(1) (2) (3) (4) (5) (6) (7) (8) (9)
Fixedrate 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
242
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
243
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
= 

(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 — « 

360
244 THE GLOBAL MONEY MARKETS
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:
92
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 3month 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:
$1
forward discount factor =  = 0.98997649
( 1.010125 )
For period 2,
$1
forward discount factor =  = 0.97969917

( 1.010125 ) ( 1.010490 )
For period 3,
$1
forward discount factor = 

( 1.010125 ) ( 1.010490 ) ( 1.011628 )
= 0.96843839
Given the ¬‚oatingrate 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 ¬‚oatingrate
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 ¬‚oatingrate payments is $14,052,917. Thus, the
245
Swaps and Caps/Floors
present value of the payments that the ¬xedrate payer will receive is
$14,052,917 and the present value of the payments that the ¬xedrate
receiver will make is $14,052,917.
Determination of the Swap Rate
The ¬xedrate payer will require that the present value of the ¬xedrate
payments that must be made based on the swap rate not exceed the
$14,052,917 payments to be received from the ¬‚oatingrate payments.
The ¬xedrate receiver will require that the present value of the ¬xedrate
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
¬xedrate payments to be $14,052,917. If that is the case, the present
value of the ¬xedrate payments is equal to the present value of the ¬‚oat
ingrate 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 ¬xedrate payments are the same
interest rates as those used to discount the ¬‚oatingrate payments.
To show how to compute the swap rate, we begin with the basic rela
tionship for no arbitrage to exist:
PV of ¬‚oatingrate payments = PV of ¬xedrate payments
We know the value for the lefthand side of the equation.
EXHIBIT 12.5 Present Value of the FloatingRate Payments
(1) (2) (3) (4) (5) (6)
Period = Forward Floatingrate PV of
Quarter Quarter End of discount payment at ¬‚oatingrate
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
246 THE GLOBAL MONEY MARKETS
If we let
SR = swap rate
and
Dayst = number of days in the payment period t
then the ¬xedrate payment for period t is equal to:
Days t
notional amount — SR — 

360
The present value of the ¬xedrate 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 ¬xedrate payment for period t is equal to:
Days
notional amount — SR — t — FDF t

360
We can now sum up the present value of the ¬xedrate payment for
each period to get the present value of the ¬‚oatingrate 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 ¬xedrate
payments can be expressed as:
N
Days
‘ notional amount — SR — t — FDF t

360
t=1
This can also be expressed as
N
Days t
SR ‘ notional amount —  — FDF t

360
t=1
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 ¬‚oatingrate payments. That is,
247
Swaps and Caps/Floors
N
Days
SR ‘ notional amount — t — FDF t = PV of floatingrate payments

360
t=1
Solving for the swap rate
PV of floatingrate payments
SR = 

N
Days
‘ notional amount — t — FDF t 
360
t=1
All of the values to compute the swap rate are known.
Let™s apply the formula to determine the swap rate for our 3year
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
¬‚oatingrate payments is $14,052,917. Therefore, the swap rate is
$14,052,917
SR =  = 0.049875 = 4.9875%

