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QF5203 Lecture 3
Interest Rate Swaps and their Risk Measures
Part 1
1. References
2. Introduction
3. FRAs and Futures
4. Expression for the Libor Rate
5. Recap
6. Construction of the Discount Function
7. Interest Rate Swap Risk Measures
8. Trader Jargon
9. Homework
10. Essay Topic
1. References
• Option, Future and Other Derivatives, John Hull
• Interest Rate Option Models, Riccardo Rebonato
• Pricing and Trading Interest Rate Derivatives, H. Darbyshire
• QuantLib Python Cookbook, Gautham Balaraman, Luigi Ballabio
• https://www.quantlib.org/quantlibxl/
• For market conventions see https://opengamma.com/wp-
content/uploads/2017/11/Interest-Rate-Instruments-and-Market-
Conventions.pdf
2. Introduction
• Interest rate swaps are probably the most important instrument to manage
interest rate risk and detailed knowledge of how this instrument is valued
provides deep insight into the associated risk sensitivities.
• An interest rate swap (IRS) is an over the counter (OTC) financial derivative
where two parties agree to exchange interest rate linked cash flows on a pre-
defined notional principal for a specified period of time and frequency.
• The simplest (plain vanilla) interest rate swap is a swap where one party pays
(receives) a fixed rate and receives (pays) a floating rate.
• The most common floating index is LIBOR which stands for London Interbank
Offered Rate, and is (supposed) to be the rate at which banks lend to each other
• As a result of the global financial crisis in 2008, LIBOR is in the process of being
discontinued, in favour of a more reliable interest rate benchmark (e.g. SOFR)
Exhibit - USD Swap Rates
Maturity Swap Rates
1y 0.66
2y 0.47
3y 0.44
4y 0.46
5y 0.49
6y 0.52
7y 0.56
8y 0.59
9y 0.62
10y 0.64
12y 0.69
15y 0.73
20y 0.78
25y 0.79
30y 0.80
Source: https://sebgroup.com/large-corporates-and-institutions/prospectuses-
and-downloads/rates/swap-rates
2. Introduction
Example – On the following slide a table of USD swap rates are shown. As in lecture
2, we will focus on a specific example, namely the 10y plain vanilla USD interest
rate swap, and explain what the number 0.64 actually means.
• A plain-vanilla interest rate swap is characterised by the following pieces of
information:
Currency: the currency in which the payments are made
Trade Date: the date on which the two parties to the swap agree to the trade
Effective Date: the date from which the cash flows are assumed to begin
Maturity Date: the date on which the last cash flow takes place and the swap ends
Notional: the currency amount on which fixed and floating interest rate payments
are calculated
2. Introduction (cont’d)
Fixed Leg Details:
Pay or Receive: specifies which party pays or receives the fixed leg
Fixed Rate: this is the fixed rate to be paid or received, and the number
quoted on an IRS broker screen (on slide 5 this rate is shown as
0.64%)
Fixed Frequency: this is the frequency on which fixed rate is payed or
received (e.g. quarterly, semi-annual, etc.) and is currency specific
Fixed Rate Day Count Fraction: this is the day count basis used to determine
the interest rate accrual (e.g. Act/360, etc.) of the fixed leg and is
currency specific
Fixed Rate Business Day Convention: this determines how dates are
adjusted for holidays (e.g. following, modified following, etc.)
Fixed Payment Calendar: this determines the calendar used for payments
2. Introduction (cont’d)
Floating Leg Details:
Pay or Receive: which party pays or receives the fixed leg
Floating Index: this is the floating index to be paid or received, and this
number (e.g. USD Libor 3m) is published daily at 11am UK time
Floating Frequency: this is the frequency on which floating rate is payed or
received (e.g. quarterly, semi-annual, etc.)
Floating Rate Day Count Fraction: this is the day count basis used to
determine the interest rate accrual (e.g. Act/360, etc.) for the
floating leg
Floating Rate Business Day Convention: this determines how dates are
adjusted for holidays (e.g. following, modified following, etc.)
Float Payment Calendar: this determines the calendar used for payments
2. Introduction (cont’d)
• Note that floating rate cash flows have 4 dates associated with them
i. Reset Date – this is the date that the (previously unknown) cash flow
becomes known (or fixed); in the case of LIBOR, this is the date that you
physically look at the screen (i.e. 11am London time)
ii. Interest Accrual Start Date – this is the date when interest begins to accrue
iii. Interest Accrual End Date – this is the date when interest stops accruing
iv. Payment Date – this is the date when the associated interest amount is paid
• Note that floating interest reset dates vary by currency (e.g. 2 business days for
USD)
• Note that there are some swaps where the floating rate is set at the end of the
floating rate period. These are called Libor in Arrears swaps, and we will cover
these later.
2. Introduction (cont’d)
• Business day Conventions
i. Following – if a cash flow takes place on a holiday then advance to next
good business day (whether or not it is in the same month)
ii. Modified Following – similar to (i) however, if the next good business day is
in the same month then use it, otherwise select the closest previous
business day
iii. Preceding – The closest previous business day
iv. Modified Preceding – if the preceding good business day is in the same
month then use it, otherwise select the closest following business day
v. End of Month – the last good business day of the month
vi. IMM – days on which International Monetary Market (IMM) futures
contracts (e.g. Eurodollar futures) settle, namely third Wednesday of
March, June, September and December
• Refer to Open Gamma market conventions guide
2. Introduction (cont’d)
• Accrual Conventions – these determine the precise cash flow
• i.e. Cash Flow = Notional * Rate * Day Count Fraction
• There are various day count fraction conventions
i. Actual/360
ii. Actual/365
iii. 30/360
iv. Actual/Actual
v. Etc.
• Fortunately bank’s have internal analytics libraries or use third party vendor
analytics which incorporate all possible accrual conventions. We will learn how
to use QuantLib to calculate accrual factors later in this lecture.
• Refer to Open Gamma market conventions guide
2. Introduction (cont’d)
• Returning to slide 5, 0.64% is the number for which the present value (PV) of the fixed
leg of a 10y USD swap (assuming semi-annual payments on a 30/360 day count basis)
equals the PV of the floating leg (assuming quarterly payments of USD Libor 3m).
• Definition – the fixed rate of a fixed versus floating interest rate swap is defined to be
the interest rate for which the PV of the fixed leg equals the PV of the floating leg
• But the floating leg of a swap makes reference to a floating (meaning not known today)
interest rate which is not known today, so how do I value these cash flows?
• On the next slide is a screen shot of a 10y USD IRS from QuantLib Excel, and we will go
through the inputs one by one
• This is followed by another screen shot of the same 10y USD IRS using QuantLib Python
In other words, if I build a swap yield curve using 0.64 as the 10y input and then use that
same yield curve to price a 10y swap then the value of that swap is 0.
