Financial Engineering
David Oakes
–
IC302
3. Swaps
Autumn Term 2020/1
Swaps
2
Swaps
• Swap: a contract in which two parties agree to exchange payments at regular intervals over an agreed period, with
a different rule used to calculate the payments that are made by each of the parties
• Interest rate swaps
– Fixed-for-floatingswap(e.g.3mUSDLIBORvsUSDfixedrate) – Tenorbasisswap(e.g.3mUSDLIBORvs6mUSDLIBOR)
– Cross-currencybasisswap(e.g.3mUSDLIBORvs3mEuribor)
• Equity swaps
• Commodity swaps
• Swaps on virtual assets (e.g. inflation, variance)
3
Interest Rate Swap
• Fixed-for-floating or plain vanilla interest rate swap:
USD fixed at 5% USD floating at LIBOR
• Notional amount (i.e. principal amount on which interest is calculated) not exchanged if both legs in same currency.
• Floating leg resets at start of each payment period to current value of floating reference rate (e.g. LIBOR).
Payer of Fixed
Receiver of Fixed
4
Swap Cash Flows
•
5-year fixed-for-floating interest rate swap with fixed swap rate 5% and notional principal USD 100 million:
LIBOR-based Floating Payments
Start Date
• •
2.5m
2.5m 2.5m
2.5m 2.5m Fixed Payments
End Date (5 years later)
2.5m 2.5m
2.5m 2.5m 2.5m
No money exchanged upfront (initial MTM value zero).
Changes in rates change relative value of fixed and floating legs, hence value to payer and receiver of fixed.
5
Source: Refinitiv Eikon
6
Source: Refinitiv Eikon
7
Source: Refinitiv Eikon
8
Source: Refinitiv Eikon
9
Swap Payment Conventions
Currency
Fixed Leg
Floating Leg
Frequency
Basis
Tenor and Frequency
Basis
USD
Semi-annual
30/360
3m LIBOR, Quarterly
Act/360
USD
Annual
Act/360
3m LIBOR, Quarterly
Act/360
EUR
Annual
30/360
6m Euribor, Semi-annually
Act/360
EUR
Annual
30/360
3m Euribor, Quarterly
Act/360
GBP
Semi-annual
Act/365
6m LIBOR, Semi-annually
Act/365
JPY
Semi-annual
Act/365
6m LIBOR, Semi-annual
Act/360
CAD
Semi-annual
Act/365
3m CDOR, Quarterly
Act/365
10
Interest Rate Risk of Swaps
11
Interest Rate Risk of Swaps
• Higher rates result in MTM profit to payer of fixed.
Interest Rate
Yield curve later Yield curve now
12345
Maturity (years)
12
Interest Rate Risk of Swaps
• Lower rates result in MTM loss to payer of fixed.
Interest Rate
Yield curve now Yield curve later
12345
Maturity (years)
13
Interest Rate Risk of Swaps
• Paying fixed is like being short a fixed-rate bond.
• Receiving fixed is like being long a fixed-rate bond.
• Swap positions expose swap participants to interest rate
risk in much the same way as bonds and futures.
• We can use swaps to express views about future interest rates or to hedge interest rate risk.
• Swap rates quoted by dealers reflect their hedging costs. 14
Hedging Swaps with Futures
• Changes in forward rates affect MTM value of swap.
• Swap dealer can hedge interest rate risk in swap by taking an offsetting position in an interest rate futures strip (i.e. a series of interest rate futures for sequential periods).
– Payeroffixedinswapmakesprofitifratesrise.
– Longpositioninfutureslosesifratesrise(increaseinforward
rates reduces futures price)
– Hedgeforpayingfixedinswapistogolongthefuturesstrip.
• Relevant for short swap maturities and in currencies with liquid interest rate futures market (e.g. US dollar, Euro).
15
Hedging Swaps with Bonds
• Swap dealers can also hedge risk with government bonds.
• Paying fixed in swap is like being short a bond, so hedge for paying fixed in swap is to go long a government bond.
• Swap dealers funds bond positions through sale and repurchase agreements (repo), funding at repo rate.
• Hedging costs are therefore also affected by changes in spread between LIBOR or Euribor and repo rate.
