Download - Applied Financial - LECTURE 5
Applied FinancialInstruments &
Risk Management(FINM7041)
Lecture 5Swap
Overview of this lecture
I. Introduction
II. Main uses of Swaps
III. Valuation of Interest Rate Swap
IV. Currency Swap
V. Credit Risk
I. Introduction
• Swap is an agreement to exchange cash flows at
specified future times according to certain specified rules.
• A swap is equivalent to a coupon-bearing asset plus a coupon-bearing liability. Note that the coupons might be fixed or floating.
• Swaps can also be thought of as a package of forward contract.
• Two most common swaps: • Plain vanilla interest rate swaps • Fixed-for-fixed currency swaps
I. Introduction
• Mechanics of Interest Rate Swap (Plain Vanilla) Party A (buyer who longs the contract) agrees to pay fixed rate and receive floating rate, from counter-party B (seller who short the contract).
There is no initial exchange of principal
Define as the fixed rate. as the floating rate.
fixrfloatr
Receives variable cash flows
0 1 2 3 4 float float float float
fix fix fix fix fix
r r r r r
r r r r r
Pays fixed cash flows
I. Introduction
• Mechanics of Interest Rate Swap (Plain Vanilla) • One cash flows based on a fixed interest rate and another
referenced to an index that varies over time. • The principal (termed the notional principal) itself is not
exchanged. • At each settlement period, the two rates are compared
and the difference (times the notional principal) is paid by one counterparty to the other.
• The time t variable cash flow is based on the time t-1 floating interest rate, therefore • the first cash flow is known • all subsequent cash flows are unknown (but always
known one period in advance).
I. Introduction
• Mechanics of Interest Rate Swap (Plain Vanilla)• Example: A 3-year swap initiated on 5 March 2004 between Microsoft
and Intel. A notional principal of $100 million. Microsoft agrees to pay
to Intel an interest rate of 5% p.a. In return, Intel agrees to pay Microsoft
the six-month LIBOR rate. Payments are to be exchanged every six months. The 5% interest rate is quoted with semi-annual compounding
( )Analysisof cash Flows CFs to Microsoft
Date LIBOR
Floating
CF
Fixed
CF
Net
CF
05-Mar -04
05-Sep- 04
05-Mar- 05
05-Sep- 05
05 -Mar-06
05-Sep -06
05-Mar -07
0.042
0.048
0.053
0.055
0.056
0.059
-
-
$2.10 m
$2.40 m
$2.65 m
$2.75 m
$2.80 m
$2.95 m
-
-$2.5 m
-$2.5 m
-$2.5 m
-$2.5 m
-$2.5 m
-$2.5 m
-
-$0.40 m
-$0.10 m
$0.15 m
$0.25 m
$0.30 m
$0.45 m
II. Main uses of Swaps
• Converting a Liability
• Example: Suppose the following:
• Both companies want to borrow $10 million for 5 years
• Company A wants the liability to be in the floating rate
• Company B wants the liability to be in the fixed rate
A B Diff
(B-A)
Fixed Rate
Floating Rate
5.2%
LIBOR
+1.7%
2.0%
1.2%
7.2%
LIBOR+0.5%
II. Main uses of Swaps
• Converting a Liability • Do nothing: • Company A borrows at the floating rate of LIBOR + 0.5% • Company B borrows at the fixed rate of 7.2%
• Using swap: • In absolute term, • Company A has absolute advantage in both floating rate and
fixed rate markets. • But in comparative term, • Company B has a comparative advantage in the floating rate market
• Company A has a comparative advantage in the fixed rate market • Hence, an interest rate swap for 5 years between A and B but
how?
• Converting a Liability • Recall:
• Company B has a comparative advantage in the floating rate market but wants to borrow fixed rate.
• Company A has a comparative advantage in the fixed rate market but wants borrow floating
rate. • Suppose the term of the swap is that Company
A agreed to pay LIBOR + 1.7% to Company B
and Company B agreed to pay 6.8% to Company A.