2. Introduction (cont’d)
General Inputs Fixed Leg Details Float Leg Details Name Object ID Error
Evalution Date3-Apr-20 Ccy USD Ccy USD Fixed Leg Schedule IDUSDFixedLegSchedule1#0008
Set Evaluation DateTRUE Notional 100,000,000 Notional 100,000,000 Fixed Leg IDUSDFixedLeg1#0008
Result Ccy USD Start Date 7-Apr-20 Start Date 7-Apr-20
Days To Spot 2 Maturity 10y Maturity 10y Float Leg ScheduleUSDF oatLegSchedule1#0008
Spot Date 7-Apr-20 End Date 8-Apr-30 End Date 8-Apr-30 Float Leg Index IDUSDLiborIndex3m#0008
Start Date 7-Apr-20 Pay/Rec REC Pay/Rec PAY Float Leg IDUSDLiborLeg1#0008
Maturity 10y Fwd Swap 3.99996% Margin 0.00%
End Date 8-Apr-30 Coupon 4.00000% Freq Quarterly Fixed Leg NPV32,695,774
Notional 100,000,000 Coupon Freq Semiannual Basis Actual/360 Float Leg NPV32,695,430
Basis 30/360 (Bond Basis) Bus Day ConvModified Following Swap NPV 344
Yield Curve Inputs Bus Day Conv Modified Following Pmt CalendarUnitedStates::Settlement
Handle USDFlatFwdYieldCurve Pmt Calendar UnitedStates::Settlement Reset CalendarU itedKingdom::Settlement Vanilla Swap IDUS VanillaSwap1#0008
Ndays 0
Calendar UnitedStates::Settlement Swap Engine IDUS DiscountingSwapEngine#0008
Rate 4.00% Pricing Engine IDTRUE
Day Count 30/360 (Bond Basis)
CompoundingCompounded NPV 344
Frequency Semiannual Npv Details
Yc IDUSDFlatFwdYieldCurve#0012 Fixed Leg 32,695,774 USD 32,695,774 32.70%
Float Leg -32,695,523 USD -32,695,523 -32.70%
Npv Deal 251 0.00%
2. Introduction (cont’d)
2. Introduction (cont’d)
• From the previous slide and accompanying Excel Spreadsheet it is clear that the
swap valuation is carried out on a leg by leg basis as follows:
• = σ=1
−1, ()
• = σ=1
−1, −1, ()
• = −
• In the above and are the IRS notional and fixed rates,
−1, and −1, are the accrual fractions associated with the fixed and
floating legs of the swap respectively, and ()is the discount factor
corresponding to date .
• But where do we get the future Libor rates −1, from since these haven’t
fixed yet?
3. FRAs and Futures
• A forward rate agreement (FRA) is an OTC financial instrument used to lock in a
fixed rate today for a forward starting date beginning 1 , and ending 2
• The fixed rate is known as the FRA rate
• The payout of the FRA payer contract at time 2 is:
2 = 1, 2 − 1, 2
• Or equivalently,
1 =
1, 2 − 1, 2
1 + 1, 2 1, 2
• Since in practice the FRA is settled at 1
• Buying a FRA corresponds to paying fixed whereas selling a FRA corresponds to
receiving fixed
3. FRAs and Futures (cont’d)
• FRAs are quoted in terms of the start date and end date of the floating rate
• So for example, a 3x6 FRA corresponds is a contract to lock in a fixed borrowing
beginning 3 months from today and ending in 6 months from today. The
underlying rate which is being locked in is a 3 month rate.
• Similarly, a 2x8 FRA corresponds is a contract to lock in a fixed borrowing
beginning 2 months from today and ending in 8 months from today. The
underlying rate which is being locked in is a 6 month rate
• Note the similarity of the FRA payout with each term in the valuation of a swap
• We sometimes refer to a plain vanilla IRS as a portfolio of FRAs
• If one buys a FRA one pays the fixed rate
• If one sells a FRA one receives the fixed rate
3. FRAs and Futures (cont’d)
• FRAs are convex instruments and therefore have gamma risk, unlike a futures
contract whose payout is linear in the forward rate.
• Suppose one has entered into a FRA receiver contract (see formula on slide 17
for the FRA payer and just interchange the forward rate and the strike)
• If rates go up then one will lose money on the FRA, but since rates are higher the
(negative) difference between the projected Libor rate and the fixed rate is being
discounted at a higher rate (i.e. being multiplied by a smaller discount factor),
and so the P&L loss is not as much
• Similarly, if rates go down then one will make money on the FRA, but since rates
are lower the (positive) difference between the projected Libor rate and the
fixed rate is being discounted at a lower rate (i.e. being multiplied by a larger
discount factor), and so the P&L gain is amplified
• Therefore there is always a benefit for the party who is receiving fixed on a FRA
and this party is said to long gamma (or long convexity)
3. FRAs and Futures (cont’d)
General Inputs FRA Details Underlying FRA Details
Ccy USD FRA # FRA Start FRA End Pay/Rec FRA Rate Notional Fwd Sensitivity PV01 NPV FRA
Quote Date3-Apr-20 1 1M 4M Pay 4.000% 100,000,000 4.000% 0.255 0
Set Evaluation DateTRUE 2 2M 5M Rec 4.000% 200,000,000 4.000% 0.249 0
Days To Spot 2 3 3M 6M Pay 4.000% 50,000,000 4.000% 0.249 0
Spot Date 7-Apr-20 4 4M 7M Rec 4.000% 100,000,000 4.000% 0.251 0
Float Freq S 5 1M 7M Pay 4.000% 100,000,000 4.000% 0.502 0
Float BasisActual/360 6 2M 8M Rec 4.000% 100,000,000 4.000% 0.502 0
Float Ref RateLIBOR3M 7 3M 6M Pay 1.157% 100,000,000 4.000% 0.249 711,678
Float Bus Day ConvModified F llowing 8 4M 7M Rec 1.303% 100,000,000 4.000% 0.251 -687,295
Pmt CalendarUnitedSt tes::Settlement 9 4M 7M Pay 1.096% 100,000,000 4.000% 0.251 740,046
Reset CalendarUnitedKing om::Settlement 10 4M 7M Rec 0.935% 100,000,000 4.000% 0.251 -781,075
11 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
Yield Curve Inputs 12 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
Handle USDFRAFlatFwdYieldCurve13 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
Ndays 0 14 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
Calendar UnitedStates::Settlement15 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
Rate 4.00% 16 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
Day CountActual/360 17 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
CompoundingCompounded 18 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
FrequencyQuarterly 19 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
Yc ID USDFRAFlatFwdYieldCurve#000320 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
21 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
22 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
23 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
24 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
25 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
3. FRAs and Futures (cont’d)
• Interest rate futures contracts are standardised exchange traded instruments
with daily margining (settlement)
• These contracts are traded on the CME, CBOT, ICE
• These contracts are standardised in the following ways:
• The contracts expire on the third Wednesday of March, June, September and
December
• The contract size is fixed (e.g. USD 1m for Eurodollars and GBP 500k for Short
Sterling)
• The tick size is fixed (e.g. USD 25 for Eurodollars and GBP 12.50 for Short
Sterling)
• The price of the contract is quoted on discount basis, namely 100-Futures Price
(so if the futures price is 99 then the implied 3 month rate is 1%)
3. FRAs and Futures (cont’d)
• Futures contracts, by virtue of their daily mark to market and margining, do not
display the same convexity and are purely linear in the projected Libor rate
• Therefore
• A useful exercise is to consider hedging a $10m receiver FRA with Eurodollar
futures
• So need to work out how many futures to buy/sell and then plot the P&L as a
function of the yield curve change
• Choose the same FRA dates as the relevant Futures contract
4. Floating Rates from Discount Function
• Note that for both the FRA and the Swap, we need to be able to value future
Libor rates and for both instruments the floating rate payments are of the form:
, +1 ( , +1)
• Where , +1 is the Libor rate observed at t (in practice the observation of
the rate is currency dependent)
• Using intuitive replication arguments, one can show that the unknown floating
rate , +1 must be equal to:
, +1 =
1
( , +1)
()
(+1)
− 1
• The proof is as follows:
4. Floating Rates from Discount Function (cont’d)
• Consider the following strategy where ; denotes a zero coupon bond of
maturity as of time
• At some time < 1
i. purchase $1 face value of a 2 maturity zero coupon bond of value ; 2
ii. sell short $
(;2)
(;1)
face value of a 1 zero coupon bond of value ; 1
• The net cash flow at is zero
• When the 1 maturity zero coupon bond matures (at 1), requiring repayment
of $
(;2)
(;1)
, borrow this same amount at the rate ; 1, 2 over the period
[1, 2].
