Lecture 3 – Interest rate derivatives IMQF, Spring Semester 2011/2012

Module: Derivatives and Fixed Income Securities

Course: Derivatives, part I

Lecturer: Miloš Božović

Lecture outline

  Interest rate forwards and futures

  Swaps

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Interest rate forwards and futures

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Types of interest rates

  Treasury rates

  Interbank rates

  Repo rates

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Treasury rates

  Rates on instruments issued by a government in its own currency

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Interbank rates

  Daily rates of interest at which a panel of banks is prepared to lend or borrow money in an interbank market   Lending -> offer rate

  Borrowing -> bid rate

  Examples:   LIBOR (London Interbank Offered Rate)

  EURIBOR (Euro Interbank Offered Rate)

  BELIBOR (Belgrade Interbank Offered Rate)

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Repo rates

  Repurchase agreement (or “repo”) is an agreement where a financial institution that owns securities agrees to sell them today for X and buy them bank in the future for a slightly higher price, Y

  The financial institution obtains a loan.

  The rate of interest is calculated from the difference between X and Y and is known as the repo rate

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Zero rates

  A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T

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Forward rates

  The forward rate is the rate of interest agreed today for borrowing that will occur in the future.

  It is implied by today’s term structure of spot interest rates.   No-arbitrage argument

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Example

  One-year spot rate = 4%

  18-month spot rate = 4.5%

  Find the forward rate between 12 and 18 months.

  Solution:

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%51.5)1)(04.1()045.1(

)1)(1()1(

18,12

2/118,12

2/3

2/118,121,0

2/318,0

=⇒+=

++=+

mm

mm

mmm

FF

FRR

Forward rates: general case

  Annual compounding:

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1+ R(0, t1)[ ]t11+ F(t

1, t2)[ ]t2!t1

= 1+ R(0, t2)[ ]t2

  Continuous compounding:

eR(0,t1 )t1e

F (t1,t2 )(t2!t1 ) = eR(0,t2 )t2

" F(t1, t2) =

R(0, t2)t2! R(0, t

1)t1

t2! t

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Forward rate agreement

  A forward rate agreement (FRA) is an OTC agreement that a certain rate will apply to a certain principal during a certain future time period

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Forward rate agreement

  Simplest OTC interest rate contracts.

  Two parties exchange cash flow:   Only once, at a predetermined date.   Based on two different rates

  Usually, one is fixed and predetermined.   The other is variable and determined during the life of the

contract.

  The difference between two rates is multiplied by the notional principal.

  Risk is symmetric, therefore no premium.

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Example

  Consider a long position in a 3-month forward on LIBOR 3x6:

  We pay F(3m,6m) = 4.85% in 6 months.

  We receive 3m spot LIBOR determined after 3 months, R(3m,6m).

  N = $ 2,000,000.

  Day count convention is 30/360.

  Cash flow depends on R(3m,6m):

  If R(3m,6m) = 4.64%, our cash flow is (4.64% – 4.85%) × $ 2,000,000 × (90/360) = – $1,050.

  If R(3m,6m) = 4.90%, our cash flow is: (4.90% – 4.85%) × $ 2,000,000 × (90/360) = + $ 250.

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Day count convention

  Defines:   The period of time to which the interest rate applies

  The period of time used to calculate accrued interest (relevant when the instrument is bought of sold)

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Day count conventions in the U.S.

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Treasury Bonds: Actual/Actual

Corporate Bonds: 30/360

Money Market Instruments: Actual/360

Treasury Bill prices in the U.S.

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price quoted is $100 per price cash is

100360

PY

Yn

P )( −=

US Treasury Bond price quotes

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Treasury Bond futures

Cash price received by party with short position =

Most recent settlement price × Conversion factor + Accrued interest

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Example

  Most recent settlement price = 90.00

  Conversion factor of bond delivered = 1.3800

  Accrued interest on bond = 3.00

  Price received for bond is 1.3800×90.00+3.00 = 127.20

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Conversion factor

  The conversion factor is the present value of cash flows generated by the bond.

