VAR
The Question Being Asked in VaR
“What loss level is such that we are X%
confident it will not be exceeded in N business
days?”
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.2
VaR and Regulatory Capital
• Regulators base the capital they require
banks to keep on VaR
• The market-risk capital is k times the 10-day
99% VaR where k is at least 3.0
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.3
VaR vs. C-VaR
• VaR is the loss level that will not be exceeded
with a specified probability
• C-VaR (or expected shortfall) is the expected
loss given that the loss is greater than the VaR
level
• Although C-VaR is theoretically more
appealing, it is not widely used
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.4
Advantages of VaR
• It captures an important aspect of risk
in a single number
• It is easy to understand
• It asks the simple question: “How bad can
things get?”
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.5
Time Horizon
• Instead of calculating the 10-day, 99% VaR directly
analysts usually calculate a 1-day 99% VaR and
assume
10 - day VaR  10  1- day VaR
• This is exactly true when portfolio changes on
successive days come from independent identically
distributed normal distributions
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.6
The Model-Building Approach
• The main alternative to historical simulation is to
make assumptions about the probability
distributions of return on the market variables and
calculate the probability distribution of the change in
the value of the portfolio analytically
• This is known as the model building approach or the
variance-covariance approach
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.7
Description of Value at Risk:
• Definition:
– Value at Risk is an estimate of the worst possible loss an investment
could realize over a given time horizon, under normal market
conditions (defined by a given level of confidence).
– To estimate Value at Risk a confidence level must be specified.
Choice of confidence level – 95%
5%
95%
Investment returns
Normal market conditions – the returns that account for 95% of the
distribution of possible outcomes.
Abnormal market conditions – the returns that account for the other 5% of
the possible outcomes.
If a 95% confidence level is used to estimate
Value at Risk for a monthly horizon;
losses greater than the Value at Risk
estimate are expected to occur one in
twenty months (5%).
Illustrate Value at Risk:
• Step 1: Transform simple monthly stock returns into continuously
compounded stock returns.
Note: Technically, log stock returns are “more likely” to be normally distributed.
• Step 2: Choose a level of confidence.
– 90%, 95%, 99%, etc.
– Banks are required to report Value at Risk estimated with a 99% level
of confidence to determine regulatory capital requirements.
• Step 3: Compute Value at Risk from sample estimates of  and .
– For example, the largest likely loss in the household industry over the
next month under normal market conditions with a 95% level of
confidence is: $18,000.
Note: It is possible to realize a loss greater than $18,000.
Other Common Interpretations of Value at
Risk:
• “an attempt to provide a single number for senior management
summarizing the total risk in a portfolio of assets”
– Hull, OF&OD
• “an estimate, with a given degree of confidence, of how much one can
lose from one’s portfolio over a given time horizon”
– Wilmott, PWOQF
Conclusions:
• Value at Risk can be used as a stand alone risk measure or be applied to a
portfolio of assets.
• Value at Risk is a dollar value risk measure, as opposed to the other
measurements of risk in the financial industry such as: beta and standard
deviation.
• “We are X percent certain that we will not lose more than V dollars in the
next N days.” – Hull
Measures of Risk
• Standard Deviation ()
• Beta (ß)
• Value at Risk (VaR)
Measured by
VAR
Stand-Alone Risk
Or
Total Risk
Measured by
ß

Systematic
Risk
+
Unsystematic
Risk

NonDiversifiable
Risk
+
Diversifiable
Risk
+
CompanySpecific Risk
 Market Risk
Dispersion of Returns –
Variances and Standard Deviations
• Variance (2) Formula:

n
2

  ( ki  k ) 2
i 1
• Variance and Standard Deviation are measures of total (or
stand-alone) risk.
• The larger the variance (or Std. Dev.), the lower the
probability that actual returns will be close to the
expected return.
Risk Measure - Beta (ß)
• Beta (ß) formula:
Cov( ki , k m )

Var ( k m )
• Beta measures the portfolio’s systematic risk, that
is, the degree to which its return is correlated
with the return on the market as a whole.
• Stock with high beta (ß>1) is more volatile than
the market taken as a whole.
Risk Measures – Value at Risk (VaR)
• VaR is a measure of risk based on a probability
of loss and a specific time horizon.
• VaR translates portfolio volatility into a dollar
value.
• Measure of Total Risk rather than Systematic
(or Non-Diversifiable Risk) measured by Beta.
Advantages of VaR
• VaR provides an measure of total risk.
• VaR is an easy number to understand and explain
to clients.
• VaR translates portfolio volatility into a dollar
value.
• VaR is useful for monitoring and controlling risk
within the portfolio.
Advantages of VaR (Cont.)
• VaR can measure the risk of many types of financial
securities (i.e., stocks, bonds, commodities, foreign
exchange, off-balance-sheet derivatives such as
futures, forwards, swaps, and options, and etc.)
• As a tool, VaR is very useful for comparing a portfolio
with the market portfolio (S&P500).
Monte Carlo Simulation
• Calculate DP
• Repeat many times to build up a probability
distribution for DP
• VaR is the appropriate fractile of the
distribution times square root of N
• For example, with 1,000 trial the 1 percentile
is the 10th worst case.
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.21
Stress Testing
• This involves testing how well a portfolio
performs under some of the most extreme
market moves seen in the last 10 to 20 years
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.22
Back-Testing
• Tests how well VaR estimates would have
performed in the past
• We could ask the question: How often was the
actual 10-day loss greater than the 99%/10
day VaR?
Options, Futures, and Other Derivatives 6th
Edition, Copyright © John C. Hull 2005
18.23
VaR is useful also for:
– Limits of risk control
– Measure return adjusted for risk  RAROC
– Pricing
– CAPITAL Allocation
24
How to calculate Var
•
VaR of a position is measured by :
–
–
–
Market Value of position (VM)
Sensitivity of price to market variations ()
Adverse price variation as the product of:
•
•
Volatility of instruments ()
A factor  corrispondin to the confidence levele desired
VaRi  VM i   i   i  
25
example
•
BTP 10y for nominal €1mln
–
–
–
–
Ptel quel = 105
DM
= 7 anni
rend.gg.
= 15 b.p. (0,15%)

= 2,326 (c = 99%)
VaRBTP  1.050.000  7  0,15%  2,326  25.644,15
26
VaR of a portfolio
• Can be estimated using portfolio theory :
– MARKET VALUES and sensitivity of all single instruments
– Volatility of single instruments
– Correlatons between the N instruments
VaRP, N 
N N
  (VM i   i     i )  (VM j   j     j )   ij
i 1 j 1
27
The limits of VaR models





They disregard exceptional events
They disregard clint relationships (short termism)
They are based on non realistic hypotheses
They amplify instability of financial markets
They arrive late
28
The major limit is maybe that ...
• They might disregard the magnitude of the
loss:
– VaR with confidence level of 99%
–  probability to lose more than VaR is only 1%…
– …but HOW MUCH CAN YOU LOSE IF EVENTS
INCLUDED IN THAT 1 % HAPPEN ?
29