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An interest rate swap is a contract between two parties to exchange a series of cash flows
interest rate
similar to those that would result if the parties instead were to exchange equal dollar values of
different types of bonds. Swaps arose originally as a means of managing interest rate risk. The
Contracts between
volume of swaps has increased from virtually zero in 1980 to about $60 trillion today. (Inter-
two parties to trade
est rate swaps do not have anything to do with the Homer“Leibowitz bond swap taxonomy set
the cash flows
corresponding to out earlier.)
different securities To illustrate how swaps work, consider the manager of a large portfolio that currently in-
without actually
cludes $100 million par value of long-term bonds paying an average coupon rate of 7%. The
exchanging the
manager believes that interest rates are about to rise. As a result, he would like to sell the
securities directly.
bonds and replace them with either short-term or floating-rate issues. However, it would be
exceedingly expensive in terms of transaction costs to replace the portfolio every time the
forecast for interest rates is updated. A cheaper and more flexible way to modify the portfolio
is for the manager to “swap” the $7 million a year in interest income the portfolio currently
generates for an amount of money that is tied to the short-term interest rate. That way, if rates
do rise, so will the portfolio™s interest income.
A swap dealer might advertise its willingness to exchange or “swap” a cash flow based on
the six-month LIBOR rate for one based on a fixed rate of 7%. (The LIBOR, or London In-
terBank Offer Rate, is the interest rate at which banks borrow from each other in the Eurodol-
lar market. It is the most commonly used short-term interest rate in the swap market.) The
portfolio manager would then enter into a swap agreement with the dealer to pay 7% on
notional principal of $100 million and receive payment of the LIBOR rate on that amount of
notional principal
notional principal.4 In other words, the manager swaps a payment of 0.07 $100 million for
Principal amount used
a payment of LIBOR $100 million. The manager™s net cash flow from the swap agreement
to calculate swap
is therefore (LIBOR 0.07) $100 million.

The participants to the swap do not loan each other money. They agree only to exchange a fixed cash flow for a vari-
able cash flow that depends on the short-term interest rate. This is why the principal is described as notional. The no-
tional principal is simply a way to describe the size of the swap agreement. In this example, the parties to the swap
exchange a 7% fixed rate for the LIBOR rate; the difference between LIBOR and 7% is multiplied by notional prin-
cipal to determine the cash flow exchanged by the parties.
Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill
Essentials of Investments, Portfolios Companies, 2003
Fifth Edition

10 Managing Bond Portfolios

Now consider the net cash flow to the manager™s portfolio in three interest rate scenarios:


6.5% 7.0% 7.5%
Interest income from bond portfolio ( 7% of
$100 million bond portfolio) $7,000,000 $7,000,000 $7,000,000
Cash flow from swap [ (LIBOR 7%)
notional principal of $100 million] (500,000) 0 500,000
Total ( LIBOR $100 million) $6,500,000 $7,000,000 $7,500,000

Notice that the total income on the overall position”bonds plus swap agreement”is now
equal to the LIBOR rate in each scenario times $100 million. The manager has in effect con-
verted a fixed-rate bond portfolio into a synthetic floating-rate portfolio.
You can see now that swaps can be immensely useful for firms in a variety of applica-
tions. For example, a corporation that has issued fixed-rate debt can convert it into synthetic
floating-rate debt by entering a swap to receive a fixed interest rate (offsetting its fixed-rate
coupon obligation) and pay a floating rate. Or, a bank that pays current market interest rates to
its depositors might enter a swap to receive a floating rate and pay a fixed rate on some
amount of notional principal. This swap position, added to its floating-rate deposit liability,
would result in a net liability of a fixed stream of cash. The bank might then be able to invest
in long-term fixed-rate loans without encountering interest rate risk.
What about the swap dealer? Why is the dealer, which is typically a financial intermediary
such as a bank, willing to take on the opposite side of the swaps desired by these participants?
Consider a dealer who takes on one side of a swap, let™s say paying LIBOR and receiving
a fixed rate. The dealer will search for another trader in the swap market who wishes to re-
ceive a fixed rate and pay LIBOR. For example, company A may have issued a 7% coupon
fixed-rate bond that it wishes to convert into synthetic floating-rate debt, while company B
may have issued a floating-rate bond tied to LIBOR that it wishes to convert into synthetic
fixed-rate debt. The dealer will enter a swap with company A in which it pays a fixed rate and
receives LIBOR, and it will enter another swap with company B in which it pays LIBOR and
receives a fixed rate. When the two swaps are combined, the dealer™s position is effectively
neutral on interest rates, paying LIBOR on one swap, and receiving it on another. Similarly,
the dealer pays a fixed rate on one swap and receives it on another. The dealer is an interme-
diary, funneling payments from one party to the other.5 The dealer finds this activity profitable
because it will charge a bid“ask spread on the transaction.
This arrangement is illustrated in Figure 10.9. Company A has issued 7% fixed-rate debt
(the leftmost arrow in the figure) but enters a swap to pay the dealer LIBOR and receive a
6.95% fixed rate. Therefore, the company™s net payment is 7% (LIBOR 6.95%)
LIBOR 0.05%. It has thus transformed its fixed-rate debt into synthetic floating-rate debt.
Conversely, company B has issued floating-rate debt paying LIBOR (the rightmost arrow), but
enters a swap to pay a 7.05% fixed rate in return for LIBOR. Therefore, its net payment is
LIBOR (7.05% LIBOR) 7.05%. It has thus transformed its floating-rate debt into syn-
thetic fixed-rate debt. The bid“ask spread in the example illustrated in Figure 10.9 is 0.1% of
notional principal each year.
Actually, things are a bit more complicated. The dealer is more than just an intermediary because it bears the credit
risk that one or the other of the parties to the swap might default on the obligation. Referring to Figure 10.9, if com-
pany A defaults on its obligation, for example, the swap dealer still must maintain its commitment to company B. In
this sense, the dealer does more than simply pass through cash flows to the other swap participants.
Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill
Essentials of Investments, Portfolios Companies, 2003
Fifth Edition

368 Part THREE Debt Securities

6.95% 7.05%

7% coupon
Company A Swap dealer Company B


Company B pays a fixed rate of 7.05% to the swap dealer in return for LIBOR.
Company A receives 6.95% from the dealer in return for LIBOR. The swap dealer
realizes a cash flow each period equal to .1% of notional principal.

