A portfoli

1
A portfolió-választási feladat instabilitása

• Kondor Imre
• Collegium Budapest és ELTE
  
• Eloadás az MTA Közgazdaságtudományi
  Intézetében
• Budapest, 2006 március 2

2
Contents

• The subject of the talk lies at the crossroads
  of finance, statistical physics and computer
  science
• I. The investment problem portfolios, rational
  portfolio selection, risk measures, the
  problem of estimation error (noise), noise
  sensitivity of risk measures
• II. A wider context algorithms, computational
  complexity, critical phenomena, estimation error
  as a critical phenomenon

3
Contents

• The subject of the talk lies at the crossroads
  of finance, statistical physics and computer
  science
• I. The investment problem portfolios, rational
  portfolio selection, risk measures, the
  problem of estimation error (noise), noise
  sensitivity of risk measures
• II. A wider context algorithms, computational
  complexity, critical phenomena, estimation error
  as a critical phenomenon

4
Contents

• The subject of the talk lies at the crossroads
  of finance, statistical physics and computer
  science
• I. The investment problem portfolios, rational
  portfolio selection, risk measures, the
  problem of estimation error (noise), noise
  sensitivity of risk measures
• II. A wider context algorithms, computational
  complexity, critical phenomena, estimation error
  as a critical phenomenon

5
A disclaimer

• This talk is not about
• human nature
• utility functions
• the structure of markets
• the nature of data
• bounded rationality.
• It is about the purely technical question of risk
  measures as they appear in practice, their noise
  sensitivity and the limitations this imposes upon
  rational decision making.

6
Coworkers

• Szilárd Pafka (ELTE PhD student ? CIB Bank,
  ?Paycom.net, California)
• Gábor Nagy (Debrecen University PhD student and
  CIB Bank, Budapest)
• Richárd Karádi (Technical University MSc student
  ?ProcterGamble)
• Nándor Gulyás (ELTE PhD student ? Budapest Bank
  ?Lombard Leasing ?private enterpreneur)
• István Varga-Haszonits (ELTE PhD student
  ?Morgan-Stanley)

7
A portfolio

• is a combination of assets or investment
  instruments (shares, bonds, foreign exchange,
  precious metals, commodities, artworks, property,
  etc.).
• In this talk I will focus on equity portfolios.
• More generally, the various business lines of a
  big firm, or even the economy as a whole, can
  also be regarded as a portfolio.

8
A portfolio

• is a combination of assets or investment
  instruments (shares, bonds, foreign exchange,
  precious metals, commodities, artworks, property,
  etc.).
• In this talk I will focus on equity portfolios.
• More generally, the various business lines of a
  big firm, or even the economy as a whole, can
  also be regarded as a portfolio.

9
A portfolio

• is a combination of assets or investment
  instruments (shares, bonds, foreign exchange,
  precious metals, commodities, artworks, property,
  etc.).
• In this talk I will focus on equity portfolios.
• More generally, the various business lines of a
  big firm, or even the economy as a whole, can
  also be regarded as a portfolio.

10
Rational portfolio selection

• The value of assets fluctuates.
• It is dangerous to invest all our money into a
  single asset.
• Investment should be diversified, distributed
  among the various assets.
• More risky assets tend to yield higher return.
• Some assets tend to fluctuate together, some
  others in an opposite way.
• Rational portfolio selection seeks a tradeoff
  between risk and reward.

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Relative price change (return)
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Rational portfolio selection

• The value of assets fluctuates.
• It is dangerous to invest all our money into a
  single asset.
• Investment should be diversified, distributed
  among the various assets.
• More risky assets tend to yield higher return.
• Some assets tend to fluctuate together, some
  others in an opposite way.
• Rational portfolio selection seeks a tradeoff
  between risk and reward.

