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Financial Risk Management
Zvi Wiener
Following
P. Jorion, Financial Risk Manager Handbook
http://pluto.huji.ac.il/~mswiener/zvi.html
FRM
972-2-588-3049
Chapter 17
VaR Methods
Following P. Jorion 2001
Financial Risk Manager Handbook
http://pluto.huji.ac.il/~mswiener/zvi.html
FRM
972-2-588-3049
Risk Factors
There are many bonds, stocks and currencies.
The idea is to choose a small set of relevant economic
factors and to map everything on these factors.
• Exchange rates
• Interest rates (for each maturity and indexation)
• Spreads
• Stock indices
Ch. 17, Handbook
Zvi Wiener
slide 3
How to measure VaR
• Historical Simulations
• Variance-Covariance
• Monte Carlo
• Analytical Methods
• Parametric versus non-parametric approaches
Ch. 17, Handbook
Zvi Wiener
slide 4
Historical Simulations
• Fix current portfolio.
• Pretend that market changes are
similar to those observed in the past.
• Calculate P&L (profit-loss).
• Find the lowest quantile.
Ch. 17, Handbook
Zvi Wiener
slide 5
Example
Assume we have $1 and our main currency is
SHEKEL. Today $1=4.30.
Historical data:
P&L
4.00
4.20
4.30*4.20/4.00 = 4.515
0.215
4.20
4.30*4.20/4.20 = 4.30
0
4.10
4.30*4.10/4.20 = 4.198
-0.112
4.15
4.30*4.15/4.10 = 4.352
0.052
Ch. 17, Handbook
Zvi Wiener
slide 6
USD
NIS
2000
100
-120
2001
200
100
2002
-300
-20
2003
20
30
today
Ch. 17, Handbook
100
200
 300
20



2
3
1  0.06 (1  0.061) (1  0.062) (1  0.063) 4
 120
100
 20
30



2
3
1  0.1 (1  0.11)
(1  0.12)
(1  0.13) 4
Zvi Wiener
slide 7
today
100
200
 300
20



2
3
1  0.06 (1  0.061) (1  0.062) (1  0.063) 4
 120
100
 20
30



2
3
1  0.1 (1  0.11)
(1  0.12)
(1  0.13) 4
Changes
in IR
USD:
NIS:
+1% +1%
+1% 0%
+1%
-1%
+1%
-1%
100
200
 300
20



2
3
1  0.07 (1  0.071) (1  0.072) (1  0.073) 4
 120
100
 20
30



2
3
1  0.11 (1  0.11)
(1  0.11)
(1  0.12) 4
Ch. 17, Handbook
Zvi Wiener
slide 8
Returns
year
1% of worst cases
Ch. 17, Handbook
Zvi Wiener
slide 9
VaR
1
0.8
0.6
0.4
VaR1%
1%
0.2
Profit/Loss
-3
Ch. 17, Handbook
-2
-1
1
Zvi Wiener
2
3
slide 10
Variance Covariance
• Means and covariances of market factors
• Mean and standard deviation of the portfolio
• Delta or Delta-Gamma approximation
• VaR1%= P – 2.33 P
• Based on the normality assumption!
Ch. 17, Handbook
Zvi Wiener
slide 11
Variance-Covariance VaR1%  V  2.33 V
1%
2.33
-2.33
Ch. 17, Handbook

