Value at Risk

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Value at Risk
By A V Vedpuriswar
February 8, 2009
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• VAR summarizes the worst loss over a target
  horizon that will not be exceeded at a given
  level of confidence.
• For example, under normal market conditions,
  the most the portfolio can lose over a month is
  about 3.6 billion at the 99 confidence level.

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• The main idea behind VAR is to consider the
  total portfolio risk at the highest level of the
  institution.
• Initially applied to market risk, it is now used
  to measure credit risk, operational risk and
  enterprise wide risk.
• Many banks can now use their own VAR models as
  the basis for their required capital for market
  risk.

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• VAR can be calculated using two broad approaches
  
• Non parametric method This is the most
  general method which does not make any
  assumption about the shape of the distribution
  of returns.
• Parametric method VAR computation becomes much
  easier if a distribution, such as normal, is
  assumed.

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Illustration
Average revenue 5.1 million per day Total
no.of observations 254. Std dev 9.2
million Confidence level 95 No. of
observations lt - 10 million 11 No. of
observations lt - 9 million 15
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• Find the point such that the no. of
  observations to the left (254) (.05) 12.7
• (12.7 11) /( 15 11 ) 1.7 / 4 .4
• So required point - (10 - .4) - 9.6
  million
• VAR E (W) (-9.6) 5.1 (-9.6) 14.7
  million
• If we assume a normal distribution,
• Z at 95 confidence interval, 1 tailed 1.645
• VAR (1.645) (9.2) 15.2 million

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VAR as a benchmark measure

• VAR can be used as company wide yardstick to
  compare risks across different markets.
• VAR can also be used to understand whether risk
  has increased over time.
• VAR can be used to drill down into risk
  reports to understand whether the higher risk is
  due to increased volatility or bigger bets.

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VAR as a potential loss measure

• VAR can also give a broad idea of the worst loss
  an institution can incur.
• The choice of time horizon must correspond to the
  time required for corrective action as losses
  start to develop.
• Corrective action may include reducing the risk
  profile of the institution or raising new
  capital.
• Banks may use daily VAR because of the
  liquidity and rapid turnover in their
  portfolios.
• In contrast, pension funds generally invest in
  less liquid portfolios and adjust their risk
  exposures only slowly.
• So a one month horizon makes more sense.

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VAR as equity capital

• The VAR measure should adequately capture all
  the risks facing the institution.
• So the risk measure must encompass market risk,
  credit risk, operational risk and other risks.
• The higher the degree of risk aversion of the
  company, the higher the confidence level chosen.
  
• If the bank determines its risk profile by
  targeting a particular credit rating, the
  expected default rate can be converted directly
  into a confidence level.
• Higher credit ratings should lead to a higher
  confidence level.

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VAR Methods

• Mapping If the portfolio consists of a large
  number of instruments, it would be too complex
  to model each instrument separately. The first
  step is mapping. Instruments are replaced by
  positions on a limited number of risk factors.
  If we have N risk factors, the positions are
  aggregated across instruments.
• Local valuation methods make use of the
  valuation of the instruments at the current
  point, along with the first and perhaps, the
  second partial derivatives. The portfolio is
  valued only once.
• Full valuation methods, in contrast, reprice the
  instruments over a broad range of values for the
  risk factors.

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• Linear models are based on the covariance matrix
  approach.
• The matrix can be simplified using factor
  models.
• Non linear models take into account the first
  and second partial derivatives (gamma/
  convexity)

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Delta normal approach

• The delta normal method assumes that the
  portfolio measures are linear and the risk
  factors are jointly normally distributed.
• The delta normal method involves a simple
  matrix multiplication.
• It is computationally fast even with a large no.
  of assets because it replaces each position by
  its linear exposure.
• The disadvantages are the existence of fat tails
  in many distributions and the inability to
  handle non linear instruments.

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First, the asset is valued at the initial
point. V0 V(S0) dv dv/ds ds ?0 ds (?0
s) ds/s s is the risk factor. Portfolio VAR
?0 x VARs ?0 x (asS0) s Std devn of
rates of change in the price a Std normal
deviate corresponding to the specified
confidence level.
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• For more complex pay offs, local valuation is
  not enough.
• Take the case of a long straddle, i.e, the
  purchase of call and a put.
• The worst pay off (sum of the two premiums) will
  be realized if the spot rate does not move at
  all.
• In general, it is not sufficient to evaluate the
  portfolio at the two extremes.
• All intermediate values must be checked.

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Delta Gamma Method

• In linear models, daily VAR is adjusted to other
  periods, by scaling by a square root of time
  factor.
• This adjustment assumes that the position is
  fixed and the daily returns are independent and
  identically distributed.
• This adjustment is not appropriate for options
  because option delta changes dynamically over
  time.
• The delta gamma method provides an analytical
  second order correction to the delta normal VAR.

