Chapter 2 fx rate risk for currrency portfolios

Managing Market Risk Under The Basel III Framework
Copyright © 2016 CapitaLogic Limited
Chapter 2
FX Rate Risk for
Currency Portfolios
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.

Risk factors and risk measure
Standing data set
Historical simulation
Monte Carlo simulation
Variance-covariance method
Theory of diversification
Outline

Value-at-risk at T-day
qth percentile confidence level
S0
0
Worst case value
Expected value
Value-at-risk
1 - q%
q%
T days
ST

Value-at-risk at 10-day
99th percentile confidence level
0
Worst case value
Expected value
Value-at-risk
1%
99%
10 days
S0
ST

Simulation based value-at-risk
Form a projected distribution of future FX rates
Based on historical % changes
Based on a normal distribution
Worst case FX rate
Percentile(FX rate distribution, 1 - q%)
Expected FX rate
Average(FX rate distribution)
Value-at-risk
Quantity × (Worst case FX rate - Expected FX rate)

Formula based value-at-risk
% changes in a normal distribution
FX rates in a normal distribution
Worst case FX rate
Expected FX rate
Value-at-risk
[ ]( )
( )
( )
( )
( )
T 0
0
0
0
0
S = S 1 + µT + σ T × Normal 0,1
Worst case FX rate = S 1 + µT + σ T × NormSInv 1 - q%
Expected FX rate = S 1 + µT
VaR = nS σ T × NormSInv 1 - q%
= nS σ 10V × -2.3263aR
 
 

Foreign currency portfolio
A collection of investments in more than one
foreign currency
Diversification effect generally reduces the
FX rate risk
A simple sum of VaRs fails to reflect the risk
reduction arising from the diversification
effect

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

Value-at-risk at T-day
qth percentile confidence level
Value0
0
Worst case value
Expected value
Value-at-risk
1 - q%
q%
T days
ValueT

Value-at-risk at 10-day
99th percentile confidence level
Value0
0
Worst case value
Expected value
Value-at-risk
1%
99%
10 days
ValueT

Standing data set
Outline

Date mis-match
Mis-match of trading dates of FX rates
Statuary holidays in different countries
National days
Golden Week holidays in China
Different thanks giving days in US and Canada
Mis-match between business days of an
investor and trading days of foreign
currencies

Business days
Business days of the country in which an
investor’ trading activities are conducted
Proxy by the trading days of an equity index
in the country

Historical equity indices on Internet
Yahoo finance
http://finance.yahoo.com
http://finance.yahoo.com/exchanges
Google finance
https://www.google.com/finance
United States, United Kingdom, Canada, China,
Hong Kong
Example 2.1

VLookUp(…) and IsNA(…)
Look up the FX rate according to a date
VLookUp(Business date, FX rates, 2, false)
If failed to look up the FX rate according to
the date
IsNA(…)
Carry forward the FX rate on the previous date
Example 2.2

Value-at-risk
Specification
At the end of a T-day holding period (10-day)
At the qth percentile confidence level (99th percentile)
Worst case value
The minimum potential value of the foreign currency portfolio at the
end of the holding period with the lowest (1 - q%) situations excluded
Expected value
The average of all potential values of the foreign currency portfolio at
the end of the holding period
Value-at-risk (“VaR”)
The maximum unexpected loss relative to the expected value with the
worst (1 - q%) situations excluded
Worst case value - Expected value

Standing data set
Outline

Modelling FX rate
For each foreign currency
S0: Current FX rate
µT: T-day % change of FX rate
ST: FX rate in T trading days
n: Quantity

Multivariate historical simulation
For k = 1 to 500
Portfolio value in T-days
Value-at-risk
T
T
k
k
Worset case value = Percentile(All Value s, 1 - q%)
Expected value = Average(All Value s)
VaR = Worst case value - Expected value
( )
T T
k
k
T 0k-T
k
k k
T
k
T
S
= - 1 S = S 1 +
S
Value = n
µ
S
µ
∑ Example 2.3

Standing data set
Outline

Correlation coefficient
Statistic
A linear relational measure of dependency between two
data sets
Between -1 and 1
1 : same direction, same magnitude
-1: opposite direction, same magnitude
0 : independent
( )
N
k Avg k Avg
k=1
xy N N
2 2
k Avg k Avg
k=1 k=1
(x - x )(y - y )
ρ = Correl x, y =
(x - x ) × (y - y )
  ∑
∑ ∑

Correlation matrix
12 13 1M
21
31
M1
xy yx
1 ρ ρ ... ρ
ρ 1 . ... .
CorrelMatrix = ρ . 1 ... .
: : : ... :
ρ . . ... 1
where ρ = ρ
 
 
 
 
 
 
  

Lower correlation matrix in Excel
Add-in
Data Analysis Toolpak
Lower correlation matrix
Data
Data Analysis
Correlation

Modelling FX rate
S0: Current FX rate
µ: % change of FX rate
σ: Standard deviation of FX rate
T: Holding period
Normal[µ,σ]: A random number drawn from a normal distribution
with
Average = µ
Standard deviation = σ
= µ + σ × Normal[0,1]
ST: FX rate in T trading days
n: Quantity

