An introduction to coherent risk measures [5mm] - unibo. · PDF fileAn introduction to...

An introduction to coherent risk measures

Giacomo Scandolo

Universita di Firenze

Risk measures: frontiers of mathematics and regulations

Universita di Bologna - 2015

Outline

I The problem of risk assessment

I Value-at-Risk (VaR) and Expected Shortfall (ES)I Quantiles, VaR, ESI Subadditivity propertyI Convexity property and portfolio allocation

I Coherent risk measuresI Definition of coherenceI Examples of coherenceI Combinations preserving coherence

I Risk estimationI Non-parametric risk estimationI Parametric risk estimationI Robustness issues in risk estimation

An introduction to coherent risk measures Giacomo Scandolo (Unifi)

The problem of risk assessment

I Assessing the ”risk” of something (an asset, a portfolio) is tricky, asthere is no precise definition of risk

I In the ’70 some psychological tests have investigated the subjectivenotion of risk (positive attitude)

I For sure ”risk” is not exactly the same as

I dispersion

I uncertainty (or ambiguity)

I Within financial regulation: ”risk” is defined by the procedure we useto quantify it (normative attitude)

An introduction to coherent risk measures Giacomo Scandolo (Unifi) 1 / 118

Horizon

I Whatever the notion we consider, (financial) ”risk” depends on thetime horizon we look at

I very short (infra-day): liquidity risk

I short (1 to 10 days): market risk

I medium (1 month to 1 year): credit and operational risk

I long (several years): longevity and other social risks

I So, a risk figure is a number attached to a portfolio and for a given timehorizon.


Profit and Loss

I Once the horizon T is fixed, the Profit&Loss (PL) of a portfolio is

PL = VT − V0

where Vt is the portfolio ”value” at time t (t = 0,T )

I positive values are Profits, negative values are Losses

I in Insurance L = −PL (Loss variable) is more often used

I today, at time t = 0, V0 is known while VT is unknown

I actually, in practice even V0 is often ”unknown” (lack of quotedprices, liquidity issues, etc.)


Risk factors and risk mapping

I Actual portfolios are often way too complex. Some simplification isintroduced by selecting some basic financial variables

Y = (Y1, . . . ,Yd)′ (d ”small”)

and writing

Vt ' v(Yt ) = v(Y1,t , . . . ,Yd,t )

I Yi are the risk factors

I typical choices are: log-prices, log-indices, interest/FX rates,credit spreads

I v = v(y) is the risk mapping

I the higher is d, the more accurate is V ' v(Y)



I If ∆Y = YT − Y0, we can write (from now on ”=” in place of ”'”)

PL = v(YT )− v(Y0) = v(Y0 + ∆Y)− v(Y0)

I ∆Y is a log-return (prices, indices), or just a variation (rates, spreads).

I We can then writePL = pl(∆Y)

for the function pl(z) = v(Y0 + z)− v(Y0).

I Note: Y0 is known and ∆Y and pl can be seen as (actual) risk factorsand risk mapping



I 4 different situations

1. 1 risk factor, linear risk mapping.

The easiest (but less realistic) case.

2. 1 risk factor, non-linear risk mapping.

Typical for option portfolios; problems in deriving thedistribution of PL.

3. more risk factors, linear risk mapping.

This is the case of realistic equity or bond portfolios; requiresmultivariate models for R.

4. more risk factors, non-linear risk mapping.

Combining the difficulties of both case 2 and 3.


Approaches for assessing risk

I 3 different approaches for quantifying portfolio risk

I sensitivities approach:

risk is the strength of dependence of the portfolio value w.r.t. anunderlying variable

I stress testing approach:

risk is the response of the PL with respect to some extremescenarios

I probabilistic approach:

a risk statistics (like VaR or ES) is applied to the PL distribution


Sensitivity approachI Consider just 1 risk factor Y . If T is short, we can write (Delta

approximation)

PL = v(Y0 + ∆Y )− v(Y0) ' v′(Y0)∆Y

I Within the sensitivity approach, risk is quantified as the strength of thedependence of PL on ∆Y , i.e.

risk = v′(Y0)

I For an options portfolio: v′(Y0) is the Delta (Y0 underlying price). For abond portfolio: v′(Y0) is proportional to the duration (Y0 referenceyield)

I Pros: 1. no probabilistic model, 2. very common in practice since along time

I Cons: 1. just 1 variable at the time, 2. only for a short horizon, 3. riskexpressed in non-monetary terms


Stress test approachI Within the stress test approach we select a certain number of

scenarios, i.e. hypothetical evolutions for Y:

∆Y(1), . . . ,∆Y(K )

and obtain corresponding K test values for PL:

PL(i) = pl(∆Y(i)) i = 1, . . . ,K

I Then we can retain all test values or consider the worst-case, i.e.

risk = mini

PL(i)

I Pros: 1. favoured by regulators, 2. for all horizons, 3. captures extremescenarios ruled out by common probabilistic models (e.g. −10% dailystock returns, 200bp yield increases, etc), 4. risk is expressed inmonetary terms

I Cons (a big one): highly dependent on the choice of scenarios


Probabilistic approachI Within the probabilistic approach we consider a risk measureρ : L → R (L is a set of r.v.) which is law-invariant, i.e.

X ∼ Y =⇒ ρ(X ) = ρ(Y )

So, ρ depends only on FX and can be called a risk statistics

I For any portfolio, we develop a probabilistic model for PL (givenhorizon) and compute

risk = ρ(PL)

I In practice, we first build a probabilistic model for ∆Y and then derivethe distribution of PL = pl(∆Y)

I Pros: 1. for all horizons, 2. expressed in monetary terms, 3. results canbe checked on a statistical ground (backtesting), 4. different ρ capturedifferent facets of risk

I Cons: 1. building a probabilistic model is not easy, 2. sometime,non-technical people find ρ difficult to understand


The three basic approaches

I Let PL = pl(∆Y). There are 3 approaches to derive the distribution ofPL:

I Analytical (or variance-covariance).

We ”know” a distribution for ∆Y and we can obtain analyticallythe distribution of pl(∆Y)

I Monte Carlo.

We ”know” a distribution for ∆Y, but we need to resort tosimulation to get a (empirical) distribution for pl(∆Y).

I Historical.

We don’t ”know” a distribution for ∆Y. We use its empirical (orhistorical) distribution and derive the corresponding empiricaldistribution of PL.

I Analytical: typically used with linear portfolios (pl linear)Monte Carlo: non linear portfoliosHistorical: can always be used (indeed, the ”standard” method)


Outline






Quantiles - continuous r.v.

I Consider X with invertible cdf F (x) = P(X 6 x) (no jumps, no flatsections).

I The quantile of order α ∈ (0, 1) is

q(α) = F−1(α)

Other notation: qX (α) and qα(X )

I By definition: it is the unique q ∈ R s.t.

F (q) = P(X 6 q) =

∫ q

−∞f (x) dx = α

I Note: q(n/100) is the n-th percentile, q(0.5) is the median,q(0.75)− q(0.25) is the interquartile range, etc.



−3 q(20%) 0 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure: Quantile of order α = 20% in terms of F



−3 q(20%) 0 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

20%

Figure: Quantile of order α = 20% in terms of f


Quantile functionI The quantile function α 7→ q(α) is increasing and q(0+) = −∞ and

q(1−) = +∞when the support of X is R (i.e. f > 0 on all R).

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure: Black: F (standard normal). Red: q. In order to invert a function,just mirror its graph w.r.t. the main bisectrix (dashed line)


Examples

I If U ∼ U (0, 1), then F (x) = x (x ∈ [0, 1]) so q(α) = α

I If X ∼ Exp(λ), then F (x) = 1− e−λx (x > 0), so that

q(α) = − 1λ

log(1− α)

If λ = 1, then q(0.9) = 2.303 and this means P(X 6 2.303) = 90%

I For X ∼ N (0, 1) and∼ t(ν), the cdf is not known explicitly. However:

zα = qN(0,1)(α) = Φ−1(α) tν,α = qt(ν)(α)

can be easily approximated with great precision (e.g. z1% ' −2.3263)


Quantiles - general case

I F not invertible when either

I F is not strictly increasing corresponding to α (flat section)

I F has a jump corresponding to α

I In both cases, define

q(α) = infx : F (x) > α

I If F is invertible, then this definition coincides with the previous one.