• The net cash flow at 1 is also zero
4. Floating Rates from Discount Function (cont’d)
• Finally, at 2 the 2-maturity zero coupon bond I bought at matures and I
receive $1
• The loan of $
(;2)
(;1)
that I took out at 1 must be repaid, and so I must pay
$
(;2)
(;1)
1 + L ; 1, 2 1, 2 at 2
• Since the net cash flows at and 1 are zero, in order to eliminate arbitrage the
net cash flow at 2 must also be zero, leading to:
• 1 −
;2
;1
1 + L ; 1, 2 1, 2 = 0
• which leads to the expression for L ; 1, 2 shown on slide 20
• Note that this important result means that knowledge of the discount function
(i.e. the table of dates and corresponding discount factors) allows me to
determine all future Libor rates
4. Floating Rates from Discount Function (cont’d)
• By substituting the formula for ( , +1) shown on slide 20 into the expression
for the PV of the floating leg of the swap shown on slide 15, one obtains a simple
expression for the PV of the swap floating leg, namely
= () − ()
• Note that this implicitly assumes that the discount factors used for projecting
the Libor rates using the expression on slide 20, are the same as the discount
factors used for discounting these expected cash flows
• As we shall see in a later lecture, this assumption no longer holds when there is
a basis between Libor rates of different tenors
5. Recap – Swap Valuation
• The current valuation of a vanilla interest rate swap is obtained by adding the
present value of the fixed leg to the present value of the floating leg, taking into
account whether one is paying or receiving the fixed/floating legs
• The leg of a vanilla interest rate swap is just an annuity and can be valued by
multiplying the fixed cash flows by the discount factors associated with the
payment dates of the cash flows
• The floating leg of a vanilla interest rate swap can be valued by projecting the
future Libor rates and discounting the associated cash flows using the discount
corresponding to the Libor cash flow payment date
• The swap rate is defined to be the value of the fixed rate for which the present
value of the swap is zero
6. Constructing the Discount Function
• As stated on the previous slide, the valuation of an interest rate swap requires
knowledge of the discount factors corresponding to the coupon and Libor
payment dates
• When we talk about yield curves in the context of swaps, we mean a set of
discount factors which allow us to reprice all relevant instruments, namely
Deposits, Futures, FRAs, Swaps, etc.
• In lecture 2 we briefly looked at present value in the context of the yield to
maturity of a bond
• Recall that we calculated discount factors (zero coupon bond prices) using the
yield of the bond
• Swap yield curves are fundamental to interest rate derivatives and
6. Constructing the Discount Function (cont’d)
• Our objective is to derive the discount curve via bootstrapping process
• We will use the market values of a collection of interest rate instruments,
including deposits, FRAs, Futures and Swaps
• The discount curve must should have 2 basic properties
(i) it must reprice all instruments to ‘par’
(ii) it must be ‘smooth’
• The resulting curve will be a list of dates T, and corresponding discount factors Z
6. Constructing the Discount Function (cont’d)
• Let’s consider a simple example, namely a 1y dollar swap with a fixed annual
coupon versus USD Libor 3m and notional N, starting from spot
• Denote the 1y par swap rate by 1
• = 1 , 1 (1)
• = − (1)
• Since the swap is at par, the PVs of the fixed and floating leg must be equal,
• This allows us to solve for (1),
1 =
1 + 1 , 1
• Note that the discount factor can be obtained from the 2 day deposit rate
for currencies with a spot date of T+2, otherwise for currencies with spot date =
today (e.g. GBP) = 1.
6. Constructing the Discount Function (cont’d)
• Now consider a 2y par swap with annual fixed rate 2
• = 2 , 1 1 + 1, 2 2
• = − (2)
• Since the 2y swap is at par, the PV of the fixed and floating legs must be equal
• This allows me to solve for (2), noting that I have already solved for
(1) using the information on the previous slide derived from the 1y swap
2 =
− 2 , 1 (1)
1 + 2(1, 2)
6. Constructing the Discount Function (cont’d)
• Next we consider a 3y par swap with annual fixed rate 3
• = 3 , 1 1 + 1, 2 2 + 2, 3 3
• = − (3)
• Equating the PV of the fixed and floating legs as before allows me to solve for
(3), noting that I have already solved for Z(1) and Z(2) using the information
on the previous two slides derived from the 1y and 2y swaps
3 =
− 3 , 1 1 + 1, 2 2
1 + 3(2, 3)
• It is obvious that this generalises to
• Z(t) = [1 – S(t) * Sum(i=1 to t-1;Alpha(i-1,i) * Z(i)] / (1 + S(t) * Alpha(t-1,t))
• The process of using discount factors calculated for previous dates to obtain
discount factors at a later date is known as bootstrapping.
6. Constructing the Discount Function (cont’d)
• It is obvious that this generalises to:
=
− σ=1
−1 −1, ()
1 + (−1, )
• The process of using discount factors calculated for previous dates to obtain
discount factors at a later date is known as bootstrapping.
• The result is a list of dates and corresponding discount factors
6. Constructing the Discount Function (cont’d)
• Let’s look at the example of GBP
• Assume evaluation date of 3 Apr 2020
and the same spot date
• Assume fixed leg day count of Act/365
• Assume business day convention of
modified following
• Note that GBP swaps are quoted on a
semi-annual Act/365 basis so fixed
coupons are paid twice a year except
1y swaps which are quoted on an
annual Act/365 basis
Depo Rate Swap Rate
O/N 0.8584 1Y 0.4640
1W 0.8558 2Y 0.4770
1M 0.8407 3Y 0.4640
2M 0.8215 4Y 0.4770
3M 0.8031 5Y 0.4940
6M 0.7135 6Y 0.5120
7Y 0.5280
8Y 0.5410
9Y 0.5540
10Y 0.5670
12Y 0.5860
15Y 0.6050
20Y 0.6090
25Y 0.5940
30Y 0.5710
6. Constructing the Discount Function (cont’d)
• Calculating the discount factors for the deposit rates can be done directly, without
bootstrapping. Specifically,
• =
1
1+ (,)
• 1 =
1
1+ 1 (,1)
• Etc
• In the above, denotes the start date of the relevant deposit
• Note that to calculate the discount factor for the 2Y point we need the discount factor
for the 18M point, since 2Y GBP swaps pay semi-annual coupons.