Three foreign currency portfolio
Foreign currency 1
Foreign currency 2
Foreign currency 3
[ ]( )
[ ]( )
[ ]( )
1 T 1 0 1 1 1
12
2 T 2 0 2 2 2 31
23
3 T 3 0 3 3 3
S = S 1 + µ T + σ T × Normal 0,1
ρ
S = S 1 + µ T + σ T × Normal 0,1 ρ
ρ
S = S 1 + µ T + σ T × Normal 0,1
↑
↓
↑
↓

Multivariate standard
normal random numbers
( )
21
31
M1
1
ρ 1
LowerCorrelMatrix = ρ . 1
: : : ...
ρ . . ... 1
MVSNRNs = MultiVarStdNormRandNos LowerCorrelM
[C
atrix
trl]-[Shift]-[Enter]
 
 
 
 
 
 
  

Multivariate Monte Carlo simulation
For k = 1 to 1,000
Portfolio value
Value-at-risk
[ ]( )T
T T
kk
0
k k
µT + σ T × MultiVarNormal 0,1S = S 1 +
Value = nS∑ Example 2.4
T
T
k
k
Worset case value = Percentile(All Value s, 1 - q%)
Expected value = Average(All Value s)
VaR = Worst case value - Expected value

Standing data set
Outline

Normal FX rate model
VaR for individual foreign currency k
VaR for two foreign currency portfolio
VaR for three foreign currency portfolio
[ ]( )
( )
T 0
k k k k
2 2 2
1 2 12 1 2
2 2 2 2
1 2 3 12 1 2
23 2 3 31 3 1
S = S 1 + µT + σ T × Normal 0,1
VaR = n S σ T × NormSInv 1 - q%
VaR = VaR + VaR + 2ρ × VaR × VaR
VaR = VaR + VaR + VaR + 2ρ × VaR × VaR
+ 2ρ × VaR × VaR + 2ρ × VaR × VaR

M foreign currency portfolio
[ ]
[ ]
[ ]( )
1 2 3 M
12 13 1M 1
21 2
31 3
M1 M
Q = VaR VaR VaR ... VaR
1 ρ ρ ... ρ VaR
ρ 1 . ... . VaR
CorrelMatrix = Transpose Q =ρ . 1 ... . VaR
: : : ... : :
ρ . . ... 1 VaR
Λ = Sum Q × CorrelMatrix × Trans [Ctrl]-[Shift]-[Epose Q
   
   
   
   
   
   
      
( )0Expected value = nS
nt
1
e
+
VaR =
µT +
]
- Λ
r
VaR∑ Example 2.5

Component VaR
For a component foreign currency k with quantity = n units
VaR Plus
Portfolio VaR with n + 0.5 units of k
VaR Minus
Portfolio VaR with n - 0.5 units of k
Component VaR
Quantity × (VaR Plus - VaR Minus)
The VaR of individual foreign currency with the diversification
effect incorporated
Euler’s theorem
Portfolio VaR = Component VaR∑
Example 2.6

Standing data set
Outline

A hypothetical
foreign currency portfolio
Number of foreign currencies M
Each foreign currency same no. of units n
Each foreign currency same FX rate S0
Portfolio value V = MnS0
Major parameters
Each foreign currency with same standard deviation σ
Each foreign currency with same holding period T
Each foreign currency with same VaR VaR0
Each pair with same correlation coefficient ρ

Portfolio VaR
( )
[ ]
[ ]
[ ]( )
0 0
0 0 0 0
0
0
0
0
VaR = nS σ T × NormSInv 1 - q%
Q = VaR VaR VaR ... VaR
VaR1 ρ ρ ... ρ
VaRρ 1 . ... .
CorrelMatrix = Transpose Q = VaRρ . 1 ... .
:: : : ... :
VaRρ . . ... 1
= Sum Q × Correl × Transpose [CtQ rl]
  
  
  
  
  
  
     
Λ -[Shift]-[Ent
VaR
er]
= - Λ

Diversification effect
VaR for the hypothetical foreign currency portfolio
When the no. of components becomes very large
When all components are independent
When BOTH
( )
( )
( )
M - 1 1
VaR = Vσ ρ + × T × NormSInv 1 - q%
M M
VaR = Vσ ρT × NormSInv 1 - q%
T
VaR = Vσ × NormSInv 1 - q%
M
VaR = 0

Systematic risk vs specific risk
Systematic risk
Specific risk
Total risk
( )
( )
Sys
Spec
2 2 2
Total Sys Spec
M - 1
VaR = Vσ ρT × NormSInv 1 - q%
M
T
VaR = Vσ × NormSInv 1 - q%
M
VaR =VaR + VaR

Major theoretical findings
FX rate risk increase with increasing
concentration of foreign currencies
% change dependency
Due to the limited number of major foreign
currencies, it is less optimal to diversify in a
pure foreign currency portfolio

Chapter 2 fx rate risk for currrency portfolios

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