The VaR Estimation in Historical Simulation Approach Open Issues and Some Practical Proposals

1
The VaR Estimation in Historical Simulation
Approach Open Issues and Some Practical
Proposals
Conference on Numerical Methods in Finance, Paris
2009

• Michele Bonollo
• michele.bonollo_at_sgsbp.it
• Tommaso Rinaldi Prometeia SPA
• tommaso.rinaldi_at_prometeia.it

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Index

• PART I Market Risk Mgt and Historical simulation
  approach
• The stylized context vs. the real world context
• The challenges of the real world numbers
• Historical Simulation approach. Review of the
  canonical steps
• PART II Historical simulation Open isues, some
  possible approaches
• Issue 1. Scenario PL. multidimensional full
  evaluation vs. marginal full evaluation
• Issue 2. VaR Estimation. Scenario weighting by l
  and quantile etimation
• Issue 3. Component VaR. Expected return approach
  vs hybrid parametric approach
• PART III Just a practical view of the reporting
  system

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PART I
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Introduction
Once (3-4 years ago) a (world famous) financial
mathematics researcher asked me What about your
work I said I work on market risk, VaR
computation. He said Again VaR ? Is is just a
quantile . While exiting from his University,
my feeling was why so hard to meet theoretical
and applied perspective?. I do not know the
exact answer in my experience the real world
problems are always cross among several fields of
knowledge asset management, financial
instruments, financial mathematics, statistics,
computation science, regulatory contraints,
reporting processes and so on. The theoretical
research is (must be) very deep on each task. In
the next slides a (vey small) step to take in to
account both them
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Market Risk Management stylized view vs. real
world

• In the usual book description, one has two keys
  concepts concerning the VaR
• The single instrument/position j
• The portfolio, i.e. the vector of weights w
  (w1, ,wj, ,wN)
• The implied underlying idea is that one has to
  compute the risk measures (VaR, ES, ..) for the
  whole portoflio, for the single
  instrument/position, and at most for a few
  number of subportoflios, following the asset
  class or other clustering variable

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Market Risk Management stylized view vs. real
world

• In the actual risk managament process, the
  portolio is a complex multilevel tree, where the
  different levels refer to
• The banks of the groups, the types of strategies,
  the families of products, the risk factors, ..

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Historical Simulation approach. The canonical
steps

• Let
• t1..T the id of scenarios T 250, 500 daily
• j 1N the number of position/instruments
• m (m1mK) the vector of market paramers
  underlying
• f( ) the pricing functions
• The steps are
• Collect time series for underlyings/market
  parameters mt
• From data to shocks/returns st. Compound, or
  continuous,
• Evaluation of Scenario PL PLj,t fj(mjt sjt)
  fj(mj)
• Aggregate scenario PLs for the required cluster
  PLCt S
• Estimate the Quantile VaR or any other risk
  measure (ES, CVaR, ..)

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The challenges of the real world numbers

• Some numbers (magnitudes) from our bank, the 4-th
  in Italy
• 100.000 elementary positions, the gt 90
  derivatives
• portfolio tree with 1.000 nodes
• 100 billions of Notional in derivatives
• gt 1.000 elementary risk factors (IR buckets,
  underlyings, )
• As concerns the number of variables for which to
  apply a possible clustering, we have gt 10
  variables, related to portfolio/desk, risk
  factor, product family, issuer/counterparty
• Each day, we deliver (at least) 892 standard
  VaR, by .txt file. Moreover the Risk Manager can
  browse the whole portoflio and to compute the VaR
  for each required cluster or risk factor (equity,
  interest, forex) class. The combinations
  (hypercube D 10) ? 8
• 20 millions of pricing each day (Instrument x
  Scenario x RFactor)

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Historical Simulation the basic schema
From the single positions j PL
Deal
PTF
to the cluster PL PTF A Deal1 Deal2
(a possible) VaR estimation
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Historical Simulation Canonical algorithm in the
IT actual architecture
Risque
Merlino
Money Mate
Provider N
Bloomberg
Reuters
Positions Front Office
Estrattori
MaPaC (1-2)
Pricing Prometeia (3)
Pricing Sophis (3)
Shocks
Shocks
Pricing Engines
Staging Area
Export
Repository (4)
Process 3
Process 1
Clustering Cleaning
Process
Process 2
Calcolo HVaR
Calcolo PL
Risk Contributions
Browsing
VaR Computation Reporting
QlikView (5)
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PART II
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Issue 1 ScenarioPL, Full evaluation and
marginal Full evaluation

