Tải bản đầy đủ
6 The Dealer’s Problem: Finding the Other Side to the Swap

6 The Dealer’s Problem: Finding the Other Side to the Swap

Tải bản đầy đủ

OTC MARKETS AND SWAPS

295

better rated firm AA, and is willing to take a lower fixed rate from AA
(a spread of 52 bps).
The corresponding spreads for BBB-type firms were 48 bps and 54 bps.
This is one way that the swaps dealer adjusts for the alternative credit risks
associated with AA and BBB.
TABLE 8.4

Dealer Swap Schedule for AA-Type Firms

Dealer Pays (BID) the

Dealer Receives (ASK) the

5-year T-note Rate+50bps (quarterly) in
exchange for LIBOR3,t ′(␻).

5-year T-note rate+52 bps (quarterly) in
exchange for LIBOR3,t ′(␻).

That is, paying fixed and receiving floating
(a short forward position).

That is, paying floating and receiving fixed
(a long forward position).

What we did for BBB applies here and there is nothing really new. The
cash flow diagrams just have different numbers attached to them. We once
again assume that the 5-year T-Note rate is 5.00% and that NP stands for
notional principal.
FIGURE 8.9 Bid Side in a Dealer-Intermediated Swap with AA

Fixed-rate payment at all times
)
tʹ = NP * ( .0550
4
Swaps Dealer

AA
Floating-rate payment at all times
LIBOR3,tʹ(␻))
tʹ = NP * (
4

A similar picture shows the asking side of the swap from the dealer’s point
of view. In this case, AA pays the swaps dealer a fixed-rate payment every
3 months equal to Notional Principal*(.0552/4) and receives a floating rate
payment at time t′=NP*(LIBOR3,t ′(␻)/4).
FIGURE 8.10 Asked Side in a Dealer-Intermediated Swap with AA

Fixed-rate payment at all times
)
tʹ = NP *( .0552
4
AA

Swaps Dealer
Floating-rate payment at all times
tʹ = NP * (LIBOR43,tʹ(␻))

296

TRADING STRUCTURES BASED ON FORWARD CONTRACTS

Now, for whatever reason as we shall soon see, AA wishes to take a sell
position in the swap (pay floating and receive fixed). Remember that the receiver
of fixed (payer of the floating rate) is said to be selling the swap.
We know from the previous discussion that selling the swap is economically
equivalent to longing a strip of ED futures contracts. Longing because the
participant who buys the underlying 3-month ED time deposit ends up
receiving the fixed rate on those ED time deposits, assuming they bought them
forward at the futures prices. Once again, since the swap reset dates are quarterly
in our example, the long strip would have to have maturity dates that coincide
with the swap’s reset dates.
Why would anyone take a position such as AA in the swap? Just as we did
for BBB, we turn hedging on its head to back out AA’s likely position in the
spot market. Since AA is long an ED futures (forward) strip AA must, if AA
is a hedger, be short something else that is correlated with future LIBOR3,t ′(␻)
rates.
Recall that a short position in a spot commodity is one that either explicitly
or implicitly plans to buy that commodity in the future and therefore worries
about price increases in the underlying commodity. That is the motive for
hedging. If the underlying commodity increases in price (rates decline), then
the corresponding hedging vehicle’s (futures) price will also increase and a
long futures position will recoup some of the losses incurred in the spot market,
subject to basis risk of course.
So we know that AA is short something. Suppose that AA has currently
(at time t=0) issued long-term (5-year) financing at its natural rate in the fixedrate market which is 5.30%. See the chart below, Table 8.5, for AA’s respective
borrowing rates in fixed and floating markets, given its credit risk.
TABLE 8.5

Credit Spreads in the Spot Market for AA-Type Firms

Credit Rating

Fixed rate

Floating rate

AA

5-year T-Note rate+30bp

LIBOR3 flat

Note that AA is a borrower in the spot fixed-rate bond market who has
locked into paying the fixed rate of 5.30% for five years. This puts AA at a
potential disadvantage if rates decrease across the board.
If that happens, then AA is locked into too high a rate for the remaining
life of the loan. An analogous situation is the homeowner with a fixed-rate
mortgage who sees the fixed rate decline, and therefore wishes to refinance.

