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LectureQuantitative Risk Management
Rudiger FreyUniversitat Leipzig
Wintersemester 2010/11 Universitat [email protected]
www.math.uni-leipzig.de/~frey
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I: Foundations
Introduction and regulatory background Risk management for a financial firm
Modelling Value Change
Risk Measurement Stylized facts of financial time series
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A. Introduction and Regulatory Background
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A1. The Road to Basel
Risk management: one of the most important innovations of the
20th century. [Steinherr, 1998]
The late 20th century saw a revolution on financial markets. Itwas an era of innovation in academic theory, product development(derivatives) and information technology and of spectacular
market growth.
Large derivatives losses and other financial incidents raised banksconsciousness of risk.
Banks became subject to regulatory capital requirements,internationally coordinated by the Basle Committee of the Bank
of International Settlements.c2010 (Frey) 3
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The Regulatory Process
1988. First Basel Accord takes first steps toward internationalminimum capital standard. Approach fairly crude and insufficiently
differentiated.
1993. The birth of VaR. Seminal G-30 report addressing for firsttime off-balance-sheet products (derivatives) in systematic way. At
same time JPMorgan introduces the Weatherstone 4.15 daily market
risk report, leading to emergence of RiskMetrics.
1996. Amendment to Basel I allowing internal VaR models formarket riskin larger banks.
2001 onwards. Second Basel Accord, focussing on credit risk but
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also puttingoperational riskon agenda. Banks may opt for a more
advanced, so-calledinternal-ratings-basedapproach to credit.
2009 onwards Discussion about regulatory consequences from thecurrent financial crisis
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A2. Basel II:
Rationale for the New Accord: More flexibility and risk sensitivity
Structure of the New Accord: Three-pillar framework:
Pillar 1: minimal capital requirements (risk measurement)
Pillar 2: supervisory review of capital adequacy
Pillar 3: public disclosure
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Basel II Continued
Two options for the measurement ofcredit risk: Standard approach
Internal rating based approach (IRB)
Pillar 1 sets out the minimum capital requirements (Cooke Ratio):total amount of capital
risk-weighted assets 8%
MRC (minimum regulatory capital) def= 8% of risk-weighted assets
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A3. QRM: the Nature of the Challenge
Extremes Matter
From the point of view of the risk manager, inappropriate use
of the normal distributioncan lead to an understatement of risk,
which must be balanced against the significant advantage ofsimplification. From the central banks corner, the consequences
are even more serious because we often need to concentrate on
the left tail of the distribution in formulating lender-of-last-resort
policies. Improving the characterization of the distribution of
extreme valuesis of paramount importance.
[Alan Greenspan, Joint Central Bank Research Conference, 1995]
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The Interdependence and Concentration of Risks
Themultivariatenature of risk presents an important challenge. Weare generally interested in some form ofaggregate riskthat depends
on high-dimensional vectors of underlyingrisk factors.
Examples:
individual asset values in market risk
credit spreads and counterparty default indicators in credit risk.
A particular concern in multivariate risk modelling is the phenomenon
of extremal dependence when many risk factors move against ussimultaneously.
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Dependent Extreme Values: LTCM
Extreme, synchronized rises and falls in financial markets occur
infrequently but they do occur. The problem with the models is
that they did not assign a high enough chance of occurrence to the
scenario in which many things go wrong at the same timethe
perfect stormscenario.
[Business Week, September 1998]
In a perfect storm scenario the risk manager discovers that thediversification he thought he had is illusory; practitioners describe this
also as a concentration of risk.
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Concentration Risk
Over the last number of years, regulators have encouraged
financial entities to use portfolio theory to produce dynamic
measures of risk. VaR, the product of portfolio theory, is used
for short-run, day-to-day profit and loss exposures. Now is the
time to encourage the BIS and other regulatory bodies to support
studies onstress test and concentration methodologies. Planning
for crises is more important than VaR analysis. And such new
methodologies are the correct response to recent crises in the
financial industry.