16
Source: Refinitiv Eikon
USD 10-year Swap Rate and USD 10-year Treasury Benchmark Yield
17
Source: Refinitiv Eikon
USD 10-year Swap Spread
18
Factors Driving Swap Spreads
• Swap spread to government bonds affected by: – differencesincreditrisk/counterpartycreditrisk
– relativeliquidity
– repomarketconditions(‘specials’)
– demandandsupplyinthecashmarketforgovernmentbonds
• Relative demand and supply for floating rate money from corporates and other swap counterparties
• This depends on interest rate expectations, steepness of yield curve (which affects carry), and other factors.
19
Swap Rate Quotation
• Swap dealers quote two-way prices (bid and offer).
5.03% (offer) 5.00% (bid) LIBOR LIBOR
• Swap market is a market for floating rate cash flows (e.g. LIBOR or Euribor).
• Payer of fixed buys floating rate, receiver of fixed sells it.
Payer of Fixed
Swap Dealer
Receiver of Fixed
20
USD Swap Rates EUR Swap Rates
Source: Refinitiv Eikon
21
Swap Pricing and Revaluation: LIBOR Discounting
22
Swap Valuation
• By convention, no money is exchanged up front in a newly initiated plain vanilla fixed-for-floating interest rate swap.
• The fixed rate is set at a level that makes present value of the fixed cash flows equal that of the floating cash flows.
• Initial mark-to-market (MTM) value of the swap is zero.
• As rates change, however, the MTM value will change.
• How should we calculate this value?
23
Swap Cash Flows
• Swap is two sets of cash flows (fixed leg and floating leg).
• Each cash flow has some present value based on the relevant discount factor for its payment date.
Payment Date
Floating Payment
Fixed Payment
Discount Factor
1
𝐹!
𝐶
𝐷!
2
𝐹”
𝐶
𝐷”
⋯
⋯
⋯
⋯
𝑁
𝐹#
𝐶
𝐷#
24
Swap Mark-to-Market Value
• Value of floating leg (i.e. floating cash flows):
𝑉$%&'()*+ = 𝐷!𝐹! + 𝐷”𝐹” + ⋯ + 𝐷#𝐹#
• Value of fixed leg (i.e. fixed cash flows): 𝑉$),-. =𝐷!𝐶+𝐷”𝐶+⋯+𝐷#𝐶
• Mark-to-market (MTM) value of swap to payer of fixed is present value of floating cash flows minus present value of fixed cash flows:
𝑉/0’1=𝑉$%&'()*+−𝑉$),-.=𝐷!𝐹!+𝐷”𝐹”+⋯+𝐷#𝐹#− 𝐷!𝐶+𝐷”𝐶+⋯+𝐷#𝐶 = 𝐷!𝐹! +𝐷”𝐹” +⋯+𝐷#𝐹# −𝐶 𝐷! +𝐷” +⋯+𝐷#
25
Unsolved Problems
• We do not know today the LIBOR rates at which future floating payments will be made. What should we do?
• What discount factors should we use to calculate the present value of the fixed and floating cash flows?
26
LIBOR and OIS Discounting
• LIBOR discounting was the market-standard method for swap valuation prior to the 2007-9 credit crisis.
• OIS discounting was developed in response to the increased use of collateralization in derivatives since 2009.
• OIS discounting is the current market standard.
27
LIBOR and OIS Discounting
• LIBOR discounting is used in Kosowski and Neftci (ch. 4), where the authors ignore counterparty risk.
• OIS discounting is explained in simplified form in Hull (ch. 7), but the explanation given there is incomplete, since it does not show how to project forward LIBOR rates.
• In this lecture, we explain both methods and show how they are related using simple but complete examples.
28
Forward LIBORs and Swap Value
• Floating payments depend on unknown future LIBORs.
• They can be hedged by agreeing to exchange unknown future floating cash flows for known cash flows based on today’s forward LIBORs through a strip of futures or FRAs.
F1 F2
F1*
F2* F1* F2* equals
plus
Floating leg of swap
F1 F2 Long futures (short FRAs)
Future floating payments fixed at today’s forward rates
29
Forward LIBORs and Swap Value
• Futures or FRAs at today’s forward rates have zero MTM value, so this does not change the value of the swap.
