Study on Statistics and Quantitative Risk Management for Banking and Insurance

Description
In the financial (banking and insurance) industry, solvency regulation has been around for a long time. The catchwords are Basel for banking and Solvency for insurance. The former derives from the Basel Committee on Banking Supervision, a supra-national institution, based in Basel, Switzerland, working out solvency guidelines for banks.

Statistics and Quantitative Risk
Management for Banking and Insurance
Paul Embrechts
RiskLab, Department of Mathematics and Swiss Finance Institute, ETH
Zurich, 8092 Zurich, Switzerland, [email protected]

Marius Hofert
RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland,
[email protected]. The author (Willis Research Fellow) thanks
Willis Re for financial support while this work was being completed.

Keywords
Statistical methods, risk measures, dependence modeling, risk aggregation,
regulatory practice

Abstract
As an emerging field of applied research, Quantitative Risk Management
(QRM) poses a lot of challenges for probabilistic and statistical modeling.
This review provides a discussion on selected past, current, and possible
future areas of research in the intersection of statistics and quantitative risk
management. Topics treated include the use of risk measures in regulation,
including their statistical estimation and aggregation properties. An extensive
literature provides the statistically interested reader with an entrance to this
exciting field.

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1. Introduction
In 2005, the first author (Paul Embrechts), together with Alexander J. McNeil
and Rüdiger Frey, wrote the by now well-established text McNeil et al. (2005).
Whereas the title read as “Quantitative Risk Management: Concepts, Techniques, Tools”, a somewhat more precise description of the content would be
reflected in “Statistical Methods for Quantitative Risk Management: Concepts, Techniques, Tools”. Now, almost a decade and several financial crises
later, more than ever, the topic of Quantitative Risk Management (QRM) is
high on the agenda of academics, practitioners, regulators, politicians, the media,
as well as the public at large. The current paper aims at giving a brief historical
overview of how QRM as a field of statistics and applied probability came to be.
We will discuss some of its current themes of research and attempt to summarize
(some of) the main challenges going forward. This is not an easy task as the
word “risk” is omnipresent in modern society and consequently its (statistical)
quantification. One therefore has to be careful not to lose sight of the forest
for the trees! Under the heading “QRM: The Nature of the Challenge”, in
McNeil et al. (2005, Section 1.5.1) it is written: “We set ourselves the task of
defining a new discipline of QRM and our approach to this task has two main
strands. On the one hand, we have attempted to put current practice onto a
firmer mathematical footing. On the other hand, the second strand of our
endeavor has been to put together material on techniques and tools which go
beyond current practice and address some of the deficiencies that have been
raised repeatedly by critics”. Especially in the wake of financial crises, it is
not always straightforward to defend a more mathematical approach on “risk”
(more on this later). For the purpose of the paper, we interpret risk at the more
mathematical level of (random) uncertainty in both frequency as well as severity
of loss events. We will not enter into the, very important, discussion around
“Risk and Uncertainty” (Knight (1921)) on the various “Knowns, Unknowns and
Unknowables” (Diebold et al. (2010)). Concerning “mathematical level”, let
the following quotes speak for themselves. First, in their (Nobel-)path breaking
paper Gale and Shapley (1962) are addressing the question “What is mathematics?” the authors wrote “. . . any argument which is carried out with sufficient
precision is mathematical,. . . ”. Lloyd Shapley (together with Alvin Roth) got
the 2012 Nobel Prize for economics; see Roth and Shapley (2012). As a second
quote on the topic we like Norbert Wiener’s; see Wiener (1923):
“Mathematics is an experimental science . . . It matters little . . .
that the mathematician experiments with pencil and paper while
the chemist uses test-tube and retort, or the biologist stains and
the microscope . . . The only great point of divergence between
mathematics and other sciences lies in the far greater permanence of
mathematical knowledge, in the circumstance that experience only
whispers ’yes’ or ’no’ in reply to our questions, while logic shouts.”
It is precisely statistics that has to straddle the shouting world of mathematical
logic with the whispering one of practical reality.

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Paul Embrechts, Marius Hofert

2. A whisper from regulatory practice
In the financial (banking and insurance) industry, solvency regulation has been
around for a long time. The catchwords are Basel for banking and Solvency for
insurance. The former derives from the Basel Committee on Banking Supervision,
a supra-national institution, based in Basel, Switzerland, working out solvency
guidelines for banks. Basic frameworks go under the names of Basel I, II, III
together with some intermediate stages. The website www.bis.org/bcbs of
the Bank for International Settlements (BIS) warrants a regular visit from
anyone interested in banking regulation. An excellent historic overview of the
Committee’s working is to be found in Tarullo (2008). Similar developments for
the insurance industry go under the name of Solvency, in particular the current
Solvency II framework. For a broader overview, see Sandström (2006). Whereas
Solvency II is not yet in force, since January 1, 2011, the Swiss Solvency Test
(SST) is. The key principles on which the SST, originating in 2003, is based are:
(a) it is risk based, quantifying market, insurance and credit risk, (b) it stresses
market consistent valuation of assets and liabilities, and (c) advocates a total
balance sheet approach. Before, actuaries were mainly involved with liability
pricing and reserve calculations. Within the SST, stress scenarios are to be
quantified and aggregated for capital requirement. Finally, as under the Basel
regime for banking, internal models are encouraged and need approval by the
relevant regulators (the FINMA in the case of Switzerland). An interesting text
exposing the main issues both from a more methodological as well as practical
point of view is SCOR (2008). As in any field of applications, if as a statistician
one really wants to have an impact one has to get more deeply involved with
practice. From this point of view, some of the above publications are must reads.
McNeil et al. (2005, Chapter 1) gives a somewhat smooth introduction for those
lacking the necessary time.
Let us concentrate on one particular example highlighting the potentially
interesting interactions between methodology and practice, and hence the title of
this section. As already stated, the Basel Committee, on a regular basis, produces
documents aimed at improving the resilience of the international financial system.
The various stakeholders, including academia, are asked to comment on these
new guidelines before they are submitted to the various national agencies to be
cast into local law. One such document is BIS (2012). In the latter consultative
document, p. 41, the following question (Nr. 8) is asked: “What are the likely
operational constraints with moving from VaR to ES, including any challenges
in delivering robust backtesting, and how might these be best overcome?”; for
VaR and ES, see Definition 2.1. This is an eminently statistical question the
meaning of which, together with possible answers, we shall discuss below. This
example has to be viewed as a blueprint for similar discussions of statistical
nature within the current QRM landscape. Some further legal clarification may
be useful at this point: by issuing consultative documents, the Basel Committee
wants to intensify its consultation and discussions with the various stakeholders
of the financial system. In particular, academia and industry are invited to
comment on the new proposals and as such may have an influence on the precise

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formulations going forward. The official replies are published on the website
of the BIS. We will not be able to enter into all relevant details but merely
highlight the issues relevant from a more statistical point of view; in doing so,
we will make several simplifying assumptions.
The first ingredient is a portfolio P of financial positions each with a welldefined value at any time t. For t (today, say) the value of the portfolio is vt .
Though crucial in practice, we will not enter into a more detailed economic
and/or actuarial discussion of the precise meaning of value: it suffices to mention
possibilities like “fair value”, “market value” (mark-to-market), “model value”
(mark-to-model), “accounting value” (often depending on the jurisdiction under
which industry is regulated) and indeed various combinations of the above!
Viewed from today, the value of the portfolio one time period in the future is
a random variable Vt+1 . In (market) risk management one is interested in the
(positive) tail of the so-called Profit-and-Loss (P&L) distribution function
(df) of the random variable (rv)
Lt+1 = ?(Vt+1 ? vt ).

