Session 1: Rate fundamentals - BCCL: Banking Control ... · Session 1: Rate fundamentals ... Compounding and par rates Zero rates and discount factors Libor ... Bootstrapping zeros

Session 1: Rate fundamentals

Patrice Robin, Beirut, October 2016

www.consultancymatters.com © 2008 Consultancy Matters LLC. All rights reserved.www.consultancymatters.com © 2008 Consultancy Matters LLC. All rights reserved.www.consultancymatters.com © 2008 Consultancy Matters LLC. All rights reserved.

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?





Stated Rate And Compounding

N 1000

r 4%

FV EAR

1 1040,00 4,000%

2

4

12

52

1000





Stated Rate And Compounding

Exercise 2: what is the semi-annually compounded

equivalent of investing at 3% on a quarterly basis for

4Y?








Libor






Case Study





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





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?








Libor






Case Study





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?”





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





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








Libor






Case Study





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





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





Forward rates

Actual FRA price was 0.47%... What’s wrong??





-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





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








Libor






Case Study






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)





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





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








Libor






Case Study





Pricing a bond - YTM

1 2 3 4 6

t = 0

5 5 5 5

105

(Dirty) market price = 104.65

5

5





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)





YTM – reinvestment risk








Libor






Case Study





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%





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





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








Libor






Case Study





Computing Forwards From Zeros

Exercise 7: Using the results from the previous 2 slides,

compute the following forwards:

• 0x12

• 12x24

• 24x36

• 36x48

• 48x60





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)





Relationship Between Par, Zero &

Forward Rates

Forward

Zero

Par

Par

Zero

Forward








Libor






Case Study





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




www.consultancymatters.com © 2008 Consultancy Matters LLC. All rights reserved.www.consultancymatters.com © 2008 Consultancy Matters LLC. All rights reserved.www.consultancymatters.com © 2008 Consultancy Matters LLC. All rights reserved.37




Documents

Session 1: Rate fundamentals - BCCL: Banking Control ... · Session 1: Rate fundamentals ... Compounding and par rates Zero rates and discount factors Libor ... Bootstrapping zeros