Download Value at Risk (VaR)

Central Bank of Egypt
Value at Risk (VaR)
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Central Bank of Egypt
Index
I. What is VaR?
II. Types of VaR
1.
2.
3.
Historical VAR
Parametric VAR
Monte Carlo VAR
III. Back-testing VaR
IV. Stress Testing
V. Scenario Analysis
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I. What is VaR?
• What is VaR?
– Estimate of the loss from a given position over a fixed
time period that will be equaled or exceeded with a
given probability
• VaR has two equivalent interpretations:
– Worst Case Loss: over one day, there is a 95%
probability that we will not lose more than $ yy
– An unlikely event: on average, in one out of every 20
days, we should expect to incur a loss greater than or
equal to a certain amount
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I. What is VaR?
Example: Suppose an investor placed $ 100,000 in a certain
type of investment with an expected annual return of 10%
and 95% VaR = -15%.
This means that the investment is expected to generate a
10% return over 1 year but :
 There is 5% probability that the investor can lose $
15,000 or more over the next year.
 The investor is 95% confident that the maximum
amount he is going to lose is not going to exceed
$15,000
VaR = 100,000 * -15% = -15,000
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I. What is VaR?
• Key aspects in VaR:
 VaR measures the minimum potential loss at the stated
probability; the actual loss that could be incurred could
be higher.
 VaR is associated with a stated degree of probability.
Lowering the probability (increasing the confidence
interval) increases the VaR.
 VaR measure is associated with a specific time period.
Increasing the time interval will increase the VaR.
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II. Types of VaR
• Different types of VAR measures:
– Historical VAR
– Analytic VAR (Variance Covariance, Parametric)
– Monte Carlo VAR
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1. Historical VaR
• Normality Assumption is not required.
• Works on historical returns.
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1. Historical VaR Calculations
• Step 1: Collect data on historical returns for asset/portfolio.
These returns should be measured over a time interval =
desired VaR time period.
• Step 2: from this info, construct a histogram of historical
return data.
• Step 3: VaR is the return associated with the cumulative
probability from the left tail of the histogram that equals
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1. Historical VaR
• Advantages:
 Because it is non-parametric, historical method does not
require normality assumption
 Easy to understand and implement.
 Based only on historical information.
• Disadvantages:
 History repeats itself (same return distribution)
 A biased estimate of forward looking VaR (taking
historical returns of 10 years bonds misleading as
duration was higher years ago)
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2. Parametric VaR
• The most common measure of risk is standard deviation
of the distribution of returns.
• Higher volatility = higher risk = potential for higher
losses.
• Using standard deviation and some assumptions about
returns, we can create a confidence interval, or a probable
distribution of returns.
• Accordingly, VaR is the chance of losing a potential
amount over the holding period with a certain confidence
level.
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4.0%
3.7%
3.4%
3.1%
20
2.8%
2.5%
30
2.2%
1.9%
1.6%
1.3%
1.0%
0.7%
0.4%
0.1%
-0.2%
-0.5%
-0.8%
-1.1%
25
-1.4%
-1.7%
-2.0%
Central Bank of Egypt
2. Parametric VaR
45
40
35
Expected Return
Value at Risk, p%
Volatility
15
10
5
-
Return
Conditional VaR
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2. Parametric VaR
Assumptions:
• Parametric VaR assumes that asset returns are normally
distributed with known mean and standard deviation
over a specified time period
• Covariances (correlations) among assets are known for the
same time interval.
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2. Parametric VaR
Assumption of Normality
• VaR is calculated assuming normal market conditions
• What do we mean by “normal”?
 Gaussian
 Normal market circumstances –extreme market
conditions are not included; they are examined
separately
• Therefore VaR is meant to show what can happen to the
portfolio on a day-to-day basis.