16
Rational portfolio selection

• The value of assets fluctuates.
• It is dangerous to invest all our money into a
  single asset.
• Investment should be diversified, distributed
  among the various assets.
• More risky assets tend to yield higher return.
• Some assets tend to fluctuate together, some
  others in an opposite way.
• Rational portfolio selection seeks a tradeoff
  between risk and reward.

17
Rational portfolio selection

• The value of assets fluctuates.
• It is dangerous to invest all our money into a
  single asset.
• Investment should be diversified, distributed
  among the various assets.
• More risky assets tend to yield higher return.
• Some assets tend to fluctuate together, some
  others in an opposite way.
• Rational portfolio selection seeks a tradeoff
  between risk and reward.

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Mean 0.0017 StdDev 0.016
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Mean 0.00028 StdDev 0.0038
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Rational portfolio selection

• The value of assets fluctuates.
• It is dangerous to invest all our money into a
  single asset.
• Investment should be diversified, distributed
  among the various assets.
• More risky assets tend to yield higher return.
• Some assets tend to fluctuate together, some
  others in an opposite way.
• Rational portfolio selection seeks a tradeoff
  between risk and reward.

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Rational portfolio selection

• The value of assets fluctuates.
• It is dangerous to invest all our money into a
  single asset.
• Investment should be diversified, distributed
  among the various assets.
• More risky assets tend to yield higher return.
• Some assets tend to fluctuate together, some
  others in an opposite way.
• Rational portfolio selection seeks a tradeoff
  between risk and reward.

30
Risk and reward

• Financial reward can be measured in terms of the
  return (relative gain)
• The characterization of risk is more controversial

31
Risk measures

• A risk measure is a quantitative characterization
  of our intuitive concept of risk (fear of loss).
• Risk is related to the stochastic nature of
  returns. Mathematically, it is (or should be) a
  convex functional of the pdf of returns.
• 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)

32
Risk measures

• A risk measure is a quantitative characterization
  of our intuitive concept of risk (fear of loss).
• Risk is related to the stochastic nature of
  returns. Mathematically, it is (or should be) a
  convex functional of the pdf of returns.
• 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)

33
Risk measures

• A risk measure is a quantitative characterization
  of our intuitive concept of risk (fear of loss).
• Risk is related to the stochastic nature of
  returns. Mathematically, it is (or should be) a
  convex functional of the pdf of returns.
• 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)

34
The most obvious choice for a risk measure
Variance

• Variance is the average quadratic deviation from
  the average a time honoured statistical tool
• Its use assumes that the probability distribution
  of the returns is sufficiently concentrated
  around the average, that there are no large
  fluctuations
• This is true in several instances, but we often
  encounter fat tails, huge deviations with a
  non-negligible probability (e.g. the Black
  Monday).

35
The most obvious choice for a risk measure
Variance

• Variance is the average quadratic deviation from
  the average a time honoured statistical tool
• Its use assumes that the probability distribution
  of the returns is sufficiently concentrated
  around the average, that there are no large
  fluctuations
• This is true in several instances, but we often
  encounter fat tails, huge deviations with a
  non-negligible probability (e.g. the Black
  Monday).

36
The most obvious choice for a risk measure
Variance

• Variance is the average quadratic deviation from
  the average a time honoured statistical tool
• Its use assumes that the probability distribution
  of the returns is sufficiently concentrated
  around the average, that there are no large
  fluctuations
• This is true in several instances, but we often
  encounter fat tails, huge deviations with a
  non-negligible probability (e.g. the Black
  Monday).