(1)

The one-period ahead value Vt+1 is determined by the structure S(t + 1, Zt+1 )
of P, where Zt+1 in Rd is a (typically) high-dimensional random vector of risk
factors with corresponding risk-factor changes Xt+1 = Zt+1 ? zt . S is also
referred to as the mapping. As an example consider for P a linear portfolio
consisting of ?j stocks St,j , j ? {1, . . . , d} (St,j denotes the value of stock j at
time t and ?j the number of shares of stock j in P). Furthermore, assume the
risk factors to be
Zt = (Zt,1 , . . . , Zt,d ),

Zt,j = log St,j

and the corresponding risk-factor changes to be the log-returns (here again, t
denotes today):
St+1,j
= log(St+1,j ) ? log(st,j ) = log
.
st,j


Xt+1



In this case, the portfolio structure is
S(t + 1, Zt+1 ) =

d
X

?j St+1,j =

j=1

d
X

?j exp(Zt+1,j ).

(2)

j=1

(1) together with Vt+1 = S(t + 1, Zt+1 ) then implies that
Lt+1 = ?

d
X

?j (exp(Zt+1,j ) ? exp(zt,j ))

j=1

=?

d
X

?j (exp(zt,j + Xt+1,j ) ? exp(zt,j )) = ?

j=1

d
X

w˜t,j (exp(Xt+1,j ) ? 1),

j=1

where w˜t,j = ?j st,j ; wt,j = w˜t,j /vt is then the relative value of stock j in P. For
this, and further examples, see McNeil et al. (2005, Section 2.1.3).

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Paul Embrechts, Marius Hofert

Based on models and statistical estimates (computed from data) for (Xt+1 ),
the problem is now clear: find the df (or some characteristic, a risk measure) of
Lt+1 . Here, “the df” can be interpreted unconditionally or conditionally on some
family of ?-algebras (a filtration of historical information). As a consequence,
the mathematical set-up can become arbitrarily involved!
Analytically, (1) becomes
Lt+1 = ?(S(t + 1, Zt+1 ) ? S(t, zt ))
= ?(S(t + 1, zt + Xt+1 ) ? S(t, zt )).

(3)

If, for ease of notation, we suppress the time dependence, (3) becomes
L = ?(S(z + X) ? S(z))

(4)

with z ? Rd and X a d-dimensional random vector denoting the one-period
changes in factor values; d is typically high, d ? 1000! In general, the function
S can be highly non-linear. This especially holds true in cases where derivative
financial instruments (like options) are involved. If time is modeled explicitly
(which in practice is necessary) then (3) is a time series (Lt )t (stationary or
not) driven by a d-dimensional stochastic process (Xt )t and a deterministic,
though mostly highly non-linear portfolio structure S. Going back (for ease
of notation) to the static case (4), one has to model the df FL (x) = P(L ? x)
under various assumptions on the input. Only for the most trivial case, like X
is multivariate normal and S linear, can this be handled analytically. In practice,
various approximation methods are used:
(A1) Replace S by a linearized version S ? using Taylor expansion and approximate FL by the df FL? of the linearized loss L? ; here one uses smoothness
of S (usually fine) combined with a condition that X in (4) is stochastically
small (a more problematic assumption). The latter implies that one-period
changes in the risk factors are small; this is acceptable in “normal” periods
but not in “extreme” periods. Note that it is especially for the latter that
QRM is needed!
(A2) Find stochastic models for (Xt )t closer to reality, in particular beyond
Gaussianity but for which FL or FL? can be calculated/estimated readily.
This is a non-trivial task in general.
(A3) Rather than aiming for a full model for FL , estimate some characteristics
of FL relevant for solvency capital calculations, that is, estimate a so-called
risk measure. Here the VaR and ES abbreviations in the above BIS
(2012) quote enter, both are risk measures; they are defined as follows.
Definition 2.1
Suppose L as given above and let 0 < ? < 1, then
i)
The Value-at-Risk of L at confidence level ? is given by
VaR? (L) = inf{x ? R : FL (x) ? ?}
(i.e. VaR? (L) is the 100?% quantile of FL ).

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ii)

The Expected Shortfall of L at confidence level ? is given by
ES? (L) =

1 Z1
VaRu (L) du.
1?? ?

Remark 2.2
1) VaR? (L) defined above coincides with the so-called generalized inverse
FL? (?) at the confidence level ?; see Embrechts and Hofert (2013a). Its
introduction around 1994, linked to (4) as a basis for solvency capital calculation, truly led to a new risk management benchmark on Wall Street; see
Jorion (2007). A simple Google search will quickly confirm this. An interesting text aiming at improving VaR-usage and warning about the various
misunderstandings and misuses is Wong (2013).
2) For FL continuous, the definition of Expected Shortfall is equivalent to
ES? (L) = E[L | L > VaR? (L)],
hence its name. Alternative names in use throughout the financial industry
are “conditional VaR” and “conditional tail expectation”.
3) Note that ES does not exist for infinite mean models, i.e. when ES? |L| = ?.
This reveals a potential problem with ES as a risk measure in case of very
heavy-tailed loss data.
4) Both risk measures are used in industry and regulation; VaR more in banking,
ES more in insurance, the latter especially under the SST. Depending on the
risk class under consideration, different values of ? (typically close to 1) and
the holding period (“the one period” above, see (1)) are in use; see McNeil
et al. (2005) for details.
5) A whole industry of papers on estimating VaR and/or ES has emerged,
and this under a wide variety of model assumptions on the underlying P&L
process (Lt )t . McNeil et al. (2005) contains a summary of results stressing the
more statistical issues. An excellent companion text with a more econometric
flavour is Daníelsson (2011). More recent statistical techniques in use are for
instance extreme quantile estimation based on regime-switching models and
lasso-technology, see Chavez-Demoulin et al. (2013a). Estimation based on
self-exciting (or Hawkes) processes is exemplified in Chavez-Demoulin and
McGill (2012). For an early use on Hawkes processes in finance, see ChavezDemoulin et al. (2005). McNeil et al. (2005, Section 7.4.3) contains an early
text-book reference to QRM. Extreme Value Theory (EVT) methodology is
for instance presented in McNeil and Frey (2000); Chavez-Demoulin et al.
(2013b) use it in combination with generalized additive models in order to
model VaR for Operational Risk based on covariates.
6) From the onset, especially VaR has been heavily criticized as it only looks
at the frequency of extremal losses (? close to 1) but not at the severity.
The important “what if” question is more addressed by ES. Furthermore,
in general VaR is not a subadditive risk measure, i.e. it is possible that

