Noise sensitivity of risk measures

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Noise sensitivity of risk measures

• Imre Kondor
• Collegium Budapest and Eötvös
• University, Budapest, Hungary
• Institute for Theoretical Sciences, a Notre Dame
  University and Argonne National Laboratory
  collaboration, August 17, 2005

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Contents

• I. Preliminaries
• the problem of noise, risk measures, noisy
  covariance matrices
• II. Noise sensitivity of Gaussian portfolios
• III. Alternative risk measures (mean absolute
  deviation, expected shortfall, worst loss),
  their sensitivity to noise,
  the feasibility problem

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Coworkers

• Szilárd Pafka and Gábor Nagy (CIB Bank, Budapest)
• Richárd Karádi (Institute of Physics, Budapest
  University of Technology, now at ProcterGamble)
• Balázs Janecskó, András Szepessy, Tünde Ujvárosi
  (Raiffeisen Bank, Budapest)
• István Varga-Haszonits (Eötvös University,
  Budapest)

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I. PRELIMINARIES
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Preliminary considerations 1

• Portfolio selection a tradoff between risk and
  reward

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Reward vs. risk
Efficient frontier
Set of all possible portfolios in the economy
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Preliminary considerations 2

• There is a more or less general agreement on
  what we mean by reward in a finance context
• relative price change
• log return
• but the status of risk measures is controversial

8

• For optimal portfolio selection we have to know
  what we want to optimize
• The chosen risk measure should respect some
  obvious mathematical requirements, must be
  stable, and easy to implement in practice

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The problem of noise

• Even if returns formed a clean, stationary
  stochastic process, we could only observe finite
  time segments, therefore we never have sufficient
  information to completely reconstruct the
  underlying process. Our estimates will always be
  noisy.
• Mean returns (on short time horizons) are
  particularly hard to measure on the market with
  any precision
• Even if we disregard returns and go for the
  minimal risk portfolio, lack of sufficient
  information will introduce noise, i. e. error,
  into our decision.
• The problem of noise is more severe for large
  portfolios (size N) and relatively short time
  series (length T) of observations, and different
  risk measures are sensitive to noise to a
  different degree.
• We have to know how the decision error depends on
  N and T for a given risk measure

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Some elementary criteria on risk measures

• A risk measure is a quantitative characterization
  of our intuitive concept of risk (fear of
  uncertainty and loss).
• Risk is related to the stochastic nature of
  returns. It is (or should be) a functional of the
  pdf of returns.
• Any reasonable risk measure must satisfy
• - convexity
• - invariance under addition of risk free asset
• - monotonicity and assigning zero risk to a zero
  position
• The appropriate choice may depend on the nature
  of data (e.g. on their asymptotics) and on the
  context (investment, risk management,
  benchmarking, tracking, regulation, capital
  allocation)

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A more elaborate set of risk measure axioms

• Coherent risk measures (P. Artzner, F. Delbaen,
  J.-M. Eber, D. Heath, Risk, 10, 33-49 (1997)
  Mathematical Finance,9, 203-228 (1999)) Required
  properties monotonicity, subadditivity, positive
  homogeneity, and translational invariance.
  Subadditivity and homogeneity imply convexity.
  (Homogeneity is questionable for very large
  positions. Multiperiod risk measures?)
• Spectral measures (C. Acerbi, in Risk Measures
  for the 21st Century, ed. G. Szegö, Wiley, 2004)
  a special subset of coherent measures, with an
  explicit representation. They are parametrized by
  a spectral function that reflects the risk
  aversion of the investor.

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Convexity

• Convexity is extremely important.
• A non-convex risk measure
• - penalizes diversification (without convexity
  risk
• can be reduced by splitting the portfolio in
  two
• or more parts)
• - does not allow risk to be correctly aggregated
• - cannot provide a basis for rational pricing of
  risk
• (the efficient set may not be not convex)
• - cannot serve as a basis for a consistent limit
  
• system
• In short, a non-convex risk measure is really not
  a risk measure at all.

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II. NOISE SENSITIVITY OF GAUSSIAN PORTFOLIOS
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A classical risk measure the variance

• When we use variance as a risk measure we assume
  that the underlying process is essentially
  multivariate normal or close to it.