OTC MARKETS AND SWAPS

297

If rates do decline, then AA could repurchase the bonds previously issued,
but they would be selling at a higher price than originally issued, creating a
loss for AA. After repurchasing the existing bonds, AA could refinance by
issuing new bonds at lower rates. However, it would be easier for AA to hedge
itself using swaps.
What AA really wants is floating rate financing rather than fixed-rate
financing, because floating-rate financing adjusts to interest rate changes. The
question is how to most cost effectively achieve it. Of course, AA can go out
into the floating-rate market and get quarterly financing at flat LIBOR3,t ′(␻)/4.
AA notices, though, that it could issue fixed-rate debt at annualized 5.30%
and simultaneously enter into a swap with the dealer in which AA pays to
the dealer annualized LIBOR3,t ′(␻) flat and receives from the dealer the fixed
rate of 5.50%.
When the coupons become due every quarter, AA uses the dealer’s fixed
payment to pay off the annualized 5.30% and keeps the extra .20% to offset
its flat LIBOR3,t ′ borrowing rate thereby reducing it to LIBOR3,t ′(␻)–.20%.
This transforms AA’s fixed-rate borrowing into floating rate borrowing at
an annualized savings of 20bp relative to direct floating rate financing. Figure
8.11 illustrates the situation.
FIGURE 8.11 Synthetic Floating-Rate Financing for AA

LIBOR3,tʹ(␻)
4

Swaps Dealer

AA
.0550
4
$100 MM

.0530
4

Lenders
(Fixed-Rate Market)

Note that the net overall quarterly cost to AA due to this strategy is,

LIBOR 3,t′ (␻ ) ⎛ .0530 .0550 ⎞ LIBOR 3,t′ (␻ ) .002
+⎜


⎟=
4
4 ⎠
4
4
⎝ 4
which is LIBOR3,t ′(␻)–.002 annualized.

298

TRADING STRUCTURES BASED ON FORWARD CONTRACTS

Therefore, AA has saved 20 basis points (.002=.2 of a percent) by issuing
floating-rate debt and swapping fixed for floating with the swaps dealer. It
has effectively achieved fixed-rate financing as well.
Figure 8.12 shows the complete set of cash flows generated by the swap
with both counterparties AA and BBB.
FIGURE 8.12 Full Set of Swap Cash Flows for BBB, AA, and the

Dealer
LIBOR3,tʹ(␻)

LIBOR3,tʹ(␻)

4

AA

4

Swaps Dealer
.0550
4

$100 MM

BBB
.0554
4

.0530
4

LIBOR3,tʹ(␻)+50bp

$100 MM

4

Lenders
(Fixed-Rate Market)

Lenders
(Variable-Rate Market)

8.7 ARE SWAPS A ZERO SUM GAME?
We see from Figure 8.12 the net gains to AA, BBB, and to the swaps dealer.
The dealer makes 4 bp on NP annualized because it receives 5.54% annualized
from BBB and pays out 5.50% annualized to AA.
BBB effectively arranges fixed-rate financing and pays out 5.54%
(annualized) plus .50%=6.04% which represents a savings of 16 bp over its
natural fixed-rate borrowing rate of 6.20%(annualized).
Finally, AA has arranged floating-rate financing at a cost of LIBOR3,t ′(␻)–
20bp which is 20 bp below what it would naturally pay in the floating-rate
market. The sum of cost savings to all parties is therefore 4bp+16bp+
20bp=40bp.
There is a nice way of seeing where these 40bps come from. To explain
this will require a few definitions. Recall the natural rates available to AA and
BBB summarized in Table 8.6.
TABLE 8.6

Credit Spreads for AA and BBB

Credit-Rated Firms

Fixed rate

Floating rate

AA

5.30% (530bps)

