Session 5

Swaps 1
Chapter 7

1
 Objectives

• Review Plain-vanilla interest rate & currency swaps

• Look at general principles for setting swap rates
• Learn how to value swaps
• Explain the pricing of off-market swaps adjusting for
number of days

2
 Swaps Introduction
• This unit:
– Extends what covered in Treasury Management
and includes currency swaps in International
Finance.
– Focus on valuing swap contracts and determining
equilibrium pricing.
• The features of plain-vanilla configuration interest
rate and currency swaps will be reviewed briefly first.

3
 Plain-Vanilla Interest Rate Swaps
“A swap is an agreement between two companies to exchange
cash flows in the future” (Hull, p.154).
Involve:
• Exchange of interest on a semi-annual basis.
• One leg of the swap is a fixed rate of interest applied to a notional
principal, whilst the other is floating rate.
• The floating rate is six-month LIBOR or The Australian bank bill
swap rate (BBSW). BBSW is the floating rate used in most Australian
interest rate swap agreements.
• Floating rate is reset at the start of each six-month period and payable
at the end.
• Swaps are offered by dealers
4
 ‘Plain vanilla’ interest rate swap
Example 1
– Wesfarmers agrees to pay Alcoa a fixed rate of
interest (5% per annum) on a principal amount
($100 million)
– In return Alcoa agrees to pay Wesfarmers a
floating rate of interest (6-month BBSW) on the
same principal amount
•
•

Interest rate swap between Wesfarmers and Alcoa

5
 ‘Plain vanilla’ interest rate swap
Example 1(cont.)
• Cash flows ($ millions) to Wesfarmers in a $100 million 3-year interest rate
swap when a fixed rate of 5% is paid and BBSW is received

Six-month BBSW Floating cash Fixed cash

Date rate (%) flow received flow paid Net cash flow
5 Mar. 2012 4.20
5 Sep. 2012 4.80 +2.10 -2.50 -0.40
5 Mar. 2013 5.30 +2.40 -2.50 -0.10
5 Sep. 2013 5.50 +2.65 -2.50 +0.15
5 Mar. 2014 5.60 +2.75 -2.50 +0.25
•5 Sep. 2014
Table 7.1, page 156 5.90 +2.80 -2.50 +0.30
5 Mar. 2015 +2.95 -2.50 +0.45

6
 Plain vanilla’ interest rate swap
Example 1 (cont.)

• Wesfarmers and Alcoa use the swap to transform a

liability

Figure 7.2, page 157

• Wesfarmers and Alcoa use the swap to transform an asset

• Figure 7.3, page 157

7
 ‘Plain vanilla’ interest rate swap
Example 1 (cont.)
• Interest rate swap from Figure 7.2 when financial institution is involved

• Figure 7.4, page 159

• Interest rate swap from Figure 7.3 when financial institution is involved

•8 Figure 7.5, page 160

 Interest Rate Swap Example 2
• Consider a dealer that enters a four-year, $100
million swap with A, to receive floating rate for fixed
rate. Libor* +1.5%
Bank

Libor*
A
4.9% Fixed

6% Fixed

*BBSW in Australia

• The swap has removed A’s interest rate exposure, the

dealer has acquired the exposure.
9
 Hedging exposure created
• There may not be matching counterparties, as is
usually diagrammed. However, dealers aim to
balance each side of their ‘swap book’ in aggregate.

• The dealer may hedge the exposure by either:

– taking a long position in a strip of Eurodollar
futures,
OR
– finding one or more counterparties to the swap.

10
 Eurodollar futures
• Eurodollar futures, are cash settled at a price
corresponding to the actual 90-day rate paid on
Eurodollar deposits (with quarterly compounding).

• A long futures position will pay-off an amount that

will allow a future lender to lock-in the receipt of a
varying series of interest payments.

11
 Eurodollar futures
• Alternatively viewed:
– If LIBOR turns out lower than agreed in the long
futures position, the long position closes with a
profit. This compensates for a lower receipt from
the floating rate leg.

– A long position in a strip of futures contracts is

used to hedge a series of floating rate receivables
(e.g. the swap dealer in the slide 9 diagram).

