Dynamic Risk Measurement, with an Application to a Pension Fund Setting

by Servaas van Bilsen 24.26.44

A thesis submitted in partial fulﬁllment of the requirements for the degree of Master in Quantitative Finance and Actuarial Sciences

Faculty of Economics and Business Administration Tilburg University

July, 2010

Supervised by: dr. R.J.A. Laeven prof. dr. J.M. Schumacher Abstract The financial crisis has highlighted that economic agents have not been successful in associating capital requirements with the risks undertaken by financial institutions. The connection between capital requirements and risk-taking has to be improved. This thesis focuses on the improvement of risk measurement. First, we provide a literature review on static and dynamic convex measures of risk. An explicit convex measure of risk is given by the entropic risk measure. We study this risk measure and its connection with stochastic differential equations. As an extension to the entropic risk measure, we also investigate a risk measure that allows for some degree of ambiguity in choosing a probabilistic model of some random variable. The literature has often focused on risk measurement for random variables. To take into account the dynamic fluctuation of intermediate cash flows, we develop an entropic risk measure for random processes. Finally, we apply some measures of risk to a pension fund setting.

Keywords: Static Measures of Risk, Dynamic Measures of Risk, Time-consistency, Entropic Risk Measure, Ambiguity, Random Pro- cesses, Pension Funds, Backward Stochastic Diﬀerential Equations

ii Contents

1 Introduction 1

2 Static Risk Measures 6 2.1 Monetary, Convex and Coherent Risk Measures ...... 6 2.2 Robust Representation Theorems ...... 9

3 Dynamic Risk Measures in Discrete Time 13 3.1 The Setup and Notation ...... 14 3.2 Risk Measures in a Conditional Framework ...... 15 3.3 Time-Consistency Properties of Risk Measures ...... 18 3.4 Conditional Risk Measures on L∞ ...... 21

4 Some Dynamic Measures of Risk 22 4.1 The Entropic Risk Measure: Deﬁnition and Properties . . . . 23 4.1.1 Modeling Ambiguity ...... 29 4.2 The Entropic Risk Measure for Random Processes ...... 31

5 A Numerical Exercise: A Pension Fund Setting 35 5.1 One Source of Uncertainty ...... 37 5.2 A More Realistic Model ...... 43

6 Final Remarks 49

7 Appendix 50 7.1 Proofs ...... 50 7.2 Solving BSDEs: A Numerical Approach ...... 58 7.2.1 One-dimensional Case ...... 59 7.2.2 Multi-dimensional Case ...... 60 7.2.3 A Transformation Method ...... 61 7.3 Figures ...... 62

iii List of Figures

1 Solvency Buﬀer: Current Situation vs. Proposals ...... 37 2 Expected Entropic Risk Measure Process - Funding level 100% ...... 62 3 Expected Entropic Risk Measure Process - Funding level 70% 62 4 Scenario Entropic Risk Measure Process ...... 63 5 Expected Risk Measure Process based on Quantile Function . 63 6 Expected Entropy Coherent Risk Measure Process (with Am- biguity) ...... 64 7 Expected Risk Measure Process - Entropic Risk Measure vs. Entropic Risk Measure for Random Processes ...... 64 8 Scenario - Entropic Risk Measure vs. Entropic Risk Measure for Random Processes ...... 65 9 Expected Entropic Risk Measure Process - Three Diﬀerent Pension Funds ...... 65 10 Correlation between Interest Rate Risk and Equity Risk . . . 66 11 Scenario - Entropic Risk Measure vs. Entropic Risk Measure for Random Processes ...... 66

iv 1 Introduction

The variability of some financial variable is traditionally regarded as an in- dicator of risk. Any measure of dispersion can then be viewed as a quantifi- cation of risk. Markowitz [54] was one of the first who adopted the standard deviation as a measure of risk. Although the standard deviation does not fully capture downside risk, it is still a widely used measure of risk in financial economics. The downside risk is, however, taken into account by other measures of risk such as Expected Shortfall or Value at Risk (in short, VaR). In situations where agents care more about downside losses, the downside risk measures are seemingly a major improvement over traditional risk measures. Some of those downside risk measures suffer, however, from being stable and do neither encourage diversification1 nor account for the size of extremely large losses. Furthermore, VaR is an important measure of risk which is part of the capital adequacy rules laid out in the Basel Accord. Especially inspired by the Basel Accords on Banking Supervision, Artzner et al. [4] [5] started an axiomatic analysis of measures of risk. They defined a measure of risk in the context of an acceptance set C. This set includes all the future net worths (i.e. monetary payoffs) that are ‘acceptable’. The measure of risk can then be interpreted as the smallest amount of money that would have to be added to the future net worth X ∈ X to make it acceptable. This naturally leads to an axiom of cash invariance (i.e. after adding some fixed amount of money to X, the amount of money needed to make X acceptable, is reduced by the same amount of money). The riskiness of X clearly depends on the choice of the acceptance set C. A risk measure of this type (i.e. a risk measure that satisfies cash invariance) often emerges in the context of a capital requirement to regulate a financial position. By way of illustration, consider the Financial Assessment Framework for Dutch pension funds. This regulatory framework specifies the acceptance set in such a way that the probability of underfunding (i.e. funding level2 below 100%) in one year from now should at all times be less than 2.5%. Dutch pension funds should keep extra capital in reserve such that the one-year ahead funding level is contained in the acceptance set. Given the definition of a risk measure in terms of an acceptance set C, Artzner et al. [4] [5] introduced the axiomatic class of coherent risk measures

1E.g. VaR is not compatible with subadditivity (diversiﬁcation) when losses are not elliptically distributed. 2The relative value of the pension fund assets and nominal liabilities, expressed as a percentage ﬁgure.