LIBOR3

BBB

6.20% (620bps)

LIBOR3+0.5%

OTC MARKETS AND SWAPS

299

Definitions
1. The Quality Spread in Fixed is
(QSFIX)=Fixed-rate differential between AA and BBB
=620bps–530bps
=90bps.
2. The Quality Spread in Floating is
(QSFLO)=Floating rate differential between AA and BBB
=(LIBOR3+0.5%)–LIBOR3
=50bps.
Then the sum of all gains to all parties is the difference between QSFIX and
QSFLO. In our example, that difference is 90bps–50bps=40bps.
The economic argument behind this result is called comparative advantage
and the process described in our extensive example was known as ‘Arbitraging
the Swaps Market’. As noted, many researchers and practitioners question
whether there is a real arbitrage opportunity here.
8.8 WHY FINANCIAL INSTITUTIONS USE SWAPS
We have skirted by the issue of why parties such as AA and BBB would want
to transform fixed to floating-rate liabilities, or floating to fixed-rate liabilities.
That is, why would AA and BBB want to restructure their balance sheets in
the first place?
As an example, let’s look at a financial intermediary (FI) such as Bank of
America (B of A). Below is a look at what B of A’s partial balance sheet might
look like.
TABLE 8.7

Bank Of America’s Simplified Balance Sheet

Assets

Liabilities

Fixed-rate 30-year mortgages earning
ROA=rA

6-month CDs
with cost of capital=rD

An immediate problem is apparent here and it goes under the title ‘Gap
Management’.
Asset Portfolio

GAP

Liability Portfolio

300

TRADING STRUCTURES BASED ON FORWARD CONTRACTS

Bank of America is in a risky position, because it has funded long-term assets
such as 30-year fixed-rate mortgages with short-term liabilities such as 6-month
certificates of deposit (CD). The latter are floating-rate securities because B
of A will have to roll them over, reborrowing the principal every 6 months,
most likely on a rate based on LIBOR6. B of A is thus exposed to potentially
increasing financing costs.
The principle that attempts to avoid this issue is called the ‘Matching
Principle’ which is to match the duration of your assets with the duration of
your liabilities.
Duration has a technical meaning in that it measures the interest rate
sensitivity of a firm’s asset and liability portfolio. If one could apply the matching
principle perfectly, then the duration of the firm’s balance sheet would be
zero and the pure interest-rate risk of the portfolio would be neutralized.
Interest-rate swaps provide a low-cost method for a firm to accomplish this
goal. Note that the relevant measure of the life of an investment is not generally
its stated maturity. Rather, duration is the correct concept which captures the
risk of an interest-rate sensitive investment. Thirty-year fixed-rate mortgages
appear to be very long-term assets but their duration is in the 7–8 year range.
This means that a 7–8 year interest rate swap would be an appropriate hedging
vehicle.
To pursue this example a little more, we first focus on the market
participants. The issuer of a mortgage is the homeowner, who is a borrower
in this case. The investor in the mortgage is Bank of America, who is the
lender in the mortgage transaction. B of A expects to earn the spread between
the rate of return on its fixed-rate mortgages and its cost of funds. This can
be written as rA–E(rD(␻)).
What is the primary risk to B of A in this scenario? The risk is that rD(␻)
increases and B of A’s spread narrows. Potential solutions to this gap
management problem include:
a. Sell the fixed-rate mortgages in the secondary mortgage market. To
reduce the risk of these 30-year mortgages, sell them to Freddie Mac or
Fannie Mae. Then buy them back as mortgage-backed securities.
b. Don’t issue fixed-rate mortgages. Use variable-rate mortgages instead.
c. Hedge with strips of Eurodollar futures (ED futures).
d. Lock in the future costs of funding by forward rate agreements (FRAs).
e. Use swaps (Fixed-for-Floating Plain Vanilla Swaps) to transform
floating-rate costs of funds into fixed-rate costs of funds.