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Portfolios

• Consider a linear combination of returns
• with weights . The
  weights add up to unity . The
  portfolios expectation value is
  with variance
• where is the covariance matrix,
  the standard deviation of return , and
  
• the correlation matrix.

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Level surfaces of risk measured in variance

• The covariance matrix is positive definite. It
  follows that the level surfaces (iso-risk
  surfaces) of variance are (hyper)ellipsoids in
  the space of weights. The convex iso-risk
  surfaces reflect the fact that the variance is a
  convex measure.
• The principal axes are inversely proportional to
  the square root of the eigenvalues of the
  covariance matrix.
• Small eigenvalues thus correspond to long
  axes.
• The risk free asset would correspond to an
  infinite axis, and the correspondig ellipsoid
  would be deformed into an elliptical cylinder.

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The Markowitz problem

• According to Markowitz classical theory the
  tradeoff between risk and reward can be realized
  by minimizing the variance
• over the weights, for a given expected return
• and budget

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• Geometrically, this means that we have to blow up
  the risk ellipsoid until it touches the
  intersection of the two planes corresponding to
  the return and budget constraints, respectively.
  The point of tangency is the solution to the
  problem.
• As the solution is the point of tangency of a
  convex surface with a linear one, the solution is
  unique.
• There is a certain continuity or stability in the
  solution A small miss-specification of the risk
  ellipsoid leads to a small shift in the solution.

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• Covariance matrices corresponding to real markets
  tend to have mostly positive elements.
• A large matrix with nonzero average elements will
  have a large (Frobenius-Perron) eigenvalue, with
  the corresponding eigenvector having all positive
  components. This will be the direction of the
  shortest principal axis of the risk ellipsoid.
• Then the solution also will have all positive
  components. Fluctuations in the small eigenvalue
  sectors may have a relatively mild effect on the
  solution.

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The minimal risk portfolio

• Expected returns are hardly possible (on
  efficient markets, impossible) to determine with
  any precision.
• In order to get rid of the uncertainties in the
  returns, we confine ourselves to considering the
  minimal risk portfolio only, that is, for the
  sake of simplicity, we drop the return
  constraint.
• Minimizing the variance of a portfolio without
  considering return does not, in general, make
  much sense. In some cases (index tracking,
  benchmarking), however, this is precisely what
  one has to do.

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The weights of the minimal risk portfolio

• Analytically, the minimal variance portfolio
  corresponds to the weights for which
• is minimal, given .
• The solution is .
• Geometrically, the minimal risk portfolio is the
  point of tangency between the risk ellipsoid and
  the plane of the budget constraint.

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Empirical covariance matrices

• The covariance matrix has to be determined from
  measurements on the market. From the returns
  observed at time t we get the estimator
• For a portfolio of N assets the covariance matrix
  has O(N²) elements. The time series of length T
  for N assets contain NT data. In order for the
  measurement be precise, we need N ltltT. Bank
  portfolios may contain hundreds of assets, and it
  is hardly meaningful to use time series longer
  than 4 years (T1000). Therefore, N/T ltlt 1 rarely
  holds in practice. As a result, there will be a
  lot of noise in the estimate, and the error will
  scale in N/T.

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Fighting the curse of dimensions

• Economists have been struggling with this problem
  for ages. Since the root of the problem is lack
  of sufficient information, the remedy is to
  inject external info into the estimate. This
  means imposing some structure on s. This
  introduces bias, but beneficial effect of noise
  reduction may compensate for this.
• Examples
• single-index models (ßs) All these help
  to
• multi-index models various degrees.
• grouping by sectors Most studies are
  based
• principal component analysis on
  empirical data
• Bayesian shrinkage estimators, etc.

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An intriguing observation

• L.Laloux, P. Cizeau, J.-P. Bouchaud, M. Potters,
  PRL 83 1467 (1999) and Risk 12 No.3, 69 (1999)
• and
• V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N.
  Amaral, H.E. Stanley, PRL 83 1471 (1999)
• noted that there is such a huge amount of noise
  in empirical covariance matrices that it may
  render them useless.
• A paradox Covariance matrices are in widespread
  use and banks still survive ?!