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Paul Embrechts, Marius Hofert

for two positions L1 and L2 , VaR? (L1 + L2 ) > VaR? (L1 ) + VaR? (L2 ) for
some 0 < ? < 1. This makes risk aggregation and diversification arguments
difficult. ES as defined in Definition 2.1 ii) above is always subadditive. These
issues played a role in the subprime crisis of 2007–2009; see Donnelly and
Embrechts (2010) and Das et al. (2013) for some discussions on this. McNeil
et al. (2005) contains an in-depth analysis together with numerous references
for further reading. Whereas industry early on (mid nineties) did not really
take notice of the warnings concerning the problem with VaR in markets
away from “normality”, by now the negative issues underlying VaR-based
regulation have become abundantly clear and the literature is full of examples
on this. Below we have included a particular example from the realm of
credit risk stressing, hopefully exposing in a pedagogically clear way, some
of the problems. It is a slightly expanded version of McNeil et al. (2005,
Example 6.7) and basically goes back to Albanese (1997). By now, there
exist numerous similar examples.
Example 2.3 (Non-subadditivity of VaR)
Assume we have given 100 bonds with maturity T equal to one year, nominal
value 100, yearly coupon 2%, and default probability 1% (no recovery). The
corresponding losses (assumed to be independent) are
?
??2,

with probability 0.99,
Li = ?
100, with probability 0.01,

i ? {1, . . . , 100}.

(Recall the choice of negative values to denote gains!) Consider two portfolios,
P1 :

100
X

Li

and

P2 : 100L1 .

(5)

i=1

P1 is a so-called diversified portfolio and P2 clearly a highly concentrated one.
As this point the reader may judge for himself/herself which of the two portfolios
he/she believes to be less risky (and thus would assign a smaller risk measure
to).
Figure 1 shows the boundaries of this decision according to VaR (the shift by
201 in y scale is for plotting the y axis in log-scale). Rather surprisingly, for
? = 0.95, for example, VaR? is superadditive, i.e.
VaR0.95 (P1 ) > VaR0.95 (P2 ).
By the interpretation of a risk measure as the amount of capital required as
a buffer against future losses, this implies that the diversified portfolio P1
can be riskier (thus requiring a larger risk capital) than the concentrated P2
according to VaR. Indeed, one can show that for two portfolios as in (5) based
on n bonds with default probability p, VaR? is superadditive if and only if
(1 ? p)n < ? ? 1 ? p. This formula holds independently of the coupon and
nominal value. For n = 100 and p = 1% we obtain that VaR? is superadditive if
and only if 0.3660 ? 0.99100 < ? ? 0.99.

www.annualreviews.org • Statistics and Quantitative Risk Management for Banking and Insurance

7

100

? VaR?(Li) for Example 2.3

i=1

10000

VaR?(L1 + … + L100) vs

VaR?(L1 + … + L100)
100

100

i=1

1

10

VaR? + 201

1000

? VaR?(Li)

0.0

0.2

0.4

0.6

0.8

1.0

?

Figure 1
V aR? as a function in ? for the two portfolios P1 and P2 .

It is clear that for ? > 0.95, say, corresponding to values typical for financial
and insurance regulation, both VaR? (L) and ES? (L) are difficult to estimate.
One needs an excellent fit for 1?FL (x), x large; at this point, EVT as summarized
in McNeil et al. (2005, Chapter 7) comes in as a useful tool. Not that EVT is a
panacea for solving such high-quantile estimation problems, more importantly,
it offers guidance on the kind of questions from industry and regulation which
are beyond the boundaries of sufficiently precise estimation. In the spirit of
McNeil et al. (2005, Example 7.25) we present Figure 2 based on the Danish fire
insurance data. For these typical loss data from non-life insurance, the linear
log-log plot for the data above a threshold of u = 5.54 (the 90%-quantile of
the empirical distribution; in million Danish Krone) clearly indicates power-tail
behavior. The solid line through the data stems from a Peaks-Over-Threshold
fit (EVT); see McNeil et al. (2005, Chapter 7) for details. The dotted curves are
the so-called profile likelihoods, the width of these at a specific confidence level
(second y axis on the right-hand side) indicates EVT-based confidence intervals
for the risk measure under consideration (VaR on the left-hand side, ES on the
right-hand side of Figure 2). Do note the considerable uncertainty around these
high-quantile estimates!
We now return to Question 8 in BIS (2012) concerning a possible regulatory
regime switch for market risk management from VaR to ES and the issue
of “robust backtesting”. A summary of our current knowledge on Question 8
we summarize below. We compare and contrast both risk measures on the
basis of four broadly defined criteria: (C1) Existence; (C2) Ease of accurate
statistical estimation; (C3) Subadditivity, and (C4) Robust forecasting and
backtesting. The “loud” answer to the above “whispering” question, coming

8

Paul Embrechts, Marius Hofert

?

2e?04

?

?

10

20

50
Exceedances x

100

200

0.5
0.8
0.95 0.9
?
?
?
?
?
?

Confidence intervals

2e?02
5e?03

?

?

0.995

0.995 0.99

1e?03

?

?

1e?03

?
?
?

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?

GPD(0.58, 4.51) tail estimate
ES0.99 estimate (= 69.02)
ES0.99 CIs

2e?04

?

^
1 ? Fn(x)

?

?
?

5

1e?01

0.5
0.9

0.8

GPD(0.58, 4.51) tail estimate
VaR0.99 estimate (= 27.45)
VaR0.99 CIs

Confidence intervals

5e?03

^
1 ? Fn(x)

2e?02

?
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Estimated tail probabilities

0.95

1e?01

Estimated tail probabilities

5

10

20

50

?

