. 8
( 10)


EXHIBIT 12.3 Fixed-Rate Payments for Several Assumed Swap Rates

(1) (2) (3) (4) (5) (6) (7) (8) (9)
Fixed-rate payment if swap rate is assumed to be
Quarter Quarter Number Period =
starts ends of days in quarter End of quarter 4.9800% 4.9873% 4.9874% 4.9875% 4.9880%

Jan 1 year 1 Mar 31 year 1 90 1 1,245,000 1,246,825 1,246,850 1,246,875 1,247,000
Apr 1 year 1 June 30 year 1 91 2 1,258,833 1,260,679 1,260,704 1,260,729 1,260,856
July 1 year 1 Sept 30 year 1 92 3 1,272,667 1,274,532 1,274,558 1,274,583 1,274,711
Oct 1 year 1 Dec 31 year 1 92 4 1,272,667 1,274,532 1,274,558 1,274,583 1,274,711

Jan 1 year 2 Mar 31 year 2 90 5 1,245,000 1,246,825 1,246,850 1,246,875 1,247,000
Apr 1 year 2 June 30 year 2 91 6 1,258,833 1,260,679 1,260,704 1,260,729 1,260,856
July 1 year 2 Sept 30 year 2 92 7 1,272,667 1,274,532 1,274,558 1,274,583 1,274,711
Oct 1 year 2 Dec 31 year 2 92 8 1,272,667 1,274,532 1,274,558 1,274,583 1,274,711
Jan 1 year 3 Mar 31 year 3 90 9 1,245,000 1,246,825 1,246,850 1,246,875 1,247,000
Apr 1 year 3 June 30 year 3 91 10 1,258,833 1,260,679 1,260,704 1,260,729 1,260,856
July 1 year 3 Sept 30 year 3 92 11 1,272,667 1,274,532 1,274,558 1,274,583 1,274,711
Oct 1 year 3 Dec 31 year 3 92 12 1,272,667 1,274,532 1,274,558 1,274,583 1,274,711
Swaps and Caps/Floors

EXHIBIT 12.4 Calculating the Forward Discount Factor

(1) (2) (3) (4) (5) (6) (7)
Number of Period = Period Forward
Quarter Quarter days in End of Forward forward discount
starts ends quarter quarter rate rate factor

Jan 1 year 1 Mar 31 year 1 90 1 4.05% 1.0125% 0.98997649
Apr 1 year 1 June 30 year 1 91 2 4.15% 1.0490% 0.97969917
July 1 year 1 Sept 30 year 1 92 3 4.55% 1.1628% 0.96843839
Oct 1 year 1 Dec 31 year 1 92 4 4.72% 1.2062% 0.95689609
Jan 1 year 2 Mar 31 year 2 90 5 4.90% 1.2250% 0.94531597
Apr 1 year 2 June 30 year 2 91 6 5.03% 1.2715% 0.93344745
July 1 year 2 Sept 30 year 2 92 7 5.15% 1.3161% 0.92132183
Oct 1 year 2 Dec 31 year 2 92 8 5.25% 1.3417% 0.90912441
Jan 1 year 3 Mar 31 year 3 90 9 5.40% 1.3500% 0.89701471
Apr 1 year 3 June 30 year 3 91 10 5.50% 1.3903% 0.88471472
July 1 year 3 Sept 30 year 3 92 11 5.65% 1.4439% 0.87212224
Oct 1 year 3 Dec 31 year 3 92 12 5.76% 1.4720% 0.85947083

Second, forward rates are derived from spot rates so that if we dis-
counted a cash ¬‚ow using forward rates rather than spot rates, we would
come up with the same value. That is, the present value of $1 to be
received in period t can be rewritten as:

present value of $1 to be received in period t
= -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(1 + forward rate for period 1) ( 1 + forward rate for period 2) ¦ (1 + forward rate for period t)

We will refer to the present value of $1 to be received in period t as
the forward discount factor. In our calculations involving swaps, we will
compute the forward discount factor for a period using the forward rates.
These are the same forward rates that are used to compute the ¬‚oating-
rate payments”those obtained from the Eurodollar CD futures contract.
We must make just one more adjustment. We must adjust the forward
rates used in the formula for the number of days in the period (i.e., the
quarter in our illustrations) in the same way that we made this adjustment
to obtain the payments. Speci¬cally, the forward rate for a period, which
we will refer to as the period forward rate, is computed using the follow-
ing equation:

days in period
period forward rate = annual forward rate — « ----------------------------------- 
 

For example, look at Exhibit 12.2. The annual forward rate for
period 4 is 4.72%. The period forward rate for period 4 is:

period forward rate = 4.72% — « --------- = 1.2062%
 360

Column (5) in Exhibit 12.4 shows the annual forward rate for all 12 peri-
ods (reproduced from Exhibit 12.3) and Column (6) shows the period
forward rate for all 12 periods. Note that the period forward rate for
period 1 is 4.05%, the known rate for 3-month LIBOR.
Also shown in Exhibit 12.4 is the forward discount factor for all 12
periods. These values are shown in the last column. Let™s show how the
forward discount factor is computed for periods 1, 2, and 3. For period 1,
the forward discount factor is:

forward discount factor = ----------------------------- = 0.98997649
( 1.010125 )

For period 2,

forward discount factor = ----------------------------------------------------------- = 0.97969917
( 1.010125 ) ( 1.010490 )

For period 3,

forward discount factor = -----------------------------------------------------------------------------------------
( 1.010125 ) ( 1.010490 ) ( 1.011628 )
= 0.96843839

Given the ¬‚oating-rate payment for a period and the forward dis-
count factor for the period, the present value of the payment can be com-
puted. For example, from Exhibit 12.2 we see that the ¬‚oating-rate
payment for period 4 is $1,206,222. From Exhibit 12.4, the forward dis-
count factor for period 4 is 0.95689609. Therefore, the present value of
the payment is:

present value of period 4 payment = $1,206,222 — 0.95689609
= $1,154,229

Exhibit 12.5 shows the present value for each payment. The total
present value of the 12 ¬‚oating-rate payments is $14,052,917. Thus, the
Swaps and Caps/Floors

present value of the payments that the ¬xed-rate payer will receive is
$14,052,917 and the present value of the payments that the ¬xed-rate
receiver will make is $14,052,917.

Determination of the Swap Rate
The ¬xed-rate payer will require that the present value of the ¬xed-rate
payments that must be made based on the swap rate not exceed the
$14,052,917 payments to be received from the ¬‚oating-rate payments.
The ¬xed-rate receiver will require that the present value of the ¬xed-rate
payments to be received is at least as great as the $14,052,917 that must
be paid. This means that both parties will require a present value for the
¬xed-rate payments to be $14,052,917. If that is the case, the present
value of the ¬xed-rate payments is equal to the present value of the ¬‚oat-
ing-rate payments and therefore the value of the swap is zero for both
parties at the inception of the swap. The interest rates that should be used
to compute the present value of the ¬xed-rate payments are the same
interest rates as those used to discount the ¬‚oating-rate payments.
To show how to compute the swap rate, we begin with the basic rela-
tionship for no arbitrage to exist:

PV of ¬‚oating-rate payments = PV of ¬xed-rate payments

We know the value for the left-hand side of the equation.

