Session 1: Rate fundamentals
Patrice Robin, Beirut, October 2016
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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Stated rate and compounding
The future value (FV), in t years’ time, of an investment
(N) will depend on both the stated rate of interest (r)
and the compounding frequency (m)
FV = N * (1 + r/m)m.t
Exercise 1: you invest $1,000 at a stated rate of 4% for
1Y. What is the value of your investment given
• Annual compounding?
• Semi-annual compounding?
• Quarterly compounding?
• Monthly compounding?
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Stated Rate And Compounding
N 1000
r 4%
FV EAR
1 1040,00 4,000%
2
4
12
52
1000
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Stated Rate And Compounding
Exercise 2: what is the semi-annually compounded
equivalent of investing at 3% on a quarterly basis for
4Y?
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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Zero Rate
PV (Present Value) denotes the value of an investment
today
FV (Future Value) denotes the value of an investment at
maturity
We have seen that: FV = PV * (1+r/m)m*t
Which is equivalent to: FV = PV * (1+EAR)t
EAR is known as a zero rate for maturity t
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Discount Factor
A discount factor for date t represents the value today of a unit flow
(1) at time t
The previous equation for FV can be rewritten as:
PV = FV / (1+EAR)t
Or
PV = FV * DFt
Where DFt = 1/ (1+EAR)t
i.e. the value today of 1 at date T
Exercise 3: the 3-year zero rate is 4.2%, what is the 3-year discount
factor? And what is the PV of $1m to be received in 3Y?
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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LIBOR
LIBOR London Interbank Offered Rate
Rate that the most creditworthy international banks
dealing in offshore currencies charge each other on
an unsecured basis for terms ranging from 1d to 12m
The banks in the panel answer (daily) the following
question: “At what rate could you borrow funds, were
you to do so by asking for and then accepting
interbank offers in a reasonable market size just prior
to 11 am London time?”
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LIBOR
LIBOR fixing = trimmed average of the contributions
E.g. for a currency panel consisting of 17 banks, the top
4 and bottom 4 contributions are discarded and the
fixing is equal to the average of the remaining 9
As such the Libor rate represents the average bank
credit for a given term.
Day count can be either A/360 (e.g. USD, EUR) or
A/365 (e.g. GBP, ZAR)
11
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ICE LIBOR USD Panel
Lloyds TSB Bank plc; Bank of Tokyo-Mitsubishi UFJ Ltd
Barclays Bank plc; Citibank N.A. (London Branch)
Cooperatieve Rabobank U.A.; Credit Suisse AG (London Branch)
Royal Bank of Canada; HSBC Bank plc
Bank of America N.A. (London Branch)
Crédit Agricole Corporate & Investment Bank
Deutsche Bank AG (London Branch)
JPMorgan Chase Bank, N.A. London Branch
Société Générale (London Branch)
Sumitomo Mitsui Banking Corporation Europe Limited
The Norinchukin Bank
The Royal Bank of Scotland plc
UBS AG
12
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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Forward rates
A measure of where an interest rate index (e.g. 6 month
LIBOR) will be on some future date (e.g. in 6 months)
Priced using an arbitrage free construction
• Borrow for 6 months & roll borrowing for a further 6 months
will be equivalent to borrowing for 12 months
Forward interest rates give an expectation as to where
interest rates may move
2 ways to trade Forwards (Libors) in the market:
• STIR (Short Term Interest Futures)
• FRAs
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Forward rates
Exercise 4: On June 19th 2012, 3m USD Libor fixed at
0.46785% and 6m Libor fixed at 0.73740% Estimate
the price for the 3x6 FRA on that day using no-
arbitrage arguments
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Forward rates
Actual FRA price was 0.47%... What’s wrong??
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-10
0
10
20
30
40
50
18/11/2005 18/11/2006 18/11/2007 18/11/2008 18/11/2009 18/11/2010
1Y 3s 6s tenor basis swap
1Y 3s 6s tenor basis swap
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USD 5Y 3s/6s Basis swaps – last 6
years
18
0
5
10
15
20
25
30
27/0
8/2
01
0
27/1
0/2
01
0
27/1
2/2
01
0
27/0
2/2
01
1
27/0
4/2
01
1
27/0
6/2
01
1
27/0
8/2
01
1
27/1
0/2
01
1
27/1
2/2
01
1
27/0
2/2
01
2
27/0
4/2
01
2
27/0
6/2
01
2
27/0
8/2
01
2
27/1
0/2
01
2
27/1
2/2
01
2
27/0
2/2
01
3
27/0
4/2
01
3
27/0
6/2
01
3
27/0
8/2
01
3
27/1
0/2
01
3
27/1
2/2
01
3
27/0
2/2
01
4
27/0
4/2
01
4
27/0
6/2
01
4
27/0
8/2
01
4
27/1
0/2
01
4
27/1
2/2
01
4
27/0
2/2
01
5
27/0
4/2
01
5
27/0
6/2
01
5
27/0
8/2
01
5
27/1
0/2
01
5
27/1
2/2
01
5
27/0
2/2
01
6
27/0
4/2
01
6
27/0
6/2
01
6
27/0
8/2
01
6
USD 3s/6s Tenor basis swap
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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Day count conventions
An interest flow is equal to: N.r.dc
where dc represents the day-count fraction (r is the rate,
N the principal)
There are 2 main conventions for computing dc:
A/360 (or Act/360): ‘money basis’
• Day count fraction = exact number of days / 360
30/360: ‘bond basis’
• Each month is considered to have 30 days (regardless of
actual length)
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Business day conventions
Business day conventions determine what happens when the
theoretical payment date falls on a non-business day.