• Actually, we need discount factors corresponding to all coupon dates but we only have
swap rates quoted for annual maturities
• What do we do?
• We create synthetic swaps with maturities corresponding to 18M, 30M, etc. using an
interpolation scheme
6. Constructing the Discount Function (cont’d)
• Standard interpolation schemes include:
(i) Linear Interpolation of the discount factors
(ii) Linear Interpolation of the log of the discount factors
(iii) Constant Instantaneous forward rates
(iv) Linear Instantaneous forward rates
(v) Splines (e.g. cubic)
• Returning to the GBP swap example, one way of creating the synthetic swaps is
just to linearly interpolate between the two adjacent par swap rates.
6. Constructing the Discount Function (cont’d)
• The result of the bootstrapping process that we have just covered leads to a table
consisting of a column of dates, which correspond to the dates implicit in the
instruments used to bootstrap the curve, and a column of associated discount
factors.
• Remember, the reason for creating the discount function is NOT for repricing the
instruments used to build it, but for repricing our existing portfolio of interest
rate products or for pricing other (non-standard) swaps.
• We will therefore require discount factors for dates which do not correspond to
dates in our table, and therefore interpolation is inevitable
7. Interest Rate Swap Risk Measures
• The 10y USD interest rate swap pricing example that we have studied used a
particularly simple yield curve for valuation, namely a flat yield curve
• The value of this IRS is therefore dependent on the single (flat) rate that was
chosen.
• In this case the only risk measure that is relevant is the PV01 risk measure,
namely the sensitivity of the valuation to a 1bp increase in this (flat) rate.
• In practice this means that should change the input rate by 1bp and revalue the
swap. The difference is PVs is the PV01.
• Refer back to the practical QuantLib Excel example.
7. Trader Jargon
• As risk managers it is important to be able to communicate using the same
language as traders in order to create credibility
• Therefore you need to be able to speak the same ‘language’
• Fundamental Principle – Everything is about Bond Prices
• Bonds and Swaps – As we already know, when interest rates go down bond
prices go up and vice versa so a rally in the bond market can be associated with
lower interest rates, or equivalently a sell-off in the bond market can be
associated with higher interest rates
• Therefore a trader will refer to rates rallying down which actually means that
bonds were higher which in turn means that rates are lower. Note that this is
unintuitive since normally we associate a rally with higher levels but if you
remember that “it is all about bond prices” then you won’t go wrong.
• Similarly a reference to rates selling off means that bonds were lower, which in
turn means that rates are higher
7. Trader Jargon (cont’d)
Long/Short Jargon:
• In the financial markets the terms long and short are used everywhere
• If I am long an asset then if that asset price goes up I make money and if the
asset price goes down I lose money.
• If I am short an asset then if that asset price goes up I lose money and if the
asset price goes down then I make money
• Remember these by heart
7. Trader Jargon (cont’d)
Long/Short (continued)
• But how do I translate long/short into an interest rate context?
• Again, the connection is with the bond market and so if I am receiving fixed on a
swap (and implicitly paying floating), then if interest rates go up I lose money,
hence I am long (bonds)
• Similarly, if I am paying fixed on a swap (and implicitly receiving floating), then if
interest rates go up then I make money, hence I am short (bonds)
• So in summary,
• Receiving Fixed on an IRS means that I am long and paying fixed on an IRS
means that I short
• Later we will study interest rate swaptions, and same logic holds, namely
• If I am long a receiver swaption I am long the (bond) market and if I am long a
payer swaption then I am short the (bond) market
7. Trader Jargon (cont’d)
Buy/Sell Jargon
• The concept of buying and selling is familiar to everyone
• Items for purchase or sale usually have a bid or offer associated with them
• So take the example of a bond trader (market maker) who is quoting a two-way
market of 99/101 in a specific bond. This means that he is prepared to buy the
bond at 99 (trader’s bid) and sell it at 101 (trader’s offer).
• If you want to buy the bond you will need to pay 101 and ‘lift’ the trader’s offer
• The trader will then say that he has been ‘lifted’ at 101
• If you have a bond to sell then you will receive 99 and ‘hit’ the trader’s bid
• The trader will then say that he/she has been ‘hit’ at 99
• You need to memorise the following: Bids are hit and offers are lifted
7. Trader Jargon (cont’d)
Mine/Yours Jargon
• This is related to the concept of buying and selling
• If you call up a market maker who is quoting 99/101 as the two-way price for a
particular bond and say “mine in 10 million” then you will have bought $10m
nominal of the bond and paid 101 (having lifted the trader’s offer)
• Similarly if you had said “yours in 10 million” then you will have sold $10m
nominal of the bond and received 99 (having hit the trader’s bid)
• The same jargon is used in the swaps market but can be a bit confusing
• If you call up a swap market maker who is quoting 1.05/1.10 for 10y USD swaps
and say “mine in $10m” then you will have agreed to pay a fixed rate of 1.10%
for 10y and receive USD Libor 3m on a notional of $10m
• Similarly, if you had said “yours in 10 million” then you will have agreed to
receive a fixed rate 1.05% for 10y and pay USD Libor 3m on a notional of $10m
8. Homework 3.1
• Using QuantLib Python/Excel and the techniques described in the previous slides
calculate the GBP discount function for the following market data, assuming an
evaluation date of 3 April 2020, and GBP swap conventions of semi-annual
A/365 for both fixed and floating legs, including deposits. The spot lag is 0 for
GBP swaps so the start date of all instruments is 3 April 2020.
• Note that the intention of this exercise is to go through the bootstrapping
process by hand, rather than relying on QuantLib to do this for you. We will use
QuantLib to bootstrap swap yield curves as part of the next lecture.
• The output should be a table of dates and associated discount factors.
• Prove that your discount function is self consistent by using the calculated
discount factors to re-imply the deposit and swap input rates.
• Plot the projected 6-month forward rates implied from your discount factors for
annual maturities from 1 year to 30 years.
8. Homework 3.1
• Use your discount function to calculate the equilibrium swap rate for a 10y GBP
swap, starting in 5 years (forward starting swap).
• Assume a notional of GBP 100m of the same forward starting (receiver) IRS
above and produce a risk report showing the sensitivity of each yield curve input
to a 1bp change. Your risk report should be presented in a table form as follows:
• Submission date to be confirmed.
Yield Curve Tenor Sensitivity (GBP)
O/N
1W
1M
2M
…
30Y
9. Essay Topic
• Topic – Please discuss the risk management implications of IBOR
discontinuation.
• Approximately 5-7 pages of A4 using 12pt font with 1.5 spacing.
• Not more than 2500 words.
• Please utilise spell and grammar checks (see www.grammarly.com) before
submitting.
• Submission date to be confirmed.