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2. Parametric VaR
• Inputs into the VaR calculation:
 Market values of all securities in the portfolio
 Their volatilities
 The correlations between them
• The assumption is that the movement of the components of
the portfolio are random, and drawn from a normal
distribution
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2. Parametric VaR
VaR α = μ - Z α * σ
A more conservative approach:
VaR α = Z α * σ
• A zero expected return is assumed when performing a VaR calculation
using the parametric method. This is generally an acceptable
assumption for daily VaR calculations because daily returns usually are
very small; it is less acceptable for VaR calculations with longer time
horizons because returns tend to deviate substantially from zero as the
time horizon increases.
• The effect of assuming a zero expected return is to produce a more
conservative (larger) projected loss.
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2. Parametric VaR
• Here is the standardized normal distribution table
Degree of confidence
99%
98%
97%
96%
95%
90%
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Number of Standard Deviation from the mean
2.326
2.054
1.881
1.751
1.645
1.282
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2. Parametric VaR
a) Measuring Risk (Individual level)
VaR α = Market Value * Z α * σ
•
Example: Compute the 5% annual VaR of a $100,000
portfolio with an annual expected return of 8% and
standard deviation of 12%.
VaR0.05,1yr = μ - Z α * σ = 8% - 1.645 (12%) = -11.74%
VaR0.05,1yr = -11.74% * $ 100,000 = -$11,740
Or
VaR0.05,1yr = Z α * σ = - 1.645 (12%) = -19.74%
VaR0.05,1yr = -19.74% * $ 100,000 = -$19,740
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2. Parametric VaR
a) Measuring Risk (Individual level)
•
VaR for periods longer than 1 day can be determined by
multiplying the daily VaR by the square root of the number
of trading days that compose the longer VaR period.
•
Example: Compute the 5% daily VaR of a $100,000
portfolio with an annual expected return of 8% and
standard deviation of 12%.
μdaily = μ annual / 250 = 8%/250 = 0.032%
σ daily = σ annual / √250 = 12% / √250 = 0.76%
VaR0.05,daily = μdaily - Z αdaily * σ = 0.032% - 1.645 (0.76%) = -1.216%
VaR0.05,daily = -1.216% * $ 100,000 = $1,216
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2. Parametric VaR
a) Measuring Risk (Individual level)
• Compute weekly and monthly VaR for the same exercise
VaR0.05,weekly = $1,216 * √5 = $2,720
VaR0.05,monthly = $1,216 * √22 = $ 5,706
•
Compute 10 days VaR from an annual VaR = $ 2,000
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2. Parametric VaR
b) Measuring Risk (Aggregate level)
• Portfolio VaR is the sum of individual VaRs only if
instruments are perfectly correlated.
• In all other cases, portfolio VaR is less than the sum of
individual VaRs.
• We have to account for correlation in calculating portfolio
VaR.
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2. Parametric VaR
•
Example: A portfolio of $ 100,000 is composed of 2
assets:
- A stock whose expected return is 10% with a standard
deviation of 20%
- A bond whose expected annual return is 5% with a
standard deviation of 12%.
If an investor puts 60% in the stock and 40% in bonds.
what is the expected annual return, standard deviation and
95% VaR assuming a correlation of 0.30.
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2. Parametric VaR
Interest rate Risk:
• We need to make a few assumptions about how yields
move through time
• In particular, we generally assume that–Changes in yields
are normally distributed with a mean of zero and a given
standard deviation
• The change in bond prices, for a given change in yields, can
be described by modified duration
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2. Parametric VaR
Interest rate Risk:
• Modified duration and DV01 tell us, reasonably accurately,
the change in the value of our portfolio for a given change
in yields.
• There is, nevertheless, a drawback
 They don’t tell us how much yields are likely to move
 What we need is some idea of the probability of a given
change in bond yields, and the likely effect of this
change on our portfolio
Worst - Case Loss  DV 01  z    yieldchange .
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2. Parametric VaR
• Advantages:
 Easy to compute
 Take into account correlations
• Disadvantages:
 Requires that individual asset returns be normally
distributed
 The method assumes expected asset returns, standard
deviation and correlations are constant.
 Not easily measured for portfolio with options (because
return distribution of such portfolios are highly skewed,
non-linear payoff)
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3. Monte Carlo VaR
• Analytic method: used when distribution of returns is
normal and the parameters of the model can be estimated
with normal accuracy.