12
 Additional Counterparties
• After entering a swap to receive floating rate with A,
a dealer may also enter into swaps with B and C to
pay floating rate.
Libor* +1.5% Libor*
Bank B
Libor 5.1%
A
4.9% Fixed Libor*
C
6% Fixed 5.1%

*BBSW in Australia 13
 Additional counterparties
• If both B and C entered into four-year swaps, where:
– the swap with B involved a principal of $80
million, and
– the swap with C a $20 million principal,
• The dealer’s position would be completely hedged.
• Every semi-annual payment date, the locked-in profit
is:
0.20% x 0.5 x $100 million = $100,000.

14
 Macrohedge
• In practice, dealers enter a number of swaps on the
dealer pays fixed-rate side and also a number of swaps
on the dealer receives fixed-rate side.

• It will also be the case that the swaps have different

maturities, and frequently involve configurations other
than plain-vanilla.

• The resultant aggregate exposure may be hedged using

Eurodollar futures. This is a macrohedge, as distinct
from a microhedge where a single swap is hedged.
15
 Swap pricing
Two things should be apparent:
• Counterparties to the swap will only be willing to
enter the swap if the dealer is paying competitive
rates.
– the fixed rate must be set to attract both fixed-rate payers
and receivers.
• Microhedging will lock-in the dealer’s future
cashflows.
– The dealer is a price-taker in the futures market, and
– will adjust the mid-rate on the fixed rate leg of the swap to
match the futures rates.
16
 Swap pricing
• The setting of the mid-rate on the fixed-rate leg of the
swap is called ‘pricing the swap’.
– The ‘indicative pricing schedule’

• The swap is priced such that it has zero value at

inception.
– This ensures that it is equally attractive to all
parties.

17
 Plain-vanilla Currency Swaps
• The plain-vanilla configuration currency swap is
similar to the plain-vanilla interest rate swap, except:
– one leg is in one currency and the other leg is in a
second currency.
– the principals are actually swapped.
• The actual exchange of principals is of no practical
importance for the initial exchange (they are equal in
value). On maturity their relative values depend on
the exchange rate.

18
 Currency Swap Example
• Consider a dealer that enters into a four-year currency
swap with A

• Dealer receives floating rate USD in exchange for

fixed rate AUD.

• The initial principles of USD 65 million and AUD

100 million are equivalent at the current exchange
rate.

19
 Currency Swap Example cont.
USD
• The swap
Libor* +1.5%
Bank transaction has
removed A’s
USD Libor* interest rate and
A exchange rate
4.9% Fixed exposure.
AUD • The dealer has
AUD acquired both.
6% Fixed

*BBSW in Australia
20
 Currency Swap Example cont.
• As with interest rate swaps, the dealer will seek to
hedge this exposure by entering into offsetting swaps
with counterparties.

• This will only be possible if they are priced in order

to be attractive to counterparties.

• Therefore, the mid-rate on the fixed rate leg is set

such that the initial value of the swap is zero.

21
 Interest Rate Parity
• The equilibrium relation between the interest rates in
two countries, and the spot and forward exchange
rates.
F=S(1+ih)/(1+if)
• In the absence of capital controls, this relationship is
enforced by arbitrage.

22
 Interest Rate Parity Example
• If:
– 6-month LIBOR in Australia is 8% p.a.
– 6-month LIBOR in the US is 6% p.a.
– Spot exchange rate is AUD/USD = 0.6500.

• Then; if interest rate parity holds, the six-month

forward rate is AUD/USD = 0.64375.

23
 Interest Rate Parity Example
For a one-period (6-month) swap:
• In 6 months:
– principal of USD 65 million plus an interest payment of
USD 65 million x 0.03 = USD66.95 million
is exchanged for
– a principal of AUD 100 million plus an interest payment of
AUD100 million x 0.04 = AUD 104 million
• Based on the IRP forward exchange rate (AUD/USD =
0.64375), USD 66.95 million is equivalent to AUD 104
million.
24
 Pricing - General principle
• A swap of LIBOR in one currency for LIBOR in
another will have zero value.
– For the one-period swap, the equivalence of the
exchanges after 6-months is because the amounts
exchanged were based on LIBOR at the start of the
period.
– For a multi-period swap,
• equivalence is not as intuitive since the interest and
principal payments are separated.
• resembles two floating rate notes; one in each currency.
FRNs always reset to face value.
25
 Pricing - General principle
• In a correctly priced interest rate swap:
– floating-rate interest payments have the same
expected value as the fixed-rate payments.
• In a fixed-floating currency swap
– since floating-floating legs in two currencies based on
their respective LIBORs have equivalent value
– the fixed rate leg in one currency has equivalent
value to the floating rate in a second currency.