37
Alternative risk measures

• There are several alternative risk measures in
  use in the academic literature, practice, and
  regulation
• Value at risk (VaR) the best among the p
  worst losses (not convex, punishes
  diversification)
• Mean absolute deviation (MAD) Algorithmics
• Coherent risk measures (promoted by academics)
• Expected shortfall (ES) average loss beyond a
  high threshold
• Maximal loss (ML) the single worst case

38
Alternative risk measures

• There are several alternative risk measures in
  use in the academic literature, practice, and
  regulation
• Value at risk (VaR) the best among the p
  worst losses (not convex, punishes
  diversification)
• Mean absolute deviation (MAD) Algorithmics
• Coherent risk measures (promoted by academics)
• Expected shortfall (ES) average loss beyond a
  high threshold
• Maximal loss (ML) the single worst case

39
Alternative risk measures

• There are several alternative risk measures in
  use in the academic literature, practice, and
  regulation
• Value at risk (VaR) the best among the p
  worst losses (not convex, punishes
  diversification)
• Mean absolute deviation (MAD) Algorithmics
• Coherent risk measures (promoted by academics)
• Expected shortfall (ES) average loss beyond a
  high threshold
• Maximal loss (ML) the single worst case

40
Alternative risk measures

• There are several alternative risk measures in
  use in the academic literature, practice, and
  regulation
• Value at risk (VaR) the best among the p
  worst losses (not convex, punishes
  diversification)
• Mean absolute deviation (MAD) Algorithmics
• Coherent risk measures (promoted by academics)
• Expected shortfall (ES) average loss beyond a
  high threshold
• Maximal loss (ML) the single worst case

41
The variance of a portfolio

• - a quadratic form of the weights. The
  coefficients of this form are the elements of the
  covariance matrix that measures the co-movements
  between the various assets.

42
Portfolios

• A portfolio is a linear combination (a weighted
  average) of assets, with a set of weights wi that
  add up to unity (the budget constraint).
• The weights are not necessarily positive short
  selling
• The legal status of short selling
• Leverage
• The fact that the weights can be negative means
  that the region over which we are trying to
  determine the optimal portfolio is not bounded

43
Portfolios

• A portfolio is a linear combination (a weighted
  average) of assets, with a set of weights wi that
  add up to unity (the budget constraint).
• The weights are not necessarily positive short
  selling
• The legal status of short selling
• Leverage
• The fact that the weights can be negative means
  that the region over which we are trying to
  determine the optimal portfolio is not bounded

44
Portfolios

• A portfolio is a linear combination (a weighted
  average) of assets, with a set of weights wi that
  add up to unity (the budget constraint).
• The weights are not necessarily positive short
  selling
• The legal status of short selling
• Leverage
• The fact that the weights can be negative means
  that the region over which we are trying to
  determine the optimal portfolio is not bounded

45
Portfolios

• A portfolio is a linear combination (a weighted
  average) of assets, with a set of weights wi that
  add up to unity (the budget constraint).
• The weights are not necessarily positive short
  selling
• The legal status of short selling
• Leverage
• The fact that the weights can be negative means
  that the region over which we are trying to
  determine the optimal portfolio is not bounded

46
Portfolios

• A portfolio is a linear combination (a weighted
  average) of assets, with a set of weights wi that
  add up to unity (the budget constraint).
• The weights are not necessarily positive short
  selling
• The legal status of short selling
• Leverage
• The fact that the weights can be negative means
  that the region over which we are trying to
  determine the optimal portfolio is not bounded

47
Markowitz portfolio selection theory

• Rational portfolio selection realizes the
  tradeoff between risk and reward by minimizing
  the risk functional over the weights, given the
  expected return, the budget constraint, and
  possibly other costraints.

48
Ambiguity of the objective function

• The non-uniqueness of risk measures is a serious
  problem (most banks use internal models different
  from what they use for regulatory reporting).
  What do we want to optimize?
• The most popular risk measure, also deeply
  embedded in regulation, is VaR, which is
  inconsistent
• The lack of a universally agreed objective
  function is not unique to finance
• Single period vs. multiperiod optimization

49
Ambiguity of the objective function

• The non-uniqueness of risk measures is a serious
  problem (most banks use internal models different
  from what they use for regulatory reporting).
  What do we want to optimize?
• The most popular risk measure, also deeply
  embedded in regulation, is VaR, which is
  inconsistent
• The lack of a universally agreed objective
  function is not unique to finance
• Single period vs. multiperiod optimization