100

200

Exceedances x

Figure 2
Estimated tail probabilities including risk measures VaR0.99 (left) and ES0.99 (right) with
likelihood-based confidence intervals for the Danish fire insurance data.

from mathematical statistics, is expertly summarized in Gneiting (2011). In
the latter paper, the author offers a theory to assess the quality of statistical
forecasts introducing the notion of elicitability. For the ease of this discussion,
we will refer to a forecast as elicitable if it is “properly” backtestable. For the
precise mathematical meaning of “properly” we refer to the latter paper and the
references therein. It is shown that in general, VaR is elicitable whereas ES is
not (note that in Gneiting (2011), ES is referred to as Conditional Value-at-Risk,
CVaR). In Section 3.4 the author states that: “This negative result may challenge
the use of the CVaR functional as a predictive measure of risk, and may provide a
partial explanation for the lack of literature on the evaluation of CVaR forecasts,
as opposed to quantile or VaR forecasts, for which we refer to Berkowitz and
O’Brien (2002), Giacomini and Komunjer (2005), Balo et al. (2006), among
others”. See also Jorion (2007, Chapter 6). The Basel II guidelines also contain
a multiplier penalty matrix for solvency capital, the severity of the multiplier
being a function of the historical statistical accuracy of the backtesting results;
see Jorion (2007, Table 6-3, p. 148). A further comment on the difficulty of
ES-forecasting is to be found in Daníelsson (2011, Remark 2, p. 89 and Section
8.4) mainly concentrating on the need for much more data for ES as compared to
VaR, data which often is not available. This leads to the interesting question to
what extent elicitability is really fundamental towards risk-measure backtesting in
insurance and finance. Whereas we are in no doubt that the notion is important,
more research on this topic is needed. A first relevant paper beyond Gneiting
(2011) is Ziegel (2013). Both VaR and ES are law invariant risk measures, in riskmeasure terminology, ES is coherent (in particular, subadditive), whereas VaR
in general is not (as we saw in Example 2.3). This raises the question whether
there exist non-trivial, law-invariant coherent risk measures. The answer, from
Ziegel (2013), is that so-called expectiles are, but the important subclass of lawinvariant spectral risk measures are not elicitable. For some more information on
expectiles, see Rémillard (2013) and the references therein. Ziegel (2013) contains

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9

an interesting discussion on how backtesting procedures based on exceedance
residuals as in McNeil and Frey (2000) or McNeil et al. (2005, Section 4.4.3)
may possibly be put into a decision-theoretic framework akin to elicitability.
As already stated above, this is no doubt an important area of future research.
Clearly, concerning (C1), VaR? (L), when properly defined as in Definition 2.1,
always exists. For ES? [L] to exist, one needs E|L| < ?. The latter condition
may seem without any problem, see however the discussion on Operational Risk
in McNeil et al. (2005, Section 10.1.4). Concerning (C2), and this especially
for ? close to 1, both measures are difficult to estimate with great statistical
accuracy. Of course, confidence intervals around ES are often much larger than
corresponding ones around VaR; see McNeil et al. (2005, Figure 7.6) or Figure 2
above. Concerning (C3), we already discussed the fact that in general VaR is
not subadditive whereas ES is. This very important property tips the balance
clearly in favor of ES; of course ES also contains by definition important loss
information of the “what if” type.
It is clear that for criteria (C1)–(C4) above one needs much more detailed
information specifying precise underlying model assumptions, data availability,
specific applications etc. The references given yield this background. The
backtesting part of (C4) we already discussed above. When robustness in Basel’s
Question 8 is to be interpreted in a precise statistical sense, like in Huber and
Ronchetti (2009) or Hampel et al. (2005), then relevant information is to be
found in Dell’Aquila and Embrechts (2006) and Mancini and Trojani (2011);
these papers concentrate on robust estimation of risk measures like VaR. In Cont
et al. (2010) the authors conclude that “. . . historical Value-at-Risk, while failing
to be subadditive, leads to a more robust procedure than alternatives such as
Expected Shortfall”. Moreover, “the conflict we have noted between robustness
of a risk measurement procedure and the subadditivity of the risk measure show
that one cannot achieve robust estimation in this framework while preserving
subadditivitiy.” This conclusion seems in line with the results in Gneiting (2011)
replacing elicitability by robustness. However also here the discussion is not yet
closed: robustness can be interpreted in several ways and it is not clear what
kind of robustness the Basel Committee has in mind. For instance, Filipovi?
and Cambou (private communication), within the context of the SST, formulate
a “robustness criterium” which is satisfied by ES but not by VaR. Hence also
here, more work is clearly needed.
For the above discussion we did not include an overview of the explosive literature on the theory of risk measures. Some early references with a more economic
background are Gilboa and Schmeidler (1989) and Fishburn (1984). A paper
linking up more with recent developments in financial econometrics is Andersen
et al. (2013). An excellent textbook treatment at the more mathematical level
is Föllmer and Schied (2011). From a more mathematical point of view, recent
research on risk measures borrows a lot from Functional Analysis; see Delbaen
(2012). Concerning the question on robustness properties of risk measures: it
is historically interesting that the mathematical content of the main result in
Artzner et al. (1999, Proposition 4.1) is contained in the classic on statistical
robustness Huber (1981, Chapter 10, Proposition 2.1).

10

Paul Embrechts, Marius Hofert

SUMMARY POINTS
The above discussion clearly shows the importance of (mathematical) statistics and probability theory already at the level of very basic questions in
QRM of a highly relevant nature. In Section 4 we will add a further example
of this important discourse, and this under the name of model uncertainty
and risk aggregation. To this end, statisticians with a substantial background in practically relevant QRM should also be able and willing to
give their guiding advice, in particular concerning questions like the above
Question 8. Given all the practical and scientific information we know of at
the moment, our advice to the regulators would be “stick with the devil”, i.e.
stay with VaR-based capital solvency laws if a risk-measure based regime
is wanted. Be very aware of all of VaR’s weaknesses (and these are many),
especially in terms of crises. We would prefer statistical estimation for lower
confidence level ? in VaR? together with regulatory defined stress factors
(already partly in place, but for ?’s too close to 1, i.e. too far out in the tails
of the underlying P&L’s where statistical, as well as model uncertainty is
considerable). Banks should be encouraged to ask, and report on the “what
if” question, as such, reporting on ES? is definitely useful (if E|L| < ?).
Finally, though we did not report on the holding period (the plus 1 in (1)),
time scaling of VaR? , like for market risk from 1-day to 10-days (= two
trading weeks), our research indicates that (overall) it is difficult to beat the
square-root-of-time rule; see Embrechts et al. (2005) and Kaufmann (2004),
and, for an introduction to this rule, Jorion (2007, p. 98). This issue also
figures prominently in BIS (2012); see for instance Section 3.3 on “Factoring
in market liquidity”.