1 on a finite probability space (i.e. the sample space of the future net worth is supposed to be finite). A coherent risk measure satisfies four basic axioms. Furthermore, the authors argued that the riskiness of a future net worth X does not depend on a fixed probability P, but on X itself. In other words, model risk - the risk that the choice of the distribution of the future values of X is not correct - is taken into account. They concluded that any coherent measure of risk appears as given by a ‘worst case method’ in a framework of probability measures. Their ideas, and especially the robust representation theorems for coherent risk measures, trace back to Huber [45]. Delbaen [27] extended the axiomatic class of coherent risk measures to arbitrary probability spaces. Föllmer& Schied [37] and Frittelli & Rosazza Gianin [40] criticized the axiom of positive homogeneity as investigated in Artzner et al. [4] [5]. They removed this axiom and introduced the general axiomatic class of convex risk measures. Under some regularity conditions, convex duality can be used to deduce a robust representation theorem in terms of the convex conjugate. The convex conjugate corresponds with the Fenchel-Legendre transform on the dual space X 0 of X . In the context of risk measurement, the convex conjugate is called a penalty function. This function gives different probability measures varying impact in the robust representation theorem (i.e. some probability measures might be more plausible than others). In the aforementioned references, the central object of study is the single-period future net worth X. Static risk measurement arising from a stream of payoffs is analyzed in for instance Cheridito et al. [19] [20], Cvitanic & Karatzas [26] and Scandolo [64]. Section 2 provides a literature review on static measures of risk. In case of static risk measurement, the riskiness of X is only monitored at the beginning of the evaluation period. So, the risk assessment can not be adjusted during the evaluation period. A natural extension would then be to evaluate the riskiness of X at a finite collection of points in time. In fact, the information flow - which is modeled by a filtration - plays a crucial role. The riskiness of X is updated at each point in time when new information is gathered. These considerations give rise to the notion of a conditional risk measure, i.e. a risk measure that only depends on that part of the future that is not ruled out by new information. Detlefsen & Scandolo [30] studied the class of convex risk measures in a conditional framework. They introduced the notion of a dynamic convex risk measure, which is basically a family of successive conditional convex risk measures. A dynamic risk measure is very relevant for the financial industry. For instance, a dynamic risk measure can be used to determine the premium

2 for an insurance contract or to assess the riskiness of a dynamic financial policy (in the absence of complete markets). Risk measurement in a multiperiod framework is also studied - among many others - by Artzner et al. [6], Cheridito et al. [21], Cheridito & Kupper [22], Föllmer& Penner [39], Riedel [62], Roorda et al. [63] and Wang [68]. A dynamic risk measure should be consistent over time (i.e. what is preferred at one point in time should be consistent with what is preferred at another point in time). The notion of time-consistency can be understood in many ways. A risk measure is generally said to be strongly time-consistent if it can be either computed directly or in two steps backwards in time (i.e. recursively). However, many other (weaker) notions of time-consistency have been studied in the literature. The time-consistency property is often characterized in terms of conditions on a set of probability measures3, see for instance Epstein & Schneider [35] on a ‘rectangular’ set of probability measures and Artzner et al. [6] and Delbaen [28] on ‘multiplicative stability’. In addition, the object of study can be a random cash balance process defined over some time interval as in Cheridito et al. [21] and Jobert & Rogers [47] or a final payoff occurring at some terminal date T as in Detlefsen & Scandolo [30]. Section 3 will be devoted to dynamic risk measures in discrete time4. The existing literature on convex measures of risk is quite technical. Un- der a mild continuity assumption, a convex risk measure is in fact a worst expected loss with respect to a set of suitable penalized prior views (i.e. a convex risk measure arises as the ‘essential supremum’ of expected losses for some collection of suitable penalized probability measures, see for instance Delbaen [27]). It is, however, a challenge on its own to compute a measure of risk. To do so, one has to specify a penalty function and a set of den- sities that are equivalent (or at least absolutely continuous) to a reference 5 measure P. In Section 4.1, we consider a special convex risk measure. The reference measure P is here the one which is taken most seriously, and the penalty function is proportional to the deviation from a prior view to P, measured by the relative entropy. This special convex risk measure, generally called the entropic risk measure, is explicitly given in terms of an expectation under P. Here, the economic agent has to specify a parameter value. The interpretation of this parameter value is twofold. On the one hand, it is the absolute risk aversion coefficient. In this case, the agent’s risk

3also commonly referred to as probabilistic models or prior views 4Risk measures in continuous time have also been studied in the literature, see for instance Barrieu & El Karoui [8], Bion-Nadel [11] and Frittelli & Rosazza Gianin [41]. 5E.g. the agent’s beliefs to the probabilistic model of the payoﬀ X.