II. Main uses of Swaps
II. Main uses of Swaps
• Converting a Liability That is, for A,
• Borrow from outside at a fixed rate of 5.2%• At the same time, swap with B
- pay LIBOR + 1.7% - receive 6.8%
→ Effectively, A borrows a floating rate contract
6.8%
LIBOR +1.7%
5.2% LIBOR + 1.7%
A B
II Main uses of Swaps• Converting a Liability
Analysisof cashFlows
Company A Company B
Pays outside
lenders
Pay under
the swap
Receives
under the
Swap
Net interest
If no swap
Net Effect
-(5.2%)
-(LIBOR + 1.7%)
6.8%
-(LIBOR + 0.1%)
-(LIBOR + 0.5%)
0.4% gain
-(LIBOR + 1.7%)
-(6.8%)
LIBOR + 1.7%
-(6.8%)
-(7.2%)
0.4% Gain
II. Main uses of Swaps
• Converting an asset
• Example: Suppose the following
• Company A wants the asset to be in the fixed rate
• Company B wants the asset to be in the floating rate
A B Diff
(A-B)
Fixed Rate
Floating Rate LIBOR
-0.25%
4.7% 0.5%
1.25%
5.2%
LIBOR -1.5%
II. Main uses of Swaps• Converting an asset
• Do nothing:
• Company A receives at the fixed rate of 5.2%
• Company B receives at the floating rate of LIBOR-1.5%
• Using Swaps:
• Company B has a comparative advantage in the fixed
rate market (but prefer floating rate)
• Company A has a comparative advantage in the floating
rate market (but prefer fixed rate)
• Enter to an interest rate swap with the following term of
the swap:
• Company A agreed to pay LIBOR-0.4% to Company
B and
• Company B agreed to pay 5.4% to Company A.
II. Main uses of Swaps
• Converting an asset
An interest benefit for both companies.
LIBOR-0.4%
LIBOR-0.25% 4.7%
A B
%5.4
II. Main uses of Swaps
• Converting an asset
Company A Company B
Receives from
outside borrowers
Pay under the swap
Receives under
The Swap
Net interest
If no swap
Net Effect
LIBOR-0.25%
-(LIBOR-0.4%)
5.4%
5.55%
5.2%
0.35% gain
4.7%
-(5.4%)
LIBOR – 0.4%
LIBOR-1.1%
LIBOR-1.5%
0.4% Gain
Analysisof cashFlows
II. Main uses of Swaps
• Role of financial intermediary
If the swap is intermediated by a swap dealer, the gains for the
counterparties will be a little bit lower.
In our case here,
• The intermediary enters into an offsetting contract with A and B
and nets 0.03% (spread)
• The spread depends on supply and demand
5.385%A B
Intermediary
(Gain=0.03%)
5.415%
%LIBOR- 0.4 %LIBOR- 0.4
%LIBOR- 0.25 4.7%
II. Main uses of Swaps
• Role of financial intermediary.
• Both parties to a swap do not contract the
intermediary at the same time.
• Intermediary can enter into the swap and hedge
its exposure until having an offsetting contract.
• This refers to as warehousing interest rate swap.
III. Valuation of Interest Rate Swap
A. Valuation in terms of bond prices
• An interest rate swap can be valued as the difference
between value of fixed and floating rate bond.
• The position of a fixed rate payer/floating rate receiver
is equivalent to long a swap:
• Interest rate swap = Long floating rate note + short fixed
rate note
• if you long a swap (fixed rate payer), the value of swap is
swap float fixV B B
III. Valuation of Interest Rate Swap
A. Valuation in terms of bond prices
Long an interest rate swap:
= long floating rate note and short fixed rate note
0 1 2 3 4
float float float float
fix fix fix fix fix
r r r r r
r r r r r
Floating rate note:
Fixed rate note:
0 1 2 3 4
float float float float
fix fix fix fix fix
r r r r r
r r r r r
Combined
Principal
Principal
III. Valuation of Interest Rate SwapA. Valuation in terms of bond prices • Swaps is then the same as an agreement in which one party
lends principal amount at the variable rate (e.g. LIBOR) and
borrow principal amount at the fixed rate, and vice versa for
another party.