𝑉/0’1 = 𝑉$%&'()*+ − 𝑉$),-. = 𝐷!𝐹! + 𝐷”𝐹” + ⋯ + 𝐷#𝐹# − 𝐶 𝐷! + 𝐷” + ⋯ + 𝐷# = 𝐷 ! 𝐹 !∗ + 𝐷 ” 𝐹 “∗ + ⋯ + 𝐷 # 𝐹 #∗ − 𝐶 𝐷 ! + 𝐷 ” + ⋯ + 𝐷 #
• It does, however, give us known cash flows that we can discount in order to calculate present values.
• Unknown future LIBORs are replaced by forward LIBORs in both LIBOR discounting and OIS discounting.
30
Source: Refinitiv Eikon
31
Source: Refinitiv Eikon
32
Vertical Decomposition
• Replacing the unknown future LIBORS with today’s forward LIBOR rates is closely related to what Kosowski and Neftci (ch. 4) call vertical decomposition of the swap.
• They treat each exchange of fixed for floating cash flows in the swap as a FRA and use this idea to show that the swap rate must be consistent with the forward LIBORs.
• Hull (ch. 7) takes the same approach that we take here, based on no-arbitrage arguments about FRA valuation that we discussed in the last lecture (see Hull, ch. 4).
33
Calculating Discount Factors
• Future LIBORs are replaced by today’s forward LIBORs in both LIBOR discounting and OIS discounting.
• But what discount factors should we use to find present values? This is where the two valuation methods diverge.
• In LIBOR discounting, we assume that swap cash flows are of LIBOR (i.e. unsecured interbank lending) credit quality.
• This allows us to calculate the discount factors directly from the observed market swap rates.
34
Discount Factors and Par Yields
• •
Imagine that we are long a floating-rate bond and short a fixed-rate bond, both of LIBOR credit quality.
If the floating-rate bond pays a LIBOR flat coupon, then it will trade at par (i.e. its price will equal its face value).
1
L0,1 L1,2
1
Long floating-rate bond
35
Discount Factors and Par Yields
• •
The fixed-rate bond will also trade at par, provided that its coupon is equal to the LIBOR-credit-quality par yield.
The par yield is the coupon required for a bond to trade at par, given its credit quality and maturity.
1
L0,1 L1,2
plus
1
1
S2 S2
1
Long floating-rate bond
Short fixed-rate bond
36
Discount Factors and Par Yields
•
Adding the two positions together gives a set of cash flows identical to those in a fixed-for-floating interest rate swap.
1
L0,1 L1,2
1
plus
equals
L0,1 L1,2 S2 S2
Fixed-for-floating swap
•
Under LIBOR discounting, the market swap rate at each maturity is the LIBOR-credit-quality par yield.
1
S2 S2 1
Long floating-rate bond
Short fixed-rate bond
37
Horizontal Decomposition
• Kosowski and Neftci (ch. 4) call this idea horizontal decomposition: the swap is equivalent to a replicating
portfolio that is long a par-valued floating-rate bond and short a par-valued fixed-rate bond.
• Note, however, that this is only true if we can treat the swap cash flows as if they were of LIBOR credit quality.
• Under current market conditions and practice, we cannot do so, and the horizontal decomposition does not apply.
38
Discount Factors and Par Yields
• The par yield is a function of the discount factors:
1=𝐷!𝑆# +𝐷”𝑆# +⋯+𝐷#𝑆# +𝐷# -1 → 𝑆# = 1−𝐷#
𝐷! +𝐷” +⋯+𝐷#
• Rearranging this gives a bootstrapping formula that can be used iteratively to find discount factors that are consistent with the observed market swap rates:
𝑆#= 1−𝐷# →𝐷#=1−𝑆# 𝐷!+𝐷”+⋯𝐷#3! 𝐷! +𝐷” +⋯+𝐷# 1+𝑆#
39
Projected Forward LIBORs
• Projected forward LIBORs can then be calculated directly from the LIBOR discount factors, using the no-arbitrage relationship we discussed in the last lecture:
𝐷43!
𝐿43!,4= 𝐷4 −1 𝑓𝑜𝑟𝑗=1,⋯,𝑁
∗
𝑤h𝑒𝑟𝑒 𝐿43!,4 𝑖𝑠 𝑡h𝑒 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝐿𝐼𝐵𝑂𝑅 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑑𝑎𝑡𝑒𝑠 𝑗 − 1 𝑎𝑛𝑑 𝑗
• These forward LIBORs are used in place of the unknown future LIBORs to calculate the floating cash flows.