Quantiles - discrete r.v.

I Applying the definition to an empirical distribution

X ∼ x1, . . . , xN/1/N . . . , 1/N

(with xk < xk+1) we have

qX (α) =

x1 if α ∈ (0, 1/N ]x2 if α ∈ (1/N , 2/N ]. . .xk if α ∈ ((k − 1)/N , k/N ]. . .

I So, if N = 200, then

q(1%) = x2 q(5%) = x10

and so on.


First key result

I If h is continuous and strictly increasing (hence, invertible), then

qα(h(X )) = h(qα(X ))

I So, for instance,

q(X 3) = q(X )3

q(eX ) = eq(X )

q(log X ) = log q(X ) (X > 0)

q(aX + b) = aq(X ) + b (a > 0)

I Instead, q(X 2) 6= q(X )2 and q(−X ) 6= −q(X ) in general

I However, if FX is invertible, then qα(−X ) = −q1−α(X )


Quantiles and location-scale familiesI Let X have (finite) mean µ and variance σ2.

If X = (X − µ)/σ (note: it is standard), then

q(X ) = σq(X ) + µ

I For instance, if X ∼ N (5, 4), then X ∼ N (0, 1) and

q5%(X ) = 2q5%(X ) + 5 = 2z5% + 5 = 1.710

I For a given standard Z , the associated location-scale family is

σZ + µ : σ > 0, µ ∈ R = X : X = Z

So, knowing q(Z) is ”enough” for computing q(X ) for X in thelocation-scale family.

I Note: a location-scale can be defined even if Z has no mean/variance(e.g. Pareto or Cauchy): σ and µ are just parameters and X makes nosense


Second key result

I Consider X and let q = qX . Then there exists UX ∼ U (0, 1) such thatX = q(UX ). In particular

X ∼ q(U ) U ∼ U (0, 1)

I It follows ∫ 1

0q(u) du = E [q(U )] = E [X ]

I More generally, if α ∈ (0, 1) and X is a.c., then∫ α

0q(u) du = αE [q(U )|U 6 α] = αE [X |X 6 q(α)]

or1α

∫ α

0q(u) du = E [X |X 6 q(α)].


Value-at-Risk

I The Value-at-Risk of order α ∈ (0, 1) is

VaRα(X ) = −qα(X )

where X is the portfolio PL over a certain horizon T

I The minus sign translates high losses (X 0) into a high risk(VaR 0)

I Order α: 1%-5% (market risk), 1% or less (credit, operative,insurance risk)

I Horizon ∆t : 1 day (internal monitoring), 10 days (market riskunder Basel III), 1 year (credit, operative, insurance risk)


Expected Shortfall

I The Expected Shortfall of order α ∈ (0, 1) is

ESα(X ) = − 1α

∫ α

0qX (u) du

or

ESα(X ) =1α

∫ α

0VaRu(X ) du

(again, X is the portfolio PL)

I Other names are used (but beware):

Average/Conditional/Tail VaR (AVaR/CVaR/TVaR)

Tail Conditional Expectation (TCE)

Expected Tail Loss (ETL)

I α and ∆t chosen similarly as for VaR


Expected Shortfall

I If X is a.c. we know that:

ESα(X ) = −E [X |X 6 qα(X )] (1)

and this better explains the name ”Expected Shortfall”

I Remind that

E [X |A]def=

E [X · IA]

P(A)

whenever A is an event with positive probability.

I No more true if X is not a.c. In that case

ESα(X ) = −E [X |X 6 qα] · F (qα)

α+ qα ·

(F (qα)

α− 1)

However, for practical applications (1) is still a good approximation.


VaR vs ES

I If PL is continuous and v = VaRα(PL):

α = P(PL 6 −v) = P(Loss > v), Loss = −PL

I VaR5% = 10000 Euro means: the probability of a loss of 10000 Euro ormore is 5%

I Regarding ES:

ESα(PL) = E [−PL|PL 6 q5%(PL)] = E [Loss|Loss > VaR5%(PL)]

I ES5% = 12000 Euro means: the average of the 5% worst losses (largerthan 10000) is 12000


Some basic inequalities

I VaR and ES are decreasing in α. So

VaR5%(X ) 6 VaR1%(X )

ES5%(X ) 6 ES1%(X )

Remarkably, when X is normal

VaR1%(X ) ' ES2.5%(X )

I As VaR is decreasing in α:

ESα(X ) > VaRα(X )

i.e. ES is more conservative than VaR.


VaR vs ESI VaR:

I introduced in the early 90’s by JP Morgan (RiskMetricsprocedure)

I the standard in the financial sector

I Basel III capital requirement is based on VaR

I ES:I increasingly used by fund managers and in the insurance sector

I current debate about replacing VaR (at 1%) with ES (at 2.5%) inBasel rules

I Also:I Estimation process is similar

I ES is coherent, VaR is not (see later)

I VaR is defined for all distribution; ES requires an ”integrable” tail,so it is +∞ for a Cauchy (possible problem only for operative andinsurance risks)


VaR and ES computation

I VaR computation: sign change; ES: a bit more work

I If X ∼ Lap(1), i.e. f (x) = e−|x|/2, then q(u) = log(2u) (u < 1/2) and

VaRα(X ) = − log(2α) α < 1/2

I Concerning ES, we compute (α < 1/2)

ESα(X ) = − 1α

∫ α

0log(2u) du

= − 1α

[u(log(2u)− 1)]α0

= −α(log(2α)− 1)

α= 1− log(2α).


VaR and ES computationI If Z ∼ N (0, 1) and α < 1/2 then (zα < 0)

VaRα(Z) = |zα|

I Concerning ES:

ESα(Z) = −E [Z |Z 6 zα] = −E [Z · IZ6zα ]

P(Z 6 zα)= − 1

α

∫ zα

−∞xϕ(x) dx

Observing that ϕ′ = −xϕ and ϕ(−∞) = 0 we readily obtain

ESα(Z) =ϕ(zα)

α

I Key values for VaR and ES for a standard normal are as follows

α 0.1% 1% 5%

VaRα 3.090 2.326 1.645ESα 3.367 2.665 2.063


VaR and ES computation

0 0.01 0.02 0.03 0.04 0.051

2

3

4

5

6

7

8

Figure: VaR (solid) and ES (dashed) depending on α. Black: N(0,1); red:Lap(1). Note: the heavier the tail (Lap) the higher the VaR and ES.


VaR and ES of transformed variables

I As q(aX + b) = a · q(X ) + b, a > 0

VaRα(aX + b) = a VaRα(X )− b

and

ESα(aX + b) =1α

∫ α

0a VaRu(X )− b du

=aα

∫ α

0VaRu(X ) du − b

α

∫ α

0du = a ESα(X )− b

I Both are

I positively homogeneous: ρ(aX ) = aρ(X ), a > 0

I translation equivariant: ρ(X + b) = ρ(X )− b


VaR and ES and location-scale families

I If X has (finite) mean µ and variance σ2

VaRα(X ) = σ VaRα(X )− µ

ESα(X ) = σ ESα(X )− µ

I If X ∼ N (0, σ2), then (α < 1/2)

VaRα(X ) = σ · |zα| ESα(X ) = σϕ(zα)

α

so that, for instance,

VaR5%(X ) = 1.645 · σ ES5%(X ) = 2.063 · σ

I Under a location-scale family assumption, estimation of VaR/ESreduces to the estimation of µ and σ.


VaR and ES for Loss variables

I In Insurance the Loss variable L = −PL is most often used.