EXHIBIT 12.5 Present Value of the Floating-Rate Payments

(1) (2) (3) (4) (5) (6)
Period = Forward Floating-rate PV of
Quarter Quarter End of discount payment at ¬‚oating-rate
starts ends quarter factor end of quarter payment

Jan 1 year 1 Mar 31 year 1 1 0.98997649 1,012,500 1,002,351
Apr 1 year 1 June 30 year 1 2 0.97969917 1,049,028 1,027,732
July 1 year 1 Sept 30 year 1 3 0.96843839 1,162,778 1,126,079
Oct 1 year 1 Dec 31 year 1 4 0.95689609 1,206,222 1,154,229
Jan 1 year 2 Mar 31 year 2 5 0.94531597 1,225,000 1,158,012
Apr 1 year 2 June 30 year 2 6 0.93344745 1,271,472 1,186,852
July 1 year 2 Sept 30 year 2 7 0.92132183 1,316,111 1,212,562
Oct 1 year 2 Dec 31 year 2 8 0.90912441 1,341,667 1,219,742
Jan 1 year 3 Mar 31 year 3 9 0.89701471 1,350,000 1,210,970
Apr 1 year 3 June 30 year 3 10 0.88471472 1,390,278 1,229,999
July 1 year 3 Sept 30 year 3 11 0.87212224 1,443,889 1,259,248
Oct 1 year 3 Dec 31 year 3 12 0.85947083 1,472,000 1,265,141
Total 14,052,917

If we let

SR = swap rate


Dayst = number of days in the payment period t

then the ¬xed-rate payment for period t is equal to:

Days t
notional amount — SR — -------------

The present value of the ¬xed-rate payment for period t is found by mul-
tiplying the previous expression by the forward discount factor. If we let
FDFt denote the forward discount factor for period t, then the present
value of the ¬xed-rate payment for period t is equal to:

notional amount — SR — -------------t — FDF t

We can now sum up the present value of the ¬xed-rate payment for
each period to get the present value of the ¬‚oating-rate payments. Using
the Greek symbol sigma, Σ, to denote summation and letting N be the
number of periods in the swap, then the present value of the ¬xed-rate
payments can be expressed as:

‘ notional amount — SR — -------------t — FDF t

This can also be expressed as

Days t
SR ‘ notional amount — ------------- — FDF t

The condition for no arbitrage is that the present value of the ¬xed-
rate payments as given by the expression above is equal to the present
value of the ¬‚oating-rate payments. That is,
Swaps and Caps/Floors

SR ‘ notional amount — -------------t — FDF t = PV of floating-rate payments

Solving for the swap rate

PV of floating-rate payments
SR = ---------------------------------------------------------------------------------------------
‘ notional amount — -------------t — FDF t -

All of the values to compute the swap rate are known.
Let™s apply the formula to determine the swap rate for our 3-year
swap. Exhibit 12.6 shows the calculation of the denominator of the for-
mula. The forward discount factor for each period shown in Column (5)
is obtained from Column (4) of Exhibit 12.5. The sum of the last column
in Exhibit 12.6 shows that the denominator of the swap rate formula is
$281,764,282. We know from Exhibit 12.5 that the present value of the
¬‚oating-rate payments is $14,052,917. Therefore, the swap rate is

SR = ----------------------------------- = 0.049875 = 4.9875%

Given the swap rate, the swap spread can be determined. For exam-
ple, since this is a 3-year swap, the convention is to use the 3-year on-the-
run Treasury rate as the benchmark. If the yield on that issue is 4.5875%,
the swap spread is 40 basis points (4.9875% ’ 4.5875%).
The calculation of the swap rate for all swaps follows the same prin-
ciple: equating the present value of the ¬xed-rate payments to that of the
¬‚oating-rate payments.

Valuing a Swap
Once the swap transaction is completed, changes in market interest rates
will change the payments of the ¬‚oating-rate side of the swap. The value
of an interest rate swap is the difference between the present value of the
payments of the two sides of the swap. The 3-month LIBOR forward
rates from the current Eurodollar CD futures contracts are used to (1) cal-
culate the ¬‚oating-rate payments and (2) determine the discount factors at
which to calculate the present value of the payments.
EXHIBIT 12.6 Calculating the Denominator for the Swap Rate Formula

(1) (2) (3) (4) (5) (6) (7)
Number of Period = Forward Forward discount factor
Quarter Quarter days in End of discount — Days/360
starts ends quarter quarter factor Days/360 — notional

Jan 1 year 1 Mar 31 year 1 90 1 0.98997649 0.25000000 24,749,412
Apr 1 year 1 June 30 year 1 91 2 0.97969917 0.25277778 24,764,618
July 1 year 1 Sept 30 year 1 92 3 0.96843839 0.25555556 24,748,981
Oct 1 year 1 Dec 31 year 1 92 4 0.95689609 0.25555556 24,454,011

Jan 1 year 2 Mar 31 year 2 90 5 0.94531597 0.25000000 23,632,899
Apr 1 year 2 June 30 year 2 91 6 0.93344745 0.25277778 23,595,477
July 1 year 2 Sept 30 year 2 92 7 0.92132183 0.25555556 23,544,891
Oct 1 year 2 Dec 31 year 2 92 8 0.90912441 0.25555556 23,233,179
Jan 1 year 3 Mar 31 year 3 90 9 0.89701471 0.25000000 22,425,368
Apr 1 year 3 June 30 year 3 91 10 0.88471472 0.25277778 22,363,622
July 1 year 3 Sept 30 year 3 92 11 0.87212224 0.25555556 22,287,568
Oct 1 year 3 Dec 31 year 3 92 12 0.85947083 0.25555556 21,964,255
Total 281,764,282
Swaps and Caps/Floors

To illustrate this, consider the 3-year swap used to demonstrate how
to calculate the swap rate. Suppose that one year later, interest rates
change as shown in Columns (4) and (6) in Exhibit 12.7. In Column (4)
shows the current 3-month LIBOR. In Column (5) are the Eurodollar
CD futures price for each period. These rates are used to compute the
forward rates in Column (6). Note that the interest rates have increased
one year later since the rates in Exhibit 12.7 are greater than those in
Exhibit 12.2. As in Exhibit 12.2, the current 3-month LIBOR and the
forward rates are used to compute the ¬‚oating-rate payments. These pay-
ments are shown in Column (8) of Exhibit 12.7.
In Exhibit 12.8, the forward discount factor is computed for each
period. The calculation is the same as in Exhibit 12.4 to obtain the for-
ward discount factor for each period. The forward discount factor for
each period is shown in the last column of Exhibit 12.8.
In Exhibit 12.9 the forward discount factor (from Exhibit 12.8) and
the ¬‚oating-rate payments (from Exhibit 12.7) are shown. The ¬xed-rate
payments need not be recomputed. They are the payments shown in Col-
umn (8) of Exhibit 12.3. These are ¬xed-rate payments for the swap rate of
4.9875% and are reproduced in Exhibit 12.9. Now the two payment
streams must be discounted using the new forward discount factors. As
shown at the bottom of Exhibit 12.9, the two present values are as follows:

Present value of ¬‚oating-rate payments $11,459,495
Present value of ¬xed-rate payments $9,473,390

The two present values are not equal and therefore for one party the
value of the swap increased and for the other party the value of the swap
decreased. Let™s look at which party gained and which party lost.
The ¬xed-rate payer will receive the ¬‚oating-rate payments. And
these payments have a present value of $11,459,495. The present value
of the payments that must be made by the ¬xed-rate payer is
$9,473,390. Thus, the swap has a positive value for the ¬xed-rate payer
equal to the difference in the two present values of $1,986,105. This is
the value of the swap to the ¬xed-rate payer. Notice, consistent with
what we said in the previous chapter, when interest rates increase (as
they did in our illustration), the ¬xed-rate payer bene¬ts because the
value of the swap increases.
In contrast, the ¬xed-rate receiver must make payments with a
present value of $11,459,495 but will only receive ¬xed-rate payments
with a present value equal to $9,473,390. Thus, the value of the swap for
the ¬xed-rate receiver is ’$1,986,105. Again, as explained earlier, the
¬xed-rate receiver is adversely affected by a rise in interest rates because it
results in a decline in the value of a swap.
EXHIBIT 12.7 Rates and Floating-Rate Payments One Year Later if Rates Increase

(1) (2) (3) (4) (5) (6) (7) (8)
Number of Current Eurodollar Period = Floating-rate
Quarter Quarter days in 3-month futures Forward End of payments at
starts ends quarter LIBOR price rate quarter end of quarter

