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Advanced Risk Management I


Advanced Risk Management I Lecture 7 Example In applications one typically takes one year of data and a 1% confidence interval If we assume to count 4 excess losses ... – PowerPoint PPT presentation

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Title: Advanced Risk Management I

Advanced Risk Management I
  • Lecture 7

  • In applications one typically takes one year of
    data and a 1 confidence interval
  • If we assume to count 4 excess losses in one
  • Since the value of the chi-square distribution
    with one degree of freedom is 6.6349, the
    hypothesis of accuracy of the VaR measure is not
    rejected ( p-value of 0.77 è 38,02).

Christoffersen extension
  • A flaw of Kupiec test isnbased on the hypothesis
    of independent excess losses.
  • Christoffersen proposed an extension taking into
    account serial dependence. It is a joint test of
    the two hypotheses.
  • The joint test may be written as
  • LRcc LRun LRind
  • where LRun is the unconditional test and LRind
    is that of indipendence. It is distributed as a
    chis-square with 2 degrees of freedom.

Value-at-Risk criticisms
  • The issue of coherent risk measures (aximoatic
    approach to risk measures)
  • Alternative techniques (or complementary)
    expected shorfall, stress testing.
  • Liquidity risk

Coherent risk measures
  • In 1999 Artzner, Delbaen-Eber-Heath addressed the
    following problems
  • Which features must a risk measure have to be
    considered well defined?
  • Risk measure axioms
  •   Positive homogeneity ?(?X) ??(X)
  •  Translation invariance ?(X ?) ?(X) ?
  •  Subadditivity ?(X1 X2) ? ?(X1) ?(X2)

Convex risk measures
  • The hypothesis of positive homogeneity has been
    criticized on the grounds that market illiquidity
    may imply that the risk increases with the
    dimension of the position
  • For this reason, under the theory of convex risk
    measures, the axioms of positive homogeneity and
    sub-additivity were substituted with that of
  • ?(?X1 (1 ?) X2) ? ? ?(X1) (1 ?) ?(X2)

  • It is diversification a property of the measure?
  • VaR is not sub-additive. Does it mean that
    information in a super-additive measure is
  • Assume that one merges two businesses for which
    VaR is not sub-additive. He uses a measure that
    is sub-additive by definition. Does he lose some
    information that may be useful for his choice?

Expected shortfall
  • Value-at-Risk is the quantile corresponding to a
    probability level.
  • Critiques
  • VaR does not give any information on the shape of
    the distribution of losses in the tail
  • VaR of two businesses can be super-additive
    (merging two businesses, the VaR of the
    aggregated business may increase
  • In general, the problem of finding the optimal
    portfolio with VaR constraint is extremely

Expected shortfall
  • Expected shortfall is the expected loss beyond
    the VaR level. Notice however that, like VaR, the
    measure is referred to the distribution of
  • Expected shortfall is replacing VaR in many
    applications, and it is also substituting VaR in
    regulation (Base III).
  • Consider a position X, the extected shortfall is
    defined as
  • ES E(X X? VaR)

Expected shortfall pros and cons
  • Pros i) it is a measure of the shape of the
    distribution ii) it is sub-additive, iii) it is
    easily used as a constraint for portfolio
  • Cons does not give information on the fact that
    merging two businesses may increase the
    probability of default.

Stress testing
  • Stress testing techniques allow to evaluate the
    riskiness of the position to specific events
  • The choice can be made
  • Collecting infotmation on particular events or
    market situations
  • Using implied expectations in financial
    instruments, i.e. futures, options, etc
  • Scenario construction must be consistent with the
    correlation structure of data

Stress testingHow to generate consistent
  • Cholesky decomposition
  • The shock assumed on a given market and/or bucket
    propagates to others via the Cholesky matrix
  • Black and Litterman
  • The scenario selected for a given market and/or
    bucket is weighted and merged with historical
    info by a Bayesian technique.

Multivariate Normal Variables
  • Cholesky Decomposition
  • Denote with X a vector of independent random
    variables each one of which is ditributed
    acccording to a standard normal, so that the
    variance-covariance matrix of X is the n ? n
    identity matrix Assume one wants to use these
    variables to generate a second set of variables,
    that will be denoted Y, that will be correlated
    with variance-covariance matrix given ?.
  • The new system of random variables can be found
    as linear combination of the independent
  • The problem is reduced to determining a matrix A
    of dimension n? n such that

Multivariate Normal Variables
  • Cholescky Decomposition
  • The solution of the previous problem is not
    unique meaning that there exost many matrices A
    that, multiplied by their transposed, give ? as a
    result. If matrix ? is positive definite, the
    most efficient method to solve the problem
    consists in Cholescky decomposition.
  • The key point consists in looking for A in the
    shape of a lower triangular matrix .

Multivariate Normal Variables
  • Cholesky Decomposition
  • It may be verified that the elements of A can be
    recoverd by a set of iterative formulas
  • In the simple two-variable case we have

Black and Litterman
  • The technique proposed in Black and Litterman and
    largely used in asset management can be used to
    make the scenarios consistent.
  • Information sources
  • Historical (time series of prices)
  • Implied (cross-section info from derivatives)
  • Private (produced in house)

  • Assume that in house someone proposes a view
    on the performance of market 1 and a view on
    that of market 3 with respect to market 2.
  • Both views have error margins ?i with
    covariance matrix ?
  • e1' r q1 ?1
  • e3' r - e2' r q2 ?2
  • The dynamics of percentage price changes r must
    be condizioned on views view qi.

Conditioning scenarios to views
  • Let us report the views in matrixform
  • and compute the joint distribution

Conditional distribution
  • The conditional distribution of r with respect to
    q is then
  • and noticed that this may be interpreted as a
    GLS regression model (generalised least squares)

Esempio costruzione di uno scenario
  • Assumiamo di costruire uno scenario sulla curva
    dei tassi a 1, 10 e 30 anni.
  • I valori di media, deviazione standard e
    correlazione sono dati da

A shock to the term structure
Stress testing analysis (1)The short rate
increases to 6 (0.1 sd)
Stress testing analysis (1)The short rate
increases to 6(1 sd)