学霸联盟
Interest Rate Swaps and their Risk Measures
Part 1
1. References
2. Introduction
3. FRAs and Futures
4. Expression for the Libor Rate
5. Recap
6. Construction of the Discount Function
7. Interest Rate Swap Risk Measures
8. Trader Jargon
9. Homework
10. Essay Topic
1. References
• Option, Future and Other Derivatives, John Hull
• Interest Rate Option Models, Riccardo Rebonato
• Pricing and Trading Interest Rate Derivatives, H. Darbyshire
• QuantLib Python Cookbook, Gautham Balaraman, Luigi Ballabio
• https://www.quantlib.org/quantlibxl/
• For market conventions see https://opengamma.com/wp-
content/uploads/2017/11/Interest-Rate-Instruments-and-Market-
Conventions.pdf
2. Introduction
• Interest rate swaps are probably the most important instrument to manage
interest rate risk and detailed knowledge of how this instrument is valued
provides deep insight into the associated risk sensitivities.
• An interest rate swap (IRS) is an over the counter (OTC) financial derivative
where two parties agree to exchange interest rate linked cash flows on a pre-
defined notional principal for a specified period of time and frequency.
• The simplest (plain vanilla) interest rate swap is a swap where one party pays
(receives) a fixed rate and receives (pays) a floating rate.
• The most common floating index is LIBOR which stands for London Interbank
Offered Rate, and is (supposed) to be the rate at which banks lend to each other
• As a result of the global financial crisis in 2008, LIBOR is in the process of being
discontinued, in favour of a more reliable interest rate benchmark (e.g. SOFR)
Exhibit - USD Swap Rates
Maturity Swap Rates
1y 0.66
2y 0.47
3y 0.44
4y 0.46
5y 0.49
6y 0.52
7y 0.56
8y 0.59
9y 0.62
10y 0.64
12y 0.69
15y 0.73
20y 0.78
25y 0.79
30y 0.80
Source: https://sebgroup.com/large-corporates-and-institutions/prospectuses-
and-downloads/rates/swap-rates
2. Introduction
Example – On the following slide a table of USD swap rates are shown. As in lecture
2, we will focus on a specific example, namely the 10y plain vanilla USD interest
rate swap, and explain what the number 0.64 actually means.
• A plain-vanilla interest rate swap is characterised by the following pieces of
information:
Currency: the currency in which the payments are made
Trade Date: the date on which the two parties to the swap agree to the trade
Effective Date: the date from which the cash flows are assumed to begin
Maturity Date: the date on which the last cash flow takes place and the swap ends
Notional: the currency amount on which fixed and floating interest rate payments
are calculated
2. Introduction (cont’d)
Fixed Leg Details:
Pay or Receive: specifies which party pays or receives the fixed leg
Fixed Rate: this is the fixed rate to be paid or received, and the number
quoted on an IRS broker screen (on slide 5 this rate is shown as
0.64%)
Fixed Frequency: this is the frequency on which fixed rate is payed or
received (e.g. quarterly, semi-annual, etc.) and is currency specific
Fixed Rate Day Count Fraction: this is the day count basis used to determine
the interest rate accrual (e.g. Act/360, etc.) of the fixed leg and is
currency specific
Fixed Rate Business Day Convention: this determines how dates are
adjusted for holidays (e.g. following, modified following, etc.)
Fixed Payment Calendar: this determines the calendar used for payments
2. Introduction (cont’d)
Floating Leg Details:
Pay or Receive: which party pays or receives the fixed leg
Floating Index: this is the floating index to be paid or received, and this
number (e.g. USD Libor 3m) is published daily at 11am UK time
Floating Frequency: this is the frequency on which floating rate is payed or
received (e.g. quarterly, semi-annual, etc.)
Floating Rate Day Count Fraction: this is the day count basis used to
determine the interest rate accrual (e.g. Act/360, etc.) for the
floating leg
Floating Rate Business Day Convention: this determines how dates are
adjusted for holidays (e.g. following, modified following, etc.)
Float Payment Calendar: this determines the calendar used for payments
2. Introduction (cont’d)
• Note that floating rate cash flows have 4 dates associated with them
i. Reset Date – this is the date that the (previously unknown) cash flow
becomes known (or fixed); in the case of LIBOR, this is the date that you
physically look at the screen (i.e. 11am London time)
ii. Interest Accrual Start Date – this is the date when interest begins to accrue
iii. Interest Accrual End Date – this is the date when interest stops accruing
iv. Payment Date – this is the date when the associated interest amount is paid
• Note that floating interest reset dates vary by currency (e.g. 2 business days for
USD)
• Note that there are some swaps where the floating rate is set at the end of the
floating rate period. These are called Libor in Arrears swaps, and we will cover
these later.
2. Introduction (cont’d)
• Business day Conventions
i. Following – if a cash flow takes place on a holiday then advance to next
good business day (whether or not it is in the same month)
ii. Modified Following – similar to (i) however, if the next good business day is
in the same month then use it, otherwise select the closest previous
business day
iii. Preceding – The closest previous business day
iv. Modified Preceding – if the preceding good business day is in the same
month then use it, otherwise select the closest following business day
v. End of Month – the last good business day of the month
vi. IMM – days on which International Monetary Market (IMM) futures
contracts (e.g. Eurodollar futures) settle, namely third Wednesday of
March, June, September and December
• Refer to Open Gamma market conventions guide
2. Introduction (cont’d)
• Accrual Conventions – these determine the precise cash flow
• i.e. Cash Flow = Notional * Rate * Day Count Fraction
• There are various day count fraction conventions
i. Actual/360
ii. Actual/365
iii. 30/360
iv. Actual/Actual
v. Etc.
• Fortunately bank’s have internal analytics libraries or use third party vendor
analytics which incorporate all possible accrual conventions. We will learn how
to use QuantLib to calculate accrual factors later in this lecture.
• Refer to Open Gamma market conventions guide
2. Introduction (cont’d)
• Returning to slide 5, 0.64% is the number for which the present value (PV) of the fixed
leg of a 10y USD swap (assuming semi-annual payments on a 30/360 day count basis)
equals the PV of the floating leg (assuming quarterly payments of USD Libor 3m).
• Definition – the fixed rate of a fixed versus floating interest rate swap is defined to be
the interest rate for which the PV of the fixed leg equals the PV of the floating leg
• But the floating leg of a swap makes reference to a floating (meaning not known today)
interest rate which is not known today, so how do I value these cash flows?
• On the next slide is a screen shot of a 10y USD IRS from QuantLib Excel, and we will go
through the inputs one by one
• This is followed by another screen shot of the same 10y USD IRS using QuantLib Python
In other words, if I build a swap yield curve using 0.64 as the 10y input and then use that
same yield curve to price a 10y swap then the value of that swap is 0.