3. Modeling interdependence
Returning to (3), a main statistical modeling task remains the (high-dimensional)
risk factor process (Zt )t or the risk factor changes (Xt )t . This either in a stationary or a non-stationary environment. Also here, a huge, mainly econometric
literature exists; interesting texts are Engle (2009) and Christian Gourieroux
(2001). From a more static modeling point of view, Genton (2004) offers several
industry models. A review of models in use within QRM is to be found in McNeil
et al. (2005, Chapter 1, 3, and Sections 4.5, 4.6). These textbooks also contain
numerous references for further reading.
Keeping in line with the overall tenor of the paper, below we single out some
areas of current research where QRM and statistics have an interesting scientific
intersection as well as offer research of practical relevance. The modeling of
interdependence between risk factors or risk-factor changes is a typical such
example, it also lies at the very basis of most relevant QRM questions. Examples
are to be found in market risk management already discussed in the previous
section. The field of credit risk offers a clear playground for interdependence modelers; indeed most credit-risk portfolios exhibit interdependence, especially under

www.annualreviews.org • Statistics and Quantitative Risk Management for Banking and Insurance

11

extreme market conditions. The various financial crises (subprime, sovereign,. . . )
give ample of proof. For instance the famous AAA-guarantee given by the
rating agencies for the (super-)senior tranches of Collateralized Debt Obligations
(CDOs) turned out to be well below this low-risk level when the American
housing marked collapsed around 2006. An early warning for the consequences
of even a slight increase in interdependence between default-events of credit
positions for the evaluation of securitized products like CDOs is to be found in
McNeil et al. (2005, Figure 8.1); see also Hofert and Scherer (2011). The bigger
methodological umbrella to be put over such examples is Model Uncertainty.
The latter is no doubt one of the most active and practically relevant research
fields in QRM at the moment. We shall only highlight some aspects below,
starting first with the notion of copula.
In the late nineties, the concept of copula took QRM by storm. A copula
is simply a multivariate df with standard uniform univariate margins. Copulas
provide a convenient and comparably simple (mainly static) way of describing
the dependence between the components X1 , . . . , Xd of a random vector X (the
risk-factor changes in (4) for example). A publication which sparked this hype
no doubt is Embrechts et al. (2002); see Figure 3 based on the data collected
by Genest et al. (2009) (unfortunately, the latter database was not maintained
beyond 2005). This paper was available as a RiskLab preprint since early 1998

200
150

?

?

100

Embrechts, McNeil, Straumann (1998/2002)
?

50

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0

Annual number of publications on copula theory

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1995

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2000

2005

Year

Figure 3
Number of publications on copula theory.

and was also published in a much abbreviated form in Embrechts et al. (1999).
For the record, Daniel Straumann summarized the main findings of Embrechts et
al. (1999) at the first ETH Risk-Day on September 25, 1998 in a talk “Tempting
fallacies in the use of correlation”. The first author, Paul Embrechts, gave a
similar talk at the 3rd Columbia-JAFEE conference at Columbia University on
March 27, 1999, this time with a title “Insurance analytics: actuarial tools in
financial risk management”. For some of this QRM-relevant background, see

12

Paul Embrechts, Marius Hofert

Donnelly and Embrechts (2010). We would like to stress, however, that the
copula concept has been around almost for as long as modern probability and
statistics emerged in the early 20th century. Related work can be traced back to
Fréchet (1935) and Hoeffding (1940), for example.
The central theorem underlying the theory and applications of copulas is
Sklar’s Theorem; see Sklar (1959). It consists of two parts. The first one
states that for any multivariate df H with margins F1 , . . . , Fd , there exists a
copula C such that
H(x1 , . . . , xd ) = C(F1 (x1 ), . . . , Fd (xd )), x ? Rd .

(6)

This decomposition is not necessarily unique, but it is in the case where all margins F1 , . . . , Fd are continuous. The second part provides a converse statement:
Given any copula C and univariate dfs F1 , . . . , Fd , H defined by (6) is a df with
margins F1 , . . . , Fd . An analytic proof of Sklar’s Theorem can be found in Sklar
(1996), a probabilistic one in Rüschendorf (2009).
In what follows we assume the margins F1 , . . . , Fd to be continuous. In the
context of finance and insurance applications, this assumption is typically not
considered restrictive, but it is, for example, when counting data plays an
important role, such as for medical or pharmacological applications. In general,
it certainly becomes a more restrictive assumption if d is large. For the case of
non-continuous margins (with all its pitfalls in thinking about dependence) we
refer the interested reader to Genest and Nešlehová (2007). For now, we focus
on the case of continuous margins and thus have a unique representation (6) for
H in terms of the copula C and the margins of H. From this representation we
see that the copula C is precisely the function which combines (or “couples”,
hence the name) the margins F1 , . . . , Fd of H to the joint df H. To understand
the importance of Sklar’s Theorem for statistics and QRM, in particular the
aforementioned hype, we now interpret (6) in terms of two data sets shown in
Figure 4. The data sets show n = 500 realizations of (X1 , X2 ) ? H with F1
and F2 being standard normal dfs on the left-hand side, and F1 and F2 being
standard exponential dfs on the right-hand side. The generated data sets could
be historical risk-factor changes, for example. Questions which immediately
come up, are:
How can the dependence between X1 and X2 be modeled based on these two
data sets?
Which of the data sets shows “stronger” dependence between X1 and X2 ?
Although graphics such as Figure 4 are used in various fields of application to
answer such questions, it is highly critical to do so. In the two plots in Figure 4,
the margins live on quite different scales and thus disguise the underlying
dependence structure, the C in (6). To make a comparison possible, we transform
the margins to U[0, 1], i.e. the standard uniform distribution. More precisely, we
consider U = (F1 (X1 ), F2 (X2 )); note that if Xj ? Fj and Fj is continuous then
Fj (Xj ) ? U[0, 1]. The results are shown in Figure 5. Although the data sets in
Figure 4 do not seem to have much in common, after removing the influence of
the margins, we suddenly see that they do! We obtained the exact same samples

www.annualreviews.org • Statistics and Quantitative Risk Management for Banking and Insurance

13

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Figure 4. They are (even exactly; not only in distribution) identical.

14

Paul Embrechts, Marius Hofert

(not only samples equal in distribution). Since they have U[0, 1] margins, we
indeed see a sample from the copula C in (6). The first part of Sklar’s Theorem
thus allows us to make C visible in this way, to decompose H (or (X1 , X2 )) into
its margins F1 , F2 (or X1 , X2 ; note the missing vector notation here) and the
underlying dependence structure C (or (U1 , U2 )).
In a practical situation, one does not know the margins F1 , F2 . In this case one
typically replaces them by their (slightly scaled) empirical dfs in order to make (an
approximation to) C visible. One obtains the so-called pseudo-observations
ˆ i , i ? {1, . . . , n}, given by
U
Uˆij =

n ˆ
Rij
,
Fn,j (Xij ) =
n+1
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j ? {1, . . . , d},