• Historical method: used when historical return distribution
can be reasonably expected to reflect future return historical
distribution whether or not the distribution is normal.
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3. Monte Carlo VaR
• However, there are circumstances when it is desired to
determine the VaR of an item for which:
 There are insufficient historical return data to construct
an accurate picture of its return distribution (ex newly
issued securities or new financial instruments)
 Future return distribution will be different from the past
return distribution.
 Parameters that define the return distribution are
unknown but they can modeled.
Use Monte Carlo simulation method to estimate
VaR.
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3. Monte Carlo VaR
• In Monte Carlo VaR, the analyst specifies a model
consisting of at least one random variable that is used to
determine the return on the asset or portfolio/distribution of
the possible values of every random variable in the model.
• Based on these inputs, the computer generates the return
distribution for this item and its VaR.
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3. Monte Carlo VaR
Example: Black Scholes Merton Model
• A portfolio manager wants to determine the weekly VaR at
the 1% probability level for a call option on a highly
volatile stock.




Price of the underlying stock.
Risk free interest rate
Option strike price
Dividend declared on the underlying stock during the time until the option
expires.
 Volatility of the underlying stock
 Time until the option expires.
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3. Monte Carlo VaR
• All other parameters are random variables that can take any
value over the forecast period.
• The manager can specify the probability distribution that
every parameter based on experience.
• When the process is complete, the computer sorts the
possible returns and selects the 1st percentile return and
multiply it by the amount invested in the option to
determine the 1% weekly VaR.
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3. Monte Carlo VaR
• Those distributions are the inputs to the Monte Carlo
Simulation.
• The computer then randomly selects a value for each
random variable in the option pricing model from a random
number generator that is conforming with the distributions
described by the manager.
• This process is repeated 100,000 times with each iteration
generating a new weekly return.
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3. Monte Carlo VaR
• Advantages:
 Flexibility: ability to determine the type of the
distribution (normal, Poisson, exponential etc..) and the
numerical parameter values of the distribution (mean,
standard deviation etc..)
 It can be used to analyze nonlinear (non-normal return
distribution) as well as linear risks (normally distributed
returns).
 More likely to generate outlier possibilities than would
historical analysis ---- which are referred to as the
disaster scenarios.
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3. Monte Carlo VaR
• Disadvantages:
 It requires the risk analyst to develop appropriate
valuation models for the assets in a portfolio and to
specify realistic values for the parameters of the random
variables contained in the models. Otherwise, “garbage
in, garbage out”.
 It requires more computer time and power and more
analyst judgment than other methods.
 Because it is based on random number generations,
different runs of the process on the same parameters
can produce different VaRs.
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III. Backtesting VaR
• Backtesting: is the process of comparing losses predicted
by the VaR model to those actually experienced over the
sample testing period.
• If a model were completely accurate, we would expect VaR
to be exceeded with the same frequency predicted by the
confidence level used in the VaR model.
• 3 desirable attributes of VaR estimates that can be evaluated
when using a backtesting approach:
» Unbiased
» Adaptable
» Robust
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IV. Stress Testing
• During time of crisis, a contagion effect often occurs where
correlations and volatility both increase and thus reduce any
diversification benefits.
• Stressing the correlation is a method used to model the
contagion effect that could occur in a crisis event.
• One approach for stress testing is to examine historical
crisis events such as the Asian crisis.
• Advantage: no assumptions of underlying asset returns or
normality are needed.
• Disadvantage: it is limited to only evaluating events that
have actually occurred.
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V. Scenario Analysis
• Analyze different pre-determined stress scenarios such as
200 bps increase in short term rates, an extreme inversion
of the yield curve or an increase in volatility.
• Advantage: It is not limited to the evaluation of risks that
have occurred historically.
• Disadvantage: the risk measure can be deceptive.
• The worst case scenario (WCS): assumes that an
unfavorable event will occur with certainty. The focus is on
the distribution of worst possible outcomes given an
unfavorable event.
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Thank you
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