OTC MARKETS AND SWAPS

301

Each of the above solutions has costs and benefits. Instead of describing those
costs and benefits, it is time to turn to the pricing swaps.
8.9 SWAPS PRICING
What does it mean to price a generic swap? The answer is that it means the
same thing as to price a forward contract, which is to determine the
equilibrium forward price. Swaps are forward strips, so pricing generic swaps
is more complicated than pricing an individual forward contract.
Pricing a generic swap means that the swaps market, and therefore the swaps
dealer, has determined the fixed rate that the fixed-rate payer pays in a generic
fixed for floating-rate swap. This amounts to ‘pricing the swap currently’.
Note that swap pricing includes pricing at origination which we will focus
on. It also includes pricing after origination. Just as a forward contract has zero
value at origination, but assumes positive and negative values to market
participants, so for swaps the swap can assume a positive or negative value as
the underlying spot price changes.
How does the swaps market come up with the base fixed rates to charge
alternative credit risks such as AA and BBB? Recall definition 13 which is
repeated here.
The Par Swap Rate is the fixed rate at which the swap has a zero present
value at initiation.
Otherwise, there would be an arbitrage opportunity.
We know from our study of forward contracts that the value of a forward
contract is zero at initiation. We also looked at the arbitrage opportunity
available if this were not true. Since a swap is just a strip of forward contracts,
the same valuation must apply to swaps. The details follow.
When a swaps dealer takes the opposite side of a swaps transaction with a
firm such as BBB who has bought the swap, it ends up receiving fixed and paying
floating periodically. To get a handle on the pricing problem, we can interpret
the swaps dealer’s positions in terms of bonds.
8.9.1 An Example
Consider a 3-year swap. Suppose that LIBOR12,0 is the spot 1-year LIBOR12
at the beginning of year 1. It applies to determine cash flows to be received
at the end of year 1 and is known at t=0.

302

TRADING STRUCTURES BASED ON FORWARD CONTRACTS

Also, LIBOR12,1(␻) is 1-year spot LIBOR12 at the beginning of year 2.
Finally, LIBOR12,2(␻) is 1-year spot LIBOR12 at the beginning of year 3. We
will assume that notional principal (NP) is $100,000,000.
The swap’s floating payments are,
FLO1=NP*LIBOR12,0,
FLO2=NP*LIBOR12,1(␻),
FLO3=NP*LIBOR12,2(␻).
The swap’s fixed payments are,
FIX1=NP*R,
FIX2=NP*R,
FIX3=NP*R,
where R is the par swap rate.
Note that all the fixed payments are the same: FIX1=FIX2=FIX3=FIX, as is
characteristic of a fixed-for-floating swap. The swap’s cash flows are illustrated
in Figure 8.13:
FIGURE 8.13 Cash Flows for an Annual Rate Swap from the Dealer’s

Point of View
FLO1 FIX1

t=0

0.5

1.0

FLO2 FIX2

1.5

2.0

FLO3 FIX3

2.5

3.0 = T

But we can write the cash flows of the swap in terms of two bonds, a
floating-rate bond and a fixed-rate bond. Bonds differ from swaps in that, for
bonds, NP must be repaid at maturity T=3.
Note that NP is a wash in the swap because it gets paid out via the floatingrate bond and it is received from the fixed-rate bond. Also note that the swap’s
cash flows are equal to the vertical sum of the two bonds’ cash flows, as illustrated
in Figure 8.14.
The swaps dealer has effectively issued (shorted) a floating-rate bond and
invested in (longed) a fixed-rate bond. Long the fixed-rate bond because he
is receiving the fixed rate from the counterparty, and short the floating-rate
bond because he is paying LIBOR12,t ′ to the counterparty at each time t′.