(7)

where Fˆn,j denotes the empirical df of the jth margin and Rij denotes the
rank of Xij among X1j , . . . , Xnj . A plot of the pseudo-observations similar to
Figure 5 is therefore referred to as rank plot. It may easily be computed with
the free statistical software R via plot(apply(x, 2, rank, na.last = "keep
") / (nrow(x) + 1)) where x denotes the (n, d)-matrix whose rows contain
realizations of X.
The first part (decomposition) of Sklar’s Theorem proves especially useful if
d is large. Assume d = 100 and each of the margins to have two parameters.
Furthermore, assume the copula to have one parameter. Computing a maximum
likelihood estimator for all parameters in such a set-up would amount to solving
an optimization problem in 201 dimensions. By estimating the parameters of the
margins individually and estimating the copula parameter based on the pseudoobservations (the scaling n/(n + 1) in (7) is used so that the copula density can
be evaluated at the pseudo-observations), one only has to solve 100 optimization
problems in two dimensions and one univariate optimization problem. Note
that all of these can be carried out in parallel as they do not depend on each
other. This approach is not a panacea for solving all estimation problems of
multivariate models since the pseudo-observations are not independent anymore
(due to the ranks). This influences the estimator, especially in high dimensions;
see Embrechts and Hofert (2013b). However, it makes parameter estimation
feasible and comparably fast even in high dimensions (depending on the problem
at hand) and is therefore widely used in practice.
The second part of Sklar’s Theorem basically reads (6) backwards. Given
any marginal dfs F1 , . . . , Fd (continuous or not) and any copula C, H in (6)
is a multivariate df with margins F1 , . . . , Fd . This provides us with a concrete
construction principle for new multivariate dfs. Important for finance and
insurance practice is that it gives us additional modeling options besides the
multivariate normal or t distribution, it thus allows us to construct more flexible,
realistic multivariate models for X. This is also used for stress testing to answer
questions like
How does a VaR estimate react if one keeps the margins fixed but change the
dependence?
How does it change if the margins vary but the dependence remains fixed?

www.annualreviews.org • Statistics and Quantitative Risk Management for Banking and Insurance

15

Another important aspect of the second part of Sklar’s Theorem is that it gives
us a sampling algorithm for dfs H constructed from copula models. This is
heavily used in practice as realistic models are rarely analytically tractable and
thus are often evaluated based on simulations. The algorithm implied by the
second part of Sklar’s Theorem for sampling X ? H where H is constructed
from a copula C and univariate dfs F1 , . . . , Fd is as follows:
Algorithm 3.1
1) Sample U ? C;
2) Return X = (X1 , . . . , Xd ) where Xj = Fj? (Uj ), j ? {1, . . . , d}.
It should by now be clear to the reader how we constructed Figures 4 and
5. In line with Algorithm 3.1, we first constructed a sample of size 500 from
U ? C for C being a Gumbel copula (with parameter such that Kendall’s tau
equals 0.5). This is shown in Figure 5. We then used the quantile function of the
standard normal distribution to map U to X having N(0, 1) margins (shown
on the left-hand side of Figure 4) and similarly for the standard exponential
margins on the right-hand side of Figure 4.
After this brief introduction to the modeling of stochastic dependence with
copulas, a few remarks are in order. Summaries of the main results partly
including historical overviews can be found in the textbooks of Schweizer and
Sklar (1983, Chapter 6), Nelsen (1999), Cherubini et al. (2004), or Jaworski et al.
(2010). The relevance for finance is summarized in Embrechts (2009). Adding
Schweizer (1991), both papers together give an idea of the key developments.
Embrechts (2009) also recalls some of the early discussions between protagonists
and antagonists of the usefulness of copulas in insurance and finance in general
and QRM in particular.
History has moved on and by now copula technology forms a well-established
tool set for QRM. Like VaR, also the copula concept is marred with warnings
and restrictions when it comes to its applications. An early telling example is the
use of the Gaussian copula (the copula C corresponding to H being multivariate
normal via (6)) in the realm of CDO-pricing, an application which for some
lay at the core of the subprime crisis; see for instance the online publication
(blog) Salmon (2009). Somewhat surprisingly, the statistical community bought
the arguments; see Salmon (2012). This despite the fact that the weakness
(even impossibility) of the Gaussian copula for modeling joint extremes was
known for almost 50 years: Sibuya (1959) showed that the Gaussian copula
does not exhibit a notion referred to as tail dependence; see McNeil et al. (2005,
Section 5.2.3). Informally, the latter notion describes the probability that one
(of two) rvs takes on a large (small) value, given that the other one takes on a
large (small) value; in contrast to the Gaussian copula, a Gumbel copula (being
part of the Archimedean class) is able to capture (upper) tail dependence and
we may already suspect that by looking at Figure 5. With the notion of tail
dependence at hand, it is not difficult to see its importance for CDO pricing
models (or more generally, intensity-based default models) such as the one based
on Gaussian copulas (see Li (2000)), Archimedean copulas (see Schönbucher
and Schubert (2001)), or nested Archimedean copulas (see Hofert and Scherer

16

Paul Embrechts, Marius Hofert

(2011)); the latter two references introduce models which are able to capture tail
dependence. Due to its importance for QRM, Sibuya (1959) and, more broadly,
tail dependence was referred to in the original Embrechts et al. (2002, Section 4.4).
We repeat the main message from the latter paper: copulas are a very useful
concept for better understanding the concept of stochastic dependence, not a
panacea for its modeling. The title of Lipton and Rennie (2007) is telling in this
context; see also the discussions initiated by Mikosch (2006).
As mentioned before, within QRM, copulas have numerous applications towards risk aggregation and stress testing, and as such have entered the regulatory
guidelines. Below we briefly address some of the current challenges; they are
mainly related to high-dimensionality, statistical estimation, and numerical
calculations.
A large part of research on copulas is conducted and presented in the bivariate
case (d = 2). Bivariate copulas are typically convenient to work with, the case
d = 2 offers more possibilities to construct copulas (via geometric approaches),
required quantities can often be derived by hand, graphics (of the df or, if it
exists, its density) can still be drawn without losing information, and numerical
issues are often not severe or negligibly small. The situation is different for d > 2
and fundamentally different for d  2 (even if we stay far below d = 1000). In
general, the larger the dimension d, the more complicated it is to come up with
a numerically tractable model which is flexible enough to account for real data
behavior. Joe (1997, pp. 84) presents several desirable properties of multivariate
distributions. However, to date, no model is known which exhibits all such
properties.
The t copula belongs to the most widely used copula models; see Demarta
and McNeil (2005) for an introduction and possible extensions. Besides an
explicit density, it admits tail dependence and can be sampled comparably fast;
it also incorporates the Gaussian copula as limiting case, which, despite all its
limitations, is still a popular model. This makes the t copula especially attractive
for practical applications even in large dimensions. From a more technical point
of view, a d-dimensional t copula with ? > 0 degrees of freedom and correlation
matrix P can be written as
?1
C(u) = t?,P (t?1
? (u1 ), . . . , t? (ud )),

u ? [0, 1]d ,

where t?,P denotes the df of a multivariate t distribution with ? degrees of
freedom and correlation matrix P (corresponding to the dispersion matrix ? in
terms of which the multivariate t distribution is defined in general; see Demarta
and McNeil (2005)) and t?1
denotes the quantile function of a univariate t
?
distribution with ? degrees of freedom. From this formula we already see that
the components involved in evaluating a t copula are non-trivial. Neither t?,P
nor t?1
? are known explicitly. Especially the former is a problem and for d > 3,
randomized quasi Monte Carlo methods are used (in the well-known R package
mvtnorm) to evaluate the d-dimensional integral; see Genz and Bretz (2009,
Section 5.5.1). Furthermore, the algorithm requires ? to be an integer, which is
a limitation for which no easy solution is available yet.
The Gaussian and t copula families belong to the class of elliptical copulas