∗
40
Swap Valuation
• The fixed and projected floating cash flows are discounted using the discount factors calculated from the swap rates.
• This allows us to calculate the MTM value of the swap.
• This valuation model is correctly calibrated to current market swap rates, since it results in a MTM value of zero for each newly initiated swap of a given tenor with fixed rate equal to the current market swap rate.
41
LIBOR Discounting Example
• Consider a simple example in which we observe the following market rates for fixed-for-floating swaps:
Swap Maturity (Years)
Swap Rate
1
1%
2
2%
3
3%
4
4%
• We interpret these as LIBOR-credit-quality par yields and calculate discount factors using the bootstrapping algorithm defined earlier.
42
LIBOR Discount Factors
• 1-year discount factor:
• 2-year discount factor:
𝐷! = 1 = 1 = 0.9901 1 + 𝑆! 1 + 0.01
𝐷” =1−𝑆”𝐷! =1−0.02 0.9901 1 + 𝑆” 1 + 0.02
• 3-year discount factor:
= 0.9610
1−𝑆6 𝐷!+𝐷” 𝐷6 = 1 + 𝑆6
= 1 − 0.03 0.9901 + 0.9610 1 + 0.03
= 0.9140
43
LIBOR Discount Factors
• 4-yeardiscountfactor: 𝐷7 =1−𝑆7 𝐷! +𝐷” +𝐷6
= 1 − 0.04 0.9901 + 0.9610 + 0.9140 1 + 0.04
= 0.8513
• Since these are LIBOR discount factors, they can be used to project the forward LIBORs, using the relationship:
𝐷43!
𝐿43!,4= 𝐷4 −1 𝑓𝑜𝑟𝑗=1,⋯,𝑁
∗
𝑤h𝑒𝑟𝑒 𝐿43!,4 𝑖𝑠 𝑡h𝑒 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝐿𝐼𝐵𝑂𝑅 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑑𝑎𝑡𝑒𝑠 𝑗 − 1 𝑎𝑛𝑑 𝑗
1 + 𝑆7
∗
44
Projected Forward LIBORs
• Forward LIBOR for period between years 1 and 2 is:
𝐷! 0.9901
𝐿!,” =𝐷” −1=0.9610−1=3.0303%
• The remaining projected forward LIBORs are:
𝐷” 0.9610
𝐿”,6 =𝐷6 −1=0.9140−1=5.1345%
𝐷6 0.9140
𝐿6,7 =𝐷7 −1=0.8513−1=7.3654%%
• (Note: These forward LIBORs are based on discount factors rounded to higher accuracy than reported here.)
∗
∗
∗
45
Swap Valuation
• We now have a complete set of discount factors and forward rates consistent with the market swap rates:
Maturity
Swap Rate
Discount Factor
Forward LIBOR
1
1%
0.9901
1.0000%
2
2%
0.9610
3.0303%
3
3%
0.9140
5.1345%
4
4%
0.8513
7.3654%
• These can be used to calculate the MTM value of a swap at any agreed fixed rate with maturity up to four years in a market in which the swap rates are as shown in the table.
46
Model Calibration
• Consider first a newly initiated pay-fixed 3-year swap at the current 3-year swap rate with notional principal 100:
Year
Discount Factor
Floating
PV Floating
Fixed
PV Fixed
1
0.9901
1.0000
0.9901
−3.0000
−2.9703
2
0.9610
3.0303
2.9121
−3.0000
−2.8829
3
0.9140
5.1345
4.6932
−3.0000
−2.7421
𝑃𝑉 𝐹𝑙𝑜𝑎𝑡𝑖𝑛𝑔
8.5954
𝑃𝑉 𝐹𝑖𝑥𝑒𝑑
−8.5954
𝑺𝒘𝒂𝒑 𝑵𝑷𝑽
0.0000
• As expected, the swap has zero net present value.
• This confirms that the model is correctly calibrated.
