the return of 6.5 percent. We receive payments 6.95485 after Year 1 and Year 2, and a payment of 106.95485 after Year 3. If interest rates are uncertain, we face an
interest rate risk for the investment of the first coupon from Year 1 to Year 2 and for the discounting of the final payment from Year 3 to Year 1.
Suppose we enter into a forward rate agreement to lend 6.95484 from Year 1 to Year 2 at the current forward rate from Year 1 to Year 2, and we enter into a forward rate agreement to borrow 106.95485 tailed by the prevailing forward rate for Year 2 to Year 3, at the prevailing forward rate. This leads to the following cash-flow table:
Transaction today t = 0 t = 1 t = 2 t = 3
Buy three-year par bond −100
Receive first coupon 0 6.95485
Enter short FRA 0 −6.95485 6.95485 × 1.0700237
Receive second coupon 0 6.95485
Enter long FRA for tailed position 0 106.95485/1.0800705 −106.95485
Receive final coupon and principal 0 = 99.025804 106.95485
TOTAL −100 0 113.4225 0
We see that we can secure the same gross return as in the previous question, 113.4225 100÷ = 1.065. By entering appropriate FRAs, we secured the desired return of 6.5 percent. Please note that we made use of the fact that we knew that we wanted to undo the position at t = 2.
Question 7.15
a) Let us follow the suggestion of the problem and buy the two-year zero-coupon bond. We will create a synthetic lending opportunity at the zero-coupon implied forward rate of 7.00238 percent, and we will finance it by borrowing at 6.8 percent, thus creating an arbitrage opportunity. In particular, we will have:
b) Let us follow the suggestion of the problem and sell the two-year zero-coupon bond. We will create a synthetic borrowing opportunity at the zero-coupon implied forward rate of 7.00238 percent, and we will lend at 7.2 percent, thus creating an arbitrage opportunity. In particular, we will have:
We see that we have created something out of nothing, without any risk involved. We have indeed found an arbitrage opportunity.
Question 7.16
a) The implied LIBOR of the September Eurodollar futures of 96.4 is: 100 96.4400− = 0.9%
b) As we want to borrow money, we want to buy protection against high interest rates, which means low Eurodollar future prices. We will short the Eurodollar contract.
c) One Eurodollar contract is based on a $1 million three-month deposit. As we want to hedge a loan of $50M, we will enter into 50 short contracts.
d) A true three-month LIBOR of 1 percent means an annualized position (annualized by market conventions) of 1% ∗ 4 = 4%. Therefore, our 50 short contracts will pay:
[96.4 − (100 − 4) × 100 × $25 × 50] = $50,000
The increase in the interest rate has made our loan more expensive. The futures position that we entered to hedge the interest rate exposure compensates for this increase. In particular, we pay $50,000,000 × 0.01 − payoff futures = $500,000 − $50,000 = $450,000, which corresponds to the 0.9 percent we sought to lock in.
Question 7.17
aa) The interest rate is higher than the rate of the forward rate agreement, therefore the lender must pay the borrower. If the FRA is settled on day 60, the payment made by the lender to the borrower is:
The lender pays the borrower because we are in the state of the world in which the lender does not need protection: Interest rates have risen, and thus he makes a payment to bring back the interest he earns to 2.5 percent.
ba) The interest rate is lower than the rate of the forward rate agreement; therefore, the lender will receive payment from the borrower. If the FRA is settled on day 60, the payment made by the borrower to the lender is:
150
150
( )1
FRAr rr−+
× notional principal = (0.022 0.025)1 0.022
−+
× $100,000,000.00 = −$293,542.07
bb) If the FRA is settled on the date the loan is repaid (or settled in arrears), the settlement amount is determined by:
The lender is paid by the borrower because we are in the state of the world in which the lender’s protection pays off: Interest rates have gone down, and thus she is compensated for the loss in investment proceeds. The payment of the borrower brings back the interest she earns to 2.5 percent.
Question 7.18
a) We face the classic problem of asset mismatch. We are interested in locking in an interest rate for a 150-day investment, 60 days from now. However, while the Eurodollar futures matures 60 days from now, it secures a lending rate for 90 days. We face the problem that the 90-day and 150-day interest rates may not be perfectly correlated. (For example, the term structure could, over the next 60 days, move from upward sloping to downward sloping).
b) As we want to lend money, we want to buy protection against low interest rates, which means high Eurodollar future prices. We will therefore long the Eurodollar contract.
c) The implied LIBOR of the September Eurodollar futures of 94 is: 100 94400− = 1.5 percent.
Under the assumption that the three-month LIBOR rate and the 150-day interest rate are based on the same annualized interest rate of 6 percent, we are able to lock in an interest rate
of: 1.5% × 15090
= 2.5 percent. Please note that this is a rather strong assumption.
d) One Eurodollar futures contract is based on a $1 million three-month deposit. As we want to hedge an investment of $100M, we will enter into 100 long contracts. Again, we are making the strong assumption that the annualized three-month LIBOR rate and the annualized 150 day rate are identical and perfectly correlated.
We see that the hedge ratios for increases and decreases of the same number of basis points differ. Note as well that the difference between the hedge ratios increases with an increase in the basis points. This is a consequence of the convexity of the bonds. It is caused by the changes in duration as the interest rate changes.
