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Transcript of 1 Derivatives & Risk Management Lecture 4: a) Swaps b) Options: properties and non- parametric...
1
Derivatives & Risk Management
Lecture 4:
a) Swaps
b) Options: properties and non-parametric bounds
2
Nature of Swaps
• A swap is an agreement to exchange cash flows at specified future times according to certain specified rules
3
Example: a “Plain Vanilla” Interest Rate Swap
• An agreement by “Company B” to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million
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---------Millions of Dollars---------
LIBOR FLOATING FIXED Net
Date Rate Cash FlowCash FlowCash Flow
Mar.1, 1998 4.2%
Sept. 1, 1998 4.8% +2.10 –2.50 –0.40
Mar.1, 1999 5.3% +2.40 –2.50 –0.10
Sept. 1, 1999 5.5% +2.65 –2.50 +0.15
Mar.1, 2000 5.6% +2.75 –2.50 +0.25
Sept. 1, 2000 5.9% +2.80 –2.50 +0.30
Mar.1, 2001 6.4% +2.95 –2.50 +0.45
Cash Flows to Company B
5
Typical Uses for an Interest Rate Swap
• Converting a liability from
– fixed rate to floating rate
– floating rate to fixed rate
• Converting an investment from
– fixed rate to floating rate
– floating rate to fixed rate
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A and B Transform a Liability
A B
LIBOR
5%
LIBOR+0.8%
5.2%
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Financial Institution is Involved
A F.I. B
LIBOR LIBORLIBOR+0.8%
4.985% 5.015%
5.2%
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A and B Transform an Asset
A B
LIBOR
5%
LIBOR-0.25%
4.7%
9
Financial Institution is Involved
A F.I. B
LIBOR LIBOR
4.7%
5.015%4.985%
LIBOR-0.25%
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The Comparative Advantage Argument
• Company A wants to borrow floating• Company B wants to borrow fixed
Fixed Floating
Company A 10.00% 6-month LIBOR + 0.30%
Company B 11.20% 6-month LIBOR + 1.00%
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The Swap
A B
LIBOR
LIBOR+1%
9.95%
10%
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The Swap when a Financial Institution is Involved
A F.I. B
10%
LIBOR LIBOR
LIBOR+1%
9.93% 9.97%
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Criticism of the Comparative Advantage Argument
• The 10.0% and 11.2% rates available to A and B in fixed rate markets are 5-year
• The LIBOR+0.3% and LIBOR+1% rates available in the floating rate market are six-month rates
• B’s fixed rate depends on the spread above LIBOR it borrows at in the future i.e. it is fixed only as long as its creditworthiness stays the same
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Alternatives
• Information asymmetry
• Flexible and liquid instruments
• Tax and regulatory arbitrage
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Valuation of an Interest Rate Swap
• Interest rate swaps can be valued as the difference between the value of a fixed-rate bond & the value of a floating-rate bond
flfix BBV
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Valuation in Terms of Bonds
• The fixed rate bond is valued in the usual way
• The floating rate bond is valued by noting that it is worth par immediately after the next payment date
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Valuation as bonds
n
i
trtrfix
nnii QekeB1
11* trfl ekQB
K* is the floating rate know from at the beginning of the period
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Example
• A financial institution pays 6 month LIBOR and receives 8% (semi-annually) on $100 million notional principal.
• The FI has sold a floater and bought a fixed rate bond
• remaining life 1.25 years
• market rates for 3, 9, 15 months to go are 10%, 10.5% and 11%
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Example II
• The 6 month LIBOR when the swap was set up 3 months ago was 10.2%.
1.5
2102.0
1
100100 *
*
k
k
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Example III
51.1021.105
238.98$
10444
12
310.0
12
1511.0
12
9105.0
12
310.0
eB
million
eeeB
fl
fix
272.451.102238.98 V
21
Forward Rate Agreement
• A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period
• An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate
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Forward Rate Agreementcontinued (Page 100)
• Capital R is the rate measured with compounding rate reflecting maturity, i.e. if the T2 – T1 is three months the rate is compounded quarterly etc.