47
Swap Revaluation
• Now suppose that we want to value a swap that has three years remaining but an ‘off-market’ fixed rate of 2%:
Year
Discount Factor
Floating
PV Floating
Fixed
PV Fixed
1
0.9901
1.0000
0.9901
−2.0000
−1.9803
2
0.9610
3.0303
2.9121
−2.0000
−1.9220
3
0.9140
5.1345
4.6932
−2.0000
−1.8281
𝑃𝑉 𝐹𝑙𝑜𝑎𝑡𝑖𝑛𝑔
8.5954
𝑃𝑉 𝐹𝑖𝑥𝑒𝑑
−5.7302
𝑺𝒘𝒂𝒑 𝑵𝑷𝑽
2.8651
• MTM value of this swap to payer of fixed is 2.8651 per 100 notional (e.g. $2.8651 million on notional $100 million).
48
Swap Revaluation
• This is the payment we should demand from our counterparty if they ask to exit or ‘tear up’ the swap.
• It is also the profit we would realize if we were to unwind the swap by receiving fixed at the current 3-year rate.
• Notice that the MTM value makes intuitive sense. We are paying 2% but the current 3-year swap rate is 3%.
• On notional amount $100 million, we are underpaying by $1 million per year over a 3-year period. Undiscounted, this would be $3 million.
49
Alternative Floating Leg Valuation
• Under LIBOR discounting, the present value of the floating leg can be calculated using just the discount factor for the tenor or maturity of the swap:
𝑉$%&'()*+ = 1 − 𝐷# × 𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 = 1 − 𝐷6 × 𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 = 1 − 0.914046 × 100 = 8.5954
where we have used higher accuracy for 𝐷X.
• This is based on the horizontal decomposition of the swap into a floating-rate bond and a fixed-rate bond that we discussed earlier, but only works under LIBOR discounting.
50
Swap DV01
• How can we measure interest rate risk in a swap?
• The DV01 of a swap can be calculated by ‘bumping’ the swap curve by one basis point and revaluing the swap.
• Horizontal composition implies that the swap DV01 must be similar to that of a fixed-coupon bond of the same maturity with a coupon rate equal to the swap rate.
• The same is true of the swap’s duration (another measure of interest rate risk).
51
LIBOR Valuation Model
• We can illustrate this and other aspects of swap pricing and revaluation under LIBOR discounting using a simple Excel-based swap valuation model.
52
Swap Pricing and Revaluation: OIS Discounting
53
Interest Rates in the Credit Crisis
• The financial crisis of 2007-9 created enormous disruption in markets, not least in short-term interest rates.
• Prior to the crisis, rates on different instruments of the same tenor (e.g. 3-month LIBOR and 3-month OIS) traded at small and stable spreads to one another.
• Market forward LIBOR rates closely reflected fair-value forward rates implied by deposit rates, and spreads on tenor and cross-currency basis swaps were usually small.
• All that changed with the crisis.
Source: Refinitiv Eikon
3-month LIBOR-OIS Spread
55
Credit and Liquidity Risk
• The crisis was a forceful reminder to the market of the
importance of credit risk and liquidity risk.
• Unsecured lending rates (e.g. LIBOR) now trade at wider
credit spreads to secured lending rates (e.g. repo).
• Rates for longer tenors now trade at wider spreads to rates for shorter tenors (e.g. 6-month vs. 3-month LIBOR).
• We can no longer treat all money market interest rates as is they were essentially the same.
56
Counterparty Risk
• The crisis also forced the market to pay much more
attention to counterparty risk in derivatives transactions.
• Counterparty risk is the risk that our counterparty will not
meet their obligations in a derivative transaction.
• We have counterparty risk exposure in any derivatives position with a positive MTM value from our point of view, since we will suffer a loss if our counterparty fails.
• Counterparty risk management practices and regulatory standards have changed dramatically since the crisis.
57
Counterparty Risk Management
• Counterparty risk is managed through netting (offsetting obligations between counterparties to reduce exposure)
and collateralization (taking and holding cash or other assets from the counterparty that can be used to offset losses on derivatives in the event of counterparty default).
• Many derivatives (including much of the swap market) are now centrally cleared, with netting and collateralization occurring multilaterally through a central counterparty).
• Most non-centrally-cleared derivatives are now subject to rules that enforce bilateral netting and collateralization.
58
Collateralization and Discounting
• Properly collateralized derivatives positions are subject to relatively little credit risk due to counterparty default.