[Scholes, 2000]
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QRM and the current crisis
What happened
Starting in late 2006 high interest rates and falling house prices inUS large scale default of sub-prime mortgages
Sub-prime defaults collapse of MBS (mortgage-backed securities)market
Collapse of MBS market collapse in CDO market
Collapse in market for securitized debt write-offs and generalnervousness in banks
Nervousness about bad debt in banks drying up of interbanklending market and increase in cost of debt financing
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Increase in cost of debt financing liquidity problems at banks
Liquidity problems bank runs and (near) default of many financialinstitutions such as Northern Rock, Bear Stearns, Lehman , AIG,
Citi, Hypo real Estate; end of traditional American investment banks
general financial malaise leading to a global recession (fuelled
further by global economic imbalances) ; nationalization of banks
and various
rescue packages for financial institutions of unprecedented size;
discussion about regulatory reform
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Critical comments on quantitative methods
The Economist: With their snappy name and flashy mathematicalformulae quants were the stars of the finance show before the credit
crisis
Colin Creevy (Europ. Kommission) the irresponsible lending, blindinvesting, bad liquidity management, excessive stretching of ratingagency brands anddefective Value at Risk Modellingthat prompted
the turmoil of recent months[the subprime credit crisis]
Financial Times It is a worry, though, that Merrill Lynch can justifya write-down of $4.5bn one week and $7.9bn just three weeks later.It seems that valuation is still a matter ofpick a number and divide
by the chief traders golf handicap . . .
From Black Scholes to black holes. . .
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Picture from H. Fllmer
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My view
Quantitative methods are here to stay: They provide importantconcepts, tools and techniques for dealing with financial risk
There is room for improved mathematical modelling and advancedstatistical techniques that can help in building better risk
management systems Nonetheless we have to be aware of the inherent limitations of
mathematical models in the financial world
RM is like driving a car through the back mirror.
(J. Longerstay)In physics there may one day be a model of everything. In
finance one is fortunate if there is a usable theory of anything
(E. Derman)
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B. Basic Concepts in Valuation and Risk
Management
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B1. Risk Management for a Financial Firm
A good way to understand the risks faced by a financial institution(bank / insurance company) is to look at a stylized balance sheet.
Key concepts.
assets (Aktiva). Describes the investments of the institution liabilities (Passiva). Describes how the institution is funding itself
equity (Eigenkapital.) Defined by thebalance sheet equation
value of assets = value of liabilities + equity
Equity consists of equity capital raised by share issues etc, augmented
by retained profits and reduced by dividends and losses.c2010 (Frey) 18
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solvency. A firm is called solvent if equity > 0 and otherwiseinsolvent. Distinguish from default that occurs if firm misses a
payment to debtholders.
Valuation principles
Fair value accounting. Value an item on the balance sheet by (anestimate of) its market value. Special case: risk neutral valuation asin mathematical finance.
Book value. In finance typically nominal value - risk provision (e.g.for a loan)
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Balance sheet of a bank
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Balance sheet of an insurer
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Risks faced by a financial firm
Risks faced by a typical bank
Decrease in value of investments on the asset side (market risk andcredit risk)
Funding and maturity mismatch. (long-term illiquid assets fundedby short-term liabilities)
Key risk of an insurer is insolvency. Sources:
asset side: decrease in value of investments.
liability side: reserves insufficient to cover future claim payments.Note that for life insurers liabilities are long-term.
We conclude that funding of positions plays a crucial role and that the
two sides of the balance sheet have to be looked at jointly.
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B2. Loss Distributions
To model risk we use language ofprobability theory. Risks arerepresented byrandom variablesmapping unforeseen future states of
the world into values representingprofits and losses.
The risks which interest us areaggregaterisks. In general we consider
aportfoliowhich might be
a collection ofstocks and bonds;
a book ofderivatives;
a collection of riskyloans;
a financial institutionsoverall positionin risky assets.c2010 (Frey) 23
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Portfolio Values and Losses
Consider a portfolio and let Vt denote itsvalueat time t; we assumethis random variable is observableat time t.
Suppose we look at risk from perspective of time t and we consider
the time period [t, t + 1]. The value Vt+1 at the end of the time
period is unknown to us.
The distribution of(Vt+1 Vt) is known as the profit-and-loss orP&Ldistribution. We denote thelossbyLt+1= (Vt+1 Vt). By thisconvention, losses will be positive numbers and profits negative.
We refer to the distribution ofLt+1 as the loss distribution.
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Risk Factors and mapping
Generally the value of the portfolio at time t will depend on time anda set of observablerisk factors Zt= (Zt,1, . . . , Z t,d)
. Formally,
Vt=f(t,Zt) forf: R+ Rd R, .
This representation is termedmapping. Examples for risk factors
include logarithmic stock prices or index values, yields and exchange
rates.