• The agreed cash flows are:
• T1: - L
• T2: L [1+ Rk (T2-T1)]
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FRA
• Note if Rf = Rk the FRA is worth 0. Why?• To value the FRA, we can compare now two payments
at time T2:• One that pays Rk and one that pays Rf
• Note: we are not assuming anything more than no arbitrage
2
2
-r 2k 2 1 f 2 1
-r 2k f 2 1
L 1+ R (T -T ) -L 1+ R (T -T ) e =
L R -R (T -T )e
T
T
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Valuing future cash flows
2r
1r fr
Hypothetical cash flowCash
settlement
2 1fr T TXe X
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Alternative Valuation of Interest Rate Swap: portfolio of FRA
• Swaps can be valued as a portfolio of forward rate agreements (FRAs)
• Each exchange of payments in an interest rate swap can be analyzed as an FRA
• The relevant interest rates are the fixed for one leg, and the forward associated with the period to be valued for the other leg
26
Swaps as FRA’s
• Suppose an interest rate swap promises fixed rate payments C and receives floating payments P1
fl at regular intervals
• We have seen that this could be valued as a portfolio of bonds
• What about valuating it as a package of FRA’s?
27
Swaps as FRA’s II
flP3flP2
flP1flP4
CCC C
t1T 4T3T2T
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Swaps as FRA’s III
• The floating rate payment is computed based on the prevailing spot rate at T1
CP fl 2
• Consider the second exchange of payments (the first is known)
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Swaps as FRA’s IV
2,12
~100 RP fl
cC 100
t1T 2T
1r
2r
1,2R
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Swaps as FRA’s V
• So we want to compute the PV of
cR 2,1
~100
• Which can be written as
cR 1100~
1100 2,1
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Swaps V
• The value of the fixed part of this payment is obvious
1221100 TTrfix ecv
• The value of the floating part less so because it involves a random interest rate
2,1
~1100 RPVv fl
32
Swaps as FRA’s VI
• We know that at T2, the floating rate payment will be worth
2,1,
~1100
2Rv Tfl
• And thus T1, it must be worth
100~1
~1100
2,1
2,1, 1
R
Rv Tfl
33
Swaps as FRA’s VII
• And thus at time t it must be worth
11100 Trfl ev
• Recall that by no arbitrage
• So that
2 2 1 1 2 1fr T rT r T T
1 1 2 2 2 1frT r T r T T
34
Swaps as FRA’s VIII
• Hence
2 2 2 1
2 1 2 2
100
100
f
f
r T r T Tfl
r T T r T
v e
e e
Changing compounding convention
2 2100 1 r TfR e
35
Swaps as FRA
2 2 2 2
2 2
1,2100
100 1 100 1
100
r T r TF
r TF
PV R c
R e c e
R c e
So to value a fixed for floating exchange, compute the present value of the exchange between the forward rate and the fixed rate
36
An Example of a Currency Swap
An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years
37
Exchange of Principal
• In an interest rate swap the principal is not exchanged
• In a currency swap the principal is exchanged at the beginning &the end of the swap
38
The Cash Flows
Years
Dollars Pounds$
------millions------
0 –15.00 +10.001 +1.20 –1.102 +1.20 –1.10
3 +1.20 –1.104 +1.20 –1.10
5 +16.20 -11.10
£
39
Typical Uses of a Currency Swap
• Conversion from a liability in one currency to a liability in another currency
• Conversion from an investment in one currency to an investment in another currency
40
Comparative Advantage Arguments for Currency Swaps
• Company A wants to borrow AUD• Company B wants to borrow USD
USD AUD
Company A 5.0% 12.6%
Company B 7.0% 13.0%
41
Valuation of Currency Swaps
• Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts
42
Swaps & Forwards(continued)
• The value of the swap is the sum of the values of the forward contracts underlying the swap
• Swaps are normally “at the money” initially– This means that it costs NOTHING to
enter into a swap– It does NOT mean that each forward
contract underlying a swap is “at the money” initially
43
Credit Risk
• A swap is worth zero to a company initially• At a future time its value is liable to be
either positive or negative• The company has credit risk exposure
only when its value is positive