Question 7.20
We will use the Excel functions Duration and Mduration to calculate the required durations. They are of the form:
where frequency determines the number of coupon payments per year. In order to use the function, we have to give Excel a start date and terminal date, but we can just pick two dates that are exactly the requested number of years apart. Plugging in the values of the exercises yields:
a) Macaulay Duration = 4.59324084
Modified Duration = 4.3983078
b) Macaulay Duration = 5.99377496
Modified Duration = 5.73566981
c) We need to find the yield to maturity of this bond first. We can do so by using the YIELD function of Excel. Plugging in the relevant values, we get: Yield = 0.07146759. Now we can again use the Mduration and Duration formulas. This yields:
Macaulay Duration = 7.6955970
Modified Duration = 7.1822957
Question 7.21
b) Let us start with part (b) because we already know the Excel function for the Macaulay duration, DURATION().
Using the equation with DURATION(01/01/1980; 01/01/2000; 0.06; 0.20; 2), and DURATION (01/01/1970; 01/01/2000; 0.06; 0.20; 2) yields for the 20-year bond:
Macaulay Duration = 6.09533079
and for the 30-year bond:
Macaulay Duration = 5.66839682
a) Now we can back out the price value of a basis point by multiplying the Macaulay duration by −B(y)/(1 + y). In order to do so, we must find the prices of the two bonds. As the yield to maturity is given, we simply have:
c) As we can see from the above example, this statement is not always true. The above example is an extreme one in which the yield to maturity is extremely high. Since the Macaulay duration is a weighted average of the time until the bond payments occur, with the weights being the percentage of the bond price accounted for by each payment, we see that with a very high yield, the last payments get significantly more discounted than the previous ones. For a coupon bond, the last payment is the largest one, being interest plus principal. Therefore, with a high yield, the largest payment of a long term bond gets a higher discount than a bond with the same characteristics but a shorter maturity. Overall, this can make the duration of a long-term bond smaller than that of a short-term one.
Question 7.22
We will exploit equation (7.13) of the main text to find the optimal hedge ratio:
We see that it is more advantageous for the short position to deliver the six-year bond, as the owner of the short position can make a small profit by delivering the six-year bond. He would lose money if he delivered the eight-year bond.
Question 7.24
a) Compute the convexity of a three-year bond paying annual coupons of 4.5 percent and selling at par. For a par bond, the yield to maturity is equal to the coupon, or 4.5 percent in our case. We can calculate the convexity based on the formula:
Convexity ( )( )
( )( )2 22 21
1 111 1
ni ni
y
i i n nC m MB m my m y m+ +=
⎡ ⎤+ += +⎢ ⎥
+ +⎢ ⎥⎣ ⎦∑
( )( )
( )( )1 2 2 22 2
1 1 1 2 2 11 4.5 1 4.5 1100 1 11 0.045 1 0.045+ +
⎡ + += +⎢
+ +⎢⎣
( )( )3 22
3 3 1 104.51 1 0.045 1 +
⎤++ ⎥
+ ⎥⎦
= 10.3680
b) Compute the convexity of a three-year 4.5 percent coupon bond that makes semiannual coupon payments and that currently sells at par.
Now, m = 2. We have n = m ∗ T = 2 ∗ 3 = 6. The convexity is:
c) Is the convexity different in the two cases? Why?
Yes, the convexity for the semiannual bond is smaller. We spread the bond payments out over more periods, which makes the bond’s duration slightly less susceptible to interest rate changes.
Question 7.25
Suppose a 10-year zero-coupon bond with a face value of $100 trades at $69.20205.
a) What is the yield to maturity and modified duration of the zero-coupon bond?
The yield of the zero-coupon bond is equal to
(1 + y)10 = ( )0
1000,10P
y = 1 10100 1
69.20205⎛ ⎞ −⎜ ⎟⎝ ⎠
= 0.03750
Calculate its modified duration.
The modified duration of the ten-year zero-coupon bond is:
modified duration ( ) ( ) ( )1
1 11 1 1
ni ni
i C m n MB y y m m my m y m=
⎡ ⎤= × +⎢ ⎥
+ + +⎢ ⎥⎣ ⎦∑
=( )10
1 1 10 10069.20205 1 0.0375 1 1 0.0375
⎡ ⎤× ⎢ ⎥
+ +⎢ ⎥⎣ ⎦
= 101.0375
=9.63855
b) Calculate the approximate bond price change for a 50 basis point increase in the yield, based on the modified duration you calculated in part (a). Also calculate the exact new bond price based on the new yield to maturity.
c) Calculate the convexity of the 10-year zero-coupon bond.
Convexity ( )
( )( )
( )( )2 22 21
1 111 1
ni ni
i i n nC m MB y m my m y m+ +=
⎡ ⎤+ += +⎢ ⎥
+ +⎢ ⎥⎣ ⎦∑
( )( )10 22
10 10 11 100069.20205 1 1 0.0375 +
⎡ ⎤+= +⎢ ⎥
+⎢ ⎥⎣ ⎦
= 102.19190
d) Now use the formula (equation 7.15) that takes into account both duration and convexity to approximate the new bond price. Compare your result to that in part (b).
B (y + ε) = B (y) − [DMod × B (y) ε] + 0.5 × Convexity × B (y) × ε2
The approximation is much better now compared to the result in part (b), but it is still somewhat off the true price. The long time to maturity and the considerable change in basis points for this bond is responsible for the deviation.
