Chapter 8 equity risk management

Managing Market Risk Under The Basel IV Framework
Copyright © 2016 CapitaLogic Limited
Chapter 8
Equity Risk
Management
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.

Equity investments
Equity risk identification
Equity risk measurement
Equity risk monitoring
Equity risk mitigation
Outline

Functional purposes
DividendIncome
Short to medium term
down trend of equity
price
Short to medium term
up trend of equity price
Speculation
Medium to longer term
up trend of equity price
Capital
appreciation
ShortLongPosition

Cash flows
- Dividend+ DividendInterim
- Equity price+ Equity priceClosing
+ Equity price- Equity priceEntering
ShortLongPosition

Valuation
Single equity portfolio
Single currency, multiple equity portfolio
International equity portfolio
M
k k
k=1
M
k k k
k=1
Value = Quantity × Equity price
Value = FX rate × Quantity × Equity price
Value = Quantity × FX rate × Equity price
∑
∑

Expected shortfall
Specification
At the end of a T-day holding period (10-day)
At the qth percentile confidence level (97.5th percentile)
Worst case value
The minimum potential portfolio value at the end of the holding period with
the lowest (1 - q%) situations excluded
Tail value
The average of potential portfolio values when the potential portfolio values
are below the worst case value
Expected value
The average of all potential portfolio values at the end of the holding period
Expected shortfall (“ES”)
The average of unexpected loss relative to the expected value during the
worst (1 - q%) situations
Tail value - Expected value

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

Expected shortfall at 10-day
97.5th percentile confidence level
Value0
0
Worst case value
Expected value
Expected shortfall
2.5%
97.5%
10 days
ValueT
Tail value

Value-at-risk vs expected shortfall
NormSInv(1 - q%)
Percentile(Values,
1 - q%)
Worst case value
99th percentile
Value-at-risk
NormDist(Critical value,
0, 1, False) /(1 - q%)
Variance-covariance
method
AverageIF(Values <
Worst case value)
Historical simulation/
Monte Carlo simulation
Tail valueExtremity
97.5th percentileConfidence level (q)
Expected shortfall
Examples 8a

Variance-covariance method
Equity price in a normal distribution
Critical value
[ ]( )
( )
0S 1 + µT + σ T × Normal 0,1
NormSInv 1 - q%

Tail equity price
Expected equity price
Expected shortfall
( )
( )
( )
0
0
0
NormDist CV,0,1,False
S 1 + µT + σ T × -
1 - q%
S 1 + µT
- nS σ T ×
1 - q%
   
  
   

Expected shortfall under Basel IV
Adopted by Basel IV
To be implemented in 2019
Taking into account the loss beyond the worst
case value
Not a robust market risk measure
Very sensitive to data error in the tail area
Back testing methodology has not be well
developed

Equity investments
Outline

International equity portfolio
A collection of investments
in many equities
listed on stock exchanges in different countries
denominated in different currencies

Historical equity prices
and equity index values
Yahoo finance
http://finance.yahoo.com
Google finance
https://www.google.com/finance
Stooq.com
http://stooq.com

Equity risk factors
for international equity portfolio
Equity risk
Value
Quantity
Holding period
dispersion
Equity price
Standard
deviation
Holding period
Diversification
effect
Concentration
of equities
% change
dependency
FX rate Examples 8b

Equity risk
Total equity risk
The risk of losses arising from movements in equity
prices
Systematic equity risk
The risk of losses arising from movements in equity index
levels
The majority of equity risk in a well diversified equity
portfolio
Specific equity risk
The equity risk not captured by the systematic equity risk
Minimize through diversification

Equity index portfolio
A collection of investments
in many well diversified equities
listed on the same stock exchange
subject to the same equity index
denominated in the same currency

Systematic equity risk
( ) ( )
Today
E
Yesterday
Today
I
Yesterday
E I
E I I E I
Equity price
µ = - 1 × 100%
Equity price
Index level
µ = - 1 × 100%
Index level
µ = βµ + α + Residual error
= Slope µ , µ × µ + Intercept µ , µ + Resi
 
  
 
 
  
 
I
dual error
Systematic % change = βµ
For each equity

Weighted average Beta
Value of index portfolio
Average % change of index portfolio
S.D. % change of index portfolio
Equity index risk
( )
( ) ( )
( ) ( )
M
k k k M
k=1
k kM
k=1
k k
k=1
T+1 T I
β n S
β = Value = n S
n S
Value = Value 1 + βµ
Average Index portfolio % changes = β × Average Index % changes
S.D. Index portfolio % changes = β × S.D. Index % changes
∑
∑
∑

Equity index risk factors
Equity index risk
Index portfolio value
Equity value
Holding period
dispersion
FX rate
Index standard
deviation
Beta
Holding period

Systematic equity risk factors
for international equity portfolio
Systematic
equity risk
Index portfolio
value
Equity value
Holding period
dispersion
FX rate
Standard
deviation
Beta
Diversification
effect
Concentration
of indices
% change
dependency
Holding period

Equity investments
Outline

Expected shortfall methodologies
Historical simulation
Monte Carlo simulation
Marginal expected shortfall
Component expected shortfall