Jan 1 year 2 Mar 31 year 2 90 5.25% 1 1,312,500

Apr 1 year 2 June 30 year 2 91 94.27 5.73% 2 1,448,417
July 1 year 2 Sept 30 year 2 92 94.22 5.78% 3 1,477,111
Oct 1 year 2 Dec 31 year 2 92 94.00 6.00% 4 1,533,333
Jan 1 year 3 Mar 31 year 3 90 93.85 6.15% 5 1,537,500
Apr 1 year 3 June 30 year 3 91 93.75 6.25% 6 1,579,861
July 1 year 3 Sept 30 year 3 92 93.54 6.46% 7 1,650,889
Oct 1 year 3 Dec 31 year 3 92 93.25 6.75% 8 1,725,000
EXHIBIT 12.8 Period Forward Rates and Forward Discount Factors One Year Later if Rates Increase

(1) (2) (3) (4) (5) (6) (7)
Number of Period = Period Forward
Quarter Quarter days in End of Forward forward discount
starts ends quarter quarter rate rate factor

Jan 1 year 2 Mar 31 year 2 90 1 5.25% 1.3125% 0.98704503

Apr 1 year 2 June 30 year 2 91 2 5.73% 1.4484% 0.97295263
July 1 year 2 Sept 30 year 2 92 3 5.78% 1.4771% 0.95879023
Oct 1 year 2 Dec 31 year 2 92 4 6.00% 1.5333% 0.94431080
Jan 1 year 3 Mar 31 year 3 90 5 6.15% 1.5375% 0.93001186
Apr 1 year 3 June 30 year 3 91 6 6.25% 1.5799% 0.91554749
July 1 year 3 Sept 30 year 3 92 7 6.46% 1.6509% 0.90067829
Oct 1 year 3 Dec 31 year 3 92 8 6.75% 1.7250% 0.88540505
EXHIBIT 12.9 Valuing the Swap One Year Later if Rates Increase

(1) (2) (3) (4) (5) (6) (7)
Forward Floating cash PV of Fixed cash PV of
Quarter Quarter discount ¬‚ow at end ¬‚oating cash ¬‚ow at end ¬xed
starts ends factor of quarter ¬‚ow of quarter cash ¬‚ow

Jan 1 year 2 Mar 31 year 2 0.98704503 1,312,500 1,295,497 1,246,875 1,230,722
Apr 1 year 2 June 30 year 2 0.97295263 1,448,417 1,409,241 1,260,729 1,226,630
July 1 year 2 Sept 30 year 2 0.95879023 1,477,111 1,416,240 1,274,583 1,222,058
Oct 1 year 2 Dec 31 year 2 0.94431080 1,533,333 1,447,943 1,274,583 1,203,603
Jan 1 year 3 Mar 31 year 3 0.93001186 1,537,500 1,429,893 1,246,875 1,159,609
Apr 1 year 3 June 30 year 3 0.91554749 1,579,861 1,446,438 1,260,729 1,154,257
July 1 year 3 Sept 30 year 3 0.90067829 1,650,889 1,486,920 1,274,583 1,147,990

Oct 1 year 3 Dec 31 year 3 0.88540505 1,725,000 1,527,324 1,274,583 1,128,523
Total 11,459,495 9,473,390

Summary Fixed-rate payer Fixed-rate receiver

PV of payments received 11,459,495 9,473,390
PV of payments made 9,473,390 11,459,495
Value of swap 1,986,105 ’1,986,105
Swaps and Caps/Floors

EXHIBIT 12.10 Swap Rates and Spreads for Various Maturities

Source: Bloomberg Financial Markets

The same valuation principle applies to more complicated swaps.
For example, there are swaps whose notional amount changes in a pre-
determined way over the life of the swap. These include amortizing
swaps, accreting swaps, and roller coaster swaps. Once the payments
are speci¬ed, the present value is calculated as described above by sim-
ply adjusting the payment amounts by the changing notional amounts”
the methodology does not change.

As we have seen, interest rate swaps are valued using no-arbitrage rela-
tionships relative to instruments (funding or investment vehicles) that
produce the same cash ¬‚ows under the same circumstances. Earlier we
provided two interpretations of a swap: (1) a package of futures/forward
contracts and (2) a package of cash market instruments. The swap spread
is de¬ned as the difference between the swap™s ¬xed rate and the rate on a
Treasury whose maturity matches the swap™s tenor.
Exhibit 12.10 displays a Bloomberg screen with interest rate swap rates
(in percent) and swap spreads (in basis points) for various maturities out to
30 on December 7, 2001. Recall, the bid price is the ¬xed rate that the bro-

ker/dealer is willing to pay in order to receive a ¬‚oating rate. Conversely,
the ask price is the ¬xed rate the broker/dealer wants to receive in order to
pay a ¬‚oating rate. Current swap rates and spreads for a number of coun-
tries can be obtained on Bloomberg with the function IRSB. Exhibit 12.11
presents a Bloomberg screen of interest rate swap rates for eight different
currencies. Bloomberg collects the spread information throughout the trad-
ing day and an average is calculated using the spreads from three market
makers. The actual swap rates can be obtained simply by adding the swap
spreads to the on-the-run U.S. Treasury yield curve. Exhibit 12.12 is a time
series plot obtained from Bloomberg for daily values of the 5-year swap
spread (in basis points) for the period December 7, 2000 to December 7,
2001. This plot can be obtained using the function USSP5 Index GP.
The swap spread is determined by the same factors that drive the
spread over Treasuries on instruments that replicate a swap™s cash ¬‚ows
i.e., produce a similar return or funding pro¬le. As discussed below, the
swap spread™s key determinant for swaps with tenors (i.e., maturities) of
¬ve years or less is the cost of hedging in the Eurodollar CD futures mar-
ket.4 Although listed contracts exist with delivery dates out to 10 years,
the liquidity of the Eurodollar CD futures market diminishes considerably
after about ¬ve years. For longer tenor swaps, the swap spread is largely
driven by credit spreads in the corporate bond market.5 Speci¬cally,
longer-dated swaps are priced relative to rates paid by investment-grade
credits in traditional ¬xed- and ¬‚oating-rate markets.
Given that a swap is a package of futures/forward contracts, the
shorter-term swap spreads respond directly to ¬‚uctuations in Eurodollar
CD futures prices. As noted, there is a liquid market for Eurodollar CD
futures contracts with maturities every three months for approximately
¬ve years. A market participant can create a synthetic ¬xed-rate security
or a ¬xed-rate funding vehicle by taking a position in a bundle of Euro-
dollar CD futures contracts (i.e., a position in every 3-month Eurodollar
CD futures contract up to the desired maturity date).

Naturally, this presupposes the reference rate used for the floating-rate cash flows
is LIBOR. Furthermore, part of swap spread is attributable simply to the fact that
LIBOR for a given maturity is higher than the rate on a comparable-maturity U.S.
The default risk component of a swap spread will be smaller than for a comparable
bond credit spread. The reasons are straightforward. First, since only net interest
payments are exchanged rather than both principal and coupon interest payments,
the total cash flow at risk is lower. Second, the probability of default depends jointly
on the probability of the counterparty defaulting and whether or not the swap has a
positive value. See John C. Hull, Introduction to Futures and Options Markets,
Third Edition (Upper Saddle River, NJ: Prentice Hall, 1998).
Swaps and Caps/Floors