2 types of adjustment:
• To compute the interest amount: Adjusted/Non-adjusted
• To work out the actual payment day: Following/Modified
Following/Preceding/ Modified Preceding
Following convention: payment made on next business day.
Modified following: same except that if using the next business
day involves crossing a month then use the previous business
day.
Preceding: Payment made on the previous business day. Modified
Preceding: use previous bd unless that involves crossing a
month in which case use next bd
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Business day/day count
conventions
Example: we have USD interest flow over the period
2012/06/29 Fri – 2012/12/29 Sat
DC fraction using A/360 adjusted, MF (Modified
Following):
• 2012/12/29 Sat not a business day. Following business day,
2012/12/31, will be used as payment day
• Adjusted basis: the period 2012/06/29– 2012/12/31 will be
used for the computation of interest
• A/360: day-count fraction= 185/360
DC fraction using 30/360 unadjusted, MF
• Theoretical period= 6 months
• Unadjusted: 2012/12/31 still used as payment date but 29th
used for calculation of interest
• 30/360: day-count fraction=180/360
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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Pricing a bond - YTM
1 2 3 4 6
t = 0
5 5 5 5
105
(Dirty) market price = 104.65
5
5
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Pricing a bond
Exercise 5:
• A 10-year bond with annual coupons of 6% trade at a price
of 93 YTM? (a coupon has just been paid)
• A 5-year bond with semi-annual coupons of 4.5% trades at a
YTM of 3.95%. Market price of the bond? (a coupon has
just been paid)
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YTM – reinvestment risk
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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Bootstrapping zeros from par rates
Consider the following 4 bonds, all annual 30/360 unadj
and all trading at par:
Par (coupon) Zero DF
1Y 4%
2Y 5%
3Y 5.5%
4Y 6%
Z1: 100 = 104 / (1+Z1) Z1= 4%
Z2: 100= 5/(1+Z1) + 105/(1+Z2)2 Z2= 5.0252%
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Bootstrapping zeros from par rates
Z3: 100 = 5.5/(1+Z1) + 5.5/(1+Z2)2 +105.5/(1+Z3)
3
Z3= 5.5470%
Z4: 100 = 6/(1+Z1) + 6/(1+Z2)2 +6/(1+Z3)
3+106/(1+Z4)4
Z4=
6.0865%
Exercise 6: Compute the Discount factors associated with the zeros
just computed
With the YTM all flows were discounted at the same rate. With a
zero curve the term structure of rates is taken into account and each
flow is discounted at a different rate.
known
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Completing The Curve:
Interpolation
Bootstrapping allows for the identification of the zero
coupon yields at different points in time (in our
example: annual granularity).
Interpolation allows for the identification of the zero
coupon rates between these times. For example:
• Linear interpolation
• Cubic spline interpolation
It is important to note that it is the zero rates that are
interpolated, not the discount factors
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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Computing Forwards From Zeros
Exercise 7: Using the results from the previous 2 slides,
compute the following forwards:
• 0x12
• 12x24
• 24x36
• 36x48
• 48x60
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Computing Forwards From Zeros
Par Zero Forward Df MP
1 4,0% 4,0000% 4,0000% 0,961538 100
2 5,0% 5,0252% 6,0606% 0,906593 100
3 5,5% 5,5470% 6,5983% 0,850477 100
4 6,0% 6,0865% 7,7217% 0,789513 100
Cannot compute the 48x60 (a 5y zero would be needed)
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Relationship Between Par, Zero &
Forward Rates
Forward
Zero
Par
Par
Zero
Forward
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1. Rates fundamentals
Compounding and par rates
Zero rates and discount factors
Libor
Money markets and Forward rates
Day count conventions
Bond pricing and YTM
Deriving the zero curve from par rates
Relationship between par, zero and forward rates
Case Study
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Case Study
Fill in the following table:
Compute the following forwards (in years):
• 1x2
• 2x4
All annual 30/360 unadjusted
T Par MP Zero Df YTM
1 4,00% 101
2 3,00% 3,53%
3 5,50% 4,41%
4 4,00% 97
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