50
Ambiguity of the objective function

• The non-uniqueness of risk measures is a serious
  problem (most banks use internal models different
  from what they use for regulatory reporting).
  What do we want to optimize?
• The most popular risk measure, also deeply
  embedded in regulation, is VaR, which is
  inconsistent
• The lack of a universally agreed objective
  function is not unique to finance
• Single period vs. multiperiod optimization

51
Ambiguity of the objective function

• The non-uniqueness of risk measures is a serious
  problem (most banks use internal models different
  from what they use for regulatory reporting).
  What do we want to optimize?
• The most popular risk measure, also deeply
  embedded in regulation, is VaR, which is
  inconsistent
• The lack of a universally agreed objective
  function is not unique to finance
• Single period vs. multiperiod optimization

52
How do we know the returns and the covariances?

• In principle, from observations on the market
• If the portfolio contains N assets, we need O(N²)
  data
• The input data come from T observations for N
  assets
• The estimation error is negligible as long as
  NTgtgtN², i.e. NltltT
• In practice T is never longer than 4 years, i.e.
  T1000, whereas in a typical banking portfolio N
  is several hundreds or thousands.
• NltltT is therefore never fulfilled in practice.

53
How do we know the returns and the covariances?

• In principle, from observations on the market
• If the portfolio contains N assets, we need O(N²)
  data
• The input data come from T observations for N
  assets
• The estimation error is negligible as long as
  NTgtgtN², i.e. NltltT
• In practice T is never longer than 4 years, i.e.
  T1000, whereas in a typical banking portfolio N
  is several hundreds or thousands.
• NltltT is therefore never fulfilled in practice.

54
How do we know the returns and the covariances?

• In principle, from observations on the market
• If the portfolio contains N assets, we need O(N²)
  data
• The input data come from T observations for N
  assets
• The estimation error is negligible as long as
  NTgtgtN², i.e. NltltT
• In practice T is never longer than 4 years, i.e.
  T1000, whereas in a typical banking portfolio N
  is several hundreds or thousands.
• NltltT is therefore never fulfilled in practice.

55
How do we know the returns and the covariances?

• In principle, from observations on the market
• If the portfolio contains N assets, we need O(N²)
  data
• The input data come from T observations for N
  assets
• The estimation error is negligible as long as
  NTgtgtN², i.e. NltltT
• In practice T is never longer than 4 years, i.e.
  T1000, whereas in a typical banking portfolio N
  is several hundreds or thousands.
• NltltT is therefore never fulfilled in practice.

56
How do we know the returns and the covariances?

• In principle, from observations on the market
• If the portfolio contains N assets, we need O(N²)
  data
• The input data come from T observations for N
  assets
• The estimation error is negligible as long as
  NTgtgtN², i.e. NltltT
• In practice T is never longer than 4 years, i.e.
  T1000, whereas in a typical banking portfolio N
  is several hundreds or thousands.
• NltltT is therefore never fulfilled in practice.

57
How do we know the returns and the covariances?

• In principle, from observations on the market
• If the portfolio contains N assets, we need O(N²)
  data
• The input data come from T observations for N
  assets
• The estimation error is negligible as long as
  NTgtgtN², i.e. NltltT
• In practice T is never longer than 4 years, i.e.
  T1000, whereas in a typical banking portfolio N
  is several hundreds or thousands.
• NltltT is therefore never fulfilled in practice.

58
Information deficit

• Thus the Markowitz problem suffers from the
  curse of dimensions, or from information
  deficit
• The estimates will contain error and the
  resulting portfolios will be suboptimal
• How serious is this effect?
• How sensitive are the various risk measures to
  this kind of error?
• How can we reduce the error?

