Chapter 5 fx forwards

Managing Market Risk Under The Basel III Framework
Copyright © 2016 CapitaLogic Limited
Chapter 5
Foreign Exchange
Forwards
Managing Market Risk Under The Basel III 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.

BIS statistics on FX derivatives
Notional amount (USD bn)
75,87974,519Total
14,60013,558Options
24,20424.724Swaps
http://stats.bis.org/statx/srs/table/d6?p=20151&c=
37,07637,238Forwards
20142015FX derivatives

Time value of money
Currency mis-match problem
FX forwards analysis
FX forwards market
CNY non-deliverable forwards
Outline

Natural exponent and natural logarithm
What is e?
Natural exponent
Natural logarithm
( )
( )
N
r r
N
N
N
a
r
lim 1+ = 2.718281828 = e
N
1
lim 1 + = 2.718281828e =
N
b = e = exp a
a = ln b
→∞
→∞
 
 
 
 
 
 

Future value and present value
Future value
Value of an asset grows to after one or more
compounding periods
Present value
Future value of an asset discounted by a periodic
interest rate
( )
( )
T
T
Future value = Present value 1 + Periodic interest rate
Future value
Present value =
1 + Periodic discount rate

Risk-free security
Functional purpose
Created by financial
economists
For theory development
Never exists in the
world
Proxy by top quality
assets, e.g. domestic
currency in a bank
account
Cash flows
Out
Principal at origination
In
Principal at maturity
Scheduled interest

Risk-free rate
Annualized return of a risk-free security
Discrete compounding
Continuous compounding
Conversion
( )
( )
( ) ( ) T
T 0
T T T
TT
0
T
0
0 T
0
0
S1
S = S exp rT S = S
S S
S = S 1
exp -rT r
+ r S = r = - 1
S1 + r
= ln
T S
 
 
 
( )Continuous compounding = ln 1 + Discrete compounding

Time value of currency
STexp(rfT)S0
Foreign
currency
exp(rdT)1
Domestic
currency
T years laterTodayTime

Time value of currency
STS0exp(-rfT)
Foreign
currency
1exp(-rdT)
Domestic
currency
T years laterTodayTime

Long position and short position
Long position
You own a foreign currency
Quantity is a positive number
When the FX rate increases, you sell the foreign currency
and make a profit
Short position
You borrow and sell a foreign currency
Quantity is a negative number
When the FX rate decreases, you buy and return the
foreign currency and make a profit

Time value of money
FX forwards market
Outline

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

FX forward
An agreement between an investor and a bank
on a foreign currency such that
The investor must buy from the bank the foreign
currency at an agreed strike rate K at maturity T
The bank must sell to the investor the foreign
currency at an agreed strike rate K at maturity T

At maturity
ST above K
The investor pays to the
bank K units of domestic
currencies
The bank
supplements ST - K
buys one foreign currency at
ST
delivers to the investor one
unit of foreign currency
ST below K
bank K units of domestic
currencies
The bank
pockets K - ST
buys one foreign currency at
ST
delivers to the investor one
unit of foreign currency

Payoff
The benefit to the investor resulting from a
FX forward at maturity
Single FX forward
Payoff = ST - K
FX forwards contract
An agreement to enter many identical FX
forwards in one transaction
Contract payoff = Quantity × (ST - K)

Payoff diagram
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1.52 1.54 1.56 1.58 1.6
FX rate at maturity (USD per GBP)
Payoff(USD)
Payoff

Moneyness
< KOut-of-the-money
KAt-the-money
> KIn-the-money
FX rate (S0)Moneyness

Long position and short position
Long position
An investor has the right and obligation to buy from the
bank the foreign currency at strike rate K at maturity T
Short position
The bank has the right and obligation to sell to the
investor a foreign currency at strike rate K at maturity T
An investor can choose to enter a short position
The bank then enters the long position

Time value of money
FX forwards market
Outline

Analysis of derivatives
Functional purposes
Hedging
Investment
Cash flows
Outflows
Inflows
Valuation
When there is market price
Value
= Quantity × Market price
When there is no market
price, how much does it
worth?
Market risk
VaR
Worst case loss