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Loss Distributions
Denote the time series ofrisk factor changesbyX
t+1=Zt+1 Zt.Then the loss can be written as
Lt+1= (f(t + 1,Zt+ Xt+1) f(t,Zt). (1)
As of time t only random part is the risk factor change Xt+1. Henceloss distribution is determined by fand by the distribution of risk
factor change. Sometimes we use a linearizedversion of (1).
Lt+1 ft(t,Zt)t + di=1
fZi(t,Zt)Xt+1,i=:Lt+1, (2)where subscripts denote partial derivatives and where t is the risk
management horizon.
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Example: Portfolio of Stocks
Considerd stocks; let i denote number of shares in stock i at time tand let St,i denote price.
The risk factors: following standard convention we take logarithmic
prices as risk factors Zt,i= log St,i, 1
i
d.
The risk factor changes: in this case these are
Xt+1,i= log St+1,i log St,i, which correspond to the so-calledlog-returnsof the stock.
The Mapping
Vt=
di=1
iSt,i=
di=1
ieZt,i. (3)
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Example Continued
The Loss
Lt+1 =
di=1
ieZt+1,i
di=1
ieZt,i
= Vtdi=1
t,i
eXt+1,i 1 (4)wheret,i=iSt,i/Vt is relative weight of stock i at time t.
Here there is no explicit time dependence in the mapping (3). The
partial derivatives with respect to risk factors are
fZi(t,Zt) =ieZt,i, 1 i d,
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and hence the linearized loss (??) is
Lt+1= di=1
ieZt,iXt+1,i= Vt d
i=1
t,iXt+1,i, (5)
wheret,i=iSt,i/Vt is relative weight of stock i at time t. This
formula may be compared with (4).
Moments of linearized loss
Assume that X has mean vector and covariance matrix . Then
E(Lt+1) = Vt var(Lt+1) =V2t
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An example with BMW and Siemens shares
Respective prices on evening 23.07.96: 844.00 and 76.9. Considerportfolio of one BMW share and 10 Siemens shares. We get the
following results
Lt+1=
(844(ex1
1) + 769(ex2
1))
Lt+1= (844X1+ 769X2)
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The dataBMW
Time
300
500
700
9
00
02.01.89 02.01.90 02.01.91 02.01.92 02.01.93 02.01.94 02.01.95 02.01.96
Siemens
Time
50
60
70
80
02.01.89 02.01.90 02.01.91 02.01.92 02.01.93 02.01.94 02.01.95 02.01.96
BMW and Siemens Data: 1972 days to 23.07.96.
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BMW
Time
-0.
10
0.
00
.05
02.01.89 02.01.90 02.01.91 02.01.92 02.01.93 02.01.94 02.01.95 02.01.96
Siemens
Time
-0.
10
0.
0
0.
05
02.01.89 02.01.90 02.01.91 02.01.92 02.01.93 02.01.94 02.01.95 02.01.96
BMW and Siemens Log Return Data: 1972 days to 23.07.96.
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Example: European Call Option
Consider portfolio consisting of one standard European call on anon-dividend payingstock Swithmaturity T andexercise price K.
We assume that Black-Scholes formula is used to value the option.
Recall that Black-Scholes price of a European call on S is given by
CBS(t, S; r, ) =S(d1) Ker(Tt)(d2), where
is standard normal df, r represents risk-free interest rate, the
volatility of underlying stock, and
d1=log(S/K) + (r+ 2/2)(T t)
T t andd2=d1
T t.
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Example Continued
Canonical risk factor: log-price of underlying asset. In reality interestrates and volatilities tend to fluctuate over time; they should be added
to the set of risk factors.
The risk factors: Zt= (log St, rt, t)
.
The risk factor changes: Xt= (log(St/St1), rt rt1, t t1). The mapping: Vt=CBS(t, St; rt, t).
Remark. In practice t would be computed as implied volatility fromobserved option prices.
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Linearized loss and Greeks
For derivative positions it is quite common to calculate linearized loss.Lt+1=
ft(t,Zt)t +
3i=1 fZi(t,Zt)Xt+1,i
.
It is more common to write the linearized loss as
Lt+1= CBS t + CBSS StXt+1,1+ CBSr Xt+1,2+ CBS Xt+1,3 ,
in terms of the derivatives of the BS formula.
CBSS is known as thedeltaof the option. CBS is thevega. CBSr is therho. CBS is thetheta.c2010 (Frey) 35
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Valuation methods and fair value
Thefair valueof an asset is an estimate of the price one would receivein selling the asset on an active market.