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17

and are given implicitly by solving (6) for C in the case where H is the df of
an elliptical distribution. A different ansatz stems from Archimedean copulas,
which are given explicitly in terms of a so-called generator ? : [0, ?) ? [0, 1]
via
C(u) = ?(? ?1 (u1 ) + · · · + ? ?1 (ud )),

u ? [0, 1]d ;

(8)

see McNeil and Nešlehová (2009). For the Gumbel copula family, ?(t) =
exp(?t1/? ), ? ? 1. Archimedean copulas allow for a fast sampling algorithm,
see Marshall and Olkin (1988), and, in contrast to elliptical copulas, are not
limited to radial symmetry. Elliptical copulas put the same probability mass
in the upper-right corner than in the lower-left corner of the distribution and
thus value joint large gains in the same way as joint large losses. On the other
hand, it directly follows from (8) that Archimedean copulas are symmetric, that
is, invariant under permutation of the arguments. This is also visible from the
symmetry with respect to the diagonal in Figure 5. Still, the flexibility in the
(upper or lower) tail of copulas such as the Gumbel are interesting for practical
applications and Archimedean copula families such as the Clayton or Gumbel
belong to the most widely used copula models (the Gumbel family additionally
belongs to the class of extreme value copulas and is one of the rare comparably
tractable examples in this class).
The rather innocent functional form (8) bears interesting numerical problems,
already in small to moderate dimensions which have even led to wrong statements
in the more recent literature. The copula density is easily seen to be
c(u) = (?1)d ? (d) (? ?1 (u1 ) + · · · + ? ?1 (ud ))

d
Y

(?? ?1 )0 (uj ),

u ? (0, 1)d .

j=1

In particular, it involves the dth generator derivative. For d large, these are
difficult to access. For the Gumbel copula, an explicit form for (?1)d ? (d) has
only recently been found; see Hofert et al. (2012). The formula is
(?1)d ? (d) (t) =

d
?(t) X
adk (?)t?k ,
td k=1
d?k

adk (?) = (?1)

d
X

k
d! X
k
? s(d, j)S(j, k) =
k! j=1 j
j=k
j

!

!

?j
(?1)d?j ,
d

where ? = 1/? ? (0, 1] and s and S denote the Stirling numbers of the first
and the second kind, respectively. The mathematical beauty disguises the
numerical problems in evaluating (?1)d ? (d) (t); see Hofert et al. (2013) and the
source code of the R package copula for solutions. Even if we can evaluate the
generator derivatives, Figure 6 shows what we try to achieve by going towards
higher dimensions; note the scale of the y axis and the steep curve near zero
even in log-scale.
Derivatives of Archimedean generators not only appear in densities of Archimedean copulas, they appear in other, more flexible dependence models based
on Archimedean copulas, such as Khoudraji-transforms, Archimedean Sibuya

18

Paul Embrechts, Marius Hofert

1e+117
1e?45

1e+09

1e+63

=2
=5
= 10
= 20
= 50

1e?99

Absolute value of the d th generator derivative

d
d
d
d
d

0

200

400

600

800

1000

t

Figure 6
Generator derivatives for Gumbel copulas with parameter such that Kendall’s tau equals 0.5.

copulas, nested Archimedean copulas, but also in statistically highly relevant
quantities such as conditional distributions or Kendall dfs. Problems as the
above also appear for example when computing the densities of copulas describing the dependence in multivariate extreme value distributions. Besides the
theoretical challenge to derive explicit mathematical formulas, it is generally an
(even larger) challenge to evaluate the formulas in a fast and numerically reliable
way in computer arithmetic. In high dimensions, one soon enters the world of
multi-precision arithmetic (mainly to test, but also to be able to compute certain
quantities at all) and parallel computing (nowadays of special interest due to
multi-core processors even in standard notebooks).
Let us now turn to another aspect of high-dimensional models in general.
Assume we fitted a d-dimensional copula C to the pseudo-observations obtained
from realizations of the vector X of risk-factor changes. Suppose we would
like to compute the probability that X falls in a non-empty hypercube (y, z]
(the most basic shapes we typically compute probabilities of). This would
require us to evaluate C at 2d many points, the corners of (a, b] ? [0, 1]d , where
a = (F1 (y1 ), . . . , Fd (yd )) and b = (F1 (z1 ), . . . , Fd (zd )). This C-volume can be
computed via
?(a,b] C =

X

Pd

(?1)

j
k=1 k

1
d
C(aj11 b1?j
, . . . , ajdd b1?j
).
1
d

j?{0,1}d

Although theoretically “just” a sum, we indeed have 2d summands of alternating
sign. If we can not exactly evaluate C (and this is even a problem if C is
explicit such as in the Gumbel case), we soon, for increasing d, end up with
a volume outside [0, 1]. Furthermore, 2d grows very fast, it is thus not only
numerically critical but also time consuming to compute probabilities of falling
in hypercubes. Indeed, for d ? 270 the number of corners of a d-dimensional

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19

hypercube is roughly equal to the number of atoms in the universe. Clearly,
for d = 1000 approximation schemes and possibly new approaches must be
developed.
New high-dimensional dependence models for X that recently became of
interest are hierarchical models. Through a matrix of parameters, elliptical
copulas such as the Gaussian or t copulas allow for d2 -many (an additional
one for the t copula) parameters. Symmetric models such as (8) typically allow
for just a handful of parameters, depending on the parametric generator under
consideration. Hierarchical models try to go somewhere
in-between, which is
 
