Value-at-Risk (VaR)
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
May-00
Risk Management
Risk

Business Risk

Financial Risk
– market risk
– credit risk
– liquidity risk
VaR

Operational Risk

Legal Risk
May-2000
slide 2
How much can we lose?
Everything
correct, but useless answer.
How much can we lose realistically?
VaR
May-2000
slide 3
Derivatives 1993-1995
Shova
Shell, Japan
Kashima Oil, Japan
Metallgesellschaft
Barings, U.K.
Codelco, Chile
Procter & Gamble, US
VaR
May-2000
($ million)
1,580
1,450
1,340
1,330
200
157
slide 4
Barings
VaR

February 26, 1995

233 year old bank

28 year old Nick Leeson

$1,300,000,000 loss

bought by ING for $1.5
May-2000
slide 5
Public Funds
Orange County
 San Diego
 West Virginia
 Florida State Treasury
 Cuyahoga County
 Texas State

VaR
May-2000
($ million)
1,640
357
279
200
137
55
slide 6
Orange County

Bob Citron, the county treasures

$7.5B portfolio (schools, cities)

borrowed $12.5B, invested in 5yr. notes

interest rates increased

reported at cost - big mistake!

realized loss of $1.64B
VaR
May-2000
slide 7
Financial Losses
Barings
 Bank Negara, Malaysia 92
 Banesto, Spain
 Credit Lyonnais
 S&L, U.S.A.
 Japan

VaR
May-2000
$1.3B
$3B
$4.7B
$10B
$150B
$500B
slide 8
Metallgesellshaft
14th largest industrial group
 58,000 employees
 offered long term oil contracts
 hedge by long-term forward contracts
 short term contracts were used (rolling hedge)
 1993 price fell from $20 to $15
 $1B margin call in cash

VaR
May-2000
slide 9
VaR
May-2000
slide 10
What is the current Risk?
Bonds
 Stocks
 Options
 Credit
 Forex
 Total

VaR
duration, convexity
volatility
delta, gamma, vega
rating
target zone
?
May-2000
slide 11
Standard Approach
VaR
May-2000
slide 12
Modern Approach
Financial Institution
VaR
May-2000
slide 13
Risk Management
VaR

Risk measurement

Reporting to board

Limits monitoring

Diversification, reinsurance

Vetting

Reporting to regulators

Decision making based on risk
May-2000
slide 14
Who manages risk?
Citibank
AIG
Nike
Bank of England General Re Sony
CIBC
Swiss Re
Dell Computers
J. P. Morgan
Aetna
Philip Morris
Bankers Trust
Zurich
Ford Motor
VaR
May-2000
slide 15
Regulators
VaR

BIS

FSA

SEC

ISDA

FASB

Bank of Israel

Galai’s committee
May-2000
slide 16
Basic Steps in RM process
VaR

Identify risks

Data base (market + position)

Risk measurement

Regulators

Risk Management

Reporting

Strategic decisions
May-2000
slide 17
Building a RM system
VaR

Initial study of risks

Decision, Risk Manager

Risk measurement system

Responsibilities and structure

Testing

Active Risk Management

Staff training and maintenance
May-2000
slide 18
Risk Management and
Risk Measurement
VaR
May-2000
slide 19
Risk Management System Can NOT
Predict future
 Identify business opportunities
 Be always right!

Risk Management System Can
Predict loss, given event
 Identify most dangerous scenarios
 Recommend how to change risk profile

VaR
May-2000
slide 20
Tool, not rule!
VaR
May-2000
slide 21
Definition
VaR is defined as the predicted worst-case
loss at a specific confidence level (e.g. 99%)
over a certain period of time.
VaR
May-2000
slide 22
VaR
1
0.8
0.6
0.4
VaR1%
1%
0.2
Profit/Loss
-3
VaR
-2
-1
1
May-2000
2
3
slide 23
Meaning of VaR
A portfolio manager has a daily VaR equal
$1M at 99% confidence level.
This means that there is only one chance in
100 that a daily loss bigger than $1M occurs,
under normal market conditions.
VaR
1%
VaR
May-2000
slide 24
History of VaR
VaR

80’s - major US banks - proprietary

93 G-30 recommendations

94 - RiskMetrics by J.P.Morgan

98 - Basel

SEC, FSA, ISDA, pension funds, dealers

Widely used and misused!
May-2000
slide 25
Risk Management Structure
Market data
Current position
Risk Mapping
Valuation
Value-at-Risk
Reporting and Risk Management
VaR
May-2000
slide 26
Value
8.25
8
7.75
7.5
7.25
10
4.3
4.25
4.2
11
4.15
12
13
Interest Rate
14
4.1
dollar
interest rates and dollar are
NOT independent
VaR
May-2000
slide 27
Risk Measuring Software
CATS, CARMA
 Algorithmics, Risk Watch
 Infinity
 J.P. Morgan, FourFifteen
 FEA, Outlook
 Reuters, Sailfish
 Kamacura
 Bankers Trust, RAROC
 INSSINC, Orchestra