• But cash flows (from derivatives or any other source) should be discounted at interest rates appropriate to their level of risk. This suggests that collateralized derivatives should be discounted at near-risk-free interest rates.
• In practice, this is done by constructing discount factors from market quotes for overnight index swaps (OIS), which reflect secured or unsecured overnight rates.
59
Overnight Index Swap (OIS)
• Overnight index swap (OIS): fixed-for-floating swap in which floating payment is based on compounding at an overnight interest rate over each settlement period.
Fixed Rate
Compounded Overnight Rate
• Overnight rate may be a secured or unsecured rate.
• Floating payment for each period is based on overnight
rates observed during that period (payment in arrears).
60
OIS Payer
OIS Receiver
OIS Discounting
• Overnight lending is subject to much less credit and liquidity risk than lending for longer terms.
• They therefore contain much smaller risk premiums, and discount factors constructed from market OIS rates are much better suited to discounting derivative cash flows.
• Revaluation of derivatives using discount factors derived from market OIS rates is called OIS discounting.
• This is the current market standard (see Hull, ch. 7).
61
USD OIS Rates
EUR €STR OIS Rates
Source: Refinitiv Eikon
62
OIS Discount Factors
• Discount factors are calculated from OIS rates, using the same bootstrapping method as in LIBOR discounting:
𝐷#=1−𝑂𝐼𝑆# 𝐷!+𝐷”+⋯𝐷#3! 1 + 𝑂𝐼𝑆#
where 𝑂𝐼𝑆Z is the N-year overnight index swap rate.
• Under LIBOR discounting, we projected forward LIBORs
directly from the swap discount factors.
• Now this won’t work, because the discount factors are OIS discount factors, not LIBOR discount factors.
63
Projecting Forward LIBORs
• Forward LIBORs must be consistent with the swap rates.
• A newly initiated swap at the current market swap rate
has zero value. The present value of its fixed cash flows is
equal to the present value of its floating cash flows:
##
∗ 𝑆#[𝐷4 =[𝐷4𝐿43!,4
48! 48!
• We can rearrange this to give a bootstrapping formula for the projected forward LIBORs:
# #3!∗
𝑆# ∑48! 𝐷4 − ∑48! 𝐷4𝐿43!,4
𝐷#
∗
𝐿#3!,# =
64
Swap Valuation
• Forward LIBORs projected using the bootstrapping formula are consistent with the OIS discount factors and the quoted fixed-for-floating LIBOR-based swap rates.
• We can use the projected forward LIBORs and the OIS discount factors to calculate the MTM value of a swap.
65
OIS Discount Factors
• Suppose that the OIS and LIBOR-based swap rates are:
Maturity (Years)
OIS Rate
LIBOR-based Swap Rate
1
0.5%
1%
2
1.5%
2%
3
2.5%
3%
4
3.5%
4%
• Use bootstrapping to calculate the OIS discount factors:
Year
OIS Rate
OIS Discount Factor
1
0.5%
0.9950
2
1.5%
0.9705
3
2.5%
0.9277
4
3.5%
0.8683
66
Projected Forward LIBORs
• Then use bootstrapping again to project forward LIBORs:
Year
OIS Rate
LIBOR-based Swap Rate
OIS Discount Factor
Forward LIBOR
1
0.5%
1%
0.9950
1.0000%
2
1.5%
2%
0.9705
3.0253%
3
2.5%
3%
0.9277
5.1188%
4
3.5%
4%
0.8683
7.3319%
• Notice that both the discount factors and the projected forward LIBORs are different than with LIBOR discounting.
67
Model Calibration
• Once again, however, the model correctly reproduces the zero MTM value of a newly initiated 3-year swap at the current market 3-year swap rate:
Year
Discount Factor
Floating
PV Floating
Fixed
PV Fixed
1
0.9950
1.0000
0.9950
−3.0000
−2.9851
2
0.9705
3.0253
2.9361
−3.0000
−2.9116
3
0.9277
5.1188
4.7486
−3.0000
−2.7830
𝑃𝑉 𝐹𝑙𝑜𝑎𝑡𝑖𝑛𝑔
8.6796
𝑃𝑉 𝐹𝑖𝑥𝑒𝑑
−8.6796
𝑺𝒘𝒂𝒑 𝑵𝑷𝑽
0.0000
• The model is properly calibrated to the OIS discount factors and the quoted rates for LIBOR-based swaps.