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• Gamma gives the rate of change in delta with
  respect to the spot price.
• Long positions in options with a positive gamma
  have less risk than with a linear model.
• Conversely, short positions in options have
  greater risk than implied by a linear model.

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Historical simulation method

• The historical simulation method consists of
  going back in time and applying current weights
  to a time series of historical asset returns.
• This method makes no specific assumption about
  return distribution, other than relying on
  historical data.
• This is an improvement over the normal
  distribution because historical data typically
  contain fat tails.
• The main drawback of this method is its reliance
  on a short historical moving window to infer
  movements in market prices.

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• The sampling variation of historical simulation
  VAR is greater than for a parametric method.
• Longer sample paths are required to obtain
  meaningful quantities.
• The dilemma is that this may involve
  observations that are no longer relevant.
• Banks use periods between 250 and 750 days.
• This is taken as a reasonable trade off between
  precision and non stationarity.
• Many institutions are now using historical
  simulation over a window of 1-4 years, duly
  supplemented by stress tests .

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Monte Carlo Simulation Method

• The Monte Carlo Simulation Method is similar to
  the historical simulation, except that movements
  in risk factors are generated by drawings from
  some pre specified distribution.
• The risk manager samples pseudo random numbers
  from this distribution and then generates pseudo
  dollar returns as before.
• Finally, the returns are sorted to produce the
  desired VAR.
• This method uses computer simulations to
  generate random price paths.

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• They are by far the most powerful approach to
  VAR.
• They can account for a wide range of risks
  including price risk, volatility risk, fat tails
  and extreme scenarios and complex interactions.
  
• Non linear exposures and complex pricing
  patterns can also be handled.
• Monte Carlo analysis can deal with time decay
  of options, daily settlements associated cash
  flows and the effect of pre specified trading or
  hedging strategies.

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• The Monte Carlo approach requires users to make
  assumptions about the stochastic process and to
  understand the sensitivity of the results to
  these assumptions.
• Different random numbers will lead to different
  results.
• A large number of iterations may be needed to
  converge to a stable VAR measure.
• When all the risk factors have a normal
  distribution and exposures are linear, the
  method should converge to the VAR produced by
  the delta-normal VAR.

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• The Monte Carlo approach is computationally
  quite demanding.
• It requires marking to market the whole
  portfolio over a large number of realisations
  of underlying random variables.
• To speed up the process, methods, have been
  devised to break the link between the number of
  Monte Carlo draws and the number of times the
  portfolio is repriced.
• In the grid Monte Carlo approach, the portfolio
  is exactly valued over a limited number of grid
  points.
• For each simulation, the portfolio is valued
  using a linear interpolation from the exact
  values at adjoining grid points.

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• The first and most crucial step consists of
  choosing a particular stochastic model for the
  behaviour of prices.
• A commonly used model in Monte carlo simulation
  is the Geometric Brownian motion model which
  assumes movements in the market price are
  uncorrelated over time and that small movements
  in prices can be described by
• dSt µt St dt st St dz
• dz is a random variable distributed normally
  with mean zero and variance dt.

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• This rules out processes with sudden jumps for
  instance.
• This process is also geometric because all the
  parameters are scaled by the current price, St.
  
• µt and st represent the instantaneous drift and
  volatility that can evolve over time.

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• Integrating ds/s over a finite interval, we have
  approximately
• ?St St-1 (µ ?t szv?t)
• z is a standard normal random variable with
  mean zero and unit variance.
• St1 St St (µ ?t sz1v?t)
• St2 St1 St1 (µ ?t sz2v?t)

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• Monte Carlo simulations are based on random
  draws z from a variable with the desired
  probability distribution.
• The first building block is a uniform
  distribution over the interval (0,1) that
  produces a random variable x.
• Good random number generators must create series
  that pass all conventional tests of
  independence.
• Otherwise, the characteristics of the simulated
  price process will not obey the underlying
  model.
• The next step is to transform the uniform random
  number x into the desired distribution through
  the inverse cumulative probability
  distribution.

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Selective Sampling

• Sample along the paths that are most important
  to the problem at hand.
• If the goal is to measure a tail quantile,
  accurately, there is no point in doing
  simulations that will generate observations in
  the centre of the distribution.
• To increase the accuracy of the VAR estimator,
  we can partition the simulation region into two
  or more zones.
• Appropriate number of observations is drawn from
  each region.

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• Using more information about the portfolio
  distribution results in more efficient
  simulations.
• The simulation can proceed in two phases.
• The first pass runs a traditional Monte Carlo.
• The risk manager then examines the region of the
  risk factors that cause losses around VAR.
• A second pass is then performed with many more
  samples from the region.