I Modified definitions of VaR and ES:

VaR(Ins)β (L) = qβ(L)

ES(Ins)β (L) =

11− β

∫ 1

β

VaR(Ins)u (X ) du

= E [L|L > qβ(L)] (L a.c.)

where β close to 1 (e.g. β = 95%, 99%, 99.9%).

I Note, when PL is a.c.

VaR(Bank)α (PL) = −qα(PL) = q1−α(L) = VaR(Ins)

1−α(L)

and similarly for ES.


Outline






Sub-additivity

I A risk measure is sub-additive if

ρ(X + Y ) 6 ρ(X ) + ρ(Y ) for all X ,Y

I There are (at least) 3 advantages choosing a sub-additive ρ:

1. Sub-additivity deters a financial institution from splitting in orderto lower the total risk (hence, the capital requirement asked by theregulator)

2. If PL = PL1 + . . .+ PLN where PLi is the PL of internal unit i, then

ρ(PL) 6 ρ(PL1) + . . .+ ρ(PLN )

Estimation of partial risks (i.e. ρ(PLi)) is usually more accurateand ρ(PL1) + . . .+ ρ(PLN ) is a reliable upper bound.

3. A sub-additive ρ is also convex and this is a key property inportfolio optimization.

I But not everybody is so fascinated with subadditivity...


VaR and elliptical returns

I Are VaR and ES sub-additive?

I We start with a particular case:

I Assume that ∆Y ∈ RN is elliptical and let

L = X = v′∆Y : v ∈ RN

be the vector space of all r.v. that are linear combinations of ∆YI If α < 0.5 there exists a constant c > 0 such that

VaRα(X ) = c · σ(X )− E [X ] for all X ∈ L

As a consequence

VaRα(X + Y ) 6 VaRα(X ) + VaRα(Y ) for all X ,Y ∈ L

A similar result holds for ES as well.

I ”Elliptical” means that level sets of the density are ”ellipsoids”. Keyexample: multivariate normal.


Beyond elliptical returns

I Put it briefly, VaR and ES are certainly sub-additive when ∆Y iselliptical and we only consider linear portfolios (linear pl).

I These assumptions are often reasonable when dealing with equityportfolios

I This is no more the case for portfolios containing options or whenconsidering other types of risks (credit, operational)

I Is VaR always sub-additive? No.

I Is ES always sub-additive? Yes.


VaR is not sub-additive - Example 1

I Consider a portfolio made by D = 100 different ZC defaultable bonds,face value 105, current price 100, maturity T = 1.

I Defaults are independent and occur with probability p = 2% for eachbond

I The PL of bond d is

PLd = 105(1− Bd)− 100 = 5− 105Bd

where Bd = 1 if bond d defaults and Bd = 0 otherwise.

I The total PL is

PL =∑

d

PLd = 500− 105∑

d

Bd ,

where the Bd are IID Ber(2%)



I For any d, VaR5%(PLd) = −5

I Using a MC simulation we estimate

VaR5%(PL) ' 25

I Therefore

VaR5%

(∑d

PLd

)' 25 > −500 =

∑d

VaR5%(PLd) = 100VaR5%(PL1)

and sub-additivity is broken at α = 5%

I Instead

VaR1%

(∑d

PLd

)' 130 < 10000 = 100VaR1%(PL1)


VaR is not sub-additive - Example 1I Consider

I Portfolio A: we invest 100 Euro in each bond

I Portfolio B: we invest 10000 Euro in bond 1

I We have just seen that VaR5%(PLA) > VaR5%(PLB)

−10000 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure: PL histogram for portfolio A (blue) and B (red)



I A Cauchy r.v. (or t-Student with ν = 1) has density

f (x) =1

π(1 + x2)

so that f ∼ x−2 and F ∼ |x|−1 for |x| → ∞

I The following is a known (and odd..) fact:

I if X ,Y are Cauchy and independent, then X + Y ∼ 2X

I Note: if X ,Y are IID standard normal (finite variance), thenX + Y ∼

√2X . A Cauchy r.v. has infinite variance

I Therefore, for IID Cauchy

VaRα(X + Y ) = VaRα(2X ) = 2VaRα(X ) = VaRα(X ) + VaRα(Y )

I Weird, but still not a counterexample for lack of sub-additivity...



I Consider a Pareto-like distribution

F (x) = |x|−1/2, x < −1,

for which f (x) ∼ |x|−3/2 (like a t-Student with ν = 1/2).

I If X ,Y both have the distribution F above and are independent then itcan be computed

P(X + Y 6 −z) =2√

z − 1z

<

√2z

= P(2X 6 −z) z > 2

I As a consequence, for any α ∈ (0, 1)

VaRα(X + Y ) > VaRα(2X ) = VaRα(X ) + VaRα(Y )

I Lack of subadditivity is uniform in α


VaR is not sub-additive - Example 3I The following problem has recently received attention:

maxVaRα(X + Y ) : X ∼ F , Y ∼ G,

where F and G are fixed distributions. We are looking for the worst VaRof the sum for given marginals

I There is no comprehensive solution, but some particular cases can betreated effectively

I For instance, if F and G are standard normal, then it has been shownthat

maxVaR5%(X + Y ) : X ∼ Y ∼ N (0, 1) ' 3.92 > 2 · 1.645

I Therefore, we can violate sub-additivity even when the marginaldistributions are simple (normal).

I However the (worst-case) dependence structure (copula) between Xand Y is far from a normal one (and highly non-elliptical)


VaR is not sub-additive

I We have seen 3 counterexamples

I Bond: IID risks with a pronounced skewnessExamples: credit risk, portfolios with highly non-linear options

I Cauchy: IID risks with null skewness, but very thick tailsExamples: operative risk, insurance products

I Copula: ID normal risks, but highly asymmetric dependencestructureExample: interactions during crisis periods


Why VaR is not sub-additive

I Assume PL = X + Y

I Consider two time series of length 100 for the vector (X ,Y ):

x1, . . . . . . , x100

y1, . . . . . . , y100

where xn and yn are the values observed n days ago

I Using the historical method amounts to use the empirical distributionfor X

FX (x) =1

100

100∑n=1

Ix>xn

and similar for Y



I For α = 2% we have

VaR2%(X ) = −x(2), VaR2%(Y ) = −y(2)

where

x(1) 6 x(2) 6 . . . 6 x(100)

y(1) 6 y(2) 6 . . . 6 y(100)

are the order statistics for (xn) and (yn)

I If zn = xn + yn, then

VaR2%(X + Y ) = −z(2),

where z(1) 6 . . . 6 z(100) is the order statistics of the sum X + Y

I Warning! It can be the case that z(2) 6= x(2) + y(2)



I Consider

n . . . 22 . . . 45 . . . 91 . . .

xn + −20 + −10 + −5 +yn + −5 + −10 + −19 +

zn = xn + yn + −25 + −20 + −24 +

where + indicates some positive values for the variables.

I We haveVaR2%(X + Y ) = −z(2) = 24

whileVaR2%(X ) + VaR2%(Y ) = −x(2) − y(2) = 20

I ThereforeVaR2%(X + Y ) > VaR2%(X ) + VaR2%(Y )


Why ES is sub-additive

I Recall

n . . . 22 . . . 45 . . . 91 . . .

xn + −20 + −10 + −5 +yn + −5 + −10 + −19 +

zn = xn + yn + −25 + −20 + −24 +

I We have

ES2%(X + Y ) = −z(1) + z(2)

2= 24.5

while

ES2%(X ) + ES2%(Y ) = −x(1) + x(2) + y(1) + y(2)

2= 29.5


Why ES is sub-additive

I The crucial point is:

I we can have(x + y)(2) < x(2) + y(2)

and this leads to lack of sub-additivity for VaR

I we always have

(x + y)(1) + (x + y)(2) > x(1) + x(2) + y(1) + y(2)

and this implies sub-additivity for ES

I It is possible to prove that ES is sub-additive for general distributions.Embrechts and Wang (WP, 2015) provide 7 (!) different (rigorous)proofs.