Zvi Wiener
slide 12
Monte Carlo
1
0.5
-1
0.5
-0.5
1
-0.5
-1
Ch. 17, Handbook
Zvi Wiener
slide 13
Monte Carlo
• Distribution of market factors
• Simulation of a large number of events
• P&L for each scenario
• Order the results
• VaR = lowest quantile
Ch. 17, Handbook
Zvi Wiener
slide 14
Monte Carlo Simulation
15
10
5
10
20
30
40
-5
-10
-15
Ch. 17, Handbook
Zvi Wiener
slide 15
Weights
Since old observations can be less relevant,
there is a technique that assigns decreasing
weights to older observations. Typically the
decrease is exponential.
See RiskMetrics Technical Document for
details.
Ch. 17, Handbook
Zvi Wiener
slide 16
Stock Portfolio
• Single risk factor or multiple factors
• Degree of diversification
• Tracking error
• Rare events
Ch. 17, Handbook
Zvi Wiener
slide 17
Bond Portfolio
• Duration
• Convexity
• Partial duration
• Key rate duration
• OAS, OAD
• Principal component analysis
Ch. 17, Handbook
Zvi Wiener
slide 18
Options and other derivatives
• Greeks
• Full valuation
• Credit and legal aspects
• Collateral as a cushion
• Hedging strategies
• Liquidity aspects
Ch. 17, Handbook
Zvi Wiener
slide 19
Credit Portfolio
• rating, scoring
• credit derivatives
• reinsurance
• probability of default
• recovery ratio
Ch. 17, Handbook
Zvi Wiener
slide 20
Reporting
Division of VaR by business units, areas of
activity, counterparty, currency.
Performance measurement - RAROC (Risk
Adjusted Return On Capital).
Ch. 17, Handbook
Zvi Wiener
slide 21
Backtesting
Verification of Risk Management models.
Comparison if the model’s forecast VaR with
the actual outcome - P&L.
Exception occurs when actual loss exceeds
VaR.
After exception - explanation and action.
Ch. 17, Handbook
Zvi Wiener
slide 22
Backtesting
Green zone - up to 4 exceptions
OK
Yellow zone - 5-9 exceptions
increasing k
Red zone - 10 exceptions or more
intervention
Ch. 17, Handbook
Zvi Wiener
slide 23
Stress
Designed to estimate potential losses in abnormal
markets.
Extreme events
Fat tails
Central questions:
How much we can lose in a certain scenario?
What event could cause a big loss?
Ch. 17, Handbook
Zvi Wiener
slide 24
Local Valuation
Worst dP  ( D * P)  (Worst dy )
Simple approach based on linear approximation.
Full Valuation
Worst dP  P( y0  Worst dy)  P( y0 )
Requires repricing of assets.
Ch. 17, Handbook
Zvi Wiener
slide 25
Delta-Gamma Method
2
dP
1d P
2
dP 
dy 
(dy)
2
dy
2 dy
dP   D * Pdy  0.5CP(dy)
2
The valuation is still local (the bond is priced
only at current rates).
Ch. 17, Handbook
Zvi Wiener
slide 26
FRM-97, Question 13
An institution has a fixed income desk and an exotic
options desk. Four risk reports were produced, each
with a different methodology. With all four
methodologies readily available, which of the
following would you use to allocate capital?
A. Simulation applied to both desks.
B. Delta-Normal applied to both desks.
C. Delta-Gamma for the exotic options desk and the
delta-normal for the fixed income desk.
D. Delta-Gamma applied to both desks.
Ch. 17, Handbook
Zvi Wiener
slide 27
FRM-97, Question 13
An institution has a fixed income desk and an exotic
options desk. Four risk reports were produced, each
with a different methodology. With all four
methodologies readily available, which of the
following would you use to allocate capital?
A. Simulation applied to both desks.
B. Delta-Normal applied to both desks.
C. Delta-Gamma for the exotic options desk and the
delta-normal for the fixed income desk.
D. Delta-Gamma applied to both desks.
Ch. 17, Handbook
Zvi Wiener
slide 28
Mapping
Replacing the instruments in the portfolio by
positions in a limited number of risk factors.
Then these positions are aggregated in a
portfolio.
Ch. 17, Handbook
Zvi Wiener
slide 29
Delta-Normal method
Assumes
• linear exposures
• risk factors are jointly normally distributed
The portfolio variance is
 (returns)  x x
2
T
Forecast of the covariance matrix for the horizon
Ch. 17, Handbook
Zvi Wiener
slide 30
Delta-normal
Valuation
linear
Distribution
normal
Extreme events
low prob.
Ease of comput. Yes
Communicability Easy
VaR precision
Bad
Major pitalls
nonlinearity
fat tails
Ch. 17, Handbook
Zvi Wiener
Histor.
full
actual
recent
intermed.
Easy
depends
unstable
MC
full
general
possible
No
Difficult
good
model
risk
slide 31
FRM-97, Question 12
Delta-Normal, Historical-Simulations, and MC are
various methods available to compute VaR. If
underlying returns are normally distributed, then
the:
A. DN VaR will be identical to HS VaR.
B. DN VaR will be identical to MC VaR.
C. MC VaR will approach DN VaR as the number
of simulations increases.
D. MC VaR will be identical to HS VaR.
Ch. 17, Handbook
Zvi Wiener
slide 32
FRM-97, Question 12
Delta-Normal, Historical-Simulations, and MC are
various methods available to compute VaR. If
underlying returns are normally distributed, then
the:
A. DN VaR will be identical to HS VaR.
B. DN VaR will be identical to MC VaR.
C. MC VaR will approach DN VaR as the number
of simulations increases.
D. MC VaR will be identical to HS VaR.
Ch. 17, Handbook
Zvi Wiener
slide 33
FRM-98, Question 6
Which VaR methodology is least effective for
measuring options risks?
A. Variance-covariance approach.
B. Delta-Gamma.
C. Historical Simulations.
D. Monte Carlo.
Ch. 17, Handbook
Zvi Wiener
slide 34
FRM-98, Question 6
Which VaR methodology is least effective for
measuring options risks?
A. Variance-covariance approach.
B. Delta-Gamma.
C. Historical Simulations.
D. Monte Carlo.
Ch. 17, Handbook
Zvi Wiener
slide 35
FRM-99, Questions 15, 90
The VaR of one asset is 300 and the VaR of
another one is 500. If the correlation between
changes in asset prices is 1/15, what is the
combined VaR?
A. 525
B. 775
C. 600
D. 700
Ch. 17, Handbook
Zvi Wiener
slide 36
FRM-99, Questions 15, 90


2
A B
2
A B
     2  A B
2
A
300  500
 300  500 2
15
2

Ch. 17, Handbook
2
B
2
2
AB
 600
Zvi Wiener
2
slide 37
Example
On Dec 31, 1998 we have a forward contract
to buy 10M GBP in exchange for delivering
$16.5M in 3 months.
St - current spot price of GBP in USD
Ft - current forward price
K - purchase price set in contract
ft - current value of the contract
rt - USD risk-free rate, rt* - GBP risk-free rate
 - time to maturity
Ch. 17, Handbook
Zvi Wiener
slide 38
1
1
*
Pt  PV ($1) 
, Pt  PV (1GBP ) 
*
1  rt
1  rt 
St
K
*
ft 

 St Pt  KPt
*
1  rt  1  rt
df t
df t
df t
*
df t  dS  dP  * dP
dS
dP
dP
*
*
 P dS  SdP  KdP
Ch. 17, Handbook
Zvi Wiener
slide 39
*
dS
dP
* dP
df  SP
 SP *  KP
S
P
P
*
The forward contract is equivalent to
a long position of SP* on the spot rate
a long position of SP* in the foreign bill
a short position of KP in the domestic bill
Ch. 17, Handbook
Zvi Wiener
slide 40
GBP10M  St $16.5M
Vt  Qf t 

*
1  rt 
1  rt
On the valuation date we have
S = 1.6595, r = 4.9375%, r* = 5.9688%
Vt = $93,581 - the current value of the
contract
Ch. 17, Handbook
Zvi Wiener
slide 41