3 preferences are implicitly characterized by an exponential utility function. On the other hand, it measures the degree of distrust the agent puts in the reference measure P. The entropic risk measure is thus compatible with expected utility theory, which means that it appears as a negative certainty equivalent payoff in the expected utility framework when the agent’s risk preferences are represented by an exponential utility function. Moreover, the entropic risk measure is strongly time-consistent, well-defined in continuous time6 and emerges as a solution to a suitable backward stochastic differential equation7, in short BSDE. Within the expected utility framework, the economic agent fully trusts the reference measure P. In Section 4.1.1, we investigate a risk measure that allows for some degree of ambiguity in choosing a probabilistic model of X. The recursive multiple priors model of Epstein & Schneider [35] is here applied to the entropic risk measure. This ‘recursive multiple priors’ risk measure is called the dynamic entropy coherent risk measure and emerges as a solution to a BSDE (see Laeven & Stadje [52]). The agent now has to specify a risk aversion coefficient and an ambiguity aversion coefficient, which implies that the entropy coherent risk measure is more flexible than the entropic risk measure. In Section 4.2, we develop an entropic risk measure for random processes in order to take into account the dynamic fluctuation of intermediate payoffs. The theory of dynamic risk measurement on (càdlàg)processes is studied in for instance Cheridito et al. [21]. We provide an explicit risk measure for random processes, which can be computed by means of BSDEs. Also, the entropic risk measure for random processes is well-defined in continuous time and satisfies strong time-consistency. In Section 5, we go through an extensive numerical exercise and investigate some problems we encountered while executing this exercise. The stability of a financial institution (such as banks, insurance companies and pension funds) relies on the ability of economic agents to take the right hedging decision when involved in risky positions. This hedging decision is often compulsory. For instance, the Financial Assessment Framework for Dutch pension funds specifies a capital buffer (or surplus) to be held as a safety buffer. This capital buffer is risk-based, which means that it is a basis of support for the degree of risk associated with the institution’s investment policy. The financial crisis has, however, highlighted that regulators have not been successful in associating capital requirements with the risks undertaken by financial institutions. For instance, the pension fund regulator in The

6under the assumption of Brownian filtration 7a stochastic differential equation with a specified terminal condition

4 Netherlands requires that the pension fund surplus should be such that the probability of underfunding in one year from now is less than 2.5%. In fact, the surplus to be held as a safety buffer is based on a VaR measure. A VaR measure does generally neither support convexity (i.e. diversification) nor account for size of extremely large losses. Furthermore, VaR is not compatible with a notion of time-consistency as stressed by Cheridito & Stadje [23]. Weakly time-consistent convex measures of risk are, therefore, more suited for risk measurement. In case of ambiguity, the risk of the financial position X is not based on P (or a unique model specification), but on the financial position itself (or a multiple of model specifications). So, convex measures of risk take into account different probabilistic models of X. In Section 5, we do a numerical exercise in a pension fund setting. To this end, we create two stochastic models of a pension fund. The object of interest is the pension fund surplus ratio8, which is generally exposed to market and non-market risks. We focus on equity risk and interest rate risk9. The first specification is very stylized and models the pension fund surplus only. On the contrary, the second specification is more elaborate and explicitly describes the pension fund assets and liabilities. One can also go one step further by modeling each component of the assets (e.g. bonds, commodities, equities) and liabilities separately. The second model distinguishes itself from other standard pension fund models, because it takes into account the interest rate sensitivity of the pension fund assets and liabilities. The pension fund invests its wealth in zero-coupon bonds (and not in a risk-free deposit, which is commonly assumed) and equities. After we have specified the stochastic models of a pension fund, we apply the measures of risk as investigated in Section 4. The risk measures are numerically computed by means of BSDEs. In order to compute a BSDE, we have to estimate a conditional expectation at each time step. This estimation procedure can be carried out in several ways. We rely on kernel smoothing regression. Furthermore, the pension fund has to specify a set of parameter values. The risk measure might be very sensitive to some of these parameter values. Therefore, we perform a sensitivity analysis on the key model parameters. Some final remarks are given in Section 6.

8Total assets minus the sum of all liabilities divided by the sum of all liabilities; not to be confused with the surplus in absolute terms. 9We assume that currency, inﬂation and longevity risk are hedged, absent or reinsured.

5 2 Static Risk Measures

We start our literature review on static risk measures. In fact, every payoff is concentrated at a terminal date T and we are interested in quantifying its risk today. Static risk measurement arising from a stream of payoffs is analyzed in for instance Cheridito et al. [19] [20], Cvitanic & Karatzas [26] and Scandolo [64]. In Section 2.1, we give axiomatic characterizations of risk measures. Risk assessment can be viewed in the context of an acceptance set of acceptable future net worths (e.g. financial positions). In Section 2.2, we present robust representation theorems for static risk measures. Most of the proofs are gathered in Appendix 7.1.