Outline






Coherence and diversification

I A good risk measure should encourage diversification, but this isdifficult to write in precise terms.

I Sub-additivity of a risk measure does not seem to be equivalent to this.

I Whileσ(X + Y ) 6 σ(X ) + σ(Y )

always holds,σ2(X + Y ) 6 σ2(X ) + σ2(Y )

only holds when corr(X ,Y ) 6 0.

I However, both are used in Markowitz portfolio theory.


Coherence and independence

I It would be tempting to think that the risks of two independentpositions should add up, i.e.

ρ(X + Y ) = ρ(X ) + ρ(Y ) X ,Y independent

I This is always true for σ2 (not for σ), but false in general:

I If X and Y are IID normal, then X + Y ∼√

2X . If ρ satisfies ispositively homogeneous (like VaR and ES), then

ρ(X + Y ) = ρ(√

2X ) =√

2ρ(X ) < 2ρ(X ) = ρ(X ) + ρ(Y )

provided ρ(X ) > 0

I We have seen before that if X ,Y are Pareto-like distributed andindependent, then

VaRα(X + Y ) > VaRα(X ) + VaRα(Y )

for any α


Coherence and comonotonicity

I X and Y are comonotonic if X = f (Z) and Y = g(Z) for two increasingfunctions f and g and a third r.v. Z .

I When this happens, none of the two r.v. is an hedge for the other. So, itseems natural that the two risks sum up in this case.

I A risk measure is comonotonic additive if

ρ(X + Y ) = ρ(X ) + ρ(Y )

for any X ,Y comonotonic.

I It is quite easy to verify that both VaR and ES are comonotonic additiveWhen f and g are increasing and invertible, just observe that

q(X ) + q(Y ) = f (q(Z)) + g(q(Z))

= (f + g)(q(Z)) = q((f + g)(Z)) = q(X + Y )

as f + g is also increasing and invertible.


Coherence and comonotonicityI Comonotonicity attains the maximum correlation for given marginals:

I If X and Y are comonotonic, then

corr(X ,Y ) > corr(X ′,Y ′)

for any X ′ ∼ X and Y ′ ∼ Y such that X ′ and Y ′ are notcomonotonic.

I Let ρ be sub-additive and comonotonic additive. If X and Y arecomonotonic and X ′ ∼ X and Y ′ ∼ Y , then of course

ρ(X ′ + Y ′) 6 ρ(X ′) + ρ(Y ′) = ρ(X ) + ρ(Y ) = ρ(X + Y )

I So, for the ES, the maximum risk of a sum X + Y with given marginalsis attained at the comonotonic case or, equivalently, at the maximumcorrelation case.

I This is not true for VaR, which is not sub-additive.


Subadditivity and convexityI If ρ is positively homogeneous (PH), then sub-additivity (Sub) is

equivalent to convexity (C)

ρ(λX + (1− λ)Y ) 6 λρ(X ) + (1− λ)ρ(Y ) λ ∈ [0, 1]

I Under (PH), (Sub) implies (C)

ρ(λX + (1− λ)Y ) 6 ρ(λX ) + ρ((1− λ)Y )

= λρ(X ) + (1− λ)ρ(Y )

I Under (PH), (C) implies (Sub)

ρ(X + Y ) = ρ

(12

2X +12

2Y)

612ρ(2X ) +

12ρ(2Y ) = ρ(X ) + ρ(Y )

I So, ES is convex, VaR is not


Subadditivity and convexity

I Note: ρ is convex if and only if, for all X and Y

λ 7→ ρ(λX + (1− λ)Y ) λ ∈ [0, 1]

is convex or, more in general

λ = (λ1, . . . , λN ) 7→ ρ

(∑n

λnXn

)λn > 0,

∑n

λn = 1

is convex.

I Important: (strict) convexity ensures uniqueness of a minimum; also,it greatly helps building numerical algorithms


Optimization with ES/VaR

I Consider two bonds A and B, both have current price 104.6, face value100 and coupon 8.

I Both can have a soft default, with probability 2%, or a hard default,with probability 3%. In a soft (resp. hard) default the coupon (resp. thecoupon and the face value) is not paid back

I The probability that A and B default together is 0

I The following table shows the PL of bond A, B and of the portfolio A+B

event prob. PLA PLB PLA+B

no default 90% +3.4 +3.4 +6.8soft for A 2% −4.6 +3.4 −1.2

hard for A 3% −104.6 +3.4 −101.2soft for B 2% +3.4 −4.6 −1.2

hard for B 3% +3.4 −104.6 −101.2



I Note that

VaR5%(PLA+B) = 101.2 > 9.2 = VaR5%(PLA) + VaR5%(PLB)

while

ES5%(PLA+B) = 101.2 < 2 · 64.6 = ES5%(PLA) + ES5%(PLB)

I For λ ∈ [0, 1] consider the portfolio λA + (1− λ)B with current value104.6. Its PL is

PLλ = λPLA + (1− λ)PLB



0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

Figure: VaR5% (black) and ES5% (red) of PLλ as a function of λ



I Consider N bonds with face value 100 and current price 97. Bonds cango to default, independently each other and with probability p = 3%.We invest 1 Euro allocating 1/N to each bond.

I The PL of each of the N partial investments is

PLi =1N

(100(1− Bi)

97− 1)

=3− 100Bi

97Ni = 1, . . . ,N

where Bi = 1 if bond i defaults and 0 otherwise. Therefore, the total PLis

PLtot,N =

N∑i=1

PLi =3N − 100

∑i Bi

97N=

397− 100X

97N,

where X =∑

i Bi ∼ Bin(N ,p) counts the number of defaults in theportfolio

I We expect that the larger is N , the lower is VaR5% (diversificationeffect). Instead...



0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure: VaR5%(PLN ) (black) and ES5%(PLN ) (red) as a function of N


Optimization with ES

I Rockafellar and Uryasev (2000) showed that

ESα(X ) = minη∈R−η + α−1E [(η − X )+]

and that the minimum is reached for η = VaRα(X )

I This is a useful result for the minimization of ES

I If X (λ) is the PL depending on the parameters λ,

minλ

ESα(X (λ)) = minη∈R, λ

−η + α−1E [(η − X (λ))+]

I This procedure allows to avoid a sort algorithm in the target function


Outline






Coherent risk measures

I A seminal paper by Artzner, Delbaen, Eber, Heath (Math. Finance,1999) introduced the following key definition.

I A risk measure ρ : L → R is coherent if it satisfies

I (TE) Translation Equivariance (originally, t. invariance)

ρ(X + b) = ρ(X )− b ∀b ∈ R

I (PH) Positive Homogeneity

ρ(aX ) = aρ(X ) ∀a > 0

I (M) Monotonicity

X > Y =⇒ ρ(X ) 6 ρ(Y )

I (Sub) Sub-additivity

ρ(X + Y ) 6 ρ(X ) + ρ(Y )


Coherent risk measures

I ADEH is basically the first theoretical paper dealing with general riskmeasures. Huge stream of literature and continuous debate thereafter.

I Three main reasons for this:

I Clear financial motivation for all 4 properties (but some disagree)

I Several connections with diverse areas in Analysis and Probabilityand many different examples of coherence

I But, above all: ES is coherent, VaR is not

I We have already seen that VaR and ES both satisfy TE and PH; they alsoboth satisfy M.

I However, only ES satisfies also Sub, so only ES is coherent.


Translation equivariance

I Risk measures satisfying TE are also called monetary or capitalrequirements. The reason is in the following results

I If ρ satisfies TE andA = X : ρ(X ) 6 0, then

ρ(X ) = minb : X + b ∈ A (2)

Indeed

minb : X + b ∈ A = minb : ρ(X + b) 6 0= minb : ρ(X ) 6 b = ρ(X )

I Viceversa, any ρ in the form (2) satisfies TE when finite.