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Swap Revaluation
• Now suppose that we want to value a swap that has three years remaining but an ‘off-market’ fixed rate of 2%:
Year
Discount Factor
Floating
PV Floating
Fixed
PV Fixed
1
0.9950
1.0000
0.9950
−2.0000
−1.9900
2
0.9705
3.0253
2.9361
−2.0000
−1.9410
3
0.9277
5.1188
4.7486
−2.0000
−1.8553
𝑃𝑉 𝐹𝑙𝑜𝑎𝑡𝑖𝑛𝑔
8.6796
𝑃𝑉 𝐹𝑖𝑥𝑒𝑑
−5.7864
𝑺𝒘𝒂𝒑 𝑵𝑷𝑽
2.8932
• MTM value of this swap to payer of fixed is 2.8932 per 100 notional (e.g. $2.8932 million on notional $100 million), a slightly different value than under LIBOR discounting.
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Source: Refinitiv Eikon
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Source: Refinitiv Eikon
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Swap DV01
• Interest rate risk is more complicated with OIS discounting because there are two underlying yield curves.
• DV01 can be calculated by ‘bumping’ the swap curve by one basis point and revaluing, as in LIBOR discounting.
• But changes in OIS rates will also affect the swap’s value, as will changes in the spread between OIS and swap rates.
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OIS Valuation Model
• We can illustrate this and other aspects of swap pricing and revaluation under OIS discounting using a simple Excel-based swap valuation model.
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Other Interest Rate Swaps
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Forward Starting Swaps
• Forward starting swaps become effective on a specified forward date rather than on the spot date.
• 1-into-2-year forward starting swap:
today
2,3
effective L date 1,2
L S1,3
• Analyze as ordinary swap with ‘missing’ cash flows.
• Forward swap rates implied by spot-starting swap rates.
S1,3
1 year
2 years
Uses of Forward Starting Swaps
• Hedge or express views about future swap rates.
• Pre-hedging of new bond issues.
• Underlying source of risk in swaptions (i.e. options on forward-starting swaps).
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Tenor Basis Swaps
• Tenor basis swaps are floating-for-floating swaps between rates of different tenor in same currency (e.g. 3-month USD LIBOR in exchange for 6-month USD LIBOR).
3m LIBOR + spread 6m LIBOR
• Used to connect markets for fixed-for-floating swaps that reference rates of different tenors.
Counterparty A
Counterparty B
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Cross-Currency Basis Swaps
• Cross-currency basis swaps are floating-for-floating swaps between floating rates for different currencies (e.g. 3- month USD LIBOR in exchange for 3-month Euribor).
3m Euribor ± spread 3m LIBOR
• Used to connect markets for fixed-for-floating swaps denominated in different currencies.
• Principal amounts are exchanged at beginning and end.
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Counterparty A
Counterparty B
Other Cross-Currency Swaps
• Cross-currency basis swaps can be combined with fixed- for floating swaps in a single currency to create many other kinds of cross-currency interest rate swap.
• Fixed-for-fixed cross-currency swap (e.g. fixed-rate USD against fixed-rate GBP).
• Fixed-for-floating cross-currency swap (e.g. fixed-rate CAD against floating-rate JPY).
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Using Cross-Currency Swaps
• Issuers use cross-currency swaps to ‘swap’ bond issues into another currency in order to obtain funds in the currency they desire at the lowest possible cost.
• Investors use cross-currency swaps to create synthetic versions of bonds denominated in other currencies and to assess relative value in international capital markets.
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Basis Swap Spreads
• Prior to the financial crisis, most basis swaps traded nearly ‘flat’ (i.e. with spreads of only a few basis points attached to one of the floating legs).
• Increased liquidity and credit risk premiums in LIBOR and Euribor rates and constraints on liquidity mean that many basis swaps now trade with significant spreads.
• Tenor basis swaps have positive spreads on the leg corresponding to the rate with the shorter tenor. Cross- currency swaps have non-zero spreads on non-USD leg.
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Source: BNP Paribas, Cross-Currency Basis Swaps: A Primer, June 2015.
Cross-Currency Basis Drivers
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Source: Thomson Reuters Eikon
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References Main reading:
• Hull, ch. 7
• Kosowski and Neftci, ch. 4
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