For many assets active markets are rare 3 different levels.
Level 1. Value is obtained from quoted price for the same instrumentin active market (typical example: stock portfolio) Level 2. Value is obtained from quoted price of similar but not
identical assets or from pricing models where all necessary inputs
are observable market data (typical example: European option withnon-standard strike or maturity).
Level 3. Value is obtained from pricing model where some inputs aresubjective estimates instead of market observables. (typical example:
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loan portfolio where credit spread has to be estimated via a subjective
scoring technique since there are no traded bonds or CDS related to
the borrower)
The three levels are also known as mark to market, mark to model
with objective inputs and mark to model with subjective inputs.
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B3. Evaluating loss distributions
Recall that
Lt+1= (f(t + 1,Zt+ Xt+1) f(t,Zt)). (6)
Hence finding the distribution ofLt+1 involves two tasks:
specify/estimate a model for risk factor changes Xt+1 evaluate distribution of the rv f(t + 1,Zt+ Xt+1).Three approaches:
Analytical methods such as variance-covariance method historical simulation method (bootstrap), Simulation methods (Monte Carlo).c2010 (Frey) 38
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Variance-Covariance Method
Assumptions
Xt+1 ismultivariate normallydistributed, Xt+1 Nd(, ). The linearized loss Lt+1 is a sufficiently accurate approximation of
Lt+1. Determine the distribution of
Lt+1=
c +di=1
wixi
= (c + wx)
(compare the stock-portfolio (5)). Recall that linear combinations of
multivariate normally distributed random vectors are multivariate
normal. HenceLt+1 N(cw,ww).
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Implementing the Method
1. The constant terms in c and w are calculated from the mapping f.
2. The mean vector and covariance matrix are estimated from
data Xtn+1, . . . ,Xt to give estimates and.3. Inference about the loss distribution is made using distribution
N(cw
,ww)
4. Estimates of risk measures such as VaR are calculated from the
estimated distribution ofL.
c
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Pros and Cons, Extensions
Pro. Variance-covariance offers analytical solution with nosimulation.
Cons. Linearization may be crude approximation. Assumption of
normality may seriouslyunderestimate tailof loss distribution.
Extensions. Instead of assuming normal risk factors, the methodcould be easily adapted to use multivariate Student t risk factors or
multivariate hyperbolic risk factors, without sacrificing tractability.(Method works for all elliptical distributions.)
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Historical Simulation Method
Instead of estimating the distribution ofLt+1) under some explicit
parametric model for Xt+1, one estimates distribution of the loss
corresponding to thecurrentportfolio usingempirical distributionof
past risk factor changes Xtn+1, . . . ,Xt (n data points):
1. Construct thehistorical simulation data{Ls= f(t,Zt+ Xs) f(t,Zt): s=t n + 1, . . . , t} (7)
2. Approximate loss distribution using historically simulated data:
P(Lt+1 ) 1n
nj=1
1{Ltj+1}.
c
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Discussion
Theoretical Justification. IfXtn+1, . . . ,Xt are iid or moregenerally stationary, convergence of empirical distribution to true
distribution is ensured by suitable version of law of large numbers.
Pros and Cons.
Pros. Easy to implement. No statistical estimation of the distributionofX necessary.
Cons. It may be difficult to collect sufficient quantities of relevant,synchronized data for all risk factors. Historical data may not containexamples of extreme scenarios. Sensitivity wrt. sample period.
c
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Monte Carlo Methods
Here one estimates the distribution ofLt+1 under some parametricmodel for Xt+1 using Monte Carlo methods, which involves
simulationof new risk factor data:
1. With the help of the historical risk factor data Xtn+1, . . . ,Xt
calibrate a suitable statistical model for risk factor changes and
simulatemnew dataX(1), . . . ,X(m) from this model.2. Construct the Monte Carlo dataLi= {f(t,Zt+X(i)) f(t,Zt) : i= 1, . . . , m.3. Make inference about loss distribution using the simulated data
L1, . . . ,
Lm.
c
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Pros and Cons
Pros. Very general. No restriction in our choice of distribution forXt+1.
Cons. Can be very time consuming if mapping f is difficult to
evaluate, which depends on size and complexity of portfolio.
Note that MC approach does not address the problem of determining
the distribution ofXt+1.
c
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B4. Risk Measures
Risk measures attempt to quantify the riskiness of a portfolio.Applications:
Determination of risk capital
Management tool eg. in limit systems Pricing, eg. premium principles in insuranceMost risk measures arestatistics of the loss distributionsuch as
variance or Value at Risk. Sometimes so-calledscenario-basedrisk
measures are used as-well.