I The setA is called the acceptance set and ρ(X ) is the amount ofmoney that has to be added to X to make it acceptable.


Translation equivariance

I The acceptance set for VaRα is

A = X : VaRα(X ) 6 0 = X : qα(X ) > 0

I Therefore a (a.c.) PL is acceptable if and only if

P(PL 6 0) = P(VT 6 V0) 6 α

I If α = 1% or 5%, a PL is very seldom acceptable with this criterion, butinjecting a suitable amount it will fall inA, that is

VaRα(PL) = minb : P(VT + b 6 V0) 6 α

I Similar arguments for ES


Positive homogeneity

I PH tells that when we change currency (or, more generally, numeraire),the risk changes accordingly.

I PH has been criticized: for instance, it would imply that

ρ(100000 · X ) = 100000 · ρ(X ).

and in illiquid markets the risk of 100000 shares may be higher then100000 times the risk of a single share.

I This critique seems wrong: in illiquid market, the link betweennumber of assets and the PL is a non-linear one (so the PL of 100000shares is not necessarily 100000 · X ).

I Remind: TE and PH together imply the very useful identity

ρ(X ) = −E [X ] + σ(X )ρ(X ).


Monotonicity

I Property M is almost trivial and, actually, rather weak

I X > Y is very strong and implies (1st order) stochastic dominance of Xover Y :

FX (x) = P(X 6 x) 6 P(Y 6 x) = FY (x) for all x

I It readily follows qu(X ) > qu(Y ) for all u, so that

X > Y =⇒ VaRα(X ) 6 VaRα(Y )

and VaR satisfies M.

I Integrating in u we see that ES too satisfies M


Outline






Standard deviationI In Markowitz Portfolio Theory standard deviation is used:

ρ(X ) = σ(X )

I σ satisfies PH and Sub, but not TE nor M. Instead, it is an example of adispersion measure, as:

I σ(−X ) = σ(X )

I σ(X + b) = σ(X )

I We can then consider

ρ(X ) = −E [X ] + a · σ(X ) (a > 0)

I it is still PH and Sub (note: E is linear)

I it becomes TE (note: −E [X + b] = −E [X ]− b)

I however, it is not M (check with a r.v. with 2 values)


Standard semi-deviation

I A variant of σ is the (lower) standard semi-deviation

ρ(X ) = σ−(X ) =√

E [(X − E [X ])2−],

where x− = −min(x, 0). It is PH and Sub.

I Plainlyρ(X ) = −E [X ] + a · σ−(X ) (a > 0)

is TE, PH and Sub. It can be proved it is also M, hence coherent iffa ∈ [0, 1]

I More generally, Fisher (2003) proved that

ρ(X ) = −E [X ] + a · E[(X − E [X ])

p−]1/p

is coherent provided p > 2 and a ∈ [0, 1]. These are calleddeviation-based risk measures.


Spectral risk measures

I We can write

ESα(X ) =1α

∫ α

0VaRu(X ) du =

∫ 1

0VaRu(X )ψ(u) du,

whereψ(u) = α−1I06u6α.

Notice that ψ is a density over [0, 1]

I We can then generalize and consider a risk measure in the form

ρψ(X ) =

∫ 1

0VaRu(X )ψ(u) du

where ψ is a generic density over [0, 1], i.e. ψ > 0 and∫ 1

0 ψ = 1.

I These measures are weighted sums of VaR of different orders



I We observe that (a > 0)

ρψ(aX + b) =

∫ 1

0VaRu(aX + b)ψ(u) du

= a∫ 1

0VaRu(X )ψ(u) du − b

∫ 1

0ψ(u) du

= aρψ(X )− b

I Moreover, if X > Y , then VaRu(X ) 6 VaRu(Y ) for any u and thereforeρψ(X ) 6 ρψ(Y )

I Concerning (Sub), we have the following result: ρψ is sub-additive ifand only if ψ is decreasing (weak sense).

I When ψ is decreasing, then ρψ is called a spectral risk measure (withspectrum ψ). This class of coherent risk measures was introduced byAcerbi (2002).



I ψ(u) = α−1I06u6α is decreasing and therefore ESα is coherent

I reasoning in a heuristic (but basically correct) way we observe thatVaRα corresponds to

VaRα(X ) =

∫ 1

0VaRu(X ) δα(u) du,

where δα is a point-mass in α, which is not a decreasing distribution.In fact VaRα is not coherent


Concave distortionsI We observe that ρ(X ) = −E [X ] is trivially coherent. A basic identity

(F = FX ) is

E [X ] = −∫ 0

−∞F (x) dx +

∫ +∞

0(1− F (x)) dx

I Consider a function h : [0, 1]→ [0, 1] such that h(0) = 0 and h(1) = 1and define

E h[X ] = −∫ 0

−∞h(F (x)) dx +

∫ +∞

0(1− h(F (x))) dx

I This is sometime called a Choquet integral or a distorted expectation(note: probabilities F are distorted by h). The risk measureρh(X ) = −E h[X ] satisfies TE, PH, M

I Remarkably: ρh satisfies (Sub), i.e. it is coherent, if and only if h is convex

I Usually, E h[X ] is expressed in terms of the dual functiong(u) = 1− h(1− u). Plainly, h is convex iff g is concave.


Generalized quantilesI It is a known fact in Statistics that

qα(X ) = argminq αE [(X − q)+] + (1− α)E [(X − q)−]

I We can then consider generalized quantiles by defining

qα,Φ(X ) = argminq αE [Φ((X − q)+)] + (1− α)E [Φ((X − q)−)]

for some function Φ : R+ → R+

I For instance, if Φ(x) = x2, we obtain the so-called α-expectiles

eα = argminq

αE [(X − q)2

+] + (1− α)E [(X − q)2−]

I It can be proved that if Φ is strictly increasing and strictly convex, thenqα,Φ is uniquely defined.

I If in addition Φ is differentiable and X is continuous, then q = qα,Φ isthe unique solution of

αE [Φ′((X − q)+)] = (1− α)E [Φ′((X − q)−)]An introduction to coherent risk measures Giacomo Scandolo (Unifi) 73 / 118

Generalized quantiles

I For given α and Φ (assume w.l.o.g. Φ(0) = 0, Φ(1) = 1) we can put

ρ(X ) = −qα,Φ(X )

and wonder whether ρ is coherent.

I A bit surprisingly:

I ρ is coherent if and only if Φ(x) = x2, i.e. qα,Φ = eα, and α > 0.5

I We can generalize further and consider risk measures in the form

ρ(X ) = −argminy E [Ψ(X , y)],

where Ψ = Ψ(x, y) is some convex function (with reasonableproperties ensuring ρ is well-defined)

I Such risk measures are called elicitable and backtesting them seems tobe ”easier”. Remarkably, VaR is elicitable, ES is not


Two remarks

I We have seen different classes of coherent r.m.: deviation-based,spectral, concave distortions, generalized quantiles.

I There is some overlapping:

I ρ(X ) = −E [X ] is deviation-based (a = 0), spectral (ψ(u) = u) andconcave distortion (h(x) = x)

I ES and all other spectral risk measures can be represented asconcave distortions (actually the two classes nearly coincide)

I Note: VaR is defined for all distributions, while all coherent r.m. wehave presented are not (they usually require X ∈ L1 at least).

I Does there exist coherent r.m. defined for all distributions?Remarkably: No (Delbaen, 2000)


Outline






Combinations preserving coherenceI If ρ1 and ρ2 are coherent risk measures, then

ρ(X ) = ρ1(X ) + ρ2(X )

is not coherent: TE is not satisfied

I However, it is easy to show that

maximumρ(X ) = maxρ1(X ), ρ2(X )

convex combination

ρ(X ) = λρ1(X ) + (1− λ)ρ2(X ), λ ∈ (0, 1)

inf-convolution

ρ(X ) = infρ1(X1) + ρ2(X2) : X1 + X2 = X

are coherent.