  Inputs:   Yield curve is flat at 6%

  Semiannual compounding

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CBOT T-Bonds and T-Notes

  Factors that affect the futures price:   Delivery can be made any time during the delivery month

  Any of a range of eligible bonds can be delivered

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Eurodollar futures

  A Eurodollar is a dollar deposited in a bank outside the United States

  Eurodollar futures are futures on the 3-month Eurodollar deposit rate (same as 3-month LIBOR rate)

  One contract is on the rate earned on $1 million

  A change of one basis point or 0.01 percentage points in a Eurodollar futures quote corresponds to a contract price change of $25

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Eurodollar futures

  A Eurodollar futures contract is settled in cash

  When it expires (on the third Wednesday of the delivery month) the final settlement price is 100 minus the actual three month Eurodollar deposit rate

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Eurodollar futures

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Date Quote

Nov 1 97.12

Nov 2 97.23

Nov 3 96.98

……. ……

Dec 21 97.42

Example

  Suppose you take a long position in a contract on November 1

  The contract expires on December 21

  The prices are as shown

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Example

  If on Nov. 1 you know that you will have $1 million to invest on for three months on Dec 21, the contract locks in a rate of

100 – 97.12 = 2.88%

  At expiry, the rate is

100 – 97.42 = 2.58%

  Total gain on the futures contract is 30×$25 =$750

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TED spread

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Swaps

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Nature of swaps

  A swap is an agreement to exchange cash flows at specified future times according to certain specified rules

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Example: A “plain vanilla” interest rate swap

  An agreement by Microsoft to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million

  Next slide illustrates cash flows that could occur (day count conventions are not considered)

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One possible outcome for cash flows to Microsoft

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Date LIBOR Floating Cash Flow

Fixed Cash Flow

Net Cash Flow

Mar 5, 2012 4.20%

Sep 5, 2012 4.80% +2.10 －2.50 －0.40

Mar 5, 2013 5.30% +2.40 －2.50 －0.10

Sep 5, 2013 5.50% +2.65 －2.50 + 0.15

Mar 5, 2014 5.60% +2.75 －2.50 +0.25

Sep 5, 2014 5.90% +2.80 －2.50 +0.30

Mar 5, 2015 +2.95 －2.50 +0.45

Typical uses of an interest rate swap

  Converting a liability from   fixed rate to floating rate

  floating rate to fixed rate

  Converting an investment from   fixed rate to floating rate

  floating rate to fixed rate

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Intel and Microsoft transform a liability

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Intel MS

LIBOR

5%

LIBOR+0.1%

5.2%

Financial institution is involved

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F.I.

LIBOR LIBOR LIBOR+0.1%

4.985% 5.015%

5.2% Intel MS

Financial Institution has two offsetting swaps

Intel and Microsoft transform an asset

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Intel MS

LIBOR

5%

LIBOR-0.2%

4.7%

Financial institution is involved

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Intel F.I. MS

LIBOR LIBOR

4.7%

5.015% 4.985%

LIBOR-0.2%

Quotes by a swap market maker

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Maturity Bid (%) Offer (%) Swap Rate (%)

2 years 6.03 6.06 6.045

3 years 6.21 6.24 6.225

4 years 6.35 6.39 6.370

5 years 6.47 6.51 6.490

7 years 6.65 6.68 6.665

10 years 6.83 6.87 6.850

Day count convention

  A day count convention is specified for for fixed and floating payment

  For example, LIBOR is likely to be actual/360 in the US because LIBOR is a money market rate

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The Comparative Advantage Argument

  AAACorp wants to borrow floating

  BBBCorp wants to borrow fixed

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Fixed Floating

AAACorp 4.0% 6 month LIBOR － 0.1%

BBBCorp 5.2% 6 month LIBOR + 0.6%

The Comparative Advantage Argument

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AAACorp F.I. BBBCorp

4%

LIBOR LIBOR

LIBOR+0.6%

4.33% 4.37%

Valuation of an interest rate swap

  Initially interest rate swaps are worth zero

  At later times they can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond

  Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs)

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Valuation in terms of bonds

  The fixed rate bond is valued in the usual way

  The floating rate bond is valued by noting that it is worth par immediately after the next payment date

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Valuation of floating-rate bond

