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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 in one year,
   4  4  246  246 
  ln 0.0140.99246
LR  2ln  



   250   250  


  0.77


• 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 ( pvalue 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 chissquare 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
convexity
• (X1+ (1 – ) X2)   (X1) + (1 – ) (X2)
Discussion
• It is diversification a property of the measure?
• VaR is not sub-additive. Does it mean that
information in a super-additive measure is
irrelevant?
• 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 complex.
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 losses.
• 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
optimization
• 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 testing
How to generate consistent scenarios
• 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 variancecovariance 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 variables
Y  AX
– The problem is reduced to determining a matrix A of dimension n n such
that
AA  
t
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 .
 A11

 A21
A


A
 n1
0
A22

An 2
 0 

 0 
  

 Ann 
Multivariate Normal Variables
• Cholesky Decomposition
– It may be verified that the elements of A can be recoverd by a set of
iterative formulas
i 1
aii   ii   aik2
k 1
i 1
1 

a ji    ij   aik a jk 
aii 
k 1

– In the simple two-variable case we have
0
 1


A
   1   2 
2
 2

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”)
Views
• 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
 e1 '  1 0 0~ 
1 
6%
P

q   


0%
 2 
e 3 'e 2 ' 0  1 1
and compute the joint distribution
r 
    V
VP'  
q  N P, PV ' PVP'  

 
  
Conditional distribution
• The conditional distribution of r with
respect to q is then
   VP ' PVP'  1 q - P ; 

r q ~ N
 V - VP ' PVP'  1 PV



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
6.00
 0.01
  5.77   0.06
6.58
0.07
•
0.04 0.03
 1
R  0.04
1
0.92
0.03 0.92
1 
A shock to the term structure
7
6.5
6
Yie
lds
Historic
Scenario
5.5
5
4.5
1
6
11
16
Maturities
21
26
Stress testing analysis (1)
The short rate increases to 6%
(0.1% sd)
Scadenza
Nominale
MtM
corrente
MtM
scenario
Perdita
Media
Scenario
VaR
1 anno
100
95.08
94.18
0.90
0.92
10 anni
100
58.35
56.16
2.17
2.93
30 anni
100
15.54
13.88
1.66
2.30
Stress testing analysis (1)
The short rate increases to 6%(1%
sd)
Scadenze
Nominale
MtM
Corrente
MtM
Scenario
Perdita
Media
Scenario VaR
1 anno
100
95.08
95.08
0.00
0.53
10 anni
100
58.35
58.35
0.00
1.49
30 anni
100
15.54
15.54
0.00
1.21