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Financial Risk Management
Following P. Jorion, Value at Risk, McGraw-Hill
Chapter 9
VaR Methods
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VaR Methods
�Local Valuation Methods
�valuing the portfolio once, using local derivatives :
�delta normal method
�delta-gamma ("Greeks") method
�Most appropriate to portfolios with with limited sources
of risk.
�Full Valuation Methods
�re-pricing the portfolios over a range of scenarios,
including:
�Historical
�Monte Carlo
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Delta Normal Methods
�Usually rely on normality assumption
�Worst loss for V is attained for extreme values of S
� If dS/S is normal, the portfolio VaR is:
� αααα is the standard normal deviate corresponding to the
confidence level, e.g. 1.645 for a 95% confidence level
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Delta Normal - Fixed Income Portfolio
The price-yield relationship:
where D* is the (modified) Duration
where σσσσ is the volatility in of change in level of yield
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Distribution with linear exposure
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Approximation depends on the optionality of the portfolio and the horizon
� For options (as well as bonds) non linearities exist,
� However, they don't necessarily invalidate the delta normal method for small changes and/or short term horizons
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Full Valuation
� Delta Normal may become inadequate:
� when the worst loss may not be obtained for extremes realizations
of the underlying
� options are near expiration and at-the-money with unstable deltas
(straddle, barriers, ...)
� The Full Valuation considers the portfolio for a wide range
of price levels:
� The new values can be generated by simulation methods
�Monte Carlo: sampling from a distribution (e.g. normal)
�Historical Simulations: sampling from historical data
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Full Valuation
�The portfolio is priced for each draw
�VAR is then calculated from the percentiles of the
full distribution of payoffs.
� it accounts for
�non linearities
� income payments
� time decay
�potentially:
� the most accurate method
� but the most computationally demanding
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Delta Gamma Approximations
�Extends the delta normal method with higher
moments
Γ Γ Γ Γ Γ second derivative of portfolio value
Γ Θ Θ Θ Θ is the time drift
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Delta Gamma - Examples
�Fixed Income
�D is the Duration, C is the convexity
�Vanilla Call Options:
�valid for long (Γ>0) Γ>0) Γ>0) Γ>0) or short (Γ<0) Γ<0) Γ<0) Γ<0)
�The second term decrease the linear VAR.
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Delta Gamma for a long call
� the downside risk for the option is less than given by deltaapproximation .... this is the "raison d'être" of option ...
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Delta Gamma for complex portfolios
� taking the variance at both side:
� then, under normal hypothesis:
0),cov( and )](variance[2)(variance 222 == dSdSdSdS
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Delta Gamma - Cornish Fisher Expansion
ξ is the Skewness
ξ Negative Skewness increases VAR
ξ the same applies for positive Excess Kurtosis
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Skewness
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Kurtosis
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Delta Gamma Monte Carlo
�also known as the partial simulation method:
�Create random simulation for risk factors
� then uses Taylor expansion (delta gamma) to
create simulated movements in option value
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Delta Gamma - Multiple risk factors
�∆∆∆∆ and dS are vectors
�computationally intensive
�requires estimates of:
�Gamma (implicit correlations)
�Covariance matrix
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Comparison of methods
� For lager portfolios where optionality is not dominant, the delta normal method provides a fast and efficient method for measuring VAR
� For portfolios exposed to few sources of risk and with substantial option components, the Greeks (delta-gamma) provides increase precision at low computational cost
� For portfolios with substantial option components or longer horizons, a full valuation method may be required
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Note on the "Root Squared Time" rule
�Normally daily VAR can be adjusted to other
period by scaling by a square root of time factor
�However, this adjustment assume:
� position is constant during the full period of time
�daily returns are independent and identically
distributed
�Hence, the time adjustment is not valid for options
positions (that can be replicated by dynamically
changing positions in underlying)
�For portfolios with large options components, the
full valuation must be implemented over the
desired horizon ...
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Example: Leeson's Straddle
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Sell Straddle payoff
Sell Straddle = sell call + sell put
Strike = at the money
Successful, only if the spot remains stable
Delta = 0
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Example: Leeson's Straddle
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Example: Leeson's Straddle
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Example: Leeson's Straddle
VaR Analysis could have prevented bankruptcy
if positions were known
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Example: Leeson's Straddle
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Example: Leeson's Straddle
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Example: Leeson's Straddle
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Example: Leeson's Straddle
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Delta Normal Implementation
�Simple porfolios
�More complex portfolios / instruments
� specifying a list of risk factors
� mapping the linear exposure of all instruments onto
these risk factors
�estimating the covariance matrix of risk exposure
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Delta Method Implementation
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Delta Normal Implementation
�Advantages
�easy to implement (matrix computation)
�fast
� simple to explain
�adequate in many situations
�Problems
� fat tails ���� under estimate risks
� inadequate for non linear instrument
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Historical Simulation Implementation
�Consist in going back in time (say 250 days), and
apply historical returns
�Hypothetical prices for scenario k provide a new
portfolio value
�Then VAR is estimated from the full sample
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Historical Simulation Implementation
�Advantages
� simple to implement (brute force)
� if historical data are available ...
� no need to estimate covariance matrix, etc ...
� model free method
�allow non linearities, capturing gamma, vega,
correlations risks
�account for fat tails
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Historical Simulation Implementation
�Problems
�assume we have sufficient historical data
� only one sample path is used
� assume that past data is representative of the future� the window may omit important data
� or n the other hand, may include not relevant data
� simple historical simulation may miss some dynamic
aspects (time varying volatility and clustering, ...)
� put the same weight on all observations, including old
data
� quickly become cumbersome for large portfolios
� note: most of the problems can be mitigated by time varying models like
GARCH, RiskMetrics, ...
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Monte Carlo Implementation
� 2 steps procedure
� specifying stochastic
processes for financial
variables
� then simulate price paths
� At each horizon
considered, the portfolio is
evaluated
� VAR is estimated from
simulated portfolio values
� similar to historical
simulation, except that
hypothetical price changes
is created by random draws
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Monte Carlo Implementation - Advantages
� by far the most powerful method to compute VAR
� account for a wide range of risk and features, including
� non linear price risk
� time varying volatility
� fat tails
� extreme scenarios
� can also be used to estimate expected loss beyond the VAR
� time decay of options
� effect of pre defined trading or hedging dynamic strategies
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Monte Carlo Implementation - Problems
�Major drawback: computation time
� ex: 10000 sample path for 100 assets => 1 million full valuations
� in addition, each valuation may require inner simulation to price derivatives, for example ! (Monte Carlo of Monte Carlo)
� too heavy to implement on a regular day to day basis
� require strong skills and infrastructure (Software & Hardware)
� Model Risk
� in case the stochastic processes and pricing formulas are wrong ���� sensitivity analysis
�Subject to (Small) Sample Variation Effects
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Empirical Comparisons
� Foreign currency portfolio
� Delta Normal is
� at 99% confidence level, slightly underestimate actual VAR
� the fatest method
� Full Monte Carlo
� most accurate
� slowest method
� for lage portfolios, bank still prefer the delta normal, however, this method may dangerously underestimate actual losses in case of optionality features
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Comparison of approaches to VAR
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Aactual Uses of Methods
�In practice all methods are used by bank:
� 42% delta normal and simple covariance
approach
� 31% use historical simulation
� 23% Monte Carlo
source Britain's FSA survey