2.1 Monetary, Convex and Coherent Risk Measures We present axiomatic characterizations of static risk measures. Our starting point is the following: given is some collection of mappings X :Ω → R of one-period discounted future net worths, where Ω denotes the sample space of X.A static risk measure assigns a real number ρ(X) to X ∈ X , where X is a linear space of bounded functions endowed with the supremum norm. So, ρ : X → R is a real-valued function on X . If ρ(X) satisfies certain basic properties, then we can define some axiomatic classes of risk measures. The axiomatizations we present here are mainly based on the works of Artzner et al. [4] [5], Föllmer & Schied [37] [38] and Frittelli & Rosazza Gianin [40]. The definitions of monetary, convex and coherent risk measures will be given below.

Deﬁnition 2.1 A family of mappings ρ : X → R is called a monetary risk measure if it satisﬁes the following two axioms10 for all X,Y ∈ X . 1. Monotonicity: If X ≤ Y , then ρ(X) ≥ ρ(Y ).

11 2. Cash invariance : If m ∈ R, then ρ(X + m) = ρ(X) − m.

The ﬁrst axiom has a natural interpretation. By monotonicity, the risk of some future net worth Y - which is at least as large as X for all ω ∈ Ω - can never be larger than the risk of X. The second axiom is under debate. By cash invariance, an addition of some ﬁxed amount of money m to X, reduces the risk of X + m by the same

10In some situations, it would be convenient to insist on the additional axiom of zero- normalization, i.e. ρ(0) = 0. 11also called translation invariance

6 amount of money m. This axiom simplifies the mathematics, but its economic rationale is less trivial. This can be illustrated by the following experiment. Consider a payoff structure Ω0 = {0, 1}, where each outcome is equally likely. Suppose that the agent’s risk preferences are characterized X1−γ CE by a CRRA utility function, i.e. u(X) = 1−γ . Take ρ(X) = −X , where CE 12 1 CE X is a certainty equivalent payoff . For γ = 2 , we find X = 0.25 (i.e. the agent is equally well-off by accepting a fixed amount of money of 0.25). Cash invariance implies that ρ(X + ρ(X)) = 0. However, the risk of the payoff structure Ω = {0 + ρ(X), 1 + ρ(X)} = {−0.25, 0.75} is not considered to be zero. So, the notion of cash invariance does not always seem to be compatible with von Neumann-Morgenstern expected utility theory (von Neumann & Morgenstern [56]). For a risk measure that satisfies the principle of equivalent utility, Goovaerts et al. [44] concluded that such a risk measure is additive for independent risks if and only if u is an exponential or linear utility function. If a monetary risk measure is convex, then it belongs to the axiomatic class of convex risk measures.

Deﬁnition 2.2 A family of mappings ρ : X → R is called a convex risk measure if it satisﬁes the following axioms for all X,Y ∈ X and all λ ∈ [0, 1].

1. The axioms of Deﬁnition 2.1.

2. Convexity: ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ).

The axiom of convexity implies that the risk of an allocation of resources between X and Y is not larger than the sum of the risks of the individual components. So, merging is never penalized.

Deﬁnition 2.3 A family of mappings ρ : X → R is called a coherent risk measure if it satisﬁes the following axioms for all λ ≥ 0.

1. The axioms of Deﬁnition 2.2.

2. Positive Homogeneity: ρ(λX) = λρ(X).

Please note that positive homogeneity combined with convexity implies subadditivity.

12 CE CE X is determined in such a way that u(X ) = E[u(X)].

7 Proof. ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) = ρ(λX) + ρ((1 − λ)Y ) ⇒ ρ(X + Y ) ≤ ρ(X) + ρ(Y ) The axiomatic class of coherent risk measures was introduced in Artzner et al. [5]. However, F¨ollmer& Schied [37] and Frittelli & Rosazza Gianin [40] argued that the axiom of positive homogeneity is quite unnatural. There is no economic reason why the risk of λX should be the same as the risk of X times λ. In particular, positive homogeneity is not compatible with the notion of liquidity risk. Consider the following situation. A large risk λX(λ > 1) cannot be sold due to lack of liquidity in the marketplace. As a consequence, this risk would be far more riskier than what would be implied by positive homogeneity (i.e. ρ(λX) > λρ(X)). Therefore, they replaced positive homogeneity by convexity and introduced the general axiomatic class of convex risk measures. The risk measure ρ(X) is often associated with a given set C of ‘acceptable’ future net worths. ρ(X) is then the smallest amount of money that would have to be added to X to make it acceptable. Formally,

ρC(X) = inf {m ∈ R|m + X ∈ C} . (2.1) In (2.1), the starting point is a given set C of ‘acceptable’ future net worths. It follows that, if X is acceptable, then ρ(X) is non-positive. However, it could be that X is acceptable and Y ≥ X is not acceptable, since Y is not in the given set C of ‘acceptable’ future net worths. To avoid such an undesirable feature, it is natural to consider the set Cρ ⊇ C of acceptable future net worths in terms of the monetary risk measure ρ(X). In particular,

Cρ = {X ∈ X |ρ(X) ≤ 0} . (2.2) The following two propositions summarize the relation between risk measures and their acceptance sets.