$281,764,282
Given the swap rate, the swap spread can be determined. For exam
ple, since this is a 3year swap, the convention is to use the 3year onthe
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 ¬xedrate payments to that of the
¬‚oatingrate payments.
Valuing a Swap
Once the swap transaction is completed, changes in market interest rates
will change the payments of the ¬‚oatingrate 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 3month LIBOR forward
rates from the current Eurodollar CD futures contracts are used to (1) cal
culate the ¬‚oatingrate 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
248
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
249
Swaps and Caps/Floors
To illustrate this, consider the 3year 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 3month 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 3month LIBOR and the
forward rates are used to compute the ¬‚oatingrate 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 ¬‚oatingrate payments (from Exhibit 12.7) are shown. The ¬xedrate
payments need not be recomputed. They are the payments shown in Col
umn (8) of Exhibit 12.3. These are ¬xedrate 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 ¬‚oatingrate payments $11,459,495
Present value of ¬xedrate 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 ¬xedrate payer will receive the ¬‚oatingrate payments. And
these payments have a present value of $11,459,495. The present value
of the payments that must be made by the ¬xedrate payer is
$9,473,390. Thus, the swap has a positive value for the ¬xedrate payer
equal to the difference in the two present values of $1,986,105. This is
the value of the swap to the ¬xedrate payer. Notice, consistent with
what we said in the previous chapter, when interest rates increase (as
they did in our illustration), the ¬xedrate payer bene¬ts because the
value of the swap increases.
In contrast, the ¬xedrate receiver must make payments with a
present value of $11,459,495 but will only receive ¬xedrate payments
with a present value equal to $9,473,390. Thus, the value of the swap for
the ¬xedrate receiver is ’$1,986,105. Again, as explained earlier, the
¬xedrate 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 FloatingRate Payments One Year Later if Rates Increase
(1) (2) (3) (4) (5) (6) (7) (8)
Number of Current Eurodollar Period = Floatingrate
Quarter Quarter days in 3month 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
250
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
251
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
252
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 Fixedrate payer Fixedrate 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
253
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.
PRIMARY DETERMINANTS OF SWAP SPREADS
As we have seen, interest rate swaps are valued using noarbitrage 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
254 THE GLOBAL MONEY MARKETS
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 ontherun U.S. Treasury yield curve. Exhibit 12.12 is a time
series plot obtained from Bloomberg for daily values of the 5year 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,
longerdated swaps are priced relative to rates paid by investmentgrade
credits in traditional ¬xed and ¬‚oatingrate markets.
Given that a swap is a package of futures/forward contracts, the
shorterterm 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 ¬xedrate security
or a ¬xedrate funding vehicle by taking a position in a bundle of Euro
dollar CD futures contracts (i.e., a position in every 3month Eurodollar
CD futures contract up to the desired maturity date).
4
Naturally, this presupposes the reference rate used for the floatingrate 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 comparablematurity U.S.
Treasury.
5
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).
255
Swaps and Caps/Floors
EXHIBIT 12.11 Swap Rates for Various Currencies
Source: Bloomberg Financial Markets
EXHIBIT 12.12 Time Series of the 5Year Swap Spread
Source: Bloomberg Financial Markets
256 THE GLOBAL MONEY MARKETS
For example, consider a ¬nancial institution that has ¬xedrate assets
and ¬‚oatingrate liabilities. Both the assets and liabilities have a maturity of
three years. The interest rate on the liabilities resets every three months
based on 3month LIBOR. This ¬nancial institution can hedge this mis
matched asset/liability position by buying a 3year bundle of Eurodollar CD
futures contracts. By doing so, the ¬nancial institution is receiving LIBOR
over the 3year 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
¬xedrate 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
¬xedrate funding arrangement by entering into a 3year swap in which it
pays ¬xed and receives 3month 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, longerdated swaps are priced using rates available for invest
mentgrade corporate borrowers in ¬xedrate and ¬‚oatingrate debt mar
kets. Since a swap can be interpreted as a package of long and short
positions in a ¬xedrate bond and a ¬‚oatingrate 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 ontherun yield curve for AArated
banks.6 Swap ¬xed rates are lower than AArated 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 ¬‚oatingrate payers in the
interest rate swap market; (3) the technical factors that affect swap deal
ers; and (4) the level of assetbased swap activity.
6
For a discussion of this point, see Andrew R. Young, A Morgan Stanley Guide to
Fixed Income Analysis (New York: Morgan Stanley, 1997).
7
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).
257
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 ¬xedrate or a ¬‚oatingrate will be based, in part,
on these expectations. The relative supply of ¬xed and ¬‚oatingrate 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 ¬xedrate 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
therun and offtherun Treasuries. As this spread widens, investors must
pay up for the relatively more liquid ontherun issues. This chain reac
tion continues and results in ontherun Treasuries going “on special” in
repo markets. When ontherun 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 assetbased 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, riskaverse investors
sold corporate bonds and ¬‚ed to the relative safety of U.S. Treasuries.
Credit spreads widened considerably and liquidity diminished. A con
traryminded ¬‚oatingrate investor (like a ¬nancial institution) could have
taken advantage of these circumstances by buying newly issued invest
8
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.
258 THE GLOBAL MONEY MARKETS
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 ¬xedrate paid in
the swap. Accordingly, the ¬nancial institution ends up with a synthetic
¬‚oatingrate asset with a sizeable spread above LIBOR.
By similar reasoning, investors can use swaps to create a synthetic