F I G U R E 10.9
Interest rate swap

9. A pension fund holds a portfolio of money market securities that the manager
believes are paying excellent yields compared to other comparable-risk short-term
CHECK securities. However, the manager believes that interest rates are about to fall.
What type of swap will allow the fund to continue to hold its portfolio of short-term
securities while at the same time benefiting from a decline in rates?

SUMMARY • Even default-free bonds such as Treasury issues are subject to interest rate risk. Longer
term bonds generally are more sensitive to interest rate shifts than short-term bonds.
A measure of the average life of a bond is Macaulay™s duration, defined as the weighted
average of the times until each payment made by the security, with weights proportional to
the present value of the payment.
• Duration is a direct measure of the sensitivity of a bond™s price to a change in its yield.
The proportional change in a bond™s price approximately equals the negative of duration
times the proportional change in 1 y.
• Immunization strategies are characteristic of passive bond portfolio management. Such
strategies attempt to render the individual or firm immune from movements in interest
rates. This may take the form of immunizing net worth or, instead, immunizing the future
accumulated value of a bond portfolio.

• Convexity refers to the curvature of a bond™s price-yield relationship. Accounting for
convexity can substantially improve on the accuracy of the duration approximation for
bond-price sensitivity to changes in yields.
• Immunization of a fully funded plan is accomplished by matching the durations of assets
and liabilities. To maintain an immunized position as time passes and interest rates
change, the portfolio must be periodically rebalanced.
• A more direct form of immunization is dedication or cash flow matching. If a portfolio is
perfectly matched in cash flow with projected liabilities, rebalancing will be unnecessary.
• Active bond management can be decomposed into interest rate forecasting techniques and
intermarket spread analysis. One popular taxonomy classifies active strategies as
Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill
Essentials of Investments, Portfolios Companies, 2003
Fifth Edition

10 Managing Bond Portfolios

substitution swaps, intermarket spread swaps, rate anticipation swaps, or pure yield pickup
• Interest rate swaps are important instruments in the fixed-income market. In these
arrangements, parties trade the cash flows of different securities without actually
exchanging any securities directly. This can be a useful tool to manage the duration of a

cash flow matching, 358 interest rate swaps, 366 pure yield pickup
contingent immuni- intermarket spread swap, 362
zation, 364 swap, 362 rate anticipation swap, 362
convexity, 359 Macaulay™s duration, 342 rebalancing, 355
dedication strategy, 358 modified duration, 345 substitution swap, 362
horizon analysis, 363 notional principal, 366 tax swap, 363
immunization, 350

1. A nine-year bond has a yield of 10% and a duration of 7.194 years. If the bond™s yield
changes by 50 basis points, what is the percentage change in the bond™s price?
2. Find the duration of a 6% coupon bond making annual coupon payments if it has three
years until maturity and a yield to maturity of 6%. What is the duration if the yield to
maturity is 10%?
3. A pension plan is obligated to make disbursements of $1 million, $2 million, and
$1 million at the end of each of the next three years, respectively. Find the duration of
the plan™s obligations if the interest rate is 10% annually.
4. If the plan in problem 3 wants to fully fund and immunize its position, how much of its
portfolio should it allocate to one-year zero-coupon bonds and perpetuities, respectively,
if these are the only two assets funding the plan?
5. You own a fixed-income asset with a duration of five years. If the level of interest rates,
which is currently 8%, goes down by 10 basis points, how much do you expect the price
of the asset to go up (in percentage terms)?
6. Rank the interest-rate sensitivity of the following pairs of bonds.
a. Bond A is an 8% coupon bond, with 20-year time to maturity selling at par value.
Bond B is an 8% coupon, 20-year maturity bond selling below par value.
b. Bond A is a 20-year, noncallable coupon bond with a coupon rate of 8%, selling at par.
Bond B is a 20-year, callable bond with a coupon rate of 9%, also selling at par.
7. Rank the following bonds in order of descending duration.

Bond Coupon Time to Maturity Yield to Maturity
A 15% 20 years 10%

B 15 15 10
C 0 20 10
D 8 20 10
E 15 15 15

8. Philip Morris has issued bonds that pay annually with the following characteristics:

Coupon Yield to Maturity Maturity Macaulay Duration
8% 8% 15 years 10 years
Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill
Essentials of Investments, Portfolios Companies, 2003
Fifth Edition

370 Part THREE Debt Securities

a. Calculate modified duration using the information above.
b. Explain why modified duration is a better measure than maturity when calculating
the bond™s sensitivity to changes in interest rates.
c. Identify the direction of change in modified duration if:
i. The coupon of the bond were 4%, not 8%.
ii. The maturity of the bond were 7 years, not 15 years.
9. You will be paying $10,000 a year in tuition expenses at the end of the next two years.
Bonds currently yield 8%.
a. What is the present value and duration of your obligation?
b. What maturity zero-coupon bond would immunize your obligation?
c. Suppose you buy a zero-coupon bond with value and duration equal to your
obligation. Now suppose that rates immediately increase to 9%. What happens to
your net position, that is, to the difference between the value of the bond and that of
your tuition obligation? What if rates fall to 7%?


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