Proposition 2.1 Assume that ρ is a monetary risk measure with acceptance set Cρ.

1. Cρ is non-empty and satisﬁes the following conditions:

inf {m ∈ R|m ∈ Cρ} > −∞. (2.3)

X ∈ Cρ,Y ∈ X ,Y ≥ X ⇒ Y ∈ Cρ. (2.4)

For X ∈ Cρ and Y ∈ X ,

{λ ∈ [0, 1]|λX + (1 − λ)Y ∈ Cρ} is closed in [0, 1]. (2.5)

8 2. ρ is a convex risk measure if and only if Cρ is convex.

3. ρ is positively homogeneous if and only if Cρ is a cone. In addition, ρ is coherent if and only if Cρ is a convex cone.

Proof. See Appendix 7.1.

Proposition 2.2 Assume that C ⊆ X \ {∅} which satisﬁes properties (2.3) and (2.4). Then the functional ρC has the following features:

1. ρC is a monetary risk measure.

2. If C is a convex set, then ρC is a convex risk measure.

3. If C is a cone, then ρC is positively homogeneous. In addition, ρC is a coherent risk measure if C is a convex cone.

4. If C satisﬁes property (2.5), then C = Cρ.

Proof. See Appendix 7.1.

2.2 Robust Representation Theorems One of the appealing features of a convex measure of risk is its robustness against model uncertainty. Under a mild continuity assumption, a risk measure can be represented as a worst expected loss with respect to a given set of probability measures, see for instance Artzner et al. [5] and Delbaen [27]. This robust representation theorem traces back to Huber [45]. A nice interpretation of this theorem is demonstrated by the Ellsberg Paradox (Ellsberg [34]). There are two urns each containing one hundred balls. The balls are either red or black. Urn one contains fifty red balls and fifty black balls. There is no information regarding urn two. One ball is drawn at random from each urn. Most agents are indifferent between a bet on the event that the ball drawn from urn one is either red or black. The same holds for urn two. There are, however, many agents who prefer every bet from urn one to every bet from urn two (i.e. many agents prefer known risks over unknown risks). This type of ambiguity is excluded by (subjective) expected utility theory. So we have a paradox on our hands. The agent probably forms a set of prior views and evaluates the risk of a bet from urn two according to the worst case scenario.

9 A robust representation form for convex risk measures was presented in Föllmer& Schied [37] and Frittelli & Rosazza Gianin [40]. Convexity is used here to deduce a dual characterization in terms of probability measures and minimal penalty functions. The original ideas date back to that of convex premium principles of Deprez & Gerber [29]. Denote by X the class of all bounded measurable functions on the measur- 1 able space (Ω, F). Mf (Ω, F) is the class of all finitely additive set functions such that Q(Ω) = 1 (see Definition 2.4 below). Further, M1(Ω, F) is the 1 set of σ-additive members of Mf (Ω, F), that is, the set of all probability measures on (Ω, F).

Definition 2.4 Let (Ω, F) be a measurable space. A collection of mappings µ : F → R is called a finitely additive set function if µ(∅) = 0 and if for any finite collection A1, ..., An ∈ F of mutually disjoint sets: n n [ X µ( Ai) = µ(Ai), i=1 i=1 with total variation ( ) n X ||µ||var = sup |µ(Ai)| A1, ..., An ∈ F,Ai ∩ Aj = ∅ ∀ i 6= j . i=1 1 Mf (Ω, F) is the set of all finitely additive set functions µ : F → [0, 1] which are normalized to µ(Ω) = 1. The set of all finitely additive set functions whose total variation ||µ||var is finite is represented by ba(Ω, F). Clearly, we 1 have that Mf (Ω, F) ⊂ ba(Ω, F).

The robust representation form for convex risk measures is summarized in the following theorem.

Theorem 2.1 Let X ∈ X . Any convex risk measure ρ on X admits a robust representation of the form

ρ(X) = sup (EQ[−X] − αmin(Q)), (2.6) 1 Q∈Mf where the minimal penalty function αmin(Q) is given by 1 αmin(Q) := sup EQ[−Y ] for Q ∈ Mf . Y ∈Cρ

1 In fact, αmin : Mf → (−∞, ∞], which gives diﬀerent probability measures varying impact in the robust representation result (2.6).

10 Proof. See Appendix 7.1.

Remark 2.1 1. The robust representation form (2.6) is characterized in terms of ﬁnitely additive set functions. If this robust representation form would be in terms of M1 (a.k.a. the class of all probability measures), then we would have to assume additional continuity properties, see F¨ollmer & Schied [38].