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0 t*

Valuation Date

First Pmt Date

Floating Pmt =k*

Second Pmt Date Maturity

Date

Value = L Value = L+k*

Value = PV of L+k* at t*

Example

  Pay six-month LIBOR, receive 8% (s.a. compounding) on a principal of $100 million

  Remaining life 1.25 years

  LIBOR rates for 3-months, 9-months and 15-months are 10%, 10.5%, and 11% (cont comp)

  6-month LIBOR on last payment date was 10.2% (s.a. compounding)

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Valuation using bonds

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Time Bfix cash flow

Bfl cash flow

Disc factor

PV Bfix

PV Bfl

0.25 4.0 105.100 0.9753 3.901 102.505

0.75 4.0 0.9243 3.697

1.25 104.0 0.8715 90.640

Total 98.238 102.505

Swap value = 98.238 － 102.505 = －4.267

Valuation in terms of FRAs

  Each exchange of payments in an interest rate swap is an FRA

  The FRAs can be valued on the assumption that today’s forward rates are realized

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Valuation using FRAs

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Time Fixed cash flow

Floating cash flow

Net Cash Flow

Disc factor

PV Bfl

0.25 4.0 -5.100 -1.100 0.9753 -1.073

0.75 4.0 -5.522 -1.522 0.9243 -1.407

1.25 4.0 -6.051 -2.051 0.8715 -1.787

Total -4.267

An example of a currency swap

  An agreement to pay 5% on a sterling principal of £10,000,000 & receive 6% on a US$ principal of $18,000,000 every year for 5 years

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Exchange of principal

  In an interest rate swap the principal is not exchanged

  In a currency swap the principal is usually exchanged at the beginning and the end of the swap’s life

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The cash flows

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Date Dollar Cash Flows

(millions)

Sterling cash flow

(millions)

Feb 1, 2011 –18.0 +10.0

Feb 1, 2012 +1.08 －0.50

Feb 1, 2012 +1.08 －0.50

Feb 1, 2014 +1.08 －0.50

Feb 1, 2015 +1.08 －0.50

Feb 1, 2016 +19.08 －10.50

Typical uses of a currency swap

  Convert a liability in one currency to a liability in another currency

  Convert an investment in one currency to an investment in another currency

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Valuation of currency swaps

  Like interest rate swaps, currency swaps can be valued either as the difference between two bonds or as a portfolio of forward contracts

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Example

  All JPY LIBOR rates are 4%

  All USD LIBOR rates are 9%

  5% is received in yen; 8% is paid in dollars. Payments are made annually

  Principals are $10 million and 1,200 million yen

  Swap will last for 3 more years

  Current exchange rate is 110 yen per dollar

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Valuation in terms of bonds

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Time Cash Flows ($) PV ($) Cash flows (yen) PV (yen)

1 0.8 0.7311 60 57.65

2 0.8 0.6682 60 55.39

3 0.8 0.6107 60 53.22

3 10.0 7.6338 1,200 1,064.30

Total 9.6439 1,230.55

Value of Swap = 1230.55/110 － 9.6439 = 1.5430

Valuation in terms of forwards

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Time $ cash flow

Yen cash flow

Forward Exch rate

Yen cash flow in $

Net Cash Flow

Present value

1 -0.8 60 0.009557 0.5734 -0.2266 -0.2071

2 -0.8 60 0.010047 0.6028 -0.1972 -0.1647

3 -0.8 60 0.010562 0.6337 -0.1663 -0.1269

3 -10.0 1200 0.010562 12.6746 +2.6746 2.0417

Total 1.5430

Swaps and forwards

  A swap can be regarded as a convenient way of packaging forward contracts

  Although the swap contract is usually worth close to zero at the outset, each of the underlying forward contracts are not worth zero

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Credit risk

  A swap is worth zero to a company initially

  At a future time its value is liable to be either positive or negative

  The company has credit risk exposure only when its value is positive

  Some swaps are more likely to lead to credit risk exposure than others

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Other types of swaps

  Floating-for-floating interest rate swaps   Amortizing swaps   Step up swaps   Forward swaps   Constant maturity swaps   Compounding swaps   LIBOR-in-arrears swaps   Accrual swaps   Diff swaps   Cross currency interest rate swaps   Equity swaps   Extendable swaps   Puttable swaps   Commodity swaps   Volatility swaps   …

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