59
Information deficit

• Thus the Markowitz problem suffers from the
  curse of dimensions, or from information
  deficit
• The estimates will contain error and the
  resulting portfolios will be suboptimal
• How serious is this effect?
• How sensitive are the various risk measures to
  this kind of error?
• How can we reduce the error?

60
Information deficit

• Thus the Markowitz problem suffers from the
  curse of dimensions, or from information
  deficit
• The estimates will contain error and the
  resulting portfolios will be suboptimal
• How serious is this effect?
• How sensitive are the various risk measures to
  this kind of error?
• How can we reduce the error?

61
Information deficit

• Thus the Markowitz problem suffers from the
  curse of dimensions, or from information
  deficit
• The estimates will contain error and the
  resulting portfolios will be suboptimal
• How serious is this effect?
• How sensitive are the various risk measures to
  this kind of error?
• How can we reduce the error?

62
Information deficit

• Thus the Markowitz problem suffers from the
  curse of dimensions, or from information
  deficit
• The estimates will contain error and the
  resulting portfolios will be suboptimal
• How serious is this effect?
• How sensitive are the various risk measures to
  this kind of error?
• How can we reduce the error?

63
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.
• Random matrix theory

64
Our approach

• 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 various 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.
• For simplicity, we use iid normal variables in
  the following.

65
Our approach

• 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 various 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.
• For simplicity, we use iid normal variables in
  the following.

66
Our approach

• 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 various 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.
• For simplicity, we use iid normal variables in
  the following.

67

• For such simple underlying processes the exact
  risk measure can be calculated.
• To construct the empirical risk measure, we
  generate long time series, and cut out segments
  of length T from them, as if making observations
  on the market.
• From these observations we construct the
  empirical risk measure and optimize our portfolio
  under it.
• The ratio qo of the empirical and the exact risk
  measure is a measure of the estimation error due
  to noise.

68

• For such simple underlying processes the exact
  risk measure can be calculated.
• To construct the empirical risk measure, we
  generate long time series, and cut out segments
  of length T from them, as if making observations
  on the market.
• From these observations we construct the
  empirical risk measure and optimize our portfolio
  under it.
• The ratio qo of the empirical and the exact risk
  measure is a measure of the estimation error due
  to noise.

69

• For such simple underlying processes the exact
  risk measure can be calculated.
• To construct the empirical risk measure, we
  generate long time series, and cut out segments
  of length T from them, as if making observations
  on the market.
• From these observations we construct the
  empirical risk measure and optimize our portfolio
  under it.
• The ratio qo of the empirical and the exact risk
  measure is a measure of the estimation error due
  to noise.

70

• For such simple underlying processes the exact
  risk measure can be calculated.
• To construct the empirical risk measure, we
  generate long time series, and cut out segments
  of length T from them, as if making observations
  on the market.
• From these observations we construct the
  empirical risk measure and optimize our portfolio
  under it.
• The ratio qo of the empirical and the exact risk
  measure is a measure of the estimation error due
  to noise.

71
The case of variance

• The relative error of the optimal portfolio
  is a random variable, fluctuating from sample to
  sample.
• The weights of the optimal portfolio also
  fluctuate.

72
The distribution of qo over the samples
73
The expectation value of qo as a function of N/T
74
The critical point N /T 1

• For a precise estimate we would need TgtgtN
• As N approaches T, the relative error is
  increasing and diverges at the critical point
  NT.
• The expectation value of the error can be shown
  to be
• This innocent formula had not been noticed in the
  literature before
• The variance of the distribution of qo diverges
  even more strongly, with an exponent -3/4.