EXHIBIT 12.11 Swap Rates for Various Currencies

Source: Bloomberg Financial Markets

EXHIBIT 12.12 Time Series of the 5-Year Swap Spread

Source: Bloomberg Financial Markets

For example, consider a ¬nancial institution that has ¬xed-rate assets
and ¬‚oating-rate liabilities. Both the assets and liabilities have a maturity of
three years. The interest rate on the liabilities resets every three months
based on 3-month LIBOR. This ¬nancial institution can hedge this mis-
matched asset/liability position by buying a 3-year bundle of Eurodollar CD
futures contracts. By doing so, the ¬nancial institution is receiving LIBOR
over the 3-year period and paying a ¬xed dollar amount (i.e., the futures
price). The ¬nancial institution is now hedged because the assets are ¬xed
rate and the bundle of long Eurodollar CD futures synthetically creates a
¬xed-rate funding arrangement. From the ¬xed dollar amount over the
three years, an effective ¬xed rate that the ¬nancial institution pays can be
computed. Alternatively, the ¬nancial institution can synthetically create a
¬xed-rate funding arrangement by entering into a 3-year swap in which it
pays ¬xed and receives 3-month LIBOR. Other things equal, the ¬nancial
institution will use the vehicle that delivers the lowest cost of hedging the
mismatched position. That is, the ¬nancial institution will compare the syn-
thetic ¬xed rate (expressed as a percentage over U.S. Treasuries) to the 3-
year swap spread. The difference between the synthetic spread and the swap
spread should be within a few basis points under normal circumstances.
For swaps with tenors greater than ¬ve years, we cannot rely on the
Eurodollar CD futures due to diminishing liquidity of such contracts.
Instead, longer-dated swaps are priced using rates available for invest-
ment-grade corporate borrowers in ¬xed-rate and ¬‚oating-rate debt mar-
kets. Since a swap can be interpreted as a package of long and short
positions in a ¬xed-rate bond and a ¬‚oating-rate bond, it is the credit
spreads in those two market sectors that will be the primary determinant
of the swap spread. Empirically, the swap curve lies above the U.S. Trea-
sury yield curve and below the on-the-run yield curve for AA-rated
banks.6 Swap ¬xed rates are lower than AA-rated bond yields because
their lower credit due to netting and offsetting of swap positions.
In addition, there are a number of other technical factors that in¬‚u-
ence the level of swap spreads.7 While the impact of some these factors is
ephemeral, their in¬‚uence can be considerable in the short run. Included
among these factors are: (1) the level and shape of the Treasury yield
curve; (2) the relative supply of ¬xed- and ¬‚oating-rate payers in the
interest rate swap market; (3) the technical factors that affect swap deal-
ers; and (4) the level of asset-based swap activity.

For a discussion of this point, see Andrew R. Young, A Morgan Stanley Guide to
Fixed Income Analysis (New York: Morgan Stanley, 1997).
See Ellen L. Evans and Gioia Parente Bales, “What Drives Interest Rate Swap
Spreads,” Chapter 13 in Carl R. Beidleman (ed.), Interest Rate Swaps (Burr Ridge,
IL: Irwin Professional Publishing, 1991).
Swaps and Caps/Floors

The level, slope, and curvature of the U.S. Treasury yield is an important
in¬‚uence on swap spreads at various maturities. The reason is that embed-
ded in the yield curve are the market™s expectations of the direction of future
interest rates. While these expectations are sometimes challenging to extract,
the decision to borrow at a ¬xed-rate or a ¬‚oating-rate will be based, in part,
on these expectations. The relative supply of ¬xed- and ¬‚oating-rate payers
in the interest rate swap market should also be in¬‚uenced by these expecta-
tions. For example, many corporate issuers”¬nancial institutions and fed-
eral agencies in particular”swap their newly issued ¬xed-rate debt into
¬‚oating using the swap market. Consequently, swap spreads will be affected
by the corporate debt issuance calendar. In addition, swap spreads, like
credit spreads, also tend to increase with the swap™s tenor or maturity.
Swap spreads are also affected by the hedging costs faced by swap
dealers. Dealers hedge the interest rate risk of long (short) swap positions
by taking a long (short) position in a Treasury security with the same
maturity as the swap™s tenor and borrowing funds (lending funds) in the
repo market. As a result, the spread between LIBOR and the appropriate
repo rate will be a critical determinant of the hedging costs. For example,
with the burgeoning U.S. government budget surpluses starting in the late
1990s, the supply of Treasury securities has diminished. One impact of
the decreased supply is an increase in the spread between the yields of on-
the-run and off-the-run Treasuries. As this spread widens, investors must
pay up for the relatively more liquid on-the-run issues. This chain reac-
tion continues and results in on-the-run Treasuries going “on special” in
repo markets. When on-the-run Treasuries go “on special,” it is corre-
spondingly more expensive to use these Treasuries as a hedge. This
increase in hedging costs results in wider swap spreads.8
Another in¬‚uence on the level of swap spreads is the volume of asset-
based swap transactions. An asset-based swap transaction involves the
creation of a synthetic security via the purchase of an existing security
and the simultaneous execution of a swap. For example, after the Russian
debt default and ruble devaluation in August 1998, risk-averse investors
sold corporate bonds and ¬‚ed to the relative safety of U.S. Treasuries.
Credit spreads widened considerably and liquidity diminished. A con-
trary-minded ¬‚oating-rate investor (like a ¬nancial institution) could have
taken advantage of these circumstances by buying newly issued invest-

Traders often use the repo market to obtain specific securities to cover short posi-
tions. If a security is in short supply relative to demand, the repo rate on a specific
security used as collateral in repo transaction will be below the general (i.e., generic)
collateral repo rate. When a particular security™s repo rate falls markedly, that secu-
rity is said to be “on special.” Investors who own these securities are able to lend
them out as collateral and borrow at bargain basement rates.

ment grade corporate bonds with relatively attractive coupon rates and
simultaneously taking a long position in an interest rate swap (pay ¬xed/
receive ¬‚oating). Because of the higher credit spreads, the coupon rate
that the ¬nancial institution receives is higher than the ¬xed-rate paid in
the swap. Accordingly, the ¬nancial institution ends up with a synthetic
¬‚oating-rate asset with a sizeable spread above LIBOR.
By similar reasoning, investors can use swaps to create a synthetic
¬xed-rate security. For example, during the mid-1980s, many banks
issued perpetual ¬‚oating-rate notes in the Eurobond market. A perpetual
¬‚oating-rate note is a security that delivers ¬‚oating-rate cash ¬‚ows for-
ever. The coupon is reset and paid usually every three months with a cou-
pon formula equal to the reference rate (e.g., 3-month LIBOR) plus a
spread. When the perpetual ¬‚oating-rate note market collapsed in late
1986, the contagion spread into other sectors of the ¬‚oaters market.9
Many ¬‚oaters cheapened considerably. As before, contrary-minded ¬xed-
rate investors could exploit this situation through the purchase of a rela-
tively cheap (from the investor™s perspective) ¬‚oater while simultaneously
taking a short position in an interest rate swap (pay ¬‚oating/receive ¬xed)
thereby creating a synthetic ¬xed-rate investment. The investor makes
¬‚oating-rate payments (say based on LIBOR) to their counterparty and
receives ¬xed-rate payments equal to the Treasury yield plus the swap
spread. Accordingly, the ¬xed rate on this synthetic security is equal to
the sum of the following: (1) the Treasury bond yield that matches the
swap™s tenor; (2) the swap spread; and (3) the ¬‚oater™s index spread.

The swap market is very ¬‚exible and instruments can be tailor-made to ¬t
the requirements of individual customers. A wide variety of swap con-
tracts are traded in the market. Although the most common reference rate
for the ¬‚oating-leg of a swap is six-month Libor for a semiannual paying
¬‚oating leg, other reference rates that have been used include three-month
Libor, the prime rate (for dollar swaps), the one-month commercial paper
rate, and the Treasury bill rate, and the municipal bond rate.
The term of a swap need not be ¬xed; swaps may be extendible or
putable. In an extendible swap, one of the parties has the right but not the
obligation to extend the life of the swap beyond the ¬xed maturity date,
while in a putable swap one party has the right to terminate the swap
prior to the speci¬ed maturity date.