I The solution of the inf-convolution problem may be interpreted as anoptimal sharing of the risk X between two agents.


Combinations preserving coherence

I Maximum and convex combinations can be much generalized

I Let (ρa)a∈A be an arbitrary family of coherent risk measures; then

ρ(X ) = supa∈A

ρa(X )

is coherent (provided is finite)I If (A,A) is a measurable space and µ is a probability measure on

it, then

ρ(X ) =

∫Aρa(X )µ(da)

is coherent. In particular, if (ρa)a∈(0,1) is a family of coherent riskmeasures and ψ is a density on [0, 1], then

ρ(X ) =

∫ 1

0ρa(X )ψ(a) da

is coherent.


Convex combinations of ESI We know that (ESα)α∈(0,1) is a family of coherent risk measures.

I Therefore, if ξ is a density on [0, 1]

ρ(X ) =

∫ 1

0ESa(X )ξ(a) da

is a coherent risk measure.

I We also have (V (u) = VaRu(X ))

ρ(X ) =

∫ 1

0

1a

∫ a

0V (u) du ξ(a) da =

∫ 1

0

∫ 1

0

V (u)

aIu6aξ(a) du da (Fubini)

=

∫ 1

0

∫ 1

0

1a

V (u)Ia>uξ(a) da du =

∫ 1

0V (u)

∫ 1

u

ξ(a)

ada du

=

∫ 1

0V (u)ψ(u) du

An easy check shows that ψ(u) =∫ 1

u a−1ξ(a) da is a decreasing densityon [0, 1]. So, ρ is a spectral risk measure.


Convex combinations of ES

I More in general consider a risk measure in the form

ρ(X ) =

∫ 1

0ESa(X )µ(da), (3)

where µ is a probability on [0, 1]

I Reasoning as before, we can prove that ρ can be written in the form

ρ(X ) =

∫ 1

0VaRu(X ) ν(du), (4)

where ν is decreasing, i.e. ν([a, b]) > ν([a + ε, b + ε]) for ε > 0

I Risk measures in the form (4) with ν decreasing, are coherent and canbe seen as generalized spectral risk measures.

I Indeed, more is true...


Kusuoka representation

I The following result is known as the Kusuoka representation(Kusuoka, 2001)

I Let ρ be a risk measure on L1 satisfying a (mild) continuity condition.Then the following are equivalent:

1. ρ is coherent and comonotonic additive, i.e.

ρ(X + Y ) = ρ(X ) + ρ(Y )

for any X ,Y comonotonic.

2. ρ is a convex combination of ES as in (3)

3. ρ is a (coherent) generalized spectral r.m. as in (4)

I So, generalized spectral r.m. are, in a sense, a large class of coherentrisk measures.


Dual representation

I Kusuoka representation exploits convex combinations and uses ES as abuilding block

I Dropping comonotonic additivity, there is even a more generalrepresentation: any coherent and law invariant risk measure can bewritten as a supremum of a family of generalized spectral risk measure

I Dual representation exploits maxima of risk measures and usesexpected values as building blocks. It holds even for non law-invariantrisk measures.


Dual representationI Consider a finite space Ω = (ω1, . . . , ωN )′, so that a random variable X

is represented by the vector

x = (x1, . . . , xN ) xn = X (ωn)

I A probability measure P on Ω is represented by

p = (p1, . . . ,pN )′, pn = P(ωn) > 0

I LetP = p ∈ [0, 1]N :

∑n

pn = 1

be the set of all probabilities.

I If P ∈ P , then the expectation of X under P is

E P [X ] =∑

n

pnxn = p′x


Dual representation

I LetQ ⊆ P be a set of probabilities and define

ρQ(X ) = sup−E Q[X ] : Q ∈ Q

I As each ρ(X ) = −E Q[X ] is coherent (immediate), ρQ is coherent aswell.

I Remarkably, the converse holds as well. This is the dualrepresentation:

I Any coherent risk measure can be written in the form ρQ for somesetQ

I This result holds also in general probability spaces provided ρ satisfiesa mild continuity condition.


Dual representation

I Here is a sketch of the proof

I The risk measure ρ may be seen as a map ρ : RN → R

I Since it is coherent, it is a convex and PH map, so that, accordingto a classical result in Convex Analysis we may write

ρ(x) = supv∈V−v′x,

for a subset V ∈ RN . In other words, any convex and PH functionis the pointwise supremum of a family of linear functionals.

I Using standard algebraic arguments, we can prove that

M =⇒ V ⊂ v ∈ RN : vn > 0 ∀n

TE =⇒ V ⊂ v ∈ RN :∑

n

vn = 1,

so that V ⊂ P and since v′x = E Q[X ] for some Q, we conclude.


Outline






The process of risk measurement

I Remind (below some notation is changed):

I A risk measure is fixed ρ, by a regulator or internally.Popular choices are VaR5% VaR0.1%, ES1%)

I A time horizon ∆t is fixed, again by a regulator or internally.Popular choices are 1 day or 2 weeks (market risk), 1 year (creditand operational risk).

I The portfolio of interest is singled out. For instance, the focus ison a derivatives trader, on the Euro bond unit, on the overallportfolio of the institution.

I The two regulatory frameworks are Basel III (banks) and Solvency II(insurance companies)


The process of risk measurement

I At date t (today):

I We assess the current portfolio ”value” Vt

I We propose a (multivariate) distribution for

∆Yt+T = Yt+T − Yt

The distribution can be conditional on It (information set at timet) or unconditional.

I We compute/estimate the distribution of PLt,t+T = pl(∆Yt+T )and

ρt = ρ(PLt,t+T | It ) (conditional)

= ρ(PLt,t+T ) (unconditional)

I Afterwards:

I After some runs of the steps above, we can check whether ourmethodology is sufficiently correct (this is called back-testing)


Conditional and unconditional risk measurementI Let I = It and X = ∆Y (i.e. d = 1 for simplicity)

I When measuring market risk, the horizon is short (2 weeks or less). So

I Recent information matters for future events: I must be exploited

I The conditional distribution FX |I(x) = P(X 6 x | I) differs fromthe unconditional one FX (x) = P(X 6 x). Example: a GARCHprocess is conditionally normal, but unconditionally thick-tailed

I The risk measurement is conditional:

ρ(X | I) = ρ(FX |I)

I When measuring credit, operational or insurance risk, the horizon islong (1 year). So

I Recent information matters less and we may assume FX |I ≡ FX

(e.g. X part of an IID sequence)

I The risk measurement is unconditional, i.e. we are interested in

ρ(X ) = ρ(FX )


The parametric approach

I Within the (plain) parametric approach we:

1. Assume that the (multivariate, conditional or unconditional)distribution of X = ∆Y is in the parametric family

FX(x) = F (x |θ), x = (x1, . . . , xd)′

Here, θ = (θ1, . . . , θK )′ is the parameter vector.

2. Estimate parameters from data:

θ = θ(x1, . . . , xN )

Here, xn is the n-th historical observation for X.

3. Analytically compute the distribution of PL = pl(X) (it dependson the estimate θ)

4. Compute ρ(PL) (or ρ(PL | It ))


The parametric approach - basic exampleI Equity portfolio: vi is invested (at time t) in asset i (6 d).

I If Ri is the return for asset i (from t to t + T ), then

PL =∑

i

viRi = v′R

with v = (v1, . . . , vd)′, R = (R1, . . . ,Rd)′. Note: pl(x) = v′x is linear

I A common (rough) assumption is:

R | It ∼ Nd(µ,Σ)

where µ and Σ are conditional on It (e.g. a multivariateARMA-GARCH). Note: θ = (µ,Σ)

I Under this assumption

PL | It ∼ N1(µPL, σ2PL) µPL = v′µ, σ2

PL = v′Σv

and risk measures are easily computed.


The Monte Carlo approach

I Within the (plain) Monte Carlo approach we:

1. As before, assumeFX(x) = F (x |θ)

for some parameter vector θ.