• Fixed rate debt is just ordinary coupon bond:
Where
is the time until ith payments are exchanged;
is the notional principal in swap agreement;
is the LIBOR zero rate corresponding to maturity ti;
is the fixed payment made on each payment date.
1
i i n n
nr t r t
fixi
B ke Le
it
L
ir
k
III. Valuation of Interest Rate Swap
A. Valuation in terms of bond prices
• Floating rate debt reprices to par immediately after each payment
• Par is the notional principal
• Next payment is known with certainty
• Hence, value of floating rate debt = PV of next cash flow + PV
of notional principal, or;
Where
is the floating rate payment that will be made on the next
the next payment date.
1 1( *) r tfloatB L k e
*K
III. Valuation of Interest RateSwap
A. Valuation in terms of bond prices
• Example: Bank has agreed to pay 6-month LIBOR and
Receive 8% (semi-annual) on $100 million.
• Remaining life: 1.25 years
• LIBOR (continuous compounding)
• 10% for 3 months,
• 10.5% for 9 months
• 11% for 15 months
• The LIBOR 6-months rate at the last payment date (3 months
ago) was 10.2% (semi-annual)
• Calculate the value of the swap.
III. Valuation of Interest RateSwap
A. Valuation in terms of bond prices
• Example
Fixed rate bond
Semi-annual Coupon payment
$100 8% / 2 $4 *m m
0.1*3/12 0.105 9/12
0.11 15/12
4 4
($100 4)
$98.24
fix
fix
B m e m e
m e
B m
x x
x
x
x
III. Valuation of Interest Rate Swap
A. Valuation in terms of bond prices
• Example
Floating rate bond
The semi-annual LIBOR used = 10.2%/2 = 5.1%
Hence,
* $100 5.1% $5.1k *m m0.1 3($100 $5.1 ) $102.51floatB m m e m x / 12
swap fix floatV B B ( as the bank shorts the swap)
98.24 102.51 $4.27 m
III. Valuation of Interest RateSwap
B. Valuation in Terms of Forward Rate Agreements • A one-year swap with semi-annual payments is just a package
of two forward contracts • one with a six-month maturity • another with a 12-month maturity • An interest rate swap can also be viewed as a convenient package of forward rate agreements (FRA). • each exchange of payments (except the first payment) is
an FRA • The value of swap is the sum of the values of the forward rate
agreements underlying the swap • Note that the fixed rate in an interest rate swap is chosen so
that the swap is worth zero initially. • it does not mean that each forward contract underlying
a swap is zero initially
III. Valuation of Interest Rate Swap
B. Valuation in Terms of Forward Rate Agreement
Recall: FRA
The value of FRA
where is the principal value; is the FRA rate; is the forward LIBOR rate for the period between and ; is the (continuously compounded) zero rate for a maturity .
Note that all the rates are measured with a compounding frequency
reflecting their maturity
2 22 1*( )*( ) R T
K FV L R R T T e
2R01T FR 2T
KR
L
RF
RK
2R 1T 2T2T
III. Valuation of Interest RateSwap
B. Valuation in Terms of Forward Rate Agreements
To value IRS in terms of FRA, we follow the following steps:
1. calculate each of the forward rates for each of the LIBOR rates that will determine swap cash flows.
2. calculate swap cash flows on the assumption that the LIBOR rates will equal to the forward rate (i.e.
expectations theory holds)
3. set swap rates equal to the PV of these cash flows
III. Valuation of Interest RateSwap
B. Valuation in Terms of Forward Rate Agreements • Example: Bank has agreed to pay 6-month LIBOR
and Receive 8% (semi-annual) on $100 million. • Remaining life: 1.25 years • LIBOR (continuous compounding)
• 10% for 3 months. • 10.5% for 9 months. • 11% for 15 months.• The LIBOR 6-month rate at the last payment date (3 months ago) was 10.2% (semi-annual)• Calculate the value of the swap.