• Each day t the market parameters mt move togehter
• But, for compliance and strategic views, one
  often want to decomposte the different sources of
  risk in a single instrument/portfolio equity,
  interest, forex
• So the straigh way is to aply marginal shocks stk
  and then to evaluate the marginal PLj,t,k
• But PLj,t ? Sk PLj,t,k
• If we suppose that the pricing functions are
  smooth and follow a taylor representation, it
  happens because Sk PLj,t,k consistes of the sum
  of all the pure derivatives, up to 8, in the
  taylor expansion, and the interaction terms are
  lost. Very often the first two terms are enough
  in the expansion, so the difference is mainly due
  to the terms ou fo the diagonal in the Hessian
  matrix of the pricing function
• How to reconciliate the two measures? We remind
  the data oriented HS schema requires to use a
  unique large table (millions of rows) and we can
  not row by row deal with different cases

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Issue 1 ScenarioPL, Full evaluation and
marginal Full evaluation
The graph below is an example of the difference
for a plain vanilla option (SPMIB call) using two
different approaches Full Evaluation and
Marginal Full Evaluation. is small
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Issue 1 ScenarioPL, Full evaluation and
marginal Full evaluation
Here we have a large, higly exotic portfolio
(napoleon, altiplano, rainbow, ..). We poit out
that the difference still are quite small
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Issue 1 ScenarioPL, Full evaluation and
marginal Full evaluation

• So what do we do? This interesting hard problem
  ha fortunately a small impact on computations. So
  depending from the different situations, we have
  some different strategies
• For linear or quasi linear portfolio (bond,
  equity) we can put by definition PL ? SUM of
  marginal PL
• In other cases we take the residual and we split
  it to the different sources in any way. Consider
  that this task is very frequent. For example an
  equity in has each day a new value Vt that is
  Vt Vt-1 x (1Rt)(1FXt), where R anf FX
  represent the share and the return. The
  interaction effect (R x FX) has to be splitted.
  The issue is well konwn also in asset management,
  as performace contribution attribution, see
  Brinson, Carino, .

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Issue 2 The quantile estimation and scenario
weighting by l

• Here we have to consider a trade off between
  some different goals
• The quantile estimation, e.g. the simplest
  empirical quantile (the 5-th worst scenario if T
  500, a 99), has a high variance behaviour.
  This is a well known problem in order statistics
  theory (see David, Huber, ..), but is forgotten
  from practitioners.
• From a risk management perspective, one has to
  optimize the back testing statistics. In other
  words, the out of sample PL that exxcee the
  ex-ante VaR must be close to the expected
  frequency. In a 1-year VaR estimation a 99, I
  would expect that only 2.5 times the daily PL is
  below the VaR prediction. The accuracy of back
  testing implies a different capital requiremet by
  the central bank. If good, the capital
  requirement is (approximately) 3 times the
  10-days 99 VaR
• The risk changes over time but it remains
  unobservable. We can use (see Boudoukh 1996,
  Finger 2008) also in the non parametric
  historical simulation approach a l weighting
  technique à la RiskMetrics. In this case, we
  weight the probability of scenario before
  estimating the quantile

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Issue 2 The quantile estimation and scenario
weighting by l
The graph below is an example of VaR calculation
using different lambda parameters for a large
exotic portfolio Structured Product. With l is
(now) more conservative

• Holding period 250 day
• Confidence level 99

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Issue 2 The quantile estimation and scenario
weighting by l

• The VaR of the portfolio has been calculated
  using different parameters Lambda and is on a
  portfolio composed of the following indexes
• Nikkey 225
• SP 500
• EUROSTOXX 50E
• The three indexes have the
• same weight in the portfolio composition

• Holding period 250 day
• Confidence level 99

The table shows expected and Effective outliers
data Portfolios Profit Loss compared with VaR
calculated with different Lambda
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Issue 2 The quantile estimation and scenario
weighting by l

• The second exercise of Lambda Weighting was
  calculated for a portfolio with the followes
  shares
• IT0001976403
• IT0003856405
• IT0001063210
• IT0001334587
• IT0000072725
• DE0005557508
• DE0005752000
• DE0005140008
• FR0000121261
• This shares have the
• same weight in the portfolio composition