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Backtesting

• Backtesting is done to check the accuracy of the
  model.
• It should be done in such a way that the
  likelihood of catching biases in VAR forecasts
  is maximized.
• Longer horizon reduces the number of independent
  observations and thus the power of the tests.
• Too high a confidence level reduces the expected
  number of observations in the tail and thus the
  power of the tests.
• For the internal models approach, the Basle
  Committee recommends a 99 confidence level over
  a 10 business day horizon.
• The resulting VAR is multiplied by a
  safety factor of 3 to arrive at the minimum
  regulatory capital.

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• As the confidence level increases, the number
  of occurrences below VAR shrinks, leading to
  poor measures of high quantiles.
• There is no simple way to estimate a 99.99 VAR
  from the sample because it has too few
  observations.
• Shorter time intervals create more data points
  and facilitate more effective back testing.

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Choosing the method

• Simulation methods are quite flexible.
• They can either postulate a stochastic process
  or resample from historical data.
• They allow full valuation on the target data.
• But they are prone to model risk and sampling
  variation.
• Greater precision can be achieved by increasing
  the number of replications but this may slow the
  process down.

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• For large portfolios where optionality is not a
  dominant factor, the delta normal method
  provides a fast and efficient method for
  measuring VAR.
• For fast approximations of option values, delta
  gamma is efficient.
• For portfolios with substantial option
  components, or longer horizons, a full valuation
  method may be required.

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• If the stochastic process chosen for the price
  is unrealistic, so will be the estimate of VAR.
  
• For example, the geometric Brownian motion
  model adequately describes the behaviour of
  stock prices and exchange rates but not that of
  fixed income securities.
• In Brownian motion models, price shocks are
  never reversed and prices move as a random
  walk.
• This cannot be the price process for default
  free bond prices which must converge to their
  face value at expiration.

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V A R Applications
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• VAR methods represent the culmination of a
  trend towards centralized risk management.
• Many institutions have started to measure
  market risk on a global basis because the
  sources of risk have multiplied and volatility
  has increased.
• A portfolio approach gives a better picture of
  risk rather than looking at different
  instruments in isolation.

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• Centralization makes sense for credit risk
  management too.
• A financial institution may have myriad
  transactions with the same counterparty, coming
  from various desks such as currencies, fixed
  income commodities and so on.
• Even though all the desks may have a reasonable
  exposure when considered on an individual basis,
  these exposures may add up to an unacceptable
  risk.
• Also, with netting agreements, the total
  exposure depends on the net current value of
  contracts covered by the agreements.
• All these steps are not possible in the absence
  of a global measurement system.

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• Institutions which will benefit most from a
  global risk management system are those which
  are exposed to
• - diverse risk
• - active positions taking / proprietary trading
• - complex instruments
• .

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• VAR is a useful information reporting tool.
• Banks can disclose their aggregated risk
  without revealing their individual positions.
• Ideally, institutions should provide summary VAR
  figures on a daily, weekly or monthly basis.
• Disclosure of information is an effective means
  of market discipline.

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• VAR is also a useful risk control tool.
• Position limits alone do not give a complete
  picture.
• The same limit on a 30 year treasury, (compared
  to 5 year treasury) may be more risky.
• VAR limits can supplement position limits.
• In volatile environments, VAR can be used as the
  basis for scaling down positions.
• VAR acts as a common denominator for comparing
  various risky activities.

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• VAR can be viewed as a measure of risk capital
  or economic capital required to support a
  financial activity.
• The economic capital is the aggregate capital
  required as a cushion against unexpected losses.
  
• VAR helps in measuring risk adjusted return.
• Without controlling for risk, traders may become
  reckless.
• If the trader makes a large profit, he receives
  a large bonus.
• If he makes a loss, the worst that can happen is
  he will get fined.

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• The application of VAR in performance
  measurement depends on its intended purposes.
• Internal performance measurement aims at
  rewarding people for actions they have full
  control over.
• The individual/undiversified VAR seems the
  appropriate choice.
• External performance measurement aims at
  allocation of existing / new capital to
  existing or new business units.
• Such decisions should be based on marginal and
  diversified VAR measures.

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• VAR can also be used at the strategic level to
  identify where shareholder value is being added
  throughout the corporation.
• VAR can help management take decisions about
  which business lines to expand, maintain or
  reduce.
• And also about the appropriate level of
  capital to hold.

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• A strong capital allocation process produces
  substantial benefits.
• The process almost always leads to improvements.
  
• Finance executives are forced to examine
  prospects for revenues, costs and risks in all
  their business activities.
• Managers start to learn things about their
  business they did not know.

