Lecture 12 interest rate derivatives

Managing Market Risk Under The Basel IV Framework
Copyright © 2016 CapitaLogic Limited
Chapter 12
Interest Rate
Derivatives
Managing Market Risk Under The Basel IV Framework
The Presentation Slides
Website : https://sites.google.com/site/quanrisk
E-mail : quanrisk@gmail.com

Copyright © 2016 CapitaLogic Limited 2
Declaration
Copyright © 2016 CapitaLogic Limited.
All rights reserved. No part of this presentation file may be
reproduced, in any form or by any means, without written
permission from CapitaLogic Limited.
Authored by Dr. LAM Yat-fai (林日林日林日林日辉辉辉辉),
Principal, Structured Products Analytics, CapitaLogic Limited,
Adjunct Professor of Finance, City University of Hong Kong,
Doctor of Business Administration (Finance),
CFA, CAIA, FRM, PRM.

Interbank rates
Forward rate agreements
Interest rate swaps
Cross currency swaps
Exotic swaps
Outline

Interbank rate
The interest rate of
Short term borrowing
Among high credit quality banks
Determined through demand and supply of short term funding
A benchmark for borrowing rate
Interest rate = Interbank rate + Profit margin to lender
In discrete compounding format
Frequently adopted as a proxy of short term risk free rate
Risk free rate = Ln(1 + Interbank rate)
In continuous time format

Currency and maturity
Currency
USD
EUR
GBP
JPY
CHF
CNY
HKD
Maturity
Overnight
1 week
1 month
2 months
3 months
6 months
12 months

Historical interbank rates
All interbank rates
http://stooq.com
London Interbank Offer Rates
Average borrowing rates with maturity up to one year among AA rated banks in U.K.
A proxy of short term risk-free rates for major currencies
https://research.stlouisfed.org/fred2/release?rid=253
Shanghai Interbank Offer Rates
Average borrowing rates with maturity up to one year among major banks in China
A proxy of short term risk-free rates for CNY
http://www.shibor.org
Hong Kong Interbank Offer Rates
Average borrowing rates with maturity up to one year among major banks in Hong
Kong
A proxy of short term risk-free rates for HKD
http://www.hkab.org.hk/DisplayInterestSettlementRatesAction.do?Submit=Detail&lan
g=en

Long position vs short position
Pay market
Receive strike
Pay strike
Receive market
Interest rate
Pay foreign
Receive domestic
Pay domestic
Receive foreign
FX rate
ShortLongPosition

Interbank rates
Interest rate swaps
Exotic swaps
Outline

An interest cost problem
A trading firm
Buys goods from the United States
Pays cost in USD in three months
USD cash insufficient in three months
To borrow from a bank
To incur interest cost
How to ensure that the interest cost arising
from the loan is affordable?

Forward rate agreement
An agreement between a borrower and a bank
on a future loan such that
The borrower must borrow from the bank an
agreed principal A at strike interest rate K at time
T1 and return the principal at time T2
The bank must lend to the borrower an agreed
principal A at strike interest rate K at time T1 and
receive the principal at time T2

Functional purposes
To speculate on
the down trend of
interbank rate
To speculate on
the up trend of
interbank rate
Speculation
To lock in the
interest income
of a future loan
To lock in the
interest cost of a
future loan
Hedging
ShortLongPosition

Modelling an FRA
A: Loan principal
K: Strike interest rate
The interest rate of a loan from T1 to T2, agreed today
T1: Time at which the loan is borrowed
T2: Time at which the loan is returned
r1 : The interest rate of a loan from today to T1
r2 : The interest rate of a loan from today to T2
r12 : The market interest rate of a loan from T1 to T2 to be
observed on T1

Payoff function
At time T1
The market interest rate is r12
The strike interest rate is K
Interest benefit at time T2
Ar12(T2 - T1) - AK(T2 - T1) = A(r12 - K)(T2 - T1)
Unit payoff at T2 = (r12 - K)(T2 - T1)
Unit payoff at T1 = (r12 - K)(T2 - T1)exp[-rd(T2 - T1)]

Payoff diagram at T2
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
1.170 1.175 1.180 1.185 1.190
Interest rate at T1 (%)
Payoff(%)
Payoff

Cash flows – physical settlement
- Principal+ Principal
Cash flow at
FRA time T1
K = (r2T2 - r1T1) / (T2 - T1)
- Principal
- Interest
None
Long
Strike interest
rate
+ Principal
+ Interest
Cash flow at
loan time T2
Cash flow at
origination
ShortPosition

Forward interest rate
The strike interest rate at which the total interest amount will be the same
if the loan is originated today
matured at T2
renewed according to K at T1 and matured at T2
In general, forward interest rate moves in the same direction as r2
Standard FRAs are originated at forward interest rate to avoid additional
cash outflow from either the borrower or the bank
The most important characteristic of FRAs
[ ]2 2 1 1 2 1
2 2 1 1
2 1
Ar T = A r T + K(T - T )
r T - r T
F = K =
T - T
Example 12.5