2. Introduction (cont’d)
General Inputs Fixed Leg Details Float Leg Details Name Object ID Error
Evalution Date3-Apr-20 Ccy USD Ccy USD Fixed Leg Schedule IDUSDFixedLegSchedule1#0008
Set Evaluation DateTRUE Notional 100,000,000 Notional 100,000,000 Fixed Leg IDUSDFixedLeg1#0008
Result Ccy USD Start Date 7-Apr-20 Start Date 7-Apr-20
Days To Spot 2 Maturity 10y Maturity 10y Float Leg ScheduleUSDF oatLegSchedule1#0008
Spot Date 7-Apr-20 End Date 8-Apr-30 End Date 8-Apr-30 Float Leg Index IDUSDLiborIndex3m#0008
Start Date 7-Apr-20 Pay/Rec REC Pay/Rec PAY Float Leg IDUSDLiborLeg1#0008
Maturity 10y Fwd Swap 3.99996% Margin 0.00%
End Date 8-Apr-30 Coupon 4.00000% Freq Quarterly Fixed Leg NPV32,695,774
Notional 100,000,000 Coupon Freq Semiannual Basis Actual/360 Float Leg NPV32,695,430
Basis 30/360 (Bond Basis) Bus Day ConvModified Following Swap NPV 344
Yield Curve Inputs Bus Day Conv Modified Following Pmt CalendarUnitedStates::Settlement
Handle USDFlatFwdYieldCurve Pmt Calendar UnitedStates::Settlement Reset CalendarU itedKingdom::Settlement Vanilla Swap IDUS VanillaSwap1#0008
Ndays 0
Calendar UnitedStates::Settlement Swap Engine IDUS DiscountingSwapEngine#0008
Rate 4.00% Pricing Engine IDTRUE
Day Count 30/360 (Bond Basis)
CompoundingCompounded NPV 344
Frequency Semiannual Npv Details
Yc IDUSDFlatFwdYieldCurve#0012 Fixed Leg 32,695,774 USD 32,695,774 32.70%
Float Leg -32,695,523 USD -32,695,523 -32.70%
Npv Deal 251 0.00%
2. Introduction (cont’d)
2. Introduction (cont’d)
• From the previous slide and accompanying Excel Spreadsheet it is clear that the
swap valuation is carried out on a leg by leg basis as follows:
• = σ=1
−1, ()
• = σ=1
−1, −1, ()
• = −
• In the above and are the IRS notional and fixed rates,
−1, and −1, are the accrual fractions associated with the fixed and
floating legs of the swap respectively, and ()is the discount factor
corresponding to date .
• But where do we get the future Libor rates −1, from since these haven’t
fixed yet?
3. FRAs and Futures
• A forward rate agreement (FRA) is an OTC financial instrument used to lock in a
fixed rate today for a forward starting date beginning 1 , and ending 2
• The fixed rate is known as the FRA rate
• The payout of the FRA payer contract at time 2 is:
2 = 1, 2 − 1, 2
• Or equivalently,
1 =
1, 2 − 1, 2
1 + 1, 2 1, 2
• Since in practice the FRA is settled at 1
• Buying a FRA corresponds to paying fixed whereas selling a FRA corresponds to
receiving fixed
3. FRAs and Futures (cont’d)
• FRAs are quoted in terms of the start date and end date of the floating rate
• So for example, a 3x6 FRA corresponds is a contract to lock in a fixed borrowing
beginning 3 months from today and ending in 6 months from today. The
underlying rate which is being locked in is a 3 month rate.
• Similarly, a 2x8 FRA corresponds is a contract to lock in a fixed borrowing
beginning 2 months from today and ending in 8 months from today. The
underlying rate which is being locked in is a 6 month rate
• Note the similarity of the FRA payout with each term in the valuation of a swap
• We sometimes refer to a plain vanilla IRS as a portfolio of FRAs
• If one buys a FRA one pays the fixed rate
• If one sells a FRA one receives the fixed rate
3. FRAs and Futures (cont’d)
• FRAs are convex instruments and therefore have gamma risk, unlike a futures
contract whose payout is linear in the forward rate.
• Suppose one has entered into a FRA receiver contract (see formula on slide 17
for the FRA payer and just interchange the forward rate and the strike)
• If rates go up then one will lose money on the FRA, but since rates are higher the
(negative) difference between the projected Libor rate and the fixed rate is being
discounted at a higher rate (i.e. being multiplied by a smaller discount factor),
and so the P&L loss is not as much
• Similarly, if rates go down then one will make money on the FRA, but since rates
are lower the (positive) difference between the projected Libor rate and the
fixed rate is being discounted at a lower rate (i.e. being multiplied by a larger
discount factor), and so the P&L gain is amplified
• Therefore there is always a benefit for the party who is receiving fixed on a FRA
and this party is said to long gamma (or long convexity)
3. FRAs and Futures (cont’d)
General Inputs FRA Details Underlying FRA Details
Ccy USD FRA # FRA Start FRA End Pay/Rec FRA Rate Notional Fwd Sensitivity PV01 NPV FRA
Quote Date3-Apr-20 1 1M 4M Pay 4.000% 100,000,000 4.000% 0.255 0
Set Evaluation DateTRUE 2 2M 5M Rec 4.000% 200,000,000 4.000% 0.249 0
Days To Spot 2 3 3M 6M Pay 4.000% 50,000,000 4.000% 0.249 0
Spot Date 7-Apr-20 4 4M 7M Rec 4.000% 100,000,000 4.000% 0.251 0
Float Freq S 5 1M 7M Pay 4.000% 100,000,000 4.000% 0.502 0
Float BasisActual/360 6 2M 8M Rec 4.000% 100,000,000 4.000% 0.502 0
Float Ref RateLIBOR3M 7 3M 6M Pay 1.157% 100,000,000 4.000% 0.249 711,678
Float Bus Day ConvModified F llowing 8 4M 7M Rec 1.303% 100,000,000 4.000% 0.251 -687,295
Pmt CalendarUnitedSt tes::Settlement 9 4M 7M Pay 1.096% 100,000,000 4.000% 0.251 740,046
Reset CalendarUnitedKing om::Settlement 10 4M 7M Rec 0.935% 100,000,000 4.000% 0.251 -781,075
11 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
Yield Curve Inputs 12 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
Handle USDFRAFlatFwdYieldCurve13 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
Ndays 0 14 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
Calendar UnitedStates::Settlement15 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
Rate 4.00% 16 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
Day CountActual/360 17 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
CompoundingCompounded 18 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
FrequencyQuarterly 19 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
Yc ID USDFRAFlatFwdYieldCurve#000320 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
21 4M 7M Pay 3.000% 100,000,000 4.000% 0.251 254,837
22 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
23 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
24 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
25 4M 7M Rec 3.000% 100,000,000 4.000% 0.251 -254,837
3. FRAs and Futures (cont’d)
• Interest rate futures contracts are standardised exchange traded instruments
with daily margining (settlement)
• These contracts are traded on the CME, CBOT, ICE
• These contracts are standardised in the following ways:
• The contracts expire on the third Wednesday of March, June, September and
December
• The contract size is fixed (e.g. USD 1m for Eurodollars and GBP 500k for Short
Sterling)
• The tick size is fixed (e.g. USD 25 for Eurodollars and GBP 12.50 for Short
Sterling)
• The price of the contract is quoted on discount basis, namely 100-Futures Price
(so if the futures price is 99 then the implied 3 month rate is 1%)
3. FRAs and Futures (cont’d)
• Futures contracts, by virtue of their daily mark to market and margining, do not
display the same convexity and are purely linear in the projected Libor rate
• Therefore
• A useful exercise is to consider hedging a $10m receiver FRA with Eurodollar
futures
• So need to work out how many futures to buy/sell and then plot the P&L as a
function of the yield curve change
• Choose the same FRA dates as the relevant Futures contract
4. Floating Rates from Discount Function
• Note that for both the FRA and the Swap, we need to be able to value future
Libor rates and for both instruments the floating rate payments are of the form:
, +1 ( , +1)
• Where , +1 is the Libor rate observed at t (in practice the observation of
the rate is currency dependent)
• Using intuitive replication arguments, one can show that the unknown floating
rate , +1 must be equal to:
, +1 =
1
( , +1)
()
(+1)
− 1
• The proof is as follows:
4. Floating Rates from Discount Function (cont’d)
• Consider the following strategy where ; denotes a zero coupon bond of
maturity as of time
• At some time < 1
i. purchase $1 face value of a 2 maturity zero coupon bond of value ; 2
ii. sell short $
(;2)
(;1)
face value of a 1 zero coupon bond of value ; 1
• The net cash flow at is zero
• When the 1 maturity zero coupon bond matures (at 1), requiring repayment
of $
(;2)
(;1)
, borrow this same amount at the rate ; 1, 2 over the period
[1, 2].