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Extreme Value Theory (EVT)

• EVT extends the central limit theorem which
  deals with the distribution of the average of
  identically and independently distributed
  variables from an unknown distribution to the
  distribution of their tails.
• The EVT approach is useful for estimating tail
  probabilities of extreme events.
• For very high confidence levels (gt99), the
  normal distribution generally underestimates
  potential losses.

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• Empirical distributions suffer from a lack of
  data in the tails.
• This makes it difficult to estimate VAR
  reliably.
• EVT helps us to draw smooth curves through the
  extreme tails of the distribution based on
  powerful statistical theory.
• In many cases the t distribution with 4-6
  degrees of freedom is adequate to describe the
  tails of financial data.

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• EVT applies to the tails
• Not appropriate for the centre of the
  distribution
• Also called semi parametric approach
• EVT theorem was proved by Gnedenko in 1943
• EVT helps us to draw smooth curves through the
  tails of the distribution

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EVT Theorem
F (y) 1 (1 y)- 1/ ? 0 F (y) 1
e-y 0 y (x - µ) / ß, ß gt 0 Normal
distribution corresponds to 0 Tails disappear
at exponential speed
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EVT Estimators
2 Normal EVT 0
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• Fitting EVT functions to recent historical data
  is fraught with the same pitfalls as VAR.
• Once in a lifetime events cannot be taken into
  account even by powerful statistical tools.
• So they need to be complemented by stress
  testing.
• The goal of stress testing is to identify
  unusual scenarios that would not occur under
  standard VAR models.
• Stress tests can simulate shocks that have never
  occurred or have been covered highly unlikely.
• Stress tests can also simulate shocks that
  reflect permanent structural breaks or
  temporarily changed statistical patterns.

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• Stress testing should be enforced, but the
  problem is the stress needs to be pertinent to
  the type of risk the institution has.
• It would be difficult to enforce a limited
  number of relevant stress tests.
• The complex portfolio models banks generally
  employ give the illusion of accurate simulation
  at the expense of substance.

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How effective are VAR models? VAR and sub prime

• The tendency of risk managers and other
  executives to describe events in terms of
  sigma tells us a lot.
• Whenever there is talk about sigma, it implies
  a normal distribution.
• Real life distributions have fat tails.
• Goldman Sachs chief financial officer David
  Viniar once described the credit crunch as a
  25-sigma event

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• The credit crisis of late 2007 was largely a
  failure of risk management.
• Risk models of many banks were unable to
  predict the likelihood , speed or severity of
  the crisis.
• Attention turned particularly to the use of
  value-at- risk as a measure of the risk involved
  in a portfolio.
• While a few VAR exceptions are expected 99, a
  properly working model would still produce two
  to three exceptions a year the existence of
  clusters of exceptions indicates that something
  is wrong.

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• Credit Suisse reported 11 exceptions at the
  99 confidence level in the third quarter,
  Lehman brothers three at 95, Goldman Sachs
  five at 95, Morgan Stanley six at 95, Bear
  Stearns 10 at 99 and UBS 16 at 99.
• Clearly, VAR is a tool for normal markets and it
  is not designed for stress situations.

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What window?

• It would have been difficult for VAR models to
  have captured all the recent market events,
  especially as the environment was emerging from
  a period of relatively benign volatility.
• A two-year window wont capture the extremes, so
  the VAR it produces will be too low.
• A longer window is a partial solution at best .
• It will improve matters a little, but it also
  swamps recent events.

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Is shorter window a better thing?

• A longer observation period may pick up a wider
  variety of market conditions, but it would not
  necessarily allow VAR models to react quickly
  to an extreme event.
• If the problem is that models are not reacting
  fast enough, some believe the answer would in
  fact be to use shorter windows.
• These models would be surprised by the first
  outbreak of volatility, but would rapidly adapt.
  

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What models work best?

• The best VAR models are those that are quicker
  to react to a step-change in volatility.
• With the benefit of hindsight, the type of VAR
  model that would actually have worked best in
  the second half of 2007 would most likely have
  been a model driven by a frequently updated
  short data history.
• Or any frequently updated short data history
  that weights more recent observations more
  heavily than more distant observations.

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• In an environment like the third quarter of
  2007, a long data series will include an
  extensive period of low volatility, which will
  mute the models reaction to a sudden increase
  in volatility.
• Although it will include episodes of volatility
  from several years ago, these will be outweighed
  by the intervening period of calm.

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The importance of updating

• In the wake of the recent credit crisis, an
  unarguable improvement seems to be increasing
  the frequency of updating.
• Monthly or even quarterly updating of the data
  series is the norm.
• Shifting to weekly or even daily updating would
  improve the responsiveness of the model to a
  sudden change of conditions.

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