Modelling equity index value
For each equity index
S0: Current index level
ST: Index value in T trading days
µT: T-day % change of index level
I0: Current index portfolio value
IT: Index portfolio value in T trading days

Multivariate historical simulation
For k = 1 to 500
For each index portfolio
Portfolio value in T-days
Value-at-risk
T
T
T
k
k
k
Worset case value = Percentile(All Value s, 1 - q%)
Tail value = AverageIf(Value < Worst case value)
Expected value = Average(All Value s)
ES = Tail value - Expected value
( )
T T
k
k k k
T T 0 Tk-T
k k
β
S
µ = - 1 I = I 1 + µ
S
Value = I∑
Example 8d.4
Example 8c.4

Modelling index portfolio value
For each equity index
S0: Current index level
ST: Index value in T trading days
µ: % change of index level
σ: Standard deviation of index level
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]
I0: Current index portfolio value
IT: Index portfolio value in T trading days

Three index portfolios
Index portfolio 1
Index portfolio 2
Index portfolio 3
[ ]( )
[ ]( )
[ ]( )
1 T 1 0 1 1 1
12
2 T 2 0 2 2 2 31
23
3 T 3
1
0 3 3 3
1
2 2
3 3
β β
β β
I = I 1 + µ T + σ T × Normal 0,1
ρ
I = I 1 + µ T + σ T × Normal 0,1 ρ
ρ
I = I 1 + µ T + σ T × Norm 0β ,β al 1
↑
↓
↑
↓

Multivariate Monte Carlo simulation
For k = 1 to 1,000
For each index potfolio
Portfolio value
Expected shortfall
[ ]( )T
T T
k k
0
k k
I = I 1 + µT + σ T × MultiVarNormal 0,1
Value = I
β β
∑
T
T
T
k
k
k
Worset case value = Percentile(All Value s, 1 - q%)
Tail value = AverageIf(Value < Worst case value)
Expected value = Average(All Value s)
ES = Tail value - Expected value
Example 8d.5
Example 8c.5

For each index portfolio
( )
( )
0
CV = NormSInv 1 - q%
ES = - I βσ T ×
1 - q%
Example 8c.6

M index portfolios
[ ]
[ ]
[ ]( )
1 2 3 M
12 13 1M 1
21 2
31 3
M1 M
Q = ES ES ES ... ES
1 ρ ρ ... ρ ES
ρ 1 . ... . ES
CorrelMatrix = Tran
[Ctrl]-[Shift]-[En
spose Q =ρ . 1 ... . ES
: : : ... : :
ρ . . ... 1 ES
Λ = Sum Q × CorrelMatrix × Transpose Q
E
ter]
S
   
   
   
   
   
   
      
( )0
=
Expected value = Value 1+
- Λ
βµT∑ Example 8d.6

Component expected shortfall
For an index portfolio with value I
ES Plus
Portfolio ES with value I + 0.5
ES Minus
Portfolio ES with value I - 0.5
Marginal ES
ES Plus - ES Minus
Component ES
I × Marginal ES
The ES of individual index portfolio with the diversification
effect incorporated
Euler’s theorem
Portfolio ES = Component ES∑
Example 8b.7 Example 8d.7

Tips for Monet Carlo simulation
for component expected shortfall
Frozen random numbers
Select columns of random numbers
Home, Copy, Paste special, Values
Data table
Data, Data Tools, What-If Analysis, Data Table

Equity investments
Outline

Systematic risk Specific risk from large exposures

FX rates monitoring
Equity indices monitoring
Large exposures monitoring
Diversification monitoring
Unrealized loss monitoring

Daily percentage changes
Early detection with % changes
What is today’s percentage change relative to the
most recent 500 trading days?
Emergency – extremities (+1%)
Warning – wing regions (+1% to +5%)
Attention – side regions (+5% to +10%)
Normal – middle region (-10% to 10%)
Applicable to
FX rates
Equity indices
Large exposures Example 8e.4

Daily percentage changes
80% = 400 days

Concentration of equity value
( )
M
2
k
k=1
2M
k
k=1
M
2
k k k
k=1
2M
k k k
k=1
Value
HHI =
Value
Quantity × FX rate × Equity price
=
Quantity × FX rate × Equity price
 
 
 
 
 
 
∑
∑
∑
∑
Example 8e.3

Unrealized loss monitoring
Portfolio value
Acquisition cost
Portfolio value at origination
Unrealized loss
Latest portfolio value - Acquisition cost
M
k k k
k=1
Value = Quantity × FX rate × Equity price∑
Example 8e.5

Equity investments
Outline

Controlled by investor
Quantity ↓
Beta ↓
Holding period ↓
Concentration ↓
Re-balancing
Not controlled by investor
FX rate
Equity price
Standard deviation
% change dependency
Derivatives

Equity derivatives
Equity futures
A standardized equity forward with exchange as
counterparty
European equity options
Call and put which can be exercised at maturity
only
American equity options
Call and put which can be exercised at any time
on or before maturity

Equity index derivatives
Equity index futures
A standardized equity index forward with
exchange as counterparty
European equity index options
Call and put which can be exercised at maturity
only

Specific equity risk
Minimized through re-balancing
Equity index risk
Hedged with equity index derivatives
FX risk
Hedged with FX derivatives

Chapter 8 equity risk management

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Chapter 8 equity risk management