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Laloux et al. 1999
The spectrum of the covariance matrix obtained
from the time series of SP 500 with N406,
T1308, i.e. N/T 0.31, compared with that of a
completely random matrix (solid curve). Only
about 6 of the eigenvalues lie beyond the random
band.
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Remarks on the paradox

• The number of junk eigenvalues may not
  necessarily be a proper measure of the effect of
  noise The small eigenvalues and their
  eigenvectors fluctuate a lot, but perhaps they
  have a relatively minor effect on the optimal
  portfolio, whereas the large eigenvalues and
  their eigenvectors are fairly stable.
• The investigated portfolio was too large compared
  to the length of the time series.
• Working with real, empirical data, it is hard to
  distinguish the effect of insufficient
  information from other parasitic effects, like
  nonstationarity.

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A filtering procedure suggested by RMT

• The appearence of random matrices in the context
  of portfolio selection triggered a lot of
  activity, mainly among physicists. Laloux et al.
  and Plerou et al. proposed a filtering method
  based on random matrix theory (RMT). This has
  been further developed and refined by many
  workers.
• The proposed filtering consists basically in
  discarding as pure noise that part of the
  spectrum that falls below the upper edge of the
  random spectrum. Information is carried only by
  the eigenvalues and their eigenvectors above this
  edge. Optimization should be carried out by
  projecting onto the subspace of large
  eigenvalues, and replacing the small ones by a
  constant chosen so as to preserve the trace. This
  would then drastically reduce the effective
  dimensionality of the problem.

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• Interpretation of the large eigenvalues The
  largest one is the market, the other big
  eigenvalues correspond to the main industrial
  sectors.
• The method can be regarded as a systematic
  version of principal component analysis, with an
  objective criterion on the number of principal
  components.
• According to our comparative studies, the method
  works consistently well

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A measure of the effect of noise

• Assume we know the true covariance matrix and
• the noisy one . Then a natural, though not
  unique,
• measure of the impact of noise is
• where w are the optimal weights corresponding
• to and , respectively.

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To test the noise sensitivity of various risk
measures we use simulated data

• The rationale behind this is that in order to be
  able to compare the sensitivity of risk measures
  to noise, we better get rid of other sources of
  uncertainty, like non-stationarity. This can be
  achieved by using artificial data where we have
  total control over the underlying stochastic
  process.

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The model-simulation approach

• Our strategy is to choose various model
  covariance matrices and generate long
  simulated time series by them. Then we cut out
  segments of length T from these time series, as
  if observing them on the market, and try to
  reconstruct the covariance matrices from them. We
  optimize a portfolio both with the true and
  with the observed covariance matrix and
  determine the measure .

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Model 1 the unit matrix

• Spectrum
• ? 1, N-fold degenerate
  
• Noise will split this
• into band

1
0
C
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The empirical covariance matrix corresponding
to Model 1 is the Wishart matrix

• If N and T go to infinity such that their ratio
  N/T is finite, lt 1, then the spectrum of this
  empirical covariance matrix is given by the
  Wishart or Marchenko-Pastur spectrum (eigenvalue
  distribution)
• 
  
  where

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Model 2 single-index

• Singlet ?11?(N-1) O(N)
• eigenvector (1,1,1,)
• ?2 1- ? O(1)
• (N-1) fold degenerate

?
1
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• The spectrum of the empirical covariance matrix
  corresponding to Model 2 is still the Marchenko
  Pastur spectrum, plus an isolated, large,
  Frobenius Perron eigenvalue (the market).

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Model 3 market sectors

singlet
- fold degenerate
1
This structure has also been studied by economists
- fold degenerate
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• The spectrum of the empirical covariance matrix
  corresponding to Model 3 consists of the
  Marchenko Pastur spectrum, the large market
  eigenvalue, and a number of eigenvalues in
  between. If the sectors are not equivalent, we
  can, by an appropriate choice of the parameters,
  tune the model so as to mimic the empirical
  covariance matrices observed on the market (Noh
  model)

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Model 4 Semi-empirical

• Suppose we have very long time series (T) for
  many assets (N).
• Choose N lt N time series randomly and derive Cº
  from these data. Generate time series of length
  T ltlt T from Cº.
• The error due to T is much larger than that due
  to T.