Hedging
Hedge
To reduce the uncertainty in the value of a foreign currency
To reduce the FX rate risk
Complete hedge
To eliminate completely the uncertainty in the value of a foreign
currency
To reduce the FX rate risk completely
Partial hedge
To eliminate the uncertainty in the value of part of a foreign currency
To reduce the FX rate risk below a tolerance level

Functional purposes
Long position
To hedge the value of a
future outflow in a foreign
currency
To speculate on the up
trend of a FX rate without
upfront cash outflow
To hedge a short position
in a foreign currency
Short position
To hedge the value of a
future inflow in a foreign
currency
To speculate on the down
trend of a FX rate without
borrowing and selling
To hedge a long position in
a foreign currency

A reverse currency mis-match problem
A trading firm
Buys goods from United states
Pays cost in USD
Sells goods to United Kingdom
Receives revenue in GBP
Both cost in USD and revenue in GBP in three
months are well estimated
How to ensure that the revenue in GBP which will
be sold and become USD, is sufficient to pay the
cost in USD, subject to the condition that the FX rate
in three months is unknown?

At maturity
ST below K
bank one foreign currency
The bank
sells the foreign currency at
ST
supplements K - ST
delivers to investor K units
of domestic currencies
ST above K
bank one foreign currency
The bank
sells the foreign currency at
ST
pockets ST - K
delivers to investor K units
of domestic currencies

Payoff of a short position
in FX forward
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1.52 1.54 1.56 1.58 1.6
FX rate at maturity (USD per GBP)
Payoff(USD)

Investment in FX rate
with FX forward
Long position
Short position
( )
T
T
T
T
Payoff = S - K > 0
if S > K
Payoff = - S - K > 0
if S < K

Investment in FX rate
with FX forward
When an investor expects that the FX rate will go up
Long a FX forward
When an investor expects that the FX rate will go
down
Short a FX forward
In either case, no upfront cash is required
Chance to create profit without principal
The most important characteristic of FX forwards

Modelling a FX forward
S0: Current FX rate
rd: Annualized risk-free rate of domestic currency
rf: Annualized risk-free rate of foreign currency
K: Strike rate
T: Maturity in years
ST: FX rate in T years
f: Value a FX forward

Cash flows – physical settlement
Foreign currencyK
Out flow at
maturity
K = S0exp[(rd - rf)T]
Foreign currency
None
Long
Strike rate
K
In flow at
maturity
Cash flow at
origination
ShortPosition

Cash flows – cash settlement
K = S0exp[(rd - rf)T]
ST - K
None
Long
Strike rate
K - ST
Cash inflow
at maturity
Cash flow at
origination
ShortPosition

Principles of derivatives valuation
Mark-to-market
Value = Quantity × Relevant and recent transaction price
But, FX forwards have no available transaction prices
Mark-to-model
Value calculated by valuation models and available market data
FX rate, interest rate
Transaction price ≠ Model value
Transaction price = Model value + Administrative cost + Profit
Subject to supply and demand in the financial market
Model value is a proxy of transaction price for risk management and
financial reporting purposes

Valuation – long position
f
Payoff
Strike rate
S0exp(-rfT) - Kexp(-rdT)
Value of a FX
forward today
ST - K
Benefit to
long position
K
Long position
pays at maturity
ST
FX rate at
maturity

Value of a FX forward
Forward value
Intrinsic value
Caused by payoff
Time value
Caused by time to maturity
( ) [ ] [ ]{ }
0 f d
0 0 f d
f = S exp(-r T) - Kexp(-r T)
= S - K + S exp(-r T) - 1 - K exp(-r T) - 1
[ ] [ ]
0
T
0 f d
Intrinsic value = S - K
= S - K at maturity
Time value = S exp(-r T) - 1 - K exp(-r T) - 1
= 0 at maturity

Value of a FX forward
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1.52 1.54 1.56 1.58 1.6
Current FX rate (USD per GBP)
Value(USD)
Intrinsic value
Time value