In the sequel we denote the loss distribution by P(L ) =FL()whereL is a generic loss variable such as Lt+1) orL
t+1.
c
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VaR
Given a confidence level 0< ) 1 } (8)
= inf{ R : FL() }. (9)
In probabilistic terms VaR is thus the -quantileq(FL), where for
an arbitrary dfF on R and
(0, 1)
q(F) = inf{x R : F(x) .}
c
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VaR in Visual Terms
Loss Distribution
probability
dens
ity
-10 -5 0
5
10
0.0
0
.05
0.
10
0.
15
0.
20
0.2
5Mean loss = -2.4
95% VaR = 1.6
5% probability
95% ES = 3.3
c
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Losses and Profits
Profit & Loss Distribution (P&L)
probability
dens
ity
-10 -5 0
5
10
0.0
0
.05
0.
10
0.
15
0.
20
0.2
5 Mean profit = 2.495% VaR = 1.6
5% probability
c
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Expected Shortfall
ProvidedE(|L|)< expected shortfallis defined asES=
1
1 1
qu(FL)du. (10)
For continuous loss distributions expected shortfall is the expected
loss, given that the VaR is exceeded:
Lemma. For any (0, 1) we have
ES=E(L; L q(L))
1 =E(L | L VaR) ,
whereE(X; A) :=E(X1A).c
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Expected Shortfall ctd.
Remark. For a discontinuous loss df we have the more complicatedexpression
ES= 1
1
E(L; L q) + q(1 P(L q)).
Advantages ofES.
ES takes the whole tail of the distribution beyond VaR into
account; in particular ES>VaR. ES has better properties regarding aggregation of risk. This is
related to so-calledcoherenceof risk measures.
c
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Expected shortfall: Examples
Normal losses. Suppose that L N(, 2) and fix (0, 1).Denote by the density of the standard normal distribution. Then
ES= + (1())
1
. (11)
Student t losses. Suppose that (L )/ t for >1, where thedensity of standard tdistribution is given by
g(x) =C(1 + x2/)(+1)/2. Then
ES(L) = + g(t
1 ())
1 + (t1 ())2
1
, (12)
c
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VaR versus ES: an example
Consider daily losses on position in some stock; current value of theposition equals Vt= 10 000.
Loss for this portfolio is given byLt+1= VtXt+1forXt+1the dailylog-returns.
Assume that Xt+1 has mean zero and standard deviation= 0.2/
250, (annualized volatility of 20%.)
Two different models for the distribution ofXt+1: (i) Xt+1 N(0, 2) (ii) Xt+1=
2Lfor
L tand >2 ( var(L) =2).
c
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Numerical results
0.90 0.95 0.975 0.99 0.995VaR (normal model) 162.1 208.1 247.9 294.3 325.8
VaR (t model) 137.1 190.7 248.3 335.1 411.8
ES (normal model) 222.0 260.9 295.7 337.2 365.8
ES (t model) 223.4 286.3 356.7 465.8 563.5VaR and ES in normal and t4 model for different values of.
Thet model is in principle more dangerous than the normal model.
UsingVaR this is seen only for very close to one; ES shows this
already for = 0.95.
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Scenario-based risk measures
Idea. Considers a number of possible future risk-factor changes calledscenarios; risk of portfolio is given asmaximal lossunder all scenarios;
extreme or implausible scenarios may be down-weighted.
Formal description. Fix a setX = {x1, . . . ,xn} of scenarios and avector w= (w1, . . . , wn) [0, 1]n of weights. Denote the portfolioloss caused the risk factor change x by
l[t](x) := (f(t + 1,Zt+ x) f(t,Zt)).
The risk of the portfolio is then measured as
[X,w]:= max{w1l[t](x1), . . . , wnl[t](xn)}. (13)
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Applications and Examples
The approach is frequently used formargin requirementsat exchangesand instress tests.
CME-example. [Artzner et al., 1999] :
simple portfolios consisting of a position in a futures contract and
options on this contract.
16 different scenarios: First 14 consist of an up move or a downmove of volatility combined with no move, an up or down move of
the futures price by 1/3, 2/3 or3/3. Moreover 2 extreme scenarios. The weights: w1 = = w14 = 1.. Extreme scenarios are down-
weighted: w15=w16= 0.35. Margin requirement is then computed
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Coherent Measures of Risk
There are many possible measures of the risk in a portfolio such as
VaR, ES or stress losses. To decide which are reasonable risk
measures a systematic approach is called for.
