Vorlesung-Ws10_Teil 1.pdf

8/10/2019 Vorlesung-Ws10_Teil 1.pdf

1/66

LectureQuantitative Risk Management

Rudiger FreyUniversitat Leipzig

Wintersemester 2010/11 Universitat [email protected]

www.math.uni-leipzig.de/~frey

c2010 (Frey)


2/66

I: Foundations

Introduction and regulatory background Risk management for a financial firm

Modelling Value Change

Risk Measurement Stylized facts of financial time series

c2010 (Frey) 1


3/66

A. Introduction and Regulatory Background

c2010 (Frey) 2


4/66

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


5/66

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

c2010 (Frey) 4


6/66

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

c2010 (Frey) 5


7/66

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

c2010 (Frey) 6


8/66

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

Explicit treatment ofoperational riskc2010 (Frey) 7


9/66

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]

c2010 (Frey) 8


10/66

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.

c2010 (Frey) 9


11/66

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.

c2010 (Frey) 10


12/66

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]

c2010 (Frey) 11


13/66

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

c2010 (Frey) 12


14/66

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

c2010 (Frey) 13


15/66

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. . .

c2010 (Frey) 14


16/66

Picture from H. Fllmer

c2010 (Frey) 15


17/66

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)

c2010 (Frey) 16


18/66

B. Basic Concepts in Valuation and Risk

Management

c2010 (Frey) 17


19/66

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


20/66

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)

c2010 (Frey) 19


21/66

Balance sheet of a bank

c2010 (Frey) 20


22/66

Balance sheet of an insurer

c2010 (Frey) 21


23/66

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.

c2010 (Frey) 22


24/66

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


25/66

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.

c2010 (Frey) 24


26/66

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.

c2010 (Frey) 25


27/66

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.

c2010 (Frey) 26


28/66

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)

c2010 (Frey) 27


29/66

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,

c2010 (Frey) 28


30/66

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

c2010 (Frey) 29


31/66

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)

c2010 (Frey) 30


32/66

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.

c2010 (Frey) 31


33/66

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.

c2010 (Frey) 32


34/66

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.

c2010 (Frey) 33


35/66

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.

c2010 (Frey) 34


36/66

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


37/66

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:

c2010 (Frey) 36


38/66

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.

c2010 (Frey) 37


39/66

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


40/66

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).

c2010 (Frey) 39


41/66

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

2010 (Frey) 40


42/66

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.)

c

2010 (Frey) 41


43/66

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

2010 (Frey) 42


44/66

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

2010 (Frey) 43


45/66

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

2010 (Frey) 44


46/66

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

2010 (Frey) 45


47/66

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

2010 (Frey) 46


48/66

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

2010 (Frey) 47


49/66

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

2010 (Frey) 48


50/66

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

2010 (Frey) 49


51/66

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

2010 (Frey) 50


52/66

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

2010 (Frey) 51


53/66

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

2010 (Frey) 52


54/66

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

2010 (Frey) 53


55/66

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.

c

2010 (Frey) 54


56/66

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)

c

2010 (Frey) 55


57/66

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

according to (13).c

2010 (Frey) 56


58/66

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

acceptableto a risk controller.c

2010 (Frey) 57


59/66

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

2010 (Frey) 58


60/66

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

c

2010 (Frey) 59


61/66

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.

c

2010 (Frey) 60


62/66

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.

c

2010 (Frey) 61


63/66

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)

c

2010 (Frey) 62


64/66

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

2010 (Frey) 63


65/66

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.

c

2010 (Frey) 64


66/66

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.

c 2010 (Frey) 65

Documents

Vorlesung-Ws10_Teil 1.pdf