2. As before, estimate the parameters θ from observed data for X.

3. Simulate a large IID sample w1, . . . ,wM with commondistribution F (·;θ)

4. Obtain the corresponding IID sample for the PL

PL1 = pl(w1), . . . ,PLM = pl(wM )

5. Compute ρ(PL) using the obtained empirical distribution


The historical approach

I Within the (plain) historical approach we:

1. Using the observed series x1, . . . , xN , obtain the correspondingseries for the PL

PL1 = pl(x1), . . . ,PLN = pl(xN )

2. Compute ρ(PL) using the obtained empirical distribution


Outline






An IID framework

I Consider the simplest framework in which PL = X , i.e. d = 1 and pl isthe identity.

I Also, assume an IID∼ X sequence X1,X2, . . . is available

I For instance, this can be the case when

I we are dealing with operational or insurance losses

I we are using a MC step to evaluate PL (then we get an IIDsequence for PL)

I In order to estimate ρ(PL), we can proceed non-parametrically, i.e. wedo not want to postulate any parametric family for X

I Instead, we work with the empirical distribution and the associatedplug-in estimators


Empirical distributionI Given a data set x ∈ RN , the empirical distribution function is defined

as

Fx(y) =1N

N∑n=1

Ixn6y

−4 −3 −2 −1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

Figure: Empirical distribution function for x = (−3, 0, 1, 1.5, 3) (obviously,the order does not count)


Glivenko-Cantelli Theorem - 1

I Let (Xn)n be an IID sequence with common distribution F . For a givenx ∈ R consider the random sequence

FN (y) = Fx(y), x = (X1, . . . ,XN )

I We expect FN (y) to approach F (y) in some sense (Prob, a.s.). Much(much) more is true

I Glivenko-Cantelli Theorem. It holds

supx∈R

∣∣∣FN (x)− F (x)∣∣∣ a.s.−→ 0 N →∞

I So, the convergence is strong (a.s.) and uniform across x and FN (x) is aconsistent estimator of F (x).


Glivenko-Cantelli Theorem - 2

I The probability of large errors is also uniformly bounded through theDKW inequalities

P(

supx∈R

∣∣∣FN (x)− F (x)∣∣∣ > ε

)6 2e−2Nε2

Note that the upper bound decays exponentially with N

I As FN (x) has (clearly) a binomial distribution, asymptotic normality iseasily established and we get

√N(

FN (x)− F (x))

Law−→ N (0, σ∞)

where σ∞ = F (x)− F (x)2

I This last result can be much generalized through the DonskerTheorem: basically the sequence of processes VN = (FN (t))t∈Rconverges in law to a gaussian process with mean 0 and a simplecovariance structure that can be written explicitly in terms of F .


Plug-in estimators

I Consider a (law-invariant) risk measure ρ.

I The plug-in estimator (or sample or empirical estimator) for ρ is

ρ(x) = ρ(Fx)

that is, we just plug the empirical distribution into ρ.

I We have seen that FN converges in a very strong sense to F and isasymptotically normal.

I If ρ displays some continuity or even differentiability properties, thenρN is a good estimator for ρ.


Plug-in estimators for risk measures - 1I Remind that

qα(Fx) = xk:N

where xk:N is the k-th least element in x, whenever

α ∈(

k − 1N

,kN

]or, equivalently, Nα ∈ (k − 1, k]

I As a consequence, if ρ(X ) = VaRα(X ), then the plug-in estimator is

ρ(x) = −xk:N , k = dNαe

I If ρ(X ) =∫

qX (u)ψ(u) du, we easily compute

ρ(x) = −N∑

k=1

ck,N xk:N ,

where

ck,N =

∫ kN

k−1N

ψ(u) du


Plug-in estimators for risk measures - 2

I In particular, for ρ = ESα, ψ(u) = α−1Iu6α and we immediatelycompute

ck,N =

1

Nαk 6 bNαc

1− bNαcNα

k = bNαc+ 1

0 k > bNαc+ 2

so that

ρ(x) = − 1Nα

bNαc∑k=1

xk:N −(

1− bNαcNα

)xbNαc+1:N


Order statistics - definition

I If x = (x1, . . . , xN ) is a vector and 1 6 k 6 N , we indicate with xk:N thek-th least element. In other words

x1:N 6 x2:N 6 . . . 6 xN :N

and xkk = xk:Nk .

I Note: x1:N = xk(1) for some k(1), x2:N = xk(2) for some k(2) 6= k(1), andso on

I If X = (X1, . . . ,XN ) is a random vector, then the random vector (Xk:N )k

is called the order statistics for X. Notice that a.s.

X1:N 6 X2:N 6 . . . 6 XN :N

I Note: X1:N = Xk(1) for some k(1), but k(1) depends on ω (i.e. it is itselfrandom)


Order statistics - finite sample distributionI Assume X has IID components with common a.c. distribution FI The distribution of XN :N = maxk Xk is easy to derive

FN :N (x) = P(Xk 6 x, ∀k) =

N∏k=1

P(Xk 6 x) = F (x)N

Concerning the distribution of X1:N = mink Xk

F1:N (x) = 1− P(Xk > x, ∀k) = 1− (1− F (x))N

I For a general k,

Fk:N (x) = P(Xk:N 6 x) = P(at least k elements are 6 x)

=

N∑i=k

P(exactly i elements are 6 x)

=N∑

i=k

P(X1, . . . ,Xi 6 x) · P(Xi+1, . . . ,XN > x) · (# permutations)

=

N∑i=k

(Ni

)F (x)i(1− F (x))N−i


Properties of the plug-in estimator for VaRI Let (Xn)n be IID with common distribution F . Assume F a.c. with

positive density f .

I From Glivenko-Cantelli it follows

qα,N = qα(FX) = XdNαe:Na.s.−→ qα(F ) N →∞,

meaning that the plug-in estimator for the quantile, and therefore forVaR, is strongly consistent.

I More generally, it can be proved that

XkN :Na.s.−→ qα(F ) N →∞

provided kN/N → α. An example is kN = bNαc

I Asymptotic normality also holds (N →∞)

√N(

qα,N − qα(F )) law−→ N (0, σ2

∞) σ∞ =

√α(1− α)

f (qα(F ))


Properties of the plug-in estimator for ES and ρψ - 1

I The key result (corollary of a Van Zwet Theorem) is

I Let ψ be piecewise continuous and satisfy a mild technicalcondition and let

ck,N =

∫ kN

k−1N

ψ(u) du

If (Xn)n are IID with common distribution F , then

N∑k=1

ck,N Xk:Na.s.−→

∫ 1

0q(u)ψ(u) du N →∞

I The spectrum of ES satisfies these conditions, so that its plug-inestimator is strongly consistent.



I Asymptotic normality of qα can be generalized:

I Let α1 < . . . < αD and qd the plug-in estimator for qd = qF (αd).Then√

N((q1,N , . . . , qD,N )− (q1, . . . , qD)

) a.s.−→ ND (0,Σ) N →∞,

where

Σi,j =αi(1− αj)

f (qi)f (qj)



I From the previous result (and some work), it is possible to deriveasymptotic normality for the plug-in estimator of ES and spectral riskmeasures.