Suresh E. Krishman, “Asset-Based Interest Rate Swaps,” Chapter 8 in Interest Rate
Swaps and Caps/Floors

It is also possible to transact options on swaps, known as swaptions.
A swaption is the right to enter into a swap agreement at some point in
the future, during the life of the option. Essentially a swaption is an
option to exchange a ¬xed-rate bond cash ¬‚ow for a ¬‚oating-rate bond
cash ¬‚ow structure. As a ¬‚oating-rate bond is valued on its principal
value at the start of a swap, a swaption may be viewed as the value on a
¬xed-rate bond, with a strike price that is equal to the face value of the
¬‚oating-rate bond. Swaptions will be described in more detail later.
Other swaps are described below.

Constant Maturity Swap
In a constant maturity swap, the parties exchange a Libor rate for a ¬xed
swap rate. For example, the terms of the swap might state that six-month
Libor is exchanged for the ¬ve-year swap rate on a semiannual basis for
the next ¬ve years, or for the ¬ve-year government bond rate. In the U.S.
market, the second type of constant maturity swap is known as a constant
maturity Treasury swap.

Accreting and Amortizing Swaps
In a plain vanilla swap, the notional principal remains unchanged during
the life of the swap. However it is possible to trade a swap where the
notional principal varies during its life. An accreting (or step-up) swap is
one in which the principal starts off at one level and then increases in
amount over time. The opposite, an amortizing swap, is one in which the
notional reduces in size over time. An accreting swap would be useful
where for instance, a funding liability that is being hedged increases over
time. The amortizing swap might be employed by a borrower hedging a
bond issue that featured sinking fund payments, where a part of the
notional amount outstanding is paid off at set points during the life of the
bond. If the principal ¬‚uctuates in amount, for example increasing in one
year and then reducing in another, the swap is known as a roller-coaster
swap. Another application of an amortizing swap is as a hedge for a loan
that is itself an amortizing one. Frequently this is combined with a for-
ward-starting swap, to tie in with the cash ¬‚ows payable on the loan. The
pricing and valuation of an amortizing swap is no different in principle to
a vanilla interest-rate swap; a single swap rate is calculated using the rele-
vant discount factors, and at this rate the net present value of the swap
cash ¬‚ows will equal zero at the start of the swap.

Zero-Coupon Swap
A zero-coupon swap replaces the stream of ¬xed-rate payments with a
single payment at the end of the swap™s life, or less common, at the begin-

ning. The ¬‚oating-rate payments are made in the normal way. Such a
swap exposes the ¬‚oating-rate payer to some credit risk because it makes
regular payments but does not receive any payment until the termination
date of the swap.

Libor-in-Arrears Swap
In a Libor-in-arrears swap (also known as a back-set swap), the reset date
is just before the end of the accrual period for the ¬‚oating-rate rather than
just before the start. Such a swap would be attractive to a counterparty
who had a different view on interest rates compared to the market con-
sensus. For instance in a rising yield curve environment, forward rates
will be higher than current market rates, and this will be re¬‚ected in the
pricing of a swap. A Libor-in-arrears swap would be priced higher than a
conventional swap. If the ¬‚oating-rate payer believed that interest rates
would in fact rise more slowly than forward rates (and the market) were
suggesting, he or she may wish to enter into an arrears swap as opposed
to a conventional swap.

Basis Swap
In a conventional swap one leg comprises ¬xed-rate payments and the
other ¬‚oating-rate payments. In a basis swap both legs are ¬‚oating-rate,
but linked to different money market indices. One leg is normally linked
to Libor, while the other might be linked to the CD rate or the commercial
paper rate. This type of swap would be used by a bank in the United
States that had made loans that paid at the prime rate and funded its loans
at Libor. A basis swap would eliminate the basis risk between the bank™s
income and interest expense. Other basis swaps are traded in which both
legs are linked to Libor, but at different maturities; for instance one leg
might be at three-month Libor and the other at six-month Libor. In such a
swap, the basis is different as is the payment frequency: one leg pays out
semiannually while the other would be paying on a quarterly basis.

Margin Swap
It is common to encounter swaps where there is a margin above or below
Libor on the ¬‚oating leg, as opposed to a ¬‚oating leg of Libor ¬‚at. Such
swaps are called margin swaps. If a bank™s borrowing is ¬nanced at
Libor+25bps, it may wish to receive Libor+25bps in the swap so that its
cash ¬‚ows match exactly. The ¬xed-rate quote for a swap must be
adjusted correspondingly to allow for the margin on the ¬‚oating side. So
in our example if the ¬xed-rate quote is say, 6.00%, it would be adjusted
to around 6.25%; differences in the margin quoted on the ¬xed leg might
arise if the day-count convention or payment frequency were to differ
Swaps and Caps/Floors

between ¬xed and ¬‚oating legs. Another reason why there may be a mar-
gin is if the credit quality of the counterparty demanded it, so that highly
rated counterparties may pay slightly below Libor, for instance.

Off-Market Swap
When a swap is transacted, its ¬xed rate is quoted at the current market
rate for that maturity. When the ¬xed rate is different from the market
rate, this type of swap is an off-market swap, and a compensating pay-
ment is made by one party to the other. An off-market rate may be used
for particular hedging requirements for example, or when a bond issuer
wishes to use the swap to hedge the bond as well as to cover the bond™s
issue costs.

Differential Swap
A differential swap is a basis swap but with one of the legs calculated in a
different currency. Typically one leg is ¬‚oating-rate, while the other is ¬‚oat-
ing-rate but with the reference rate stated in another currency but denomi-
nated in the domestic currency. For example, a differential swap may have
one party paying six-month sterling Libor, in sterling, on a notional princi-
pal of £10 million, and receiving euro-Libor minus a margin, payable in
sterling and on the same notional principal. Differential swaps are not very
common and are the most dif¬cult for a bank to hedge. The hedging is usu-
ally carried out using what is known as a quanto option.

Forward-Start Swap
A forward-start swap is one where the effective date is not the usual one
or two days after the trade date but a considerable time afterwards, for
instance say six months after trade date. Such a swap might be entered
into where one counterparty wanted to ¬x a hedge or cost of borrowing
now, but for a point some time in the future. Typically this would be
because the party considered that interest rates would rise or the cost of
hedging would rise. The swap rate for a forward-starting swap is calcu-
lated in the same way as that for a vanilla swap.

When ¬nancial institutions enter into a swap contract in order to hedge
interest-rate liabilities, the swap will be kept in place until its expiration.
However, circumstances may change or a ¬nancial institution may alter
its view on interest rates, and so circumstances may arise such that it may

be necessary to terminate the swap. The most straightforward option is
for the corporation to take out a second contract that negates the ¬rst.
This allows the ¬rst swap to remain in place, but there may be residual
cash ¬‚ows unless the two swaps cancel each other out precisely. The terms
for the second swap, being non-standard (and unlikely to be a exactly
whole years to maturity, unless traded on the anniversary of the ¬rst),
may also result in it being more expensive than a vanilla swap. As it is
unlikely that the second swap will have the same rate, the two ¬xed legs
will not net to zero. And if the second swap is not traded on an anniver-
sary, payment dates will not match.
For these reasons, an entity may wish to cancel the swap entirely. To
do this it will ask a swap market maker for a quotation on a cancellation
fee. The bank will determine the cancellation fee by calculating the net
present value of the remaining cash ¬‚ows in the swap, using the relevant
discount factor for each future cash ¬‚ow. In practice just the ¬xed leg will
be present valued, and then netted with Libor. The net present value of all
the cash ¬‚ows is the fair price for canceling the swap. The valuation prin-
ciples we established earlier will apply; that is, if the ¬xed rate payer is
asking to cancel the swap when interest rates have fallen, he will pay the
cancellation fee, and vice-versa if rates have risen.