13 2. According to (2.6), αmin(Q) can also be represented by

αmin(Q) = sup (EQ[−X] − ρ(X)), X∈X which corresponds with the Fenchel-Legendre transform of the convex function ρ on X . In fact, αmin(Q) is a family of mappings from the 0 dual X of X to R ∪ {∞}. 3. Any penalty function α(Q) that can be represented by

ρ(X) = sup (EQ[−X] − α(Q)), 1 Q∈Mf

deﬁnes a convex measure of risk and satisﬁes α(Q) ≥ αmin(Q) for all 1 Q ∈ Mf . In particular,

α(Q) = sup (EQ[−X] − ρ(X)) 1 Q∈Mf

≥ sup (EQ[−X] − ρ(X)) 1 Q∈Mf |X∈Cρ

≥ sup EQ[−X] 1 Q∈Mf |X∈Cρ

:= αmin(Q).

4. The ‘supremum’ operator in (2.6) captures uncertainty or ambiguity aversion.

The following theorem states that the robust representation form for coherent risk measures is a special case of the robust representation form for convex risk measures. 13The regularity conditions are fulﬁlled, i.e. X is a locally convex vector space and ρ is lower semi-continuous with respect to the weak* topology σ(X , X 0). Theorem 2.2 The minimal penalty function of a coherent risk measure takes only the values 0 and ∞. In particular, ρ can now be represented as follows: ρ(X) = sup EQ[−X], (2.7) Q∈Qmax

n 1 o where Qmax := Q ∈ Mf αmin(Q) = 0 is the largest convex set for which robust representation result (2.7) holds.

Proof. According to Proposition 2.1.3, Cρ is a convex cone if ρ(X) is a coherent risk measure. In other words, if X ∈ X , then also λX ∈ X for every λ ≥ 0. It follows that αmin(Q) = λαmin(Q). Hence, αmin(Q) = 0 or αmin(Q) = ∞. The latter case is excluded, because −∞ < ρ(X) < ∞. In sum, coherent risk measures can be represented by (2.7). We now fix a probability measure P on the measurable space (Ω, F). We can identify X with the set of equivalence classes of measurable functions ∞ 14 1 L (Ω, F, P) . In fact, any measurable function Q ∈ Mf (Ω, F) should be 15 absolutely continuous with respect to P (i.e. Q P). Further, denote 1 by M (Ω, F, P) the set of all probability measures which are absolutely continuous with respect to P. The following theorem characterizes convex ∞ risk measures on the Banach space L (Ω, F, P). In contrast with Theorem 2.1, a probability measure P is specified a priori. Throughout, equalities and inequalities between random variables are always understood in a P-almost sure sense and the ‘supremum’ a of f(x) is defined in such a way that the set {x|f(x) > a} is contained in a set of measure zero and f(x) ≤ a for almost all x (the notion of ‘infimum’ is understood in a similar way).

∞ Theorem 2.3 Let ρ : L (Ω, F, P) → R be a convex risk measure. Then the following assertions are equivalent. 1 1. ρ can be represented by some penalty function on M (Ω, F, P). 2. ρ can be represented by the restriction of the minimal penalty function 1 αmin(Q) to M (Ω, F, P).

ρ(X) = sup (EQ[−X] − αmin(Q)). 1 Q∈M (Ω,F,P)

14 0 The study of risk measures on the set L (Ω, F, P) can be found in Delbaen [27] and Cheridito et al. [20]. 15 If they are not absolutely continuous, then αmin(Q) = ∞. To see this, take some A ∈ F such that Q[A] > 0 and P[A] = 0. For any X ∈ Cρ, X − nIA is also contained in Cρ, because P[A] = 0. However, αmin(Q) ≥ EQ[−X + nIA] = EQ[−X] + nEQ[IA] → ∞ as n ↑ ∞. This case is excluded, because ρ(X) is a real-valued function. 3. ρ is continuous from above. If Xn & X P-almost surely, then ρ(Xn) % ρ(X).

4. ρ has the Fatou property. For any bounded sequence (Xn)n∈N which converges P-almost surely to some X:

ρ(X) ≤ lim inf ρ(Xn). n↑∞

5. ρ is lower semi-continuous for the weak* topology16 σ(L∞,L1).

∞ 6. The acceptance set Cρ of ρ is weak* closed in L , i.e. Cρ is closed with respect to the topology σ(L∞,L1).

Proof. See Appendix 7.1.

3 Dynamic Risk Measures in Discrete Time

As we remarked in Section 2, the riskiness of X is here only monitored at the beginning of the evaluation period. Suppose now that the evaluation period runs from S to T ≥ S, where S ∈ N and T ∈ N ∪ {∞}. In this section, we evaluate the riskiness of X at a finite collection of points in time. The riskiness of X is updated at each point in time when new information is collected. We describe the underlying information flow by a filtration. A dynamic risk measure is basically a sequence of conditional risk measures adapted to the underlying filtration. This implies that if we know that an event is prevailing, then the risk monitoring of X should only depend on that part of the future that is not excluded by this new information. So, the original list of axioms needs to include a regularity condition. In addition, the input of analysis can be discounted random cash balance processes defined over some time interval or a final discounted payoff occurring at some terminal date T . For the latter case, see for instance Detlefsen & Scandolo [30]. We will mainly focus on discounted random cash balance processes. More precisely, we consider bounded discrete-time processes and we try to measure their risk in a dynamic consistent way. Time-consistency can be formalized in many ways. One of the most used formalizations is strong