III. Valuation of Interest Rate SwapB. Valuation in Terms of Forward Rate
Agreements
1. The first cash flow (in 3 months):
The cash flows for the payments 3 months have
already been set.
A rate of 8% will be the exchanged for a rate of 10.2%.
The value of the exchange is
0.1 3/120.5 $100 (0.08 0.102) $1.07NPV m e xx x
III. Valuation of Interest Rate Swap B. Valuation in Terms of Forward Rate Agreements
2. The second cash flow (in 9 months):
• Forward rate corresponding to 3 and 9 months is :
The rate of 10.75% obtained is a rate with continuous compounding
Need to convert it to a rate with semi-annual compounding.
• The value of the exchange is :
2 2 1 1
2 1
0.105 (9 /12) 0.10 3/120.1075
9 /12 3/12F
R T RTR
T T
x x
/ 0.1075/ 21 2 1 11.044%cR mmR m e e x
0.105 (9 /12)0.5 100 (0.08 0.11044) $1.41e m xx x
III. Valuation of Interest Rate Swap
B. Valuation in Terms of Forward Rate Agreements
3. The third cash flows The value of the exchange is
4. Total value of the swap
0.11 15/120.5 100 (0.08 0.12102) $1.79e mxx x
($1.07 ) ( $1.41 ) ( $1.79 ) m m m
$4.27 m
IV. Currency Swap
A. Introduction • Currency Swap (CCS)
Exchanging principal and interest payments in one currency for principal and interest payments in another currency.
• In a currency swap, • two different currencies are periodically exchanged. • the principal is exchanged at the beginning and the
end of the swap • There are four types of basic currency swaps: • fixed for fixed • fixed for floating • floating for fixed • floating for floating
IV. Currency Swap
A. Introduction • Example of Currency Swap
• A five-year currency swap agreement between A and B. A pays a fixed rate of 11% in
sterling and receives a fixed rate of 8% in dollar from B (a fixed for fixed currency swap). Interest payments once a year. The principal amounts are $15m and £10m.
At origination:
At each annual settlement date:
At Maturity:
A B
10£ m
$15m
A B
*$15 0.08 $1.2m m
*10 11% 1.1£ £m m
A B
$15m
10£ m
IV. Currency Swap
A. Instruction
• Example: Suppose the following:
• GM has a comparative advantage in the USD market
but want to borrow in AUS.
• Qantas has a comparative advantage in the AUD market
but want to borrow in USD.
US AUD
GM 5.0% 12.6%
Qantas 7.0% 13.0%
Diff 2.0% 0.4%
IV. Currency Swap
A. Introduction
US6.3%
GM Bank QF
US 5%
AUD 11.9% AUD13%US 5% AUD 13%
IV. Currency Swap B. Valuation CCS in terms of Bond Prices
Long a currency swap: the domestic currency is received and a foreign currency
is paid
The value (in domestic currency) of a long position in swap is:
where is the value of the foreign-denominated bond underlying the swap (in the foreign currency);
is the value of home-denominated bond (in the home currency); is the spot exchange rate (number of units of domestic
currency per units of foreign currency).
0swap D FV B S B
FB
DB
0S
IV. Currency Swap B. Valuation CCS in terms of Bond Prices
The value of a swap where the foreign currency is received and
a domestic currency is paid is:
where
is the value of the foreign-denominated bond underlying
the swap (in the foreign currency);
is the value of home-denominated bond (in the home
currency);
is the spot exchange rate (number of units of domestic
currency per units of foreign currency).