I For general ρψ (under mild conditions on ψ):

σ2∞ = 2

∫ 1

0

∫ t

0ψ(s)ψ(t)

s(1− t)

f (q(s))f (q(t))ds dt

I In particular, for ESα, where ψ(s) = α−1Is6α, this is

σ2∞ =

2α2

∫(s,t) : 06s6t6α

s(1− t)

f (q(s))f (q(t))ds dt


Outline






Parametric estimation - 1

I Within the simplified setting (PL = X ), the parametric estimationconsists in:

1. Postulating that FX (x) = F (x |θ), where θ = (θ1, . . . , θK ) is theparameter vector (the value of some of the θk may be fixed inadvanced)

2. Estimating θ = θ(x) where x = (x1, . . . , xN ) is the sample ofobserved values for X

3. Computing ρ(F (· | θ))


Parametric estimation - 2

I Having fixed F (· |θ), θ ∈ Θ, and ρ (defined at least on the parametricfamily), let

h(θ) = ρ(F (· |θ))

I We know that if h is well-behaved and θ is a good estimator for θ (i.e.consistent and asymptotically normal), then h(θ) is a good estimatorfor ρ (at any F in the parametric family). This is a consequence of tworesults:

I if Yn → c a.s. and h is continuous, then h(Yn)→ h(c) a.s.I Delta-method: if θ is asymptotically normal for θ and h is

differentiable with h′(θ) 6= 0, then h(θ) is asymptotically normalfor h(θ) and moreover, the asymptotic variances satisfy

σh(θ)∞ = h′(θ) · σθ∞

(and similarly for dimension > 2)


Maximum Likelihood estimators - 1

I A general recipe to obtain a good parameter estimator is by way ofmaximizing the likelihood.

I Assume F (· |θ) has density f (· |θ). The log-likelihood at a given dataset x ∈ RN is

l(θ | x) =

N∑n=1

log f (xn |θ) θ ∈ Θ

that has to be viewed as a function of θ

I The Maximum Likelihood estimator (MLE) of θ is then defined as

θmle

(x) = arg.maxθl(θ | x)

I The MLE, when it is well defined, is consistent, asymptotically normaland has, in a sense, the lowest possible asymptotic variance


Maximum Likelihood estimators - 2

I The normal parametric family has density

f (x |µ, σ) =1

σ√

2πe−

(x−µ)2

2σ2

where θ = (µ, σ) ∈ R× R+ are parameters.

I A simple computation shows that the MLE for µ and σ are

µmle(x) =1N

N∑n=1

xn, σmle(x) =

√√√√ 1N

N∑n=1

(xn − µmle)2

I Notice that they are the usual (or sample) estimators for the mean andstandard deviation.


Maximum Likelihood estimators - 3I For the Laplace parametric family

f (x |µ, λ) =1

2λe−|x−µ|

λ

we get instead

µmle(x) =1N

N∑n=1

xn, λmle(x) =1N

N∑n=1

|xn − µmle |

I For this family, the mean is µ and the standard deviation is λ√

2

I In this case, σ is better estimated through√

2λmle than through thesample (plug-in) estimator.

I In general, the MLE may be difficult to compute (for instance fort-Student distributions), calling for numerical methods. It may not beeven well defined (because the maximization problem has no solutionor has multiple solutions)


Location-scale families

I A 2-parametric family of distributions is called a location-scale familyif it comes in the form

F (x | a, b) = F0

(x − b

a

), a > 0, b ∈ R

for some given reference distribution F0. In this case, a is the scaleparameter, b is the location parameter.

I The reason is: if F0 is the distribution of X0, then F (· | a, b) is thedistribution of aX0 + b

I If F0 is standard (mean 0, variance 1), then F (· | a, b) has mean b andvariance a2

I In terms of densities

f (x | a, b) =1a

f0

(x − b

a

), a > 0, b ∈ R


Location-scale families - basic examplesI Normal family, where a = σ and b = µ

I Laplace family, where a = λ and b = µ. Note that F0 (corresponding toµ = 0, λ = 1) is not standard as σ(F0) =

√2

I t-Student family with a fixed ν > 0 (degrees of freedom),corresponding to

f0(x) = cν

(1 +

x2

ν

)− ν+12

where cν is the normalizing constant. This is the classical form for thedensity: however σ(F0) > 1.

I When ν = 1 we have the Cauchy family, corresponding to

f0(x) =1

π(1 + x2)

This distribution does not have a mean, so location and scale have tobe interpreted in a loose sense here.


Location-scale families and risk measures - 2

I Then, if F (x | a, b) is a location-scale family and ρ is VaR, ES, ρψ or anyother risk measure such that ρ(aX + b) = aρ(X )− b, we have

h(a, b) = ρ(F (· | a, b)) = a · ρ(F0)− b

where ρ(F0) can be considered as a constant

I If a and b are good estimators (possibly MLE) for a and b, then

r = aρ(F0)− b

is a good estimator for ρ(F ) and its asymptotic variance may becomputed.

I However (a big however) this is the case only if F is indeed in thelocation-scale family.


A comparisonI Assume (Xn)n are IID standard normal.

I Postulating data come from the parametric family N (0, σ), the MLE forVaR5% is

VaRmleα,N (x) = 1.645

√√√√ 1N

N∑k=1

x2k

The estimator is consistent and asymptotically normal with σ∞ ' 1.16;however if data come from a non-normal distribution, this estimator isnot consistent

I Using a non-parametric approach, the plug-in estimator is

VaRα,N (x) = −xdN ·0.05e:N

The estimator is consistent and asymptotically normal with

σ∞ =

√0.05 · 0.95ϕ(z0.05)

' 2.11 ( 1.16)

This estimator remains consistent even if data are not normal.


Outline






Robust Statistics

I Robust Statistics is concerned with statistical procedures, primarilyestimation, for which the transition from good to bad behaviour is nottoo fast

I Loosely speaking, a parametric estimator is robust if it retains goodproperties even in a neighborhood of the parametric family

I There are several notions of robustness for an estimator, among which

I Qualitative robustness. An estimator is qualitatively robust if,changing a bit the distribution generating data, the samplingdistribution does not change too much. It is a continuity property.

I Quantitative robustness. An estimator is quantitatively robust ifthe addition of an outlier to a large data set does not change toomuch the estimate.


Qualitative robustness - 1

I Consider an estimator r and a sequence of IID (Xn)n with commondistribution F .

I The sampling distribution is defined as

Hr,F ,N = Law(r(X1, . . . ,XN )) Xn ∼ F

I We know that for an asymptotically normal estimator r at F , thesampling distribution resembles more and more to a normal one.

I In general, qualitative robustness is concerned with the (asymptotic inN ) dependence of G on F



I Consider 2 distributions, G and F . The Levy distance between G and His defined as

dL(G,F ) = infε > 0 : G(x − ε)− ε 6 F (x) 6 G(x + ε) + ε

I It is indeed a distance. Geometrically, it measures the maximumdistance between the graphs of G and F , in the direction NW-SE.

I Also, if (Gn)n is a sequence of distributions, then Gn tends to F in theweak sense if and only if dL(Gn, F )→ 0.

I Weak convergence is the classical convergence criterion amongdistributions: it corresponds to convergence in law of r.v. distributed asGn and F .



I The estimator r is (qualitatively) robust at F if for any ε there exist δand n such that

dL(G, F ) < δ implies dL(Hr,G,n,Hr,F ,n) < ε n > n

I In other words, we require the sampling distribution to dependcontinuously (uniformly in n) on the distribution generating data.

I If C is a set of distributions containing F , we may define C-robustnessby requiring the previous fact just for G ∈ C. It is a weaker, and morerealistic, notion of robustness.

I Also, we may use distances other than the Levy one. We can even usetwo different distances in the two sides of the implication. Severalrecent works in this direction.



I A crucial result is

I Hampel Theorem. Let ρ be the plug-in estimator of ρ and assumeit is consistent with ρ at G for any G in a neighborhood of F .Then ρ is qualitatively robust at F if and only if ρ is continuous(w.r.t. Levy distance) at F .

I If F is invertible at α, then ρ = −qα is continuous at F . Under the sameassumptions, ρψ is continuous at F if and only if ψ(u) = 0 around 0and 1. As both−qα and ρψ are consistent:

I the plug-in estimator for VaRα is qualitatively robustI the plug-in estimator for ρψ is not qualitatively robust whenever ψ

is decreasing (note: ψ(0) > 0)I so: the plug-in estimator for ESα is not qualitatively robust

I We can adapt Hampel Theorem to other estimators r (not necessarilyplug-in). In general MLE of VaR or ρψ for location-scale families are notrobust. See Cont, Deguest, Scandolo 2010.


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