The rate quoted for swaps in the interbank market assumes that the coun-
terparty to the transaction has a lending line with the swap bank, so the
swap rate therefore re¬‚ects the credit risk associated with interbank qual-
ity counterparty. This credit risk is re¬‚ected in the spread between the
swap rate and the equivalent-maturity government bond, although, as
noted, the spread also re¬‚ects other considerations such as liquidity and
supply and demand. The credit risk of a swap is separate from its interest-
rate risk or market risk, and arises from the possibility of the counter-
party to the swap defaulting on its obligations. If the present value of the
swap at the time of default is net positive, then a bank is at risk of loss of
this amount. While market risk can be hedged, it is more problematic to
hedge credit risk. The common measures taken include limits on lending
lines, collateral, and diversi¬cation across counterparty sectors, as well as
a form of credit value-at-risk to monitor credit exposures.
A bank therefore is at risk of loss due to counterparty default for all
its swap transactions. If at the time of default, the net present value of the
swap is positive, this amount is potentially at risk and will probably be
written off. If the value of the swap is negative at the time of default, in
Swaps and Caps/Floors

theory this amount is a potential gain to the bank, although in practice
the counterparty™s administrators will try to recover the value for their cli-
ent. In this case then, there is no net gain or loss to the swap bank. The
credit risk management department of a bank will therefore often assess
the ongoing credit quality of counterparties with whom the swap transac-
tions are currently positive in value.

So far we have discussed swap contracts where the interest payments are
both in the same currency. A cross-currency swap is similar to an interest-
rate swap, except that the currencies of the two legs are different. Like
interest-rate swaps, the legs are usually ¬xed- and ¬‚oating-rate, although
again it is common to come across both ¬xed-rate or both ¬‚oating-rate
legs in a currency swap. On maturity of the swap, there is an exchange of
principals, and usually (but not always) there is an exchange of principals
at the start of the swap. Where currencies are exchanged at the start of
the swap, at the prevailing spot exchange rate for the two currencies, the
exact amounts are exchanged back on maturity.
During the time of the swap, the parties make interest payments in
the currency that they have received when principals are exchanged. It
may seem that exchanging the same amount at maturity gives rise to some
sort of currency risk, in fact it is this feature that removes any element of
currency risk from the swap transaction.
Currency swaps are widely used in association with bond issues by bor-
rowers who seek to tap opportunities in different markets but have no
requirement for that market™s currency. By means of a currency swap, a
corporation can raise funds in virtually any market and swap the proceeds
into the currency that it requires. Often the underwriting bank that is
responsible for the bond issue will also arrange for the currency swap trans-
action. In a currency swap, therefore, the exchange of principal means that
the value of the principal amounts must be accounted for, and is dependent
on the prevailing spot exchange rate between the two currencies.
The same principles we established earlier in the chapter for the pric-
ing and valuation of interest rate swaps may also be applied to currency
swaps. A generic currency swap with ¬xed-rate payment legs would be
valued at the fair value swap rate for each currency, which would give a
net present value of zero. A ¬‚oating-¬‚oating currency swap may be valued
in the same way, and for valuation purposes the ¬‚oating-leg payments are
replaced with an exchange of principals, as we observed for the ¬‚oating
leg of an interest rate swap. A ¬xed-¬‚oating currency swap is therefore

valued at the ¬xed-rate swap rate for that currency for the ¬xed leg, and
at Libor or the relevant reference rate for the ¬‚oating leg.

A bank or corporation may enter into an option on a swap, which is
called a swaption. The buyer of a swaption has the right but not the obli-
gation to enter into an interest rate swap at any time during the option™s
life. The terms of the swaption will specify whether the buyer is the ¬xed-
or ¬‚oating-rate payer; the seller of the option (the writer) becomes the
counterparty to the swap if the option is exercised. In the market, the
convention is that if the buyer has the right to exercise the option as the
¬xed-rate payer, the buyer has purchased a call swaption, while if by
exercising the buyer of the swaption becomes the ¬‚oating-rate payer he
has bought a put swaption. The writer of the swaption is the party that
has an obligation to establish the other leg.
Swaptions are up to a point similar to forward start swaps, but the
buyer has the option of whether or not to commence payments on the
effective date. A bank may purchase a call swaption if it expects interest
rates to rise, and will exercise the option if indeed rates do rise as the
bank has expected. This is shown in the pro¬t/loss diagrams in Exhibit
12.13. The pro¬t/loss (P/L) diagram on the left is for a long swap position
while the one on the right is for a long swaption.
A corporation will use swaptions as part of an interest-rate hedge for
an anticipated future exposure. For example, assume that a corporation
will be entering into a ¬ve-year bank loan in three months™ time. Interest
on the loan is charged on a ¬‚oating-rate basis, but the corporation
intends to swap this to a ¬xed-rate liability after it has entered into the
loan. As an added hedge, the corporation may choose to purchase a
swaption that gives it the right to receive Libor and pay a ¬xed rate, say
6%, for a ¬ve-year period beginning in three months™ time. When the
time comes for the corporation to engage in a swap contract and
exchange its interest-rate liability in three months™ time (having entered
into the loan), if the ¬ve-year swap rate is below 6%, the corporation will
transact the swap in the normal way and the swaption will expire worth-
less. However, if the ¬ve-year swap rate is above 6%, the corporation will
instead exercise the swaption, giving it the right to enter into a ¬ve-year
swap and paying a ¬xed rate of 6%. Essentially the corporation has taken
out “insurance” that it does not have to pay a ¬xed rate of more than
6%. Hence swaptions can be used to guarantee a maximum swap rate lia-
bility. They are similar to forward-starting swaps, but differ because they
Swaps and Caps/Floors

represent an option (as opposed to an obligation) to enter into a swap on
¬xed terms. The swaption enables a corporation to hedge against unfa-
vorable movements in interest rates but also to gain from favorable move-
ments, although there is of course a cost associated with this, which is the
premium paid for the swaption.

In both the U.S. dollar and euro markets, the position of the government
bond yield curve as the benchmark instrument for pricing, valuation, and
hedging purposes is eroding. In the U.S. dollar market this has been the
result of the decreasing supply of U.S. Treasury securities, due to continu-
ing federal government budget surpluses, leading to illiquidity particu-
larly at the long end of the curve.10 In Europe, the introduction of the
euro in 1999 resulted in a homogeneous euro swap curve replacing indi-
vidual government bond yield curves as the benchmark. The nominal vol-
umes of swap contacts far outstrip that of government bonds in both
currency areas. For instance in June 2000 there was $22.9 trillion of swap
contracts outstanding, which was over ¬ve times the combined size of the
German, French, and Italian government bond markets.11 The falling
issuance of government bonds has placed pressure on government bonds
as benchmark instruments, which has resulted in greater basis risk for
market participants using exchange-traded government bond futures con-
tracts as hedging tools.

EXHIBIT 12.13 Pro¬t/Loss Diagrams for an Interest Rate Swap and a Swaption

On October 31, 2001, the U.S. Treasury announced it would no longer issue 30-
year bonds.
The source is the LIFFE. The authors would like to thank Nimmish Thakker at
LIFFE for assistance with statistics and information on the Swapnote contract.