¬xedrate security. For example, during the mid1980s, many banks
issued perpetual ¬‚oatingrate notes in the Eurobond market. A perpetual
¬‚oatingrate note is a security that delivers ¬‚oatingrate 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., 3month LIBOR) plus a
spread. When the perpetual ¬‚oatingrate note market collapsed in late
1986, the contagion spread into other sectors of the ¬‚oaters market.9
Many ¬‚oaters cheapened considerably. As before, contraryminded ¬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 ¬xedrate investment. The investor makes
¬‚oatingrate payments (say based on LIBOR) to their counterparty and
receives ¬xedrate 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.
NONVANILLA INTERESTRATE SWAPS
The swap market is very ¬‚exible and instruments can be tailormade 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 ¬‚oatingleg of a swap is sixmonth Libor for a semiannual paying
¬‚oating leg, other reference rates that have been used include threemonth
Libor, the prime rate (for dollar swaps), the onemonth 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.
9
Suresh E. Krishman, “AssetBased Interest Rate Swaps,” Chapter 8 in Interest Rate
Swaps.
259
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 ¬xedrate bond cash ¬‚ow for a ¬‚oatingrate bond
cash ¬‚ow structure. As a ¬‚oatingrate bond is valued on its principal
value at the start of a swap, a swaption may be viewed as the value on a
¬xedrate bond, with a strike price that is equal to the face value of the
¬‚oatingrate 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 sixmonth
Libor is exchanged for the ¬veyear swap rate on a semiannual basis for
the next ¬ve years, or for the ¬veyear 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 stepup) 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 rollercoaster
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
wardstarting 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 interestrate 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.
ZeroCoupon Swap
A zerocoupon swap replaces the stream of ¬xedrate payments with a
single payment at the end of the swap™s life, or less common, at the begin
260 THE GLOBAL MONEY MARKETS
ning. The ¬‚oatingrate payments are made in the normal way. Such a
swap exposes the ¬‚oatingrate payer to some credit risk because it makes
regular payments but does not receive any payment until the termination
date of the swap.
LiborinArrears Swap
In a Liborinarrears swap (also known as a backset swap), the reset date
is just before the end of the accrual period for the ¬‚oatingrate 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 Liborinarrears swap would be priced higher than a
conventional swap. If the ¬‚oatingrate 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 ¬xedrate payments and the
other ¬‚oatingrate payments. In a basis swap both legs are ¬‚oatingrate,
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 threemonth Libor and the other at sixmonth 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 ¬xedrate quote for a swap must be
adjusted correspondingly to allow for the margin on the ¬‚oating side. So
in our example if the ¬xedrate 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 daycount convention or payment frequency were to differ
261
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.
OffMarket 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 offmarket swap, and a compensating pay
ment is made by one party to the other. An offmarket 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 ¬‚oatingrate, while the other is ¬‚oat
ingrate 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 sixmonth sterling Libor, in sterling, on a notional princi
pal of £10 million, and receiving euroLibor 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.
ForwardStart Swap
A forwardstart 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 forwardstarting swap is calcu
lated in the same way as that for a vanilla swap.
CANCELLING A SWAP
When ¬nancial institutions enter into a swap contract in order to hedge
interestrate 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
262 THE GLOBAL MONEY MARKETS
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 nonstandard (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 viceversa if rates have risen.
CREDIT RISK
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 equivalentmaturity 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 valueatrisk 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
263
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.
CROSSCURRENCY SWAPS
So far we have discussed swap contracts where the interest payments are
both in the same currency. A crosscurrency swap is similar to an interest
rate swap, except that the currencies of the two legs are different. Like
interestrate swaps, the legs are usually ¬xed and ¬‚oatingrate, although
again it is common to come across both ¬xedrate or both ¬‚oatingrate
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 ¬xedrate 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 ¬‚oatingleg 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
264 THE GLOBAL MONEY MARKETS
valued at the ¬xedrate swap rate for that currency for the ¬xed leg, and
at Libor or the relevant reference rate for the ¬‚oating leg.
SWAPTIONS
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 ¬‚oatingrate 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
¬xedrate payer, the buyer has purchased a call swaption, while if by
exercising the buyer of the swaption becomes the ¬‚oatingrate 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 interestrate hedge for
an anticipated future exposure. For example, assume that a corporation
will be entering into a ¬veyear bank loan in three months™ time. Interest
on the loan is charged on a ¬‚oatingrate basis, but the corporation
intends to swap this to a ¬xedrate 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 ¬veyear period beginning in three months™ time. When the
time comes for the corporation to engage in a swap contract and
exchange its interestrate liability in three months™ time (having entered
into the loan), if the ¬veyear 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 ¬veyear swap rate is above 6%, the corporation will
instead exercise the swaption, giving it the right to enter into a ¬veyear
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 forwardstarting swaps, but differ because they
265
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.
SWAPNOTE®”AN EXCHANGETRADED
INTERESTRATE SWAP CONTRACT
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 exchangetraded government bond futures con
tracts as hedging tools.
EXHIBIT 12.13 Pro¬t/Loss Diagrams for an Interest Rate Swap and a Swaption
10
On October 31, 2001, the U.S. Treasury announced it would no longer issue 30
year bonds.
11
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.