• Holding period 401 day
• Confidence level 99

The table shows Expected and Effective outliers
data Portfolios Profit Loss compared with VaR
calculated with different Lambda
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Issue 2 The quantile estimation and scenario
weighting by l

• So what do we do? At now (the project is in
  progress and fine tuning)
• As concerns the high variance of the estimator,
  the system allows to smooth the system by simple
  L-estimator, e.g. we do not take the 5-th worst
  case, but we average in a neighbor. We have not
  yet applied more sophisticated technique, such as
  Harrel-Davis estimator (the weight of the
  scneario is given by its frequency in a bootsrap
  sampling), Cornish-Fischer and so on. Here, for
  auditing reasosn and reporting cotraints,
  converge to simplicity
• As concerns l, from april the official VaR is
  computed with l 0.98., but each day we compute
  as a check/warning also the the plain vanilla
  VaR, the is the simplest quantile estimation.
  Consider that the computation effort is very
  hard. So we compute 2 x 892 VaR (Portoflio,
  clustering, ). Each one of them requires sum
  of 10.000 ? 100.000 PL, sort them, estimate VaR.
  With a 48 (!!) GB RAM server, 20-30 minutes.

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Issue 3 Component VaR

• The additive decomposition of risk is veru
  useful. In the actual risk management process
• The risk limits are given as VaR limits or greeks
  limits (delta equivalent, basis point value, vega
  for 1 shift, ..). The strategic analysis of risk
  need to split the effect of the different desk /
  porfolios on the risk VaR S
• A good measure is the ComponentVaR (see Garman,
  Mausser, ..). Using for simplicity the return
  notation, not the PL notation, if we have a
  partition of the portoflio indexed by i, the
  portfolio return is RP, the VaR is RP, then
• CVARi ? E(Ri VaRP) E (Ri RP RP)
• In the gaussian context, this reduces (see
  Garman, 96) to
• CVaRi RP x bi x c, b between porfolio and
  subporfolio
• If we apply the definition for the Historical
  case, if t is the VaR scenario, we take the PL
  for the t for the subportfolio as CVaRi (see
  Hallerbach). Nevertheless this simple tecnique
  may not be used, becaues of high variance, low
  reliability.

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Issue 3 Component VaR
Here we see the weakness of the described pure
non parametric estimation, based of the
expectation definition. The portfolio model is of
european blue chips, higly correlated (avg r ?
0.7)
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Issue 3 Component VaR
A simple robust idea is to plug-in the
beta-Garman formula in the non parametric
approach. We compute b over the T PL scenario
and then fixed the t VaR-scenario, apply the
formula to decompose it. Below the same
portfolio, the same compute date. More reliable!
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Issue 3 Component VaR
The chart below shows VaR Contribution (Component
VaR) of each share in portfolio to Total VaR, by
appling the hybrid Beta technique over 400 days

• The portfolio is composed by the followes shares
  
• IT0001976403
• IT0003856405
• IT0001063210
• IT0001334587
• IT0000072725
• DE0005557508
• DE0005752000
• DE0005140008
• FR0000121261

• Holding period 401 days
• Confidence level 99

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Issue 3 Component VaR

• So what do we do? At now (the project is in
  progress and fine tuning)
• Differently from l, the CVaR is not yet published
  in the daily reporting, it is computed in order
  to make software test.
• Consider that we could deliver a number of CVaR
  very high (gt 1000) depending of all interesting
  clustering and partitioning of portfolios.
• I think we will use hybrid approach or some way
  very cloe to it. To measure risk is importante,
  but even more important is that the top
  management believes the the risk meausers. An
  irregular or strange risk measure over time
  makes is useless

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PART III
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Distribution of the Scenario PL of a large
exotic portfolio
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The Tableau de Bord
Bank filter
Portfolio / Desk
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Reporting Historical VaR
In relazione a una serie di anomalie
fisiologiche o determinate da errori del batch
Sophis sono èstato messo a punto un sistema di
controllo, denominato Outliers, che permette di
navigare e visualizzare tali casi fair value
nulli, PL rilevanti Le soglie di rilevazione
delle anomalie sono parametriche, modificabili
dallutente