• The net cash flow at 1 is also zero
4. Floating Rates from Discount Function (cont’d)
• Finally, at 2 the 2-maturity zero coupon bond I bought at matures and I
receive $1
• The loan of $
(;2)
(;1)
that I took out at 1 must be repaid, and so I must pay
$
(;2)
(;1)
1 + L ; 1, 2 1, 2 at 2
• Since the net cash flows at and 1 are zero, in order to eliminate arbitrage the
net cash flow at 2 must also be zero, leading to:
• 1 −
;2
;1
1 + L ; 1, 2 1, 2 = 0
• which leads to the expression for L ; 1, 2 shown on slide 20
• Note that this important result means that knowledge of the discount function
(i.e. the table of dates and corresponding discount factors) allows me to
determine all future Libor rates
4. Floating Rates from Discount Function (cont’d)
• By substituting the formula for ( , +1) shown on slide 20 into the expression
for the PV of the floating leg of the swap shown on slide 15, one obtains a simple
expression for the PV of the swap floating leg, namely
= () − ()
• Note that this implicitly assumes that the discount factors used for projecting
the Libor rates using the expression on slide 20, are the same as the discount
factors used for discounting these expected cash flows
• As we shall see in a later lecture, this assumption no longer holds when there is
a basis between Libor rates of different tenors
5. Recap – Swap Valuation
• The current valuation of a vanilla interest rate swap is obtained by adding the
present value of the fixed leg to the present value of the floating leg, taking into
account whether one is paying or receiving the fixed/floating legs
• The leg of a vanilla interest rate swap is just an annuity and can be valued by
multiplying the fixed cash flows by the discount factors associated with the
payment dates of the cash flows
• The floating leg of a vanilla interest rate swap can be valued by projecting the
future Libor rates and discounting the associated cash flows using the discount
corresponding to the Libor cash flow payment date
• The swap rate is defined to be the value of the fixed rate for which the present
value of the swap is zero
6. Constructing the Discount Function
• As stated on the previous slide, the valuation of an interest rate swap requires
knowledge of the discount factors corresponding to the coupon and Libor
payment dates
• When we talk about yield curves in the context of swaps, we mean a set of
discount factors which allow us to reprice all relevant instruments, namely
Deposits, Futures, FRAs, Swaps, etc.
• In lecture 2 we briefly looked at present value in the context of the yield to
maturity of a bond
• Recall that we calculated discount factors (zero coupon bond prices) using the
yield of the bond
• Swap yield curves are fundamental to interest rate derivatives and
6. Constructing the Discount Function (cont’d)
• Our objective is to derive the discount curve via bootstrapping process
• We will use the market values of a collection of interest rate instruments,
including deposits, FRAs, Futures and Swaps
• The discount curve must should have 2 basic properties
(i) it must reprice all instruments to ‘par’
(ii) it must be ‘smooth’
• The resulting curve will be a list of dates T, and corresponding discount factors Z
6. Constructing the Discount Function (cont’d)
• Let’s consider a simple example, namely a 1y dollar swap with a fixed annual
coupon versus USD Libor 3m and notional N, starting from spot
• Denote the 1y par swap rate by 1
• = 1 , 1 (1)
• = − (1)
• Since the swap is at par, the PVs of the fixed and floating leg must be equal,
• This allows us to solve for (1),
1 =
1 + 1 , 1
• Note that the discount factor can be obtained from the 2 day deposit rate
for currencies with a spot date of T+2, otherwise for currencies with spot date =
today (e.g. GBP) = 1.
6. Constructing the Discount Function (cont’d)
• Now consider a 2y par swap with annual fixed rate 2
• = 2 , 1 1 + 1, 2 2
• = − (2)
• Since the 2y swap is at par, the PV of the fixed and floating legs must be equal
• This allows me to solve for (2), noting that I have already solved for
(1) using the information on the previous slide derived from the 1y swap
2 =
− 2 , 1 (1)
1 + 2(1, 2)
6. Constructing the Discount Function (cont’d)
• Next we consider a 3y par swap with annual fixed rate 3
• = 3 , 1 1 + 1, 2 2 + 2, 3 3
• = − (3)
• Equating the PV of the fixed and floating legs as before allows me to solve for
(3), noting that I have already solved for Z(1) and Z(2) using the information
on the previous two slides derived from the 1y and 2y swaps
3 =
− 3 , 1 1 + 1, 2 2
1 + 3(2, 3)
• It is obvious that this generalises to
• Z(t) = [1 – S(t) * Sum(i=1 to t-1;Alpha(i-1,i) * Z(i)] / (1 + S(t) * Alpha(t-1,t))
• The process of using discount factors calculated for previous dates to obtain
discount factors at a later date is known as bootstrapping.
6. Constructing the Discount Function (cont’d)
• It is obvious that this generalises to:
=
− σ=1
−1 −1, ()
1 + (−1, )
• The process of using discount factors calculated for previous dates to obtain
discount factors at a later date is known as bootstrapping.
• The result is a list of dates and corresponding discount factors
6. Constructing the Discount Function (cont’d)
• Let’s look at the example of GBP
• Assume evaluation date of 3 Apr 2020
and the same spot date
• Assume fixed leg day count of Act/365
• Assume business day convention of
modified following
• Note that GBP swaps are quoted on a
semi-annual Act/365 basis so fixed
coupons are paid twice a year except
1y swaps which are quoted on an
annual Act/365 basis
Depo Rate Swap Rate
O/N 0.8584 1Y 0.4640
1W 0.8558 2Y 0.4770
1M 0.8407 3Y 0.4640
2M 0.8215 4Y 0.4770
3M 0.8031 5Y 0.4940
6M 0.7135 6Y 0.5120
7Y 0.5280
8Y 0.5410
9Y 0.5540
10Y 0.5670
12Y 0.5860
15Y 0.6050
20Y 0.6090
25Y 0.5940
30Y 0.5710
6. Constructing the Discount Function (cont’d)
• Calculating the discount factors for the deposit rates can be done directly, without
bootstrapping. Specifically,
• =
1
1+ (,)
• 1 =
1
1+ 1 (,1)
• Etc
• In the above, denotes the start date of the relevant deposit
• Note that to calculate the discount factor for the 2Y point we need the discount factor
for the 18M point, since 2Y GBP swaps pay semi-annual coupons.