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• We look for the minimal risk portfolio for both
  the true and the empirical covariances and
  determine the measure

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Numerically we get the following scaling result
for Model 1
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This confirms the expected scaling in N/T. The
corresponding analytic result

• can easily be derived for Model 1. It is valid
  within O(1/N) corrections also for more general
  models. This simple result does not seem to have
  been noticed earlier

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The derivation
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III.

• III. ALTERNATIVE RISK MEASURES

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Risk measures in practice VaR

• VaR (Value at Risk) is a high (95, or 99)
  quantile, a threshold beyond which a given
  fraction (5 or 1) of the statistical weight
  resides.
• Its merits (relative to the Greeks, e.g.)
• - universal can be applied to any portfolio
• - probabilistic content associated to the
  distribution
• - expressed in money
• Widespread across the whole industry and
  regulation. Has been promoted from a diagnostic
  tool to a decision tool.
• Its lack of convexity promted search for coherence

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Mean absolute deviation (MAD)
Some methodologies (e.g. Algorithmics) use the
mean absolute deviation rather than the standard
deviation to characterize the fluctuation of
portfolios. The objective function to minimize is
then
instead of
The iso-risk surfaces of MAD are polyhedra. This
is a feature MAD shares with some regulatory risk
measures.
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Effect of noise on absolute deviation-optimized
portfolios
We generate artificial time series (say iid
normal), determine the true abs. deviation and
compare it to the measured one
We get
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Noise sensitivity of MAD

• The result scales in T/N (same as with the
  variance). The optimal portfolio other things
  being equal - is more risky than in the
  variance-based optimization.
• Geometrical interpretation The level surfaces of
  the variance are ellipsoids.The optimal portfolio
  is found as the point where this risk-ellipsoid
  first touches the plane corresponding to the
  budget constraint. In the absolute deviation case
  the ellipsoid is replaced by a polyhedron, and
  the solution occurs at one of its corners. A
  small error in the specification of the
  polyhedron makes the solution jump to another
  corner, thereby increasing the fluctuation in the
  portfolio.

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Expected shortfall (ES) optimization

• ES is the mean loss beyond a high threshold
  defined in probability (not in money). For
  continuous pdfs it is the same as the
  conditional expectation beyond the VaR quantile.
  ES is coherent (in the sense of Artzner et al.)
  and as such it is strongly promoted by a group of
  academics. In addition, Uryasev and Rockefellar
  have shown that its optimizaton can be reduced to
  linear programming for which extremely fast
  algorithms exist.
• ES-optimized portfolios tend to be much noisier
  than either of the previous ones. One reason is
  the instability related to the (piecewise) linear
  risk measure, the other is that a high quantile
  sacrifices most of the data. The noise
  sensitivity of ES appears to be non-monotonous as
  function of the threshold.
• In addition, ES optimization is not always
  feasible!

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Before turning to the discussion of the
feasibility problem, let us compare the noise
sensitivity of the following risk measures
standard deviation, absolute deviation and
expected shortfall (the latter at 95). For the
sake of comparison we use the same (Gaussian)
input data of length T for each, determine the
minimal risk portfolio under these risk measures
and compare the error due to noise.
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The next slides show

• plots of wi (porfolio weights) as a function of i
• display of q0 (ratio of risk of optimal portfolio
  determined from time series information vs full
  information)
• results show that the effect of estimation noise
  can be significant and more advanced risk
  measures pose a higher demand for input
  information (in a portfolio optimization context)

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• the suboptimality (q0) scales in T/N (for large N
  and T)

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The feasibility problem

• For T lt N, there is no solution to the portfolio
  optimization problem under any of the risk
  measures considered here.
• For T gt N, there always is a solution under
  the variance and MAD, even if it is bad for T not
  large enough. In contrast, under ES (and WL to be
  considered later), there may or may not be a
  solution for T gt N, depending on the sample. The
  probability of the existence of a solution goes
  to 1 only for T/N going to infinity.
• The problem does not appear if short selling is
  banned