16We can consider the space L1 as a set of linear functionals on the dual space L∞ by letting x(l) := l(x) for l ∈ L∞ and x ∈ L1. The L1-topology obtained in this way is called the weak* topology on L∞. If L∞ is endowed with the weak* topology σ(L∞,L1), then L1 is the dual of L∞. time-consistency, which corresponds with a dynamic programming principle. There were several notions and characterizations of time-consistency introduced and studied in the literature. The characterization is often done in terms of conditions on a set of probability measures. Epstein & Schnei- der [35] represented the consistency property in terms of a ‘rectangular’ set of probability measures (i.e. they based their work on the atemporal multiple-priors model of Gilboa & Schmeidler [43]). Artzner et al. [6] spoke of ‘multiplicative stability’, Delbaen [28] made use of a condition called ‘fork convexity’ (a term introduced by Zitkovic [70]), Roorda et al. [63] used the term ‘product property’, and Riedel [62] used a different kind of cash invariance property. Föllmer& Penner [39] gave a characterization in terms of a ‘supermartingale property’ on the penalty function process. Our characterization is based on the work of Cheridito et al. [21]. They characterized strong time-consistency by a ‘concatenation condition’ on adapted increasing processes and by additivity of acceptance sets. Dynamic risk measurement, in a discrete time setting, is extensively studied in the literature, see for instance Artzner et al. [6], Cheridito et al. [21], Cheridito & Kupper [22], Föllmer& Penner [39], Riedel [62] and Roorda et al. [63]. The plan of this section is as follows. In Section 3.1, we introduce the setup and some notation. The primary objects of interest are bounded discrete-time processes. In Section 3.2, we investigate risk measures in a conditional framework. Time-consistency properties of risk measures are studied in Section 3.3. Finally, we give some brief review of conditional risk measures on the space of bounded random variables. Some of the proofs have been collected in Appendix 7.1.

3.1 The Setup and Notation

Denote by (Ω, F, (Ft)t∈N, P) a fixed, filtered probability space which is supposed to satisfy the usual assumptions. More specifically, the probability 17 space (Ω, F, P) is complete , the σ-algebras (Ft)t∈N contain all the sets in 18 F of zero probability and the filtration (Ft)t∈N is right continuous . 0 We call R the space of all adapted (or optional) stochastic processes (Xt)t∈N on a filtered probability space. The space of all bounded adapted stochastic processes R∞ is given by ∞ 0 R := X ∈ R ||X||R∞ < ∞ ,

17The probability space is said to be complete if every subset of a set of measure 0 is measurable and has - as a consequence - measure 0 18 T for any t ∈ N, the σ-algebra Ft+ = s>t Fs is equal to Ft where ||X|| ∞ = inf m ∈ sup |X | < m . R R t∈N t 1 1 Furthermore, we deﬁne the spaces A and A+. In particular,

1 0 1 1 A := a ∈ R ||a||A1 < ∞ and A+ := a ∈ A ∆at ≥ 0 ∀t ∈ N , P where ||a|| 1 := [ |∆at|], ∆at := at − at−1 and a−1 := 0. A E t∈N Deﬁne also the bilinear form on R∞ × A1, i.e.: X hX, ai := E[ Xt∆at]. (3.1) t∈N

In a conditional framework, the riskiness of X ∈ R∞ is evaluated at a fixed stopping time τ (0 ≤ τ < ∞) given information available at time τ. For a fixed time period θ − τ (τ ≤ θ < ∞), we define the projection ∞ 1 πτ,θ(X)t := 1{τ≤t}Xt∧θ. Accordingly, the bilinear form (3.1), R , A and 1 A+ can be written in a conditional setting. In particular, X hX, aiτ,θ := E[ Xt∆at|Fτ ], t∈[τ,θ]∩N ∞ ∞ Rτ,θ := πτ,θR , 1 1 Aτ,θ := πτ,θA , 1 1 (Aτ,θ)+ := πτ,θA+. ∞ ∞ A real-valued map ρ : Rτ,θ → L (Fτ ) is called a conditional risk measure. 1 Finally, we introduce the set Qτ,θ ⊂ (Aτ,θ)+ of density processes. It is defined as follows:

n 1 o Qτ,θ := a ∈ (Aτ,θ)+ h1, aiτ,θ = 1 .

3.2 Risk Measures in a Conditional Framework We formalize the theory of conditional risk measures. The deﬁnitions and robust representation theorems are, in fact, a generalization of the results presented in Section 2. For the sake of completeness, we give the deﬁnitions of conditional monetary, convex and coherent risk measures below.