New approach ([Artzner et al., 1999], [Follmer and Schied, 2004]):
Give a list of properties (Axioms) that a reasonable risk measureshould have; such risk measures are calledcoherentorconvex.
Study coherence of standard risk measures (VaR, ES, etc.).
More theoretical: characterize all convex/coherent risk measures.Here we view a risk measure as amount of capital that needs to be
added to a position with loss L, so that the position becomes
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The Axioms
Acoherent risk measureis a realvalued function on some spaceMof rvs (representing losses) that fulfills the following 4 axioms:
1. Monotonicity. For two rvs with L1 L2 we have (L1) (L2).
2. Translation invariance. Fora R we have (L + a) =(L) + a.
3. Subadditivity. For anyL1,L2we have(L1 + L2) (L1) + (L2).Most debated, sinceVaRis in general not subadditive. Justifications:
Reflects idea that risk can be reduced by diversificationand thata merger creates no extra risk.
Makesdecentralizedrisk management possible.c
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The Axioms II
4. Positive homogeneity. For 0 we have that (L) = (L). Ifthere is no diversification we should have equality in subadditivity
axiom.
Sometimes subadditivity and positive homogeneity are replaced by theweaker axiom of convexity [Follmer and Schied, 2004]:
5. Convexity. (L1+ (1 )L2) (L1) + ( 1 )(L2) for all [0, 1].
A risk measure that satisfies monotonicity, translation invariance
and convexity is called aconvex measure of risk
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Comments
VaR is in general not subadditve and hence not coherent (otheraxioms are satisfied)
ES is coherent, in particular subadditive.
coherent convex. The converse is wrong. If is positive homogeneous, subadditivity and convexity are
equivalent.
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Non-Coherence of VaR: an Example
Setup. Consider portfolio of 2 defaultable bonds with independentdefaults. Default probability identical and equal to p= 0.9%. Current
price of bonds equal to 100, face value equal to 105, recovery rate =0.
Li loss of one unit in bond i. We have
Li=
(105 100) = 5 (no default, probability1p= 0.991)(0 100) = 100 (default, probabilityp= 0.009) .
Set= 0.99. We have P(Li< 5) = 0 andP(Li 5) = 0.991> so that VaR(Li) = 5.
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Non-Coherence of VaR: an Example ctd.
Consider now L=L1+ L2, i.e. a portfolio of one bond from eachfirm. Since defaults are independent we get
L=
10 (no default, probability(1p)2 = 0.982)
(105
200) = 95 (exactly 1 default, probability 2p(1
p) )
200 (2 defaults, probabilityp2)
In particular P(L 10) = 0.9820.99sothat VaR(L) = 95 Hence VaR is non-coherent in this example.
Remark. In the example VaR punishes diversification, as
VaR(0.5L1+ 0.5L2) = 0.5 VaR(L) = 47.5>VaR(L1)
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Dual Representation
Theorem Consider a general probability space (,F, P) and takeMto be the set of all bounded measurable functions on (,F, P).Suppose that : M R is a risk measure with the followingcontinuity property:
For Ln M with Ln L M one has (Ln) (L). (14)
Suppose moreover that iscoherent. Then it has the representation
(L) = sup{EQ(L) : Q Q} (15)for some set Q= Q() S1(,F) (the set of all probabilitymeasures on (,F)).c
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Example: expected shortfall
Recall that ES(L) = 11 1qu(L)du, L L1(,F, P) and that ESis coherent.
The dual representation is given by
ES(L) = maxEQ(L) :Q VaR(L)}+ L1{L=VaR(L)}, (17)
for some constant L such that E(dQLdP ) = 1.
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Bibliography
[Artzner et al., 1999] Artzner, P., Delbaen, F., Eber, J., and Heath, D.(1999). Coherent measures of risk. Math. Finance, 9:203228.
[Follmer and Schied, 2004] Follmer, H. and Schied, A. (2004).
Stochastic Finance An Introduction in Discrete Time. Walter de
Gruyter, Berlin New York, 2nd edition.
[Scholes, 2000] Scholes, M. (2000). Crisis and risk management.
Amer. Econ. Rev., pages 1722.
[Steinherr, 1998] Steinherr, A. (1998). Derivatives. The Wild Beast of
Finance. Wiley, New York.
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