75
Instability of the weigthsThe weights of a
portfolio of N10 iid normal variables for a
given sample, T500
76
Instability of the weigthsThe weights of a
portfolio of N100 iid normal variables for a
given sample, T500
77
The distribution of weights in a given sample

• The optimization hardly determines the weights
  even far from the critical point!
• The standard deviation of the weights relative to
  their exact average value also diverges at the
  critical point

78
Fluctuations of a given weight from sample to
sample, non-overlapping time-windows, N100, T500
79
Fluctuations of a given weight from sample to
sample, time-windows shifted by one step at a
time, N100, T500
80
If short selling is banned

• If the weights are constrained to be positive,
  the instability will manifest itself by more and
  more weights becoming zero the portfolio
  spontaneously reduces its size!
• Explanation the solution would like to run away,
  the constraints prevent it from doing so,
  therefore it will stick to the walls.
• Similar effects are observed if we impose any
  other linear constraints, like bounds on sectors,
  etc.
• It is clear, that in these cases the solution is
  determined more by the constraints than the
  objective function.

81
If short selling is banned

• If the weights are constrained to be positive,
  the instability will manifest itself by more and
  more weights becoming zero the portfolio
  spontaneously reduces its size!
• Explanation the solution would like to run away,
  the constraints prevent it from doing so,
  therefore it will stick to the walls.
• Similar effects are observed if we impose any
  other linear constraints, like bounds on sectors,
  etc.
• It is clear, that in these cases the solution is
  determined more by the constraints than the
  objective function.

82
If short selling is banned

• If the weights are constrained to be positive,
  the instability will manifest itself by more and
  more weights becoming zero the portfolio
  spontaneously reduces its size!
• Explanation the solution would like to run away,
  the constraints prevent it from doing so,
  therefore it will stick to the walls.
• Similar effects are observed if we impose any
  other linear constraints, like bounds on sectors,
  etc.
• It is clear, that in these cases the solution is
  determined more by the constraints than the
  objective function.

83
If short selling is banned

• If the weights are constrained to be positive,
  the instability will manifest itself by more and
  more weights becoming zero the portfolio
  spontaneously reduces its size!
• Explanation the solution would like to run away,
  the constraints prevent it from doing so,
  therefore it will stick to the walls.
• Similar effects are observed if we impose any
  other linear constraints, like bounds on sectors,
  etc.
• It is clear, that in these cases the solution is
  determined more by the constraints than the
  objective function.

84
If the variables are not iid

• Experimenting with various market models
  (one-factor, market plus sectors, positive and
  negative covariances, etc.) shows that the main
  conclusion does not change.
• Overwhelmingly positive correlations tend to
  enhance the instability, negative ones decrease
  it, but they do not change the power of the
  divergence, only its prefactor

85
If the variables are not iid

• Experimenting with various market models
  (one-factor, market plus sectors, positive and
  negative covariances, etc.) shows that the main
  conclusion does not change.
• Overwhelmingly positive correlations tend to
  enhance the instability, negative ones decrease
  it, but they do not change the power of the
  divergence, only its prefactor

86
All is not lost after filtering the noise is
much reduced, and we can even penetrate into the
region below the critical point TltN
87
Similar studies under mean absolute deviation,
expected shortfall and maximal loss

• Lead to similar conclusions, except that the
  effect of estimation error is even more serious
• In addition, no convincing filtering methods
  exist for these measures
• In the case of coherent measures the existence of
  a solution becomes a probabilistic issue,
  depending on the sample
• Calculation of this probability leads to some
  intriguing problems in random geometry

88
Similar studies under mean absolute deviation,
expected shortfall and maximal loss

• Lead to similar conclusions, except that the
  effect of estimation error is even more serious
• In addition, no convincing filtering methods
  exist for these measures
• In the case of coherent measures the existence of
  a solution becomes a probabilistic issue,
  depending on the sample
• Calculation of this probability leads to some
  intriguing problems in random geometry

89
Similar studies under mean absolute deviation,
expected shortfall and maximal loss

• Lead to similar conclusions, except that the
  effect of estimation error is even more serious
• In addition, no convincing filtering methods
  exist for these measures
• In the case of coherent measures the existence of
  a solution becomes a probabilistic issue,
  depending on the sample
• Calculation of this probability leads to some
  intriguing problems in random geometry