26
 General Principle Example
• Suppose for two different plain-vanilla four-year
interest rate swaps:
– AUD LIBOR is paid against (AUD) 5% p.a. fixed, and
– USD LIBOR is paid against (USD) 7% p.a. fixed.
• The following four-year plain-vanilla currency swap
rates are implied:
i) AUD 5% against USD LIBOR,
ii) USD 7% against AUD LIBOR
iii) AUD 5% against USD 7%
iv) AUD LIBOR against USD LIBOR.

27
 Valuing Swaps
On trade date:
• A swap has zero value on the ‘trade date’ since both
sides of the swap have equivalent value. The value will
change however as interest rates and exchange rates
change.

After trade date:

• The value of a swap after the trade date indicates:
– the incentive for a party to default, and
– the current cost of unwinding a swap position.
28
 Valuing swaps – secondary market

• Value of the swap in the secondary market is the

difference between the present values of:
– the floating rate side, and
– the fixed rate side.

• These are valued in the same manner as:

– a floating rate note (FRN), and
– a coupon bond, respectively.

29
 Valuing swaps – secondary market

• The FRN returns to par value at the time of the next

interest payment (at which time, the next coupon’s
rate is fixed).

• The value is thus, the present value of the next

interest payment plus the notional principal, when
discounted using the current interest rate.

30
 Interest Rate Swap – Example
• Supposing 62 days ago:
Firm A entered a four year plain-vanilla interest rate
swap to:
– receive 9.45% p.a. fixed against LIBOR flat
– for a USD1million principal.
The first floating rate coupon was set at 8% p.a., and
will be paid in 120 days.

• Now:
– Similar swaps are quoted at the fixed rate of
9% p.a.
– Libor is 7.5%. 31
 Interest Rate Swap – Example
• The present value of the floating rate payments is:

180
(1  (0.08  ))  $1,000,000
PV  360
120
(1  (0.075  ))
360
 $1,014,634

32
 Interest Rate Swap Example (cont.)
• The present value of the fixed rate payments is:

1 0.0945 1,000,000
PV   [(  $1,000,000  (1  A7  0.09 ))  ]
0.09 120 2 0.09 7
(1   ) 2 (1  )
2 183 2
1
  [$47,250  6.8927  $734,828]
1.0295
 $1,030,111

33
Interest Rate Swap Example (cont.)

• The decrease in interest rates has increased the value

of the fixed rate payments by a greater amount.

• The swap is now worth

• $1,030,111 - $1,014,634 = $15,477 to Firm A
(the fixed rate receiver).

34
 Currency Swap – Example

• Supposing that the swap on slide 27 was:

– fixed rate AUD against USD LIBOR.
– the principal was AUD 1.00 m = USD 0.65 m at
inception, and
– the exchange rate was AUD1 = USD 0.63 after 62
days. (i.e. AUD depreciates)

35
 Currency Swap – Example (Cont.)
• Present value of the floating rate USD payments is:

180
(1  (0.08  ))  USD 650,000
PV  360
120
(1  (0.075 ))
360
 USD659,512
converting @ AUD / USD  0.6300
 AUD1,046,845

36
 Currency Swap Example (cont.)
• The present value of the fixed rate AUD payments is:

1 0.0945 1,000,000
PV   [(  $1,000,000 (1  A7  0.09 ))  ]
0.09 120 2 0.09
(1   ) 2 (1  )7
2 183 2
1
  [$47,250 6.8927  $734,828]
1.0295
 AUD 1,030,111

37
 Currency Swap Example (cont.)
• The increased value of the fixed rate AUD receipts
caused by the decline in interest rates was more than
offset by the increased value of the floating rate USD
payments.

• The swap is now worth:

• AUD1,030,111 - AUD1,046,845 = -AUD16,734
to Firm A (the fixed rate receiver).