EXHIBIT 12.14 Yield Curves for French and German Government Bonds,
Pfandbriefe Securities and Euro Interest-Rate Swaps, February 2001

The increasing importance of interest rate swaps as hedging and even
benchmark instruments was a primary motivation behind the develop-
ment of an exchange-traded contract referenced against the swap curve.
The swap curve is the inter-bank curve, derived from inter-bank deposits,
short-term interest rate futures and interest-rate swaps. Swapnote®, intro-
duced by LIFFE in 2001, is a standardized contract that allows market
participants to put on an exposure to the interest-rate swap curve, but
with the ease of access of an exchange-traded future. It is the ¬rst such
contract in the world. It may be that the euro swap curve becomes the ref-
erence not only for valuing non-government securities, but also for Euro-
pean government bonds. In that case, the euro swap curve will transform
into the cornerstone for the entire euro-area debt capital market, which
will deteriorate further the relationship between government bonds and
non-government bonds. An indication of this is given in Exhibit 12.14
which shows the yield curves for the swap curve as well as two govern-
ment curves and a AAA-rated security. The non-government security mir-
rors the swap curve much more closely than the government bonds.
Swapnote may be thought of as combining the features of an exchange-
traded futures contract and an OTC FRA contract. Alternatively, it may be
viewed as a cash-settled bond futures contract in which the delivery basket
consists of a single bond. It is referenced to the euro interest-rate swap
curve, and contracts are provided for two-, ¬ve-, and ten-year maturities.
The contract can be used for speculative purposes, or for hedging purposes
of credit exposures such as corporate bonds or an interest-rate swap book.
In theory, it provides a closer correlation between the hedging instrument
Swaps and Caps/Floors

and the exposure, thus reducing basis risk. By using an exchange-traded
contract rather than swaps themselves, users also gain from the advantages
associated with exchange-based trading and central clearing. This includes
lower regulatory capital requirements, removal of counterparty risk, and
elimination of administration requirements of actual swap contracts, which
can stretch out to many years. Market participants will compare this to
hedging using conventional interest-rate swaps, which involve credit line
issues, documentation issues, and bid-offer spreads which can make the
swap market dif¬cult and/or expensive to access.
Market participants can gain exposure to the yield curve out to ten
years; beyond that, government bonds must continue to be used.

Contract Speci¬cation
The Swapnote contract speci¬cation provides for a standardized exchange-
traded futures contract referenced to the swap curve. It is a price-based
contract, similar in concept to a forward-starting swap, and is cash set-
tled against the swap curve. The contract consists of a series of notional
cash ¬‚ows representing the cash ¬‚ows of a bond, with a ¬xed-rate cash
¬‚ow and a principal repayment. The ¬xed-rate cash ¬‚ow is set at 6%,
and the price quotation is per 100 euro just like a bond future. When the
contract expires its price re¬‚ects the market price at the time, re¬‚ecting
supply and demand, and other economic and market fundamentals. The
settlement price is calculated using the standard exchange delivery settle-
ment price methodology (EDSP). For Swapnote the EDSP is given by

EDSP = 100 d m + C ‘ A i d i (1)

C = the notional coupon for the contract, which is ¬xed at 6%
m = he maturity of the contract in years, either 2, 5 or 10
Ai = the notional accrued interest between coupon dates, given as
the number of days between the i-1 and i notional cash ¬‚ows
and divided by 360. Day counts use the 30/360 basis.
di = is the zero-coupon discount factor, calculated from the swap
rate is ¬xed for each period from the delivery date to the ith
notional cash ¬‚ow.

The zero-coupon yield curve is constructed by LIFFE from ISDA
benchmark swap ¬xes as at the expiry date of the contract. The ¬rst dis-
count factor d1 is given by

d 1 = ------------------------ (2)
1 + A 1 rs 1

where rs is the swap rate and rs1 is the one-year swap rate. The full set
of discount factors is then calculated using the bootstrapping technique,
and is given by
1 “ rs i ‘ A j d j
d i = ------------------------------------
- (3)
1 + A 1 rs i

Equation (1) states that the EDSP is the sum of the discounted notional
cash ¬‚ows, with the present value of each notional cash ¬‚ow calculated
using zero-coupon discount factors that have been derived from the ISDA
benchmark swap curve as at the expiration date. The fair price of the con-
tract is the sum of the present values of the notional cash ¬‚ows, valued to
the trade date and then forward valued to the contract delivery date. For-
ward valuing to the delivery date can be regarded as funding the position
(were it a coupon bond) from trade date to delivery date. Exhibit 12.15
presents a summary of the ten-year Swapnote contract speci¬cations.
EXHIBIT 12.15 Ten-Year Euro Swapnote Contract Speci¬cation

Unit of trading 100,000 notional principal amount
Notional ¬xed rate 6.0%
Maturity Notional principal amount due ten years from deliv-
ery day
Delivery months March, June, September, December
Delivery day Third Wednesday of delivery month
Last trading day 10:00 London time
Two business days prior to the delivery day
Price quote Per 100 nominal value
Minimum price movement 0.01
Tick size and value 10
Trading hours 07:00“18:00
(LIFFE Connect)

The contract is cash settled, therefore “principal” and “coupon” payments are no-
tional and do not actually occur.
The maturity of a Swapnote contract is defined as the time from the delivery month
to the maturity of the last notional cash flow.
Source: LIFFE
Swaps and Caps/Floors

EXHIBIT 12.16 Price Trading History, Ten-Year Swapnote (LIFFE) and
Ten-Year Bund (Eurex), September-October 2001

Trade Spread History
To illustrate the similarity in market price movements, Exhibit 12.16
shows the price trading history of the ten-year Swapnote contract against
the ten-year Bund contract as traded on Eurex during September and
October 2001. The exhibit indicates that the Swapnote is behaving as a
benchmark to the market, similar to the Bund contract, with a narrowing
spread between the contracts over time.

The Chicago Board of Trade (CBOT) introduced a swap futures contract
in late October 2001. The underlying instrument is the notional price of
the ¬xed-rate side of a 10-year interest rate swap that has a notional prin-
cipal equal to $100,000 and that exchanges semi-annual interest pay-
ments at a ¬xed annual rate of 6% for ¬‚oating interest rate payments
based on 3-month LIBOR. This swap futures contract is cash-settled with
a settlement price determined by the ISDA benchmark 10-year swap rate
on the last day of trading before the contract expires. This benchmark
rate is published with a one day lag in the Federal Reserve Board™s statis-
tical release H.15. Contracts expire the third month of each quarter

(March, June, September and December) just like the other CBOT inter-
est rate futures contracts. The last trading day is the second London busi-
ness day preceding the third Wednesday of the expiration month.
The swap futures contract will be priced just as a forward-start swap
discussed earlier in this chapter. For example, the December 2001 swap
futures contract will be for a new 10-year interest rate swap beginning on
December 17, 2001. It is anticipated that this contract will be a valuable
tool to hedge spread product.

An important option combination in debt markets is the cap and ¬‚oor, which
are used to control interest-rate risk exposure. Caps and ¬‚oors are combina-
tions of the same types of options (calls or puts) with identical strike prices
but arranged to run over a range of time periods. In the last chapter, we
reviewed the main instruments used to control interest-rate risk, including
short-dated interest-rate futures and FRAs. For example, a corporation that
desires to protect against a rise in future borrowing costs could buy FRAs or
sell futures. These instruments allow the user to lock in the forward interest
rate available today. However, such positions do not allow the hedger to gain
if market rates actually move as feared/anticipated. Hedging with FRAs or
futures can prevent loss but at the expense of any extra gain. To overcome
this, the hedger might choose to construct the hedge using options. For inter-
est rate hedges, primary instruments are the cap and ¬‚oor.12
Caps and ¬‚oors are agreements between two parties whereby one
party for an upfront fee agrees to compensate the other if a designated
interest rate (called the reference rate) is different from a predetermined
level. The party that bene¬ts if the reference rate differs from a perdeter-
mined level is called the buyer and the party that must potentially make
payments is called the seller. The predetermined interest rate level is called
the strike rate. An interest rate cap speci¬es that the seller agrees to pay
the buyer if the reference rate exceeds the strike rate. An interest rate ¬‚oor
speci¬es that the seller agrees to pay the buyer if the reference rate is
below the strike rate.
The terms of an interest rate agreement include: (1) the reference rate;
(2) the strike rate that sets the cap or ¬‚oor; (3) the length of the agree-
ment; (4) the frequency of reset; and (5) the notional amount (which
determines the size of the payments). If a cap or a ¬‚oor are in-the-money
on the reset date, the payment by the seller is typically made in arrears.