Forward rate
The strike rate at which the value of a FX forward is zero
Interest rate parity
In general, forward rate moves in the same direction as current FX rate
Standard FX forwards are originated at forward rate to avoid cash outflow
from either party
The most important characteristic of FX forwards
( )
0 f d
0 d f
f = S exp(-r T) - Kexp(-r T) = 0
K = S exp r - r T
Forward FX rate = Current FX rate
× Interest rate differential effect
  
Example 5.3

Delta
The sensitivity of a FX forward to its
underlying foreign currency
Change in FX forward value if the FX rate
increases by 1
[ ]
[ ] ( )
[ ] [ ]
0 0 f d
0 0 f d
0 0 f
f S = S exp(-r T) - Kexp(-r T)
f S +1 = S +1 exp(-r T) - Kexp(-r T)
Delta = f S +1 - f S = exp(-r T) 1≈

Replicating portfolio
A portfolio comprising
A long position in exp(-rfT) units of foreign currency
A short position in exp(-rdT) units of domestic currency
Replication
Long a FX forward is equivalent to long Delta units of
foreign currency and short a risk-free security
Short a FX forward is equivalent to short Delta units of
foreign currency and long a risk-free security
0 f df = S exp(-r T) - Kexp(-r T)

Hedge the risk of a foreign currency
Long position in
ONE FX forward
Short position in
ONE FX forward
Approximation
Long position in
exp(rfT) quantity
of FX forward
Short position in
exp(rfT) quantity
of FX forward
Forward hedge
ShortLongFX position

Approximations to risk calculations
Value and forward rate
Direct impact to profit and loss
Must look accurate
Value-at-risk
A projection of potential loss in the future
Model error always exists
Convenience is over accuracy
Approximations
For FX forwards where the maturity in general less then one year,
under the current low interest environment, the quantities exp(rdT),
exp(-rdT), exp(rfT), exp(-rfT), exp[(rd-rf)T)] and exp[(rf-rd)T)] are
close to ONE

Hedged worst case loss
Value0
0
Worst case value
Expected value
Value-at-risk
1 - q%
q%
T days
ValueT
Unrealized
loss
Worst case loss
Acquisition cost
Hedging
Hedged
worst case loss
Example 5.4
Example 5.5

FX rate risk of a FX forward
Foreign currency has FX rate risk
Domestic currency has no risk
FX rate risk
Equivalent to Delta units of foreign currency
Approximately equivalent to ONE unit of foreign
currency
Example 5.6
Example 5.7
0 f df = S exp(-r T) - Kexp(-r T)

FX rate risk factors for
FX forwards portfolio
FX rate risk
Value
Quantity
Holding period
dispersion
FX rate
Standard
deviation
Holding period
Diversification
effect
Concentration of
foreign currencies
% change
dependency

Hedge the risk of a FX forward
Long position in
another FX forward
Short position in
another FX forward
Forward
hedge
Long position in a
foreign currency
Short position in a
foreign currency
Foreign
currency
hedge
ShortLong
Forward
position

Time value of money
FX forwards market
Outline

Hypothetical sale of a FX forward
A long position at time 0
Forward rate at time τ
Value of a long position at time τ
Sell and deposit cash until maturity T
[ ]
[ ]
[ ] [ ]
[ ] [ ]
[ ] [ ]
( ) [ ]
( )
1
0 0 f 1 d
1 0 d f
2 0 d f
τ f 2 d
1
τ τ f 1 d
2 d 1 d
2 1 d
2 1
f = S exp(-r T) - K exp(-r T) = 0
K = S exp (r -r )T
K = S exp (r -r )(T-τ)
S exp -r (T-τ) = K exp -r (T-τ)
f = S exp -r (T-τ) - K exp -r (T-τ)
K exp -r (T-τ) - K exp -r (T-τ)
= K - K exp -r (T-τ)
K - K exp -
=
[ ] [ ]d d 2 1r (T-τ) exp r (T-τ) = K - K