38
 Pricing Off-market Swaps
• Clients often request swaps in which the interest
payments are made at a rate which differs from that
on the dealer’s indicative pricing schedule.
• Most commonly this arises when the clients wish to
hedge an existing liability which was issued when
interest rates were different from present.
• An ‘on-market’ or ‘par’ swap would still leave some
residual unhedged risk as in the example in slide 20.

39
 Pricing Off-market swaps (Cont.)
• The swap dealer’s mid-rate for the fixed-rate leg is
obtained as:
– that rate which will equate the present value of the
payments made on the fixed-rate leg with those on
the floating rate leg, where the floating rate leg is
set to LIBOR.

• For an off-market swap, the same principle must be

applied
– the present value of the incremental payments on
one leg of the off-market swap must be matched
by a payment (or payments) of equal value.
40
 Pricing Off-market swaps
• Calculate present value of incremental cash flows
– Cashflows paid to the client under the requested rate minus
the cashflows that would be paid on a par swap
– discount at the par swap rate.
• If this amount is positive, then it will either:
– be paid as an up-front fee to buy-up to the off-market swap,
OR
– an increased amount equal in present value will be charged for
the floating rate leg.
• For a negative amount, there will be a buy-down, or
alternatively a reduced floating rate.
41
 Pricing Off-market Interest Rate
Swap
• The interest rate adjustment to the floating rate leg of
an interest rate swap is almost trivial.
• The present values of the fixed and floating rate
payments
– are obtained by discounting with the same rate
– the incremental rate on one leg is therefore the same as the
incremental rate on the other.
• Currencies where LIBOR is specified using a 360-
day year MMY convention require conversion to a
365-day BEY convention and vice-versa.
42
Pricing Off-market Currency Swaps
• For a currency swap, the fixed rate is paid in one
currency, and the floating rate in another.

• The present value of the incremental interest

payments:
– on the fixed rate leg must equal the present value of the
incremental interest paid on the floating rate leg.
– determined by discounting the incremental payments using
the interest rate for each currency
– these rates are the mid-rates for on-market interest rate
swaps in each currency.
43
 Off-market Currency Swap
Pricing Example
• Suppose:
• Firm A in the slide 20 example wished instead to
receive AUD 6.0% p.a.
• Mid-rates for four-year on-market swaps were:
– AUD 5.0 % p.a,
– USD 7.0% p.a.
• The principals were
AUD100 million = USD 65 million.
• The off-market swap would be priced as follows.
44
 Off-market Currency Swap
Pricing Example
USD
Libor +1.5%
Bank

USD Libor
A
4.9% Fixed
AUD
AUD
6% Fixed

• Interest rate differential = 6.00% - 4.90%

= 1.10% p.a.
= 0.55% s.a.
45
 Off-market Currency Swap
Pricing Example (Cont.)
• Recall AUD mid-rate was 5.0% p.a. = 2.5% s.a.

• Present value =0.0055 x AUD100m x A8 ┌ 2.5%

=AUD550,000 x 7.1701
=AUD3,943,575
=USD2,563,324

• For an up-front fee of USD2,563,324 the dealer

would pay AUD @ 6.0% p.a. against USD LIBOR
flat. 46
 Off-market Currency Swap
Pricing Example (Cont.)

• Alternatively, the dealer could require an increased

floating rate payment.

• The following calculation is generalised such that it is

independent of the actual principle involved.

47
 Off-market Currency Swap Pricing Example

• Present value (% of principal) = 0.55 x A8 ┌ 2.5%

= AUD3.943575
• Present value (% of principal) = increment x A8 ┌3.5%
= increment x 6.8740 (USD)
(Recall USD mid-rate was 7.0% p.a. = 3.5% s.a.)
• Equating and solving for increment:
Increment = 3.9436 / 6.8740
= 0.5737 % (for USD) s.a.
= 1.1474 % p.a.
48
 Off-market Currency Swap Pricing Example

• Before this can be added to USD LIBOR a correction

for day counts must be made (same as for the interest
rate swap).

• BEY differential /365 = MMY differential / 360

• Thus, MMY increment = 1.1474 % x 360/365
= 1.1317 % p.a.
• Dealer would pay AUD @ 6.0% p.a.
against USD LIBOR plus 1.1317% p.a.

49