• Actually, we need discount factors corresponding to all coupon dates but we only have
swap rates quoted for annual maturities
• What do we do?
• We create synthetic swaps with maturities corresponding to 18M, 30M, etc. using an
interpolation scheme
6. Constructing the Discount Function (cont’d)
• Standard interpolation schemes include:
(i) Linear Interpolation of the discount factors
(ii) Linear Interpolation of the log of the discount factors
(iii) Constant Instantaneous forward rates
(iv) Linear Instantaneous forward rates
(v) Splines (e.g. cubic)
• Returning to the GBP swap example, one way of creating the synthetic swaps is
just to linearly interpolate between the two adjacent par swap rates.
6. Constructing the Discount Function (cont’d)
• The result of the bootstrapping process that we have just covered leads to a table
consisting of a column of dates, which correspond to the dates implicit in the
instruments used to bootstrap the curve, and a column of associated discount
factors.
• Remember, the reason for creating the discount function is NOT for repricing the
instruments used to build it, but for repricing our existing portfolio of interest
rate products or for pricing other (non-standard) swaps.
• We will therefore require discount factors for dates which do not correspond to
dates in our table, and therefore interpolation is inevitable
7. Interest Rate Swap Risk Measures
• The 10y USD interest rate swap pricing example that we have studied used a
particularly simple yield curve for valuation, namely a flat yield curve
• The value of this IRS is therefore dependent on the single (flat) rate that was
chosen.
• In this case the only risk measure that is relevant is the PV01 risk measure,
namely the sensitivity of the valuation to a 1bp increase in this (flat) rate.
• In practice this means that should change the input rate by 1bp and revalue the
swap. The difference is PVs is the PV01.
• Refer back to the practical QuantLib Excel example.
7. Trader Jargon
• As risk managers it is important to be able to communicate using the same
language as traders in order to create credibility
• Therefore you need to be able to speak the same ‘language’
• Fundamental Principle – Everything is about Bond Prices
• Bonds and Swaps – As we already know, when interest rates go down bond
prices go up and vice versa so a rally in the bond market can be associated with
lower interest rates, or equivalently a sell-off in the bond market can be
associated with higher interest rates
• Therefore a trader will refer to rates rallying down which actually means that
bonds were higher which in turn means that rates are lower. Note that this is
unintuitive since normally we associate a rally with higher levels but if you
remember that “it is all about bond prices” then you won’t go wrong.
• Similarly a reference to rates selling off means that bonds were lower, which in
turn means that rates are higher
7. Trader Jargon (cont’d)
Long/Short Jargon:
• In the financial markets the terms long and short are used everywhere
• If I am long an asset then if that asset price goes up I make money and if the
asset price goes down I lose money.
• If I am short an asset then if that asset price goes up I lose money and if the
asset price goes down then I make money
• Remember these by heart
7. Trader Jargon (cont’d)
Long/Short (continued)
• But how do I translate long/short into an interest rate context?
• Again, the connection is with the bond market and so if I am receiving fixed on a
swap (and implicitly paying floating), then if interest rates go up I lose money,
hence I am long (bonds)
• Similarly, if I am paying fixed on a swap (and implicitly receiving floating), then if
interest rates go up then I make money, hence I am short (bonds)
• So in summary,
• Receiving Fixed on an IRS means that I am long and paying fixed on an IRS
means that I short
• Later we will study interest rate swaptions, and same logic holds, namely
• If I am long a receiver swaption I am long the (bond) market and if I am long a
payer swaption then I am short the (bond) market
7. Trader Jargon (cont’d)
Buy/Sell Jargon
• The concept of buying and selling is familiar to everyone
• Items for purchase or sale usually have a bid or offer associated with them
• So take the example of a bond trader (market maker) who is quoting a two-way
market of 99/101 in a specific bond. This means that he is prepared to buy the
bond at 99 (trader’s bid) and sell it at 101 (trader’s offer).
• If you want to buy the bond you will need to pay 101 and ‘lift’ the trader’s offer
• The trader will then say that he has been ‘lifted’ at 101
• If you have a bond to sell then you will receive 99 and ‘hit’ the trader’s bid
• The trader will then say that he/she has been ‘hit’ at 99
• You need to memorise the following: Bids are hit and offers are lifted
7. Trader Jargon (cont’d)
Mine/Yours Jargon
• This is related to the concept of buying and selling
• If you call up a market maker who is quoting 99/101 as the two-way price for a
particular bond and say “mine in 10 million” then you will have bought $10m
nominal of the bond and paid 101 (having lifted the trader’s offer)
• Similarly if you had said “yours in 10 million” then you will have sold $10m
nominal of the bond and received 99 (having hit the trader’s bid)
• The same jargon is used in the swaps market but can be a bit confusing
• If you call up a swap market maker who is quoting 1.05/1.10 for 10y USD swaps
and say “mine in $10m” then you will have agreed to pay a fixed rate of 1.10%
for 10y and receive USD Libor 3m on a notional of $10m
• Similarly, if you had said “yours in 10 million” then you will have agreed to
receive a fixed rate 1.05% for 10y and pay USD Libor 3m on a notional of $10m
8. Homework 3.1
• Using QuantLib Python/Excel and the techniques described in the previous slides
calculate the GBP discount function for the following market data, assuming an
evaluation date of 3 April 2020, and GBP swap conventions of semi-annual
A/365 for both fixed and floating legs, including deposits. The spot lag is 0 for
GBP swaps so the start date of all instruments is 3 April 2020.
• Note that the intention of this exercise is to go through the bootstrapping
process by hand, rather than relying on QuantLib to do this for you. We will use
QuantLib to bootstrap swap yield curves as part of the next lecture.
• The output should be a table of dates and associated discount factors.
• Prove that your discount function is self consistent by using the calculated
discount factors to re-imply the deposit and swap input rates.
• Plot the projected 6-month forward rates implied from your discount factors for
annual maturities from 1 year to 30 years.
8. Homework 3.1
• Use your discount function to calculate the equilibrium swap rate for a 10y GBP
swap, starting in 5 years (forward starting swap).
• Assume a notional of GBP 100m of the same forward starting (receiver) IRS
above and produce a risk report showing the sensitivity of each yield curve input
to a 1bp change. Your risk report should be presented in a table form as follows:
• Submission date to be confirmed.
Yield Curve Tenor Sensitivity (GBP)
O/N
1W
1M
2M
…
30Y
9. Essay Topic
• Topic – Please discuss the risk management implications of IBOR
discontinuation.
• Approximately 5-7 pages of A4 using 12pt font with 1.5 spacing.
• Not more than 2500 words.
• Please utilise spell and grammar checks (see www.grammarly.com) before
submitting.
• Submission date to be confirmed.
学霸联盟