90
Similar studies under mean absolute deviation,
expected shortfall and maximal loss

• Lead to similar conclusions, except that the
  effect of estimation error is even more serious
• In addition, no convincing filtering methods
  exist for these measures
• In the case of coherent measures the existence of
  a solution becomes a probabilistic issue,
  depending on the sample
• Calculation of this probability leads to some
  intriguing problems in random geometry

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Probability of finding a solution for the minimax
problem
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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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For ES the critical value of N/T depends on the
threshold ß
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With increasing N, T ( N/T fixed) the transition
becomes sharper and sharper
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until in the limit N, T ?8 with N/T fixed we
get a phase boundary
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The mean relative error in portfolios optimized
under various risk measures blows up as we
approach the phase boundary
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Distributions of qo for various risk measures
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Instability of portfolio weights

• Similar trends can be observed if we look into
  the weights of the optimal portfolio the weights
  display a high degree of instability already for
  variance optimized portfolios, but this
  instability is even stronger for mean absolute
  deviation, expected shortfall, and maximal loss.

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Instability of weights for various risk measures,
non-overlapping windows
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Instability of weights for various risk measures,
overlapping weights
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A wider context

• Hard computational problems (combinatorial
  optimization, random assignment, graph
  partitioning, satisfiability) the length of the
  algorithm grows exponentially with the size of
  the problem. These are practically untractable.
• Their difficulty may depend on some internal
  parameter (e.g. the density of constraints in a
  satisfiability problem)
• Recently it has been observed that the difficulty
  does not change gradually with the variations of
  this parameter, but there is a critical value
  where the problem becomes hard abruptly
• This critical point is preceded by a number of
  critical phenomena

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A wider context

• Hard computational problems (combinatorial
  optimization, random assignment, graph
  partitioning, and satisfiability problems ) the
  length of the algorithm grows exponentially with
  the size of the problem. These are practically
  untractable.
• Their difficulty may depend on some internal
  parameter (e.g. the density of constraints in a
  satisfiability problem)
• Recently it has been observed that the difficulty
  does not change gradually with the variations of
  this parameter, but there is a critical value
  where the problem becomes hard abruptly
• This critical point is preceded by a number of
  critical phenomena

106
A wider context

• Hard computational problems (combinatorial
  optimization, random assignment, graph
  partitioning, and satisfiability problems ) the
  length of the algorithm grows exponentially with
  the size of the problem. These are practically
  untractable.
• Their difficulty may depend on some internal
  parameter (e.g. the density of constraints in a
  satisfiability problem)
• Recently it has been observed that the difficulty
  does not change gradually with the variations of
  this parameter, but there is a critical value
  where the problem becomes hard abruptly
• This critical point is preceded by a number of
  critical phenomena

107
A wider context

• Hard computational problems (combinatorial
  optimization, random assignment, graph
  partitioning, and satisfiability problems ) the
  length of the algorithm grows exponentially with
  the size of the problem. These are practically
  untractable.
• Their difficulty may depend on some internal
  parameter (e.g. the density of constraints in a
  satisfiability problem)
• Recently it has been observed that the difficulty
  does not change gradually with the variations of
  this parameter, but there is a critical value
  where the problem becomes hard abruptly
• This critical point is preceded by a number of
  critical phenomena

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• The critical phenomena we observe in portfolio
  selection are analogous to these, they represent
  a new random Gaussian universality class within
  this family, where a number of modes go soft in
  rapid succession, as one approaches the critical
  point.
• Filtering corresponds to discarding these soft
  modes.

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• The critical phenomena we observe in portfolio
  selection are analogous to these, they represent
  a new random Gaussian universality class within
  this family, where a number of modes go soft in
  rapid succesion, as one approaches the critical
  point.
• Filtering corresponds to discarding these soft
  modes.

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Similar examples from everyday life, and