266 THE GLOBAL MONEY MARKETS
EXHIBIT 12.14 Yield Curves for French and German Government Bonds,
Pfandbriefe Securities and Euro InterestRate 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 exchangetraded contract referenced against the swap curve.
The swap curve is the interbank curve, derived from interbank deposits,
shortterm interest rate futures and interestrate swaps. Swapnote®, intro
duced by LIFFE in 2001, is a standardized contract that allows market
participants to put on an exposure to the interestrate swap curve, but
with the ease of access of an exchangetraded 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 nongovernment securities, but also for Euro
pean government bonds. In that case, the euro swap curve will transform
into the cornerstone for the entire euroarea debt capital market, which
will deteriorate further the relationship between government bonds and
nongovernment 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 AAArated security. The nongovernment 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 cashsettled bond futures contract in which the delivery basket
consists of a single bond. It is referenced to the euro interestrate swap
curve, and contracts are provided for two, ¬ve, and tenyear maturities.
The contract can be used for speculative purposes, or for hedging purposes
of credit exposures such as corporate bonds or an interestrate swap book.
In theory, it provides a closer correlation between the hedging instrument
267
Swaps and Caps/Floors
and the exposure, thus reducing basis risk. By using an exchangetraded
contract rather than swaps themselves, users also gain from the advantages
associated with exchangebased 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 interestrate swaps, which involve credit line
issues, documentation issues, and bidoffer 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 pricebased
contract, similar in concept to a forwardstarting 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 ¬xedrate cash
¬‚ow and a principal repayment. The ¬xedrate 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
m
EDSP = 100 d m + C ‘ A i d i (1)
i=1
where
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 i1 and i notional cash ¬‚ows
and divided by 360. Day counts use the 30/360 basis.
di = is the zerocoupon discount factor, calculated from the swap
rate is ¬xed for each period from the delivery date to the ith
notional cash ¬‚ow.
The zerocoupon 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
268 THE GLOBAL MONEY MARKETS
1
d 1 =  (2)
1 + A 1 rs 1
where rs is the swap rate and rs1 is the oneyear swap rate. The full set
of discount factors is then calculated using the bootstrapping technique,
and is given by
i“1
1 “ rs i ‘ A j d j
j=1
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 zerocoupon 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 tenyear Swapnote contract speci¬cations.
EXHIBIT 12.15 TenYear 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)
Notes
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
269
Swaps and Caps/Floors
EXHIBIT 12.16 Price Trading History, TenYear Swapnote (LIFFE) and
TenYear Bund (Eurex), SeptemberOctober 2001
Trade Spread History
To illustrate the similarity in market price movements, Exhibit 12.16
shows the price trading history of the tenyear Swapnote contract against
the tenyear 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.
CBOT SWAP FUTURES CONTRACT
The Chicago Board of Trade (CBOT) introduced a swap futures contract
in late October 2001. The underlying instrument is the notional price of
the ¬xedrate side of a 10year interest rate swap that has a notional prin
cipal equal to $100,000 and that exchanges semiannual interest pay
ments at a ¬xed annual rate of 6% for ¬‚oating interest rate payments
based on 3month LIBOR. This swap futures contract is cashsettled with
a settlement price determined by the ISDA benchmark 10year 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
270 THE GLOBAL MONEY MARKETS
(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 forwardstart swap
discussed earlier in this chapter. For example, the December 2001 swap
futures contract will be for a new 10year interest rate swap beginning on
December 17, 2001. It is anticipated that this contract will be a valuable
tool to hedge spread product.
CAPS AND FLOORS
An important option combination in debt markets is the cap and ¬‚oor, which
are used to control interestrate 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 interestrate risk, including
shortdated interestrate 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 inthemoney
on the reset date, the payment by the seller is typically made in arrears.
12
The term cap and floor is not to be confused with floatingrate note products that
have caps and/or floors which restrict how much a floater™s coupon rate can float.
271
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.
Caps
A cap is essentially a strip of options. A borrower with an existing inter
estrate 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 interestrate 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 oneyear cap might be composed of
a strip of three individual caplets, each providing protection for succes
sive threemonth periods. The ¬rst threemonth period in the oneyear
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 oneyear cap on
threemonth 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 threemonth 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:
272 THE GLOBAL MONEY MARKETS
$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, threemonth 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 BlackScholes
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
273
Swaps and Caps/Floors
EXHIBIT 12.18 Bloomberg Screen with the Valuation of a Hypothetical Cap
Source: Bloomberg Financial Markets
Floors
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 ¬‚oatingrate
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 oneyear
¬‚oor on threemonth 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 threemonth 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:
274 THE GLOBAL MONEY MARKETS
$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, threemonth 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.
Collars
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
13
CHAPTER
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 shortterm, highquality 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 interestrate risk, and how these are hedged. In essence
the ALM policy of a commercial bank will be to keep this risk at an
275
276 THE GLOBAL MONEY MARKETS
acceptable level, given the institution™s appetite for risk and expectations
of future interest rate levels. Liquidity and interestrate risk are interde
pendent issues, although the risks they represent are distinct.
FOUNDATIONS OF ALM
One of the major areas of decisionmaking in a bank involves the matu