0swap F DV S B B
FB
DB
0S
IV. Currency Swap
B. Valuation CCS in terms of Bond Prices
• Example: The Japanese interest rate is 4% p.a. and the US interest rate is 9% p.a. A US company enters a swap where it receives 5% p.a. in Yen with a principal of Yen 1200 million, and pays 8% p.a. with a principal of $10 million. The swap lasts for three years and the exchange rate is $0.00909/yen (or 110 yen=$1). Assume annual interest payments. What is the value of this swap?
IV. Currency SwapB. Valuation CCS in terms of Bond Prices
• Example:
• The value of foreign bond is:
• The value of domestic bond is:
• In our case, foreign currency is received and domestic
currency is paid short
• Hence,
0.04 1 0.04 2 0.04*360 60 (1200 60) 1230.55FB e e e Yen x x m
0.09 1 0.09 2 0.09*30.8 0.8 10 0.8 $9.644DB e e e x x m
0 1230.55*0.00909 9.644 $1.54swap F DV S B B m
IV. Currency SwapC. Valuation in terms of forward contracts
• The currency swap can also be viewed a series
of forward contracts.
• Hence, the value of the swap is the sum of the values of
the values of the forward contracts underlying the swap.
Example: The Japanese interest rate is 4% p.a. and the US interest rate is 9% p.a. A US company enters a swap where it
receives 5% p.a. in Yen with a principal of yen 1200 million, and pays 8% p.a. With a principal of $10 million. The swap lasts for three year and the exchange rate is $0.00909/yen. Assume annual interest payments.
What is the value of this swap?
IV. Currency Swap
C. Valuation in terms of forward contracts
• Recall :
where
is the value of the contract today;
is the forward price today;
is delivery price in the contract.
0( ) rTf F K e
f
0F
K
1
1 2 3
2
4
3
IV. Currency Swap Valuation in terms of forward contracts
To determine the we need forward exchange rate,
Recall:
The relationship between forward and spot exchange
rate is:
where
is the forward exchange rate;
is the spot exchange rate;
is the domestic risk-free interest rate;
is the foreign risk-free interest rate.
,tF
0FXF10S
r
fr
( )
0 0fr rFX FXF S e T
IV. Currency Swap
c. Valuation in terms of forward contracts
• One-year forward exchange rate:
• two-year forward exchange rate:
• three-year forward exchange rate:
• The exchange of interests involves: • Receive: Yen 1200 m * 0.05 = 60 million
Yen • Pay: USD 10 million * 0.08 = 0.8 million
USD
(0.09 0.04) 10.00909 0.009557e xx
(0.09 0.04) 20.00909 0.010047e xx
(0.09 0.04) 30.00909 0.010562e xx
IV. Currency Swap C. Valuation in terms of forward contracts
The values of forward contracts corresponding to
three exchanges of interests are:
The value of forward contract corresponding to the
exchange of principal at maturity is
Total = - 0.2071 – 0.1647 – 0.1269 + 2.0416
= 1.543 million USD
0.09 1
0.09 2
0.09 3
(60 0.009557 0.8 )
0.2071
(60 0.010047 0.8 )
0.1647
(60 0.010562 0.8 )
0.1269
e
m
e
m
e
m
m Yen m USD
USD
m Yen m USD
USD
m Yen m USD
USD
x
x
x
x
x
x
0.09 3(1200 0.010562 10 )
2.0416
e
m
xxm Yen m USD
USD
V. Credit RiskC. The value of swap is normally zero when it is first negotiated.
• this means that it costs nothing to enter into a swap
• it does not mean that the each forward contract underlying
a swap is worth zero initially.
• At a future time, its value is liable to be either positive or negative.
• Credit risk is risk resulting from uncertainty in a counterparty’s
ability or willingness to meet its contractual obligation.
• A financial intermediary has credit risk exposure from a
swap only when the value of the swap to the financial
intermediary is positive.
• Note that there is greater credit risk with a currency swap when
there will be a final exchange of principal, because there is a higher
probability of a large buildup in value, given one of the counter- parties the incentive to default.
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