Throughout, τ and θ denote two fixed (Fτ )-stopping times such that 0 ≤ τ < ∞ and τ ≤ θ < ∞. ∞ ∞ Definition 3.1 A family of mappings ρ : Rτ,θ → L (Fτ ) is called a conditional monetary risk measure if it satisfies the following three axioms for ∞ all X,Y ∈ Rτ,θ.

1. Fτ -Regularity condition: ρτ,θ(1AX) = 1Aρτ,θ(X) (A ∈ Fτ ).

2. Monotonicity: If X ≤ Y , then ρτ,θ(X) ≥ ρτ,θ(Y ).

∞ 3. Fτ -Cash invariance: If m ∈ L (Fτ ), then ρτ,θ(X +m) = ρτ,θ(X)−m.

The regularity condition implies that if A is (Fτ )-measurable, then the agent should know at time τ if A has happened and adjust the risk evaluation accordingly. The risk assessment only depends on that part of the future that is not ruled out by new information. Detlefsen & Scandolo [30] showed that this condition is equivalent to ρτ,θ(0) = 0.

∞ ∞ Definition 3.2 A family of mappings ρ : Rτ,θ → L (Fτ ) is called a conditional convex risk measure if it satisfies the following axioms for all X,Y ∈ ∞ ∞ Rτ,θ and all λ ∈ L (Fτ ) such that 0 ≤ λ ≤ 1. 1. The axioms of Definition 3.1.

2. Fτ -Convexity: ρτ,θ(λX + (1 − λ)Y ) ≤ λρτ,θ(X) + (1 − λ)ρτ,θ(Y ).

∞ ∞ Definition 3.3 A family of mappings ρ : Rτ,θ → L (Fτ ) is called a conditional coherent risk measure if it satisfies the following axioms for all ∞ λ ∈ L (Fτ ) such that λ ≥ 0. 1. The axioms of Definition 3.2.

2. Fτ -Positive Homogeneity: ρτ,θ(λX) = λρτ,θ(X).

To sum up, a conditional risk measure is a real-valued map assigning to ∞ every X ∈ Rτ,θ a (Fτ )-measurable function ρτ,θ(X), where the additional information is thus described by a sub-σ-algebra Fτ of the total information F.

In a conditional setting, we can deﬁne an acceptance set Cρ of all bounded adapted stochastic processes that are ‘acceptable’. In order to be ‘acceptable’, X should feature a non-positive conditional risk measure. Formally,

∞ Cρ = X ∈ Rτ,θ ρτ,θ(X) ≤ 0 . (3.2) By construction, X + ρτ,θ(X) always belongs to Cρ. One can also deﬁne C ρτ,θ(X) in terms of a given set C of ‘acceptable’ future net worths.

C ∞ ρτ,θ(X) = inf f ∈ L (Fτ ) X + f1[τ,θ] ∈ C . (3.3)

Comparable to the propositions as presented in Section 2, it is possible to derive propositions in a conditional setting. We refer the reader to for instance Cheridito et al. [21] and Detlefsen & Scandolo [30] for detailed information regarding conditional risk measures.

Deﬁnition 3.4 A penalty function α on Qτ,θ is a family of mappings from the space of measurable functions f: (Ω, Fτ ) → [0, ∞] with the following features:

1. infa∈Qτ,θ ατ,θ(a) = 0.

2. ατ,θ(1Aa + 1Ac b) = 1Aατ,θ(a) + 1Ac ατ,θ(b) for every a, b ∈ Qτ,θ and A ∈ Fτ .

It is worthwhile to present the following theorem for conditional convex risk measures.

Theorem 3.1 (Cheridito et al. [21]) . The following assertions are equivalent.

∞ 1. ρ is a family of mappings deﬁned on Rτ,θ that can be represented by n o ρτ,θ(X) = sup h−X, aiτ,θ − (αmin(a))τ,θ , (3.4) a∈Qτ,θ

where the minimal penalty function (αmin(a))τ,θ is given by

(αmin(a))τ,θ := sup h−Y, aiτ,θ for a ∈ Qτ,θ. Y ∈Cρ

∞ ∞ 1 2. ρ is a convex risk measure on Rτ,θ whose acceptance set Cρ is σ(R , A )- closed.

∞ 3. ρ is a convex risk measure on Rτ,θ that is continuous for bounded decreasing sequences (“continuous from above”).

Proof. See Appendix 7.1. Remark 3.1 Any penalty function ατ,θ(a) that can be represented through n o ρ (X) = sup h−X, ai − α (a) satisﬁes α (a) ≥ (α (a)) . τ,θ a∈Qτ,θ τ,θ τ,θ τ,θ min τ,θ

A conditional convex monetary risk measure can thus be written as a worst conditional expected loss - corrected with some random penalty function - with respect to a given set of probability measures. In Section 3.3, we will study time-consistency properties of conditional monetary risk measures. Basically, risk assessments at diﬀerent time instances should be related in some way. Therefore, we investigate the notion of a strongly time-consistent conditional monetary risk measure.

3.3 Time-Consistency Properties of Risk Measures We study time-consistency properties of risk measures. It should be noticed that time-consistency can be characterized in many ways. We follow the approach of Cheridito et al. [21]. To