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Feasibility of optimization under ES
Probability of the existence of an optimum under
CVaR. F is the standard normal distribution. Note
the scaling in N/vT.
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A pessimistic risk measure worst loss

• In order to better understand the feasibility
  problem, select the worst return in time and
  minimize this over the weights
• subject to
• This risk measure is coherent, one of Acerbis
  spectral measures.
• For T lt N there is no solution
• The existence of a solution for T gt N is a
  probabilistic issue again, depending on the time
  series sample

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Why is the existence of an optimum a random event?

• To get a feeling, consider NT2.
• The two planes
• intersect the plane of the budget constraint in
  two straight lines. If one of these is
  decreasing, the other is increasing with ,
  then there is a solution, if both increase or
  decrease, there is not. It is easy to see that
  for elliptical distributions the probability of
  there being a solution is ½.

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Probability of the feasibility of the minimax
problem

• For TgtN the probability of a solution (for an
  elliptical underlying pdf) is
• (The problem is isomorphic to some problems in
  operations research and random geometry Todd,
  M.J. (1991), Probabilistic models for linear
  programming, Math. Oper. Res. 16, 671-693. )
• For N and T large, p goes over into the error
  function and scales in N/vT. For T? infinity, p
  ?1.

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Probability of the existence of a solution under
maximum loss. F is the standard normal
distribution. Scaling is in N/vT again.
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Concluding remarks

• Due to the large number of assets in typical bank
  portfolios and the limited amount of data, noise
  is an all pervasive problem in portfolio theory.
• It can be efficiently filtered by a variety of
  techniques from portfolios optimized under
  variance.
• Unfortunately, variance is not an adequate risk
  measure for fat-tailed pdfs.
• Piecewise linear risk measures show instability
  (jumps) in a noisy environment.
• Risk measures focusing on the far tails show
  additional sensitivity to noise, due to loss of
  data.
• The two coherent measures we have studied display
  large sample-to-sample fluctuations and
  feasibility problems under noise. This may cast a
  shade of doubt on their applications.

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Some references

• Physica A 299, 305-310 (2001)
• European Physical Journal B 27, 277-280 (2002)
• Physica A 319, 487-494 (2003)
• Physica A 343, 623-634 (2004)
• submitted to Quantitative Finance, e-print
  cond-mat/0402573

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Benchmark tracking

• The goal can be (e.g. in benchmark tracking or
  index replication) to minimize the risk (e.g.
  standard deviation) relative to a benchmark
• Portfolio
• Benchmark
• Relative portfolio

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• Therefore the relevant problems are of similar
  structure but with returns relative to the
  benchmark
• For example, to minimize risk relative to the
  benchmark means minimizing the standard deviation
  of
• with the usual budget contraint (no condition on
  expected returns!)

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The economic content of the single-index model

• return market return with
• standard deviation s
• The covariance matrix implied by the above
• The assumed structure reduces of parameters to
  N.
• If nothing depends on i then this is just the
  caricature Model 2.

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Risk measures implied by regulation

• Banks are required to set aside capital as a
  cushion against risk
• Minimal capital requirements are fixed by
  international regulation (Basel I and II, Capital
  Adequacy Directive of the EEC) the magic 8
• Standard model vs. internal models
• Capital charges assigned to various positions in
  the standard model purport to cover the risk in
  those positions, therefore, they must be regarded
  as some kind of implied risk measures
• These measures are trying to mimic variance by
  piecewise linear approximants. They are quite
  arbitrary, sometimes concave and unstable

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An example Specific risk of bonds
CAD, Annex I, 14 The capital requirement of
the specific risk (due to issuer) of bonds is
Iso-risk surface of the specific risk of bonds
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Another example Foreign exchange
According to Annex III, 1, (CAD 1993, Official
Journal of the European Communities, L14, 1-26)
the capital requirement is given as
,
,
in terms of the gross
.
and the net position
The iso-risk surface of the foreign exchange
portfolio
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