The term cap and floor is not to be confused with floating-rate note products that
have caps and/or floors which restrict how much a floater™s coupon rate can float.
Swaps and Caps/Floors

Some commercial banks and investment banks now write options on
interest rate caps and ¬‚oors for customers. Options on caps are called
captions. Options on ¬‚oors are called ¬‚otions.

A cap is essentially a strip of options. A borrower with an existing inter-
est-rate liability can protect against a rise in interest rates by purchasing a
cap. If rates rise above the cap, the borrower will be compensated by the
cap payout. Conversely, if rates fall the borrower gains from lower fund-
ing costs and the only expense is the upfront premium paid to purchase
the cap. The payoff for the cap buyer at a reset date if the value of the ref-
erence rate exceeds the cap rate on that date is as follows:

Notional amount — (Value of the reference rate ’ Cap rate)
— (Number of days in settlement period/Number of days in year)

Naturally, if the reference rate is below the cap rate, the payoff is zero.
A cap is composed of a series of individual options or caplets. The
price of a cap is obtained by pricing each of the caplets individually. Each
caplet has a strike interest-rate that is the rate of the cap. For example, a
borrower might purchase a 3% cap (Libor reference rate), which means
that if rates rise above 3% the cap will pay out the difference between the
cap rate and the actual Libor rate. A one-year cap might be composed of
a strip of three individual caplets, each providing protection for succes-
sive three-month periods. The ¬rst three-month period in the one-year
term is usually not covered, because the interest rate for that period, as it
begins immediately, will be known already. A caplet runs over two peri-
ods, the exposure period and the protection period. The exposure period
runs from the date the cap is purchased to the interest reset date for the
next borrowing period. At this point, the protection period begins and
runs to the expiration of the caplet. The protection period is usually three
months, six months or one year, and will be set to the interest rate reset
liability that the borrower wishes to hedge. Therefore, the protection
period is usually identical for all the caplets in a cap.
As an illustration, let™s utilize Bloomberg™s Cap, Floor, Collar Calcula-
tor presented in Exhibit 12.17. Consider a hypothetical one-year cap on
three-month LIBOR with a strike rate of 3%. The settlement date for the
agreement is November 30, 2001 and the expiration date is November
30, 2002. The ¬rst reset date is February 28, 2002, which is labelled
"Start" in the top center of the screen. If three-month LIBOR is above the
strike rate on this date, say, 3.5%, the payoff of the cap assuming the
notional principal is $1,000,000 is computed as follows:

$1,000,000 — (3.5% ’ 3.0%) — 92/360 = $1,277.78

This payment is made on May 31, 2002. Note that the day count conven-
tion is Actual/360 in the US markets and Actual/365 in the UK. The sec-
ond reset date is May 31, 2002 for which payment is made, if necessary,
on August 31, 2002. Finally, the third reset date is August 31, 2002 for
which payment is made, if necessary, on November 30, 2002.
As noted above, each cap can be thought of a series of call options or
caplets on the underlying reference rate in this case, three-month LIBOR.
The ¬rst caplet expires on the next reset date, February 28, 2002; the sec-
ond caplet expires on May 31, 2002, and so forth. Accordingly, the value
of the cap is the sum of the values of all the caplets. In the "PRICING"
box, the "Premium" represents the value of our hypothetical cap as a per-
centage of the notional amount. For our hypothetical cap, the premium is
0.1729% or approximately $1,729. Exhibit 12.18 presents Bloomberg™s
Caplet Valuation screen that shows the value of caplet in the column
labelled “Component Value.” Bloomberg uses a modi¬ed Black-Scholes
model to value each caplet and users can choose whether to use the same
volatility estimate for each caplet or allow the volatility for each caplet to
differ. Binomial lattice models are also extensively in practice to value caps.

EXHIBIT 12.17 Bloomberg™s Cap/Floor/Collar Calculator

Source: Bloomberg Financial Markets
Swaps and Caps/Floors

EXHIBIT 12.18 Bloomberg Screen with the Valuation of a Hypothetical Cap

Source: Bloomberg Financial Markets

It is possible to protect against a drop in interest rates by purchasing a ¬‚oor.
This is exactly opposite of a cap in that a ¬‚oor pay outs when the reference
rate falls below the stike rate. This would be used by an institution that
wished to protect against a fall in income caused by a fall in interest rate”
for example, a commercial bank with a large proportion of ¬‚oating-rate
assets. For the ¬‚oor buyer, the payoff at a reset date is as follows if the
value of the reference rate at the reset date is less than the ¬‚oor rate:

Notional amount — (Floor rate ’ Value of the reference rate)
— (Number of days in settlement period/Number of days in a year)

The ¬‚oor™s payoff is zero if the reference rate is higher than the ¬‚oor rate.
To illustrate, let™s once again utilize Bloomberg™s Cap, Floor, Collar
Calculator presented in Exhibit 12.19. Consider a hypothetical one-year
¬‚oor on three-month LIBOR with a strike rate of 2.5%. The settlement
date for the agreement is November 30, 2001 and the expiration date is
November 30, 2002. If three-month LIBOR is below the strike rate on
this date, say, 2%, the payoff of the ¬‚oor assuming the notional amount is
$1,000,000 is computed as follows:

$1,000,000 — (2.5% ’ 2.0%) — 92/360 = $1,277.78

This payment is made on May 31, 2002. Note that the day count conven-
tion is Actual/360 one again.
A ¬‚oor can be thought of as a series of put options on the underlying
reference rate in this case, three-month LIBOR. The value of the ¬‚oor is
the sum of the values of all the individual put options. In the "PRICING"
box, the "Premium" for our hypothetical cap, the premium is 0.2140%
or approximately $2,140.

The combination of a cap and a ¬‚oor creates a collar, which is a corridor
that ¬xes interest payment or receipt levels. A collar is sometimes advan-
tageous for borrowers because it is a lower cost than a straight cap. A col-
lar protects against a rise in rates, and provides some gain if there is a fall
down to the ¬‚oor rate. The cheapest structure is a collar with a narrow
spread between cap and ¬‚oor rates.

EXHIBIT 12.19 Bloomberg™s Cap/Floor/Collar Calculator

Source: Bloomberg Financial Markets

Asset and Liability Management

he activity of commercial and investment banks in the money market
T centers around what is termed asset and liability management of the
main banking book. This book (also known as the liquidity book) is
comprised of the net position of the bank™s deposits and loans as well as
other short-term, high-quality debt instruments (e.g., certi¬cates of
deposit, Treasury bills, etc.). The major players in the money markets
must manage their exposure to the risk of adverse movements in interest
rates as part of their daily operations in these markets. Accordingly, an
understanding of asset and liability management, as a branch of bank-
ing risk management, is essential for a full understanding of the money
markets as a whole.
In this chapter we present an introduction to asset and liability man-
agement. Asset and liability management (ALM) is the term covering
tools and techniques used by a bank to minimize exposure to market risk
and liquidity risk while achieving its pro¬t objectives, through holding
the optimum combination of assets and liabilities. In the context of a
banking book, in theory pure ALM would attempt to match precisely the
timing and value of cash in¬‚ows of assets with the cash out¬‚ows of liabil-
ities. Given the nature of a bank™s activities, however, this would be dif¬-
cult, if not impossible, to structure. Moreover, it would be expensive in
terms of capital and opportunities foregone. For this reason a number of
other approaches are followed to manage the risks of the banking book in
a way that maximizes potential revenue. ALM also covers banking proce-
dures dealing with balance sheet structure, funding policy, regulatory and
capital issues, and pro¬t target; we do not discuss these facets of ALM
here. The aspect of ALM we are interested in is that dealing with policy
on liquidity and interest-rate risk, and how these are hedged. In essence
the ALM policy of a commercial bank will be to keep this risk at an


acceptable level, given the institution™s appetite for risk and expectations
of future interest rate levels. Liquidity and interest-rate risk are interde-
pendent issues, although the risks they represent are distinct.

One of the major areas of decision-making in a bank involves the matu-


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