A long-short portfolio
Enter a long position at time 0
Enter a short position at time τ
At maturity
Selling of a FX forward = Entering a reverse position
[ ]
[ ] [ ]
[ ]
( ) ( )
1
0 0 f 1 d
1 0 d f
2
τ τ f 2 d
2 0 d f
1 2
T T T 1 T 2 2 1
f = S exp(-r T) - K exp(-r T) = 0
K = S exp (r -r )T
f = S exp -r (T-τ) - K exp -r (T-τ) = 0
K = S exp (r -r )(T-τ)
f - f = S - K - S - K = K - K

Reverse position
Long position
Entering a short position with the same maturity day
At maturity
Net payoff = (ST - K1) - (ST - K2) = K2 - K1
Equivalent to close a long position
Short position
Entering a long position with the same maturity day
At maturity
Net payoff = (ST - K2) - (ST - K1) = K1 - K2
Equivalent to close a short position

Closing a position
Standard FX forwards contract
Customized for investor in terms of foreign currency, maturity and quantity
A bilateral agreement between an investor and a bank
Cannot be transferred
No secondary market
An investor can NOT sell his FX forwards contract
Market maker
A bank which is always ready to quote a forward rate to enter a FX forwards contract
with an investor
An investor can always enter a position or a reverse position with a market marker
By entering a reverse position, an investor can always reverse and close his existing
FX forwards position
A liquid FX forwards market
Selling a FX forwards contract in fact means closing a FX forward position by entering
a reverse position
Market markers create a liquid FX forwards market

Close out netting agreement
An agreement between an investor and a bank
Covering all FX forwards entered between the
two parties
Allowing FX forward positions with same
underlying currency and maturity to offset
among one another for settlement at maturity

Spot market vs forward market
Spot market
Domestic currency and foreign currency are exchanged on
spot in accordance with the current FX rate
Forward market
A market in which FX forward transactions are entered by
quoting the forward rates directly without any upfront
cash flows
Forward market is compatible in size with spot market
As such, the forward FX rates take an active role to drive
the current FX rates through the interest rate parity

Over-the-counter
An investor and a bank enter a FX forward through
private negotiation
High degree of customization
High confidentiality
Loosely regulated
Subject to default risk
Primarily protected by contract law through civil litigation
Resulting unknown systemic risk to regulators
Nowadays, banks in many developed countries must
report their FX forwards positions to the Central Trade
Repository

Bank business model – service based
Revenue
A bank enters a FX forward with an investor
Charges a service fee
Due to the strong competition among banks,
very slim profit is made for providing services
on FX forwards

Bank business model – position based
If the bank’s position is consistent with its
expectation on the trend of FX rate
Keep the position
If the bank’s position is consistent with its
expectation on the trend of FX rate
Enter a reverse position with foreign currency or
another FX forward

Default risk
Default
At maturity, an investor obligated to pay does not pay the
bank
Regular settlement
Before maturity
The long position deposits K - S0 to the short position when S0 <
K
The short position deposits S0 - K to the long position when S0 >
K
In almost all cases, for a period of less than one year
| S0 - K | < 25% of K

Time value of money
FX forwards market
Outline

Developed by the HKMA, TMA and major retail
banks in Hong Kong
Launched in late 2005
To facilitate the growing market of CNY due to the
hedging and investment objectives of small and
medium size investors in retail market
Small transaction size USD 10,000
Short maturities 1, 2, 3, 6, 12 months
Available from retail banking services
Lower service fee due to strong competition

Product description
Indirect quote for FX ratesDirect quote for FX rates
ShortBank payoffLongInvestor payoff
Quantity × K
Notional
amount
CNYQuantity
1 / KForward rateKStrike rate
1 / ST
Settlement
rate
ST
FX rate at
maturity
Product descriptionLecture convention

Payoff
( )
T
T
Forward rate
Payoff = Notional amount × 1 -
Settlement rate
1
K= Quantity × K× 1 -
1
S
= - Quantity × S - K
 
 
 
 
 
 
 
 
 

Other features
No explicit service charge
Embedded in strike rate
For a long position, strike rate higher than theoretical rate
For a short position, strike rate lower than theoretical rate
Market marker
When there is an enquiry, the bank must propose a strike
rate
No secondary market
Hold to maturity
Collaterals
Notional amount of liquid assets Example 5.8

Chapter 5 fx forwards

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Chapter 5 fx forwards