OTC MARKETS AND SWAPS

303

FIGURE 8.14 Decomposing the Swap’s Cash Flows into its Implicit

Bonds
FLO1

t=0

0.5

1.0

FLO2

2.0

1.5

FLO3

2.5

3.0 = T

+
FIX1

t=0

0.5

1.0

FIX2

1.5

2.0

FIX3

2.5

3.0 = T

=
FLO1 + FIX1

t=0

0.5

1.0

FLO2 + FIX2

1.5

2.0

FLO3 + FIX3

2.5

3.0 = T

The conclusion here is that the swap, from the dealer’s point of view, is
economically equivalent to short the floating-rate bond and long the fixed-rate bond.
The next step is to value each of these bonds and thereby to value the swap.
To do so, we need the appropriate discount rates.
8.9.2 Valuation of the Fixed-Rate Bond
In order to value the fixed-rate bond in which the dealer is long, we need
the appropriate discount rates to apply to the bond’s cash flows. We are in
the world of interest-rate swaps which is a LIBOR world.
So we need the current (t=0) spot LIBOR yield curve. It gives the rates
to be applied to zero-coupon Eurobonds for alternative maturities. Assume
that it is as in Table 8.8.
Our long position in the fixed-rate bond can be decomposed as the sum
of three zero-coupon bonds, and one NP repayment bond as indicated in the
multi-level cash flow diagram, Figure 8.15.

304

TRADING STRUCTURES BASED ON FORWARD CONTRACTS

TABLE 8.8

LIBOR Yield Curve (Spot Rates)

Maturity

Zero-Coupon Bond Yields

1 year

6.0%

2 years

6.5%

3 years

7.0%

FIGURE 8.15 The Implicit Fixed-Rate Bond in a Swap, Written in

Terms of Zero-Coupon Bonds
Bond 4 NP
Bond 3 R * NP
Bond 2 R * NP
Bond 1 R * NP

t=0

1.0

2.0

3.0 = T

The LIBOR zero-yield curve says that the appropriate discount rate to
apply to the cash flow from Bond 1 is 6.00%, the appropriate discount rate
to apply to the cash flow from Bond 2 is 6.5%, and the appropriate discount
rate to apply to the cash flows from Bond 3 and from Bond 4 is 7.0%.
This leads to the pricing formulas for the three zero-coupon bonds indicated:
B0,1, B0,2, B0,3, and the notional principal bond NPB0,3. The first three bonds
pay exactly the same coupon, R*NP where R is the par swap rate. The fourth
bond represents the return of the principal NP and is therefore denoted as
NPB0,3.
The time at which we have to value these bonds is t=0, as indicated by the
notation. Given this preliminary setup work, it is now easy to value each bond.

B0,1 =

R * NP
1.0600

1.0
1.0600
= R * NP * B0′ ,1
= R * NP *

OTC MARKETS AND SWAPS

B0,2 =

305

R * NP
(1.065)2

1.0
(1.065)2
= R * NP * B0′ ,2
= R * NP *

B0,3 =

R * NP
(1.070)3

1.0
(1.070)3
= R * NP * B0′ ,3
= R * NP *

NP
(1.070)3
= NP * B0′ ,3

NPB0,3 =

where we have written these bond values in terms of B′0,1, B′0,2, and B′0,3 which
are the unit ($1 payoff) discount bonds maturing at times t ′=1, 2, and 3.
Therefore, the current value of the fixed-rate bond is,

R * NP R * NP R * NP
NP
+
+
+
2
3
1.0600 (1.065) (1.070) (1.070)3
= B0,1 + B0,2 + B0,3 + NPB0,3

B0,Fixed Rate =

= R * NP * B0′ ,1 + R * NP * B0′ ,2 + R * NP * B0′ ,3 + NP * B0′ ,3

8.9.3 Valuation of the Floating-Rate Bond
The floating-rate bond is described by its cash flow diagram in Figure 8.16.
In order to price such a set of cash flows with random variable payoffs, we
have to take expected values and then discount them appropriately. This means
that, in order to come up with a price for the variable-rate bond, we have to
replace its cash flows by its expected cash flows, where the expectation is with
respect to information currently available.