VaR
May-2000
slide 28
Qualitative Requirements
 An
independent risk management unit
 Board of directors involvement
 Internal model as an integral part
 Internal controller and risk model
 Backtesting
 Stress test
VaR
May-2000
slide 29
Quantitative Requirements
99% confidence interval
 10 business days horizon
 At least one year of historic data
 Data base revised at least every quarter
 All types of risk exposure
 Derivatives

VaR
May-2000
slide 30
Types of Assets and Risks
VaR

Real projects - cashflow versus financing

Fixed Income

Optionality

Credit exposure

Legal, operational, authorities
May-2000
slide 31
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
VaR

Stock indices
May-2000
slide 32
How to measure VaR

Historical Simulations

Variance-Covariance

Monte Carlo
 Analytical
VaR
Methods
May-2000
slide 33
Historical Simulations

Fix current portfolio.

Pretend that market changes are
similar to those observed in the past.
VaR

Calculate P&L (profit-loss).

Find the lowest quantile.
May-2000
slide 34
Example
Assume we have $1 and our main currency is
SHEKEL. Today $1=4.30.
Historical data:
P&L
4.00
VaR
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
4.10*4.10/4.20 = 4.198
-0.112
4.15
4.15*4.15/4.10 = 4.352
0.052
May-2000
slide 35
USD
NIS
2000
100
-120
2001
200
100
2002
-300
-20
2003
20
30
today
VaR
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
May-2000
slide 36
today
Changes
in IR
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
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
VaR
May-2000
slide 37
Returns
year
1% of worst cases
VaR
May-2000
slide 38
VaR
1
0.8
0.6
0.4
VaR1%
1%
0.2
Profit/Loss
-3
VaR
-2
-1
1
May-2000
2
3
slide 39
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.
VaR
May-2000
slide 40
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!
VaR
May-2000
slide 41
Variance-Covariance VaR1%  V  2.33 V
1%
2.33
-2.33
VaR

May-2000
slide 42
Monte Carlo
1
0.5
-1
0.5
-0.5
1
-0.5
-1
VaR
May-2000
slide 43
Monte Carlo
VaR

Distribution of market factors

Simulation of a large number of events

P&L for each scenario

Order the results

VaR = lowest quantile
May-2000
slide 44
Monte Carlo Simulation
15
10
5
10
20
30
40
-5
-10
-15
VaR
May-2000
slide 45
Real Projects
Most daily returns are invisible.
Proper financing should be based on risk
exposure of each specific project.
Note that accounting standards not always
reflect financial risk properly.
VaR
May-2000
slide 46
Example
 You

are going to invest in Japan.
Take a loan in Yen.
Financial statements will reflect your
investment according to the exchange rate
at the day of investment and your liability
will be linked to yen.

 Actually
VaR
there is no currency risk.
May-2000
slide 47
Airline company
fuel - oil prices and $
 purchasing airplanes - $ and Euro
 salaries - NIS, some $
 tickets $
 marketing - different currencies
 payments to airports for services

VaR
May-2000
slide 48
Airline company
VaR

loans

equity

callable bonds
May-2000
slide 49
Airline company
Base currency - by major stockholder.
Time horizon - by time of possible price change.
Earnings at risk, not value at risk, since there is
too much optionality in setting prices.
One can create a one year cashflow forecast and
measure its sensitivity to different market events.
VaR
May-2000
slide 50
Reporting
Division of VaR by business units, areas of
activity, counterparty, currency.
Performance measurement - RAROC (Risk
Adjusted Return On Capital).
VaR
May-2000
slide 51
How VaR is used
VaR

Internal Risk Management

Reporting

Regulators
May-2000
slide 52
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.
VaR
May-2000
slide 53
Backtesting
Green zone - up to 4 exceptions
OK
Yellow zone - 5-9 exceptions
increasing k
Red zone - 10 exceptions or more
intervention
VaR
May-2000
slide 54
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?
VaR
May-2000
slide 55
Unifying Approach
VaR

One number

Based on Statistics

Portfolio Theory

Verification

Widely Accepted

Easy Comparison
May-2000
slide 56
Board of Directors
(Basle, September 1998)
periodic discussions with management concerning
the effectiveness of the internal control system
 a timely review of evaluations of internal controls
made by management, internal and external auditors
 periodic efforts to ensure that management has
promptly followed up on recommendations and
concerns expressed by auditors and supervisory
authorities on internal control weaknesses
 a periodic review of the appropriateness of the
bank’s strategy and risk limits.

VaR
May-2000
slide 57
Open Questions
VaR

Risks related to cashflow

Non-traded assets

Credit information

Global Database

Liquidity problem
May-2000
slide 58
Issues Specific to Israel
VaR

Indexation

Exchange Band

Shallow Markets
May-2000
slide 59