Valuation – long position
A(F - K)(T2 - T1)Net cash flow
KFInterest rate
A(F - K)(T2 - T1)exp(-rdT2)
A[1 + F(T2 - T1)]
Market lending
Value today
- A[1 + K(T2 - T1)]
Cash flow at
loan time T2
FRA borrowingPosition
Example 12.6

Moneyness
< KOut-of-the-money
= KAt-the-money
> KIn-the-money
FMoneyness

Cash flows – cash settlement
A(K - rT2)(T2 - T1)
exp[-rd(T2 - T1)]
A(rT2 - K)(T2 - T1)
exp[-rd(T2 - T1)]
Cash flow
at time T1
None
Long
Cash flow at
origination
ShortPosition

Interest rate risk
( )
( )
2 1 d 2
2 2 1 1
2 1
1 1
2 2
1
2d
2
FX rate
Interbank ra
Value = × A F - K (T - T )exp(-r T )
r T - r T
F =
T - T
r = + Profit margin
r = + Profit margin
r = ln 1 +
te
Interbank rate
Interbank rate

Interest rate risk factors
for international FRA portfolio
Interest rate risk
Value
Principal
Holding period
dispersion
Interbank rates
Standard
deviation
Holding period
Diversification
effect
Concentration
of equities
% change
dependency
FX rate

Interbank rates
Interest rate swaps
Exotic swaps
Outline

Bank’s deposits taking
and lending businesses
Deposits
1 day to
1 year
Bank
1 year to
30 years
Lending

A typical interbank rate curve
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6 8 10 12 14
Maturity (months)
USDinterbankrate(%)

Long-short interest rate strategy
Investment strategy
Borrow short term at lower interest rate
Lend longer term at higher interest rate
Renew regularly the short term borrowing
Risky assumption on interest rates
Short term interest rates remain at lower level

An interest cost problem
A bank
Borrow short term funding from a lender at a
Cost = Fixed interest rate
Lend medium term funding to a borrower with an
Income = Fixed interest rate + Profit margin
The borrowed short term funding will be renewed
several times during the medium term lending period
to the borrower
How to ensure that the interest cost arising from the
fluctuation of the short term interest rate can be
covered by the income from the borrower?

Cash flow diagram
Borrower
Bank
Lender
Variable interest
Fixed interest

Interest rate swap
An agreement between bank A and bank B
such that
Bank A borrows a loan from Bank B and pays
Strike interest rate
Bank B borrows the same loan from Bank A and
pays
Market interest rate = Interbank rate

Functional purposes
To speculate on
the down trend of
interbank rate
To speculate on
the up trend of
interbank rate
Speculation
To lock in the
income of a
future loan
To lock in the
cost of a future
loan
Hedging
ShortLongPosition

Cash flow diagram
Borrower
Bank A
Lender
Bank B
Variable interest
Fixed interest
Fixed interest
Variable interest

Payoff function for each fixing
Cash inflow
Cash outflow
Contract payoff
Unit payoff
( )
Principal × Market interest rate
Interest frequency
Principal × Strike interest rate
Interest frequency
Principal × Market interest rate - Strike interest rate
Interest frequency
Market interest rate - Strike interest rate
Interest frequency

Cash flows
Fixed amount at
strike interest rate
Variable amount at
market interest rate
Cash in flow
on interest
payment dates
Swap rateStrike interest rate
Variable amount at
market interest rate
Fixed amount at
strike interest rate
Cash out flow
on interest
payment dates
None
Long
Cash flow at
origination
ShortPosition

( )
( )
( )
( )
( )
d,1 1
d,1 1
N
d,k k
k=1
d,N
Principal × Market interest rate × exp -r T
PV Variable =
Interest frequency
+ Principal × exp -r T
Principal × Strike interest rate × exp -r T
PV Fixed =
Interest frequency
+ Principal × exp -r
∑
( )
( ) ( )
( ) ( )
NT
Value = PV Cash inflows - PV Cash outflows
= PV Variable - PV Fixed
Example 12.8

Swap rate
The strike interest rate such that
( ) ( )
( )
( ) ( )
( )
d,1 1
d,1 1 d,N N
N
d,k k
k=1
Value = PV Variable - PV Fixed = 0
Swap rate = Strike interest rate
Market interest rate × exp -r T
Interest frequency
+ exp -r T - exp -r T
=
exp -r T
Interest frequency
∑
Example 12.7

Interest rate risk
( )
1
kd,k
InterbankMarket interest rate =
r =
rate
Interbankln 1 rate+

Interest rate risk factors for
interest rate swap portfolio
Interest rate risk
Value
Principal
Holding period
dispersion
Interbank rates
Standard
deviation
Holding period
Diversification
effect
Concentration
of equities
% change
dependency

Interbank rates
Interest rate swaps
Exotic swaps
Outline

A currency mis-match problem
A trading firm
Buys goods from France
Pays regular cost in EUR
Sells goods to United States
Receives regular income in USD
Both regular cost in EUR and income in USD in the
recent future are well estimated
How to ensure that the regular income in USD which
will be used to buy EUR, is sufficient to pay the cost
in EUR, subject to the situation that the FX rate in
the recent future is unknown?