d
often a good strategy for d large. While estimating 2 -many parameters may be
time-consuming and, for example in the case of a small sample size, not justifiable
by the data at hand or task in mind, using the assumption of symmetry for blocks
or groups of variables among X allows to reduce the number of parameters to
fall in between a fully symmetric model and a model considering parameters for
all d2 pairs of variables. Extending (8) to allow for hierarchies can be achieved
in various ways, see for example McNeil (2008) or Hofert (2012). Reducing
the parameters in elliptical models can be done by considering block matrices;
see Torrent-Gironella and Fortiana (2011) which is one of the rare publications
carefully considering numerical issues behind the scenes as well. For a different
idea to incorporate groups in t copulas, see Demarta and McNeil (2005).
SUMMARY POINTS
Overall, to our experience, there are many challenges in the area of high
dimensional dependence modeling. It does not seem to be a pleasing
task to go there as numerics will heavily enter soon after one leaves the
“comfort zone” of d being as small as two or three. However, from a
practical point of view it is necessary. Working in high dimensions may
require starting from scratch, model building already has to take numerics
into account, not only mathematical beauty or convenience. The same
applies to the (even more complicated) case of time dependent models not
discussed here. To give an example, sampling dependent Lévy processes
with the notion of Lévy copulas to model dependence between jumps is
typically also numerically a non-trivial task and limited to rather small
dimensions. Although mathematically d is just a natural number and can
be increased arbitrarily in the above discussion and formulas, solutions
to the problems mentioned above heavily depend on d. In the end of the
(bank’s) day, models, estimators, stress tests etc. have to be computed and
thus, additionally, require reliable software behind the scenes. It is to be
encouraged that academia contributes here as well (and gets scientifically
rewarded for entering this time-consuming process), as new statistical models
and procedures (including their limitations) are typically best understood
by those who invented them. Being pressed for time in business practice
often requires to implement a model in a small amount of time, basically
until it “works”. However, proper testing of statistical software has to go
far beyond this point and extensive test suites have to be put in place.

20

Paul Embrechts, Marius Hofert

Open source statistical software (slowly finding its way into business practice
but still not used to a sufficient degree) typically guarantees more stable and
reliable procedures as anyone can report bugs, for example. Furthermore,
open source increases the understanding of new models and the recognition
of their limitations. Software development also has to be communicated
on an academic level as it is already part of (but also limited to) statistics
courses. Preparing students for (or at least to be aware of) the computational
challenges they will meet in practice should rather become the rule than
the exception.

4. Risk aggregation and model uncertainty
The subadditivity property of a risk measure is often defended in order to
achieve a coherent, consistent way to aggregate risks across different entities
(business lines, say) and also in order to find methodologically consistent ways to
disaggregate, also referred to risk allocation. Further, in the background lies the
concept of risk diversification. In this section we will give a very brief overview
of some of the underlying mathematical issues. Given a general risk measure R
(examples treated so far include R = VaR, R = ES), for d risks X1 , . . . , Xd , we
define the coefficient of diversification as
Pd

Xj )
.
j=1 R(Xj )

R(

DR (X) = Pd

j=1

In the superadditive case, DR (X) may be larger than 1; an important question is
“by how much?”, and this in particular as a function of the various probabilistic
assumptions on the joint df H of X. Model Uncertainty in particular enters if we
only have partial information on the marginal dfs F1 , . . . , Fd and a/the copula C;
see our discussion in the previous section. The mathematical problem underlying
the calculation of DR (X) has a long history. Below we will content ourselves
with a very brief summary pointing the reader to some of the basic literature;
we will also present some examples aimed at understanding the range of values
for DR (X) if only information on F1 , . . . , Fd is available (the unconstrained
problem).
The reference on the topic, from a more mathematical point of view is
Rüschendorf (2013). As can be seen from the title, it contains all relevant topics
to be addressed in this section. It also contains an extensive list of historically
relevant references. We know that in the case of R = VaR, DR (X) > 1 typically
happens for the Fj ’s being very heavy tailed (e.g. independent and infinite
mean), very skew (see Example 2.3) or for the risks X1 , . . . , Xd exhibiting a
special dependence structure (even with F1 = · · · = Fd being N(0, 1)); McNeil
et al. (2005, Example 6.22). Interesting questions for practice now are: (Q1)
Given a specific model leading to superadditivity of DR (X), by how much can
DR (X) > 1, and (Q2) which dependence structures between X1 , . . . , Xd lead
to such extreme cases. Are these realistic from a practical point of view? And

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21

finally, (Q3) under what kind of dependence information between X1 , . . . , Xd
can such bounds be computed. Besides the above textbook reference, questions
(Q1)–(Q3) are discussed in detail in Embrechts et al. (2013). From the latter
paper we borrow the example below.

Example 4.1
Suppose Xj ? Pareto(2), j ? {1, . . . , d}, i.e. 1 ? Fj (x) = P(Xj > x) = (1 + x)?2 ,
P
x ? 0. For d = 8 and ? = 0.99, VaR? (Xj ) = 9 resulting in 8j=1 VaR? (Xj ) = 72,
the so-called comonotonic case for C; see McNeil et al. (2005, Proposition 6.15).
An exact upper bound on DR (X) can be given in this case, it equals about 2;
see Table 4 in Embrechts et al. (2013). The latter paper also contains lighter
tailed examples like log-normal and gamma; see Figure 4 in the latter paper.
Here the upper bound for DR (X) for ? = 0.99 is between 1.15 and 1.5. The
main lesson learned is that DR (X) can be considerably larger than 1 in the
unconstrained case. Constraining the rvs to e.g. positive quadrant dependence
does not change the bounds by much. For X being elliptically distributed, or
for R = ES, the upper bound is always 1! Note that for DR (X) > 1, we enter
P
the world of non-coherence, i.e., the risk of the overall portfolio position dj=1 Xj
is not covered, in a possibly conservative way, by the sum of the marginal
risk contributions. In finance parlance, people would say that in such a case,
“diversification does not work”.

For the above kind of problems, techniques from the realm of Operations
Research will no doubt turn out to be relevant. Key tools to look for are Robust
and Convex Optimization; see for instance Ben-Tal et al. (2009) and Boyd
and Vandenberghe (2004). Early work, stressing the link between Operations
Research and QRM, especially in the context of the calculation of ES is by Stan
Uryasev and R. Tyrell Rockafellar; see for instance Uryasev and Rockafellar
(2013) and Chun et al. (2012).

SUMMARY POINTS
In this final section, we only briefly entered into the field of Model Uncertainty. Going forward, and in the wake of the recent financial crisis, this
area of research will gain importance. Other areas of promising research,
which for reasons of space limitations we were not able to discuss, are High
Frequency (Inter Day) Data in finance, and Systemic Risk (Network Theory).
For the former, the classic “that started it all” is Dacorogna et al. (2001).
By now, the field is huge! A more recent edited volume is Viens et al. (2012).
For a start on some of the more mathematical work on Systemic Risk and
Network Theory, we suggest interested readers to start with papers on the
topic by Rama Cont (Imperial College, London) and Tom Hund (McMaster
University, Hamilton, Canada).

22

Paul Embrechts, Marius Hofert

Acknowledgements
We apologize in advance to all the investigators whose research could not be
appropriately cited owing to space limitations.

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23

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