Cash flow diagram
Income
Firm
Cost
Foreign currency
Domestic currency

Cross currency swap
An agreement between an investor and a bank such
that
The investor deposits to the bank an amount in foreign
currency
Receives fixed interest amount in foreign currency
The investor borrows from the bank a loan in domestic
currency
Pays fixed interest amount in domestic currency
Value of the deposit in foreign currency
= Value of the loan in domestic currency

Functional purposes
To speculate on
the down trend of
FX rate
To speculate on
the up trend of
FX rate
Speculation
To hedge future
regular cash
inflows in foreign
currency
To hedge future
regular cash
outflows in
foreign currency
Hedging
ShortLongPosition

Foreign interest
Cash flow diagram
Income
Firm
Cost
Bank
Foreign currency
Domestic currency
Domestic interest

Payoff function for each fixing
Cash inflow
Cash outflow
Contract payoff
Foreign principal × Foreign interest rate
Interest frequency
Domestic principal × Domestic interest rate
Interest frequency
FX rate × Foreign principal × Foreign interest rate
- Domestic principal × Domestic interest rate
Interest frequency

Cash flows
+ Foreign principal
- Domestic principal
- Foreign principal
+ Domestic principal
Cash flow at
origination
- Foreign principal
+ Domestic principal
+ Foreign principal
- Domestic principal
Cash flow
on at maturity
- Foreign interest
+ Domestic interest
+ Foreign interest
- Domestic interest
Cash flow on
interest payment
dates
Long ShortPosition

( )
( )
( )
( )
( )
N
f,k k
k=1
f,N N
N
d,k k
k=1
Principal × Foreign interest rate × exp -r T
PV Foreign =
Interest frequency
+ Principal × exp -r T
Principal × Domestic interest rate × exp -r T
PV Domestic =
Interest frequency
+ Principal
∑
∑
( )
( ) ( )
( ) ( )
d,N N× exp -r T
= FX rate × PV Foreign - PV Domestic
Example 12.9

FX risk factors for
a cross currency swap portfolio
FX risk
Value
Principal
Holding period
dispersion
FX rate
Standard
deviation
Holding period
Diversification
effect
Concentration
of equities
% change
dependency
Interbank rates

Interbank rates
Interest rate swaps
Exotic swaps
Outline

Interest rate swap
Foreign
currency
Receive
Market
interest rate
Pay
Domestic
currency
Strike
interest rate
Interest rate
up

Cross currency swap
Receive
Foreign
currency
Market
interest rate
Pay
Domestic
currency
Strike
interest rate
FX rate
up

Exotic swap 1
Receive
Foreign
currency
Market
interest rate
Pay
Domestic
currency
Strike
interest rate
IR up
FX rate up

Exotic swap 2
Receive
Foreign
currency
Pay
Market
interest rate
Domestic
currency
Strike
interest rate
IR down
FX rates up

Market interest rate expectations
DownUp
Market
FX rate
DownUp
Market
interest rate
Pay
= Short
Receive
= Long
Position

Exotic swap 3
PayReceive
Foreign
currency
Market
interest rate
Domestic
currency
Strike
interest rate

Exotic swap 4
Receive
Foreign
currency
Pay
Market
interest rate
Domestic
currency
Strike
interest rate

Exotic swap 5
Receive
Foreign
currency 1
Market
interest rate
Pay
Foreign
currency 2
Strike
interest rate

Exotic swap 6
Receive
Foreign
currency 1
Pay
Market
interest rate
Foreign
currency 2
Strike
interest rate

Exotic swap 7
Receive
Foreign
currency 1
Market
interest rate
Pay
Foreign
currency 2
Strike
interest rate

Exotic swap 8
Receive
Foreign
currency 1
Pay
Market
interest rate
Foreign
currency 2
Strike
interest rate

Valuation
( )
( )
( )
( )
( )
1 1
1 1
k k
k=1
PV Variable cash flows = FX rate × Principal
Market interest rate × exp -r T
× + exp -r T
Interest frequency
PV Fixed cash flows = FX rate × Principal
Strike interest rate
× exp -r T
Interest frequency
 
 
 
( )
( ) ( )
N
N N+ exp -r T
 
 
 
∑

Interest rate risk factors for
international swap portfolio
Interest rate risk
Value
Principal
Holding period
dispersion
Interbank rates
Standard
deviation
Holding period
Diversification
effect
Concentration
of equities
% change
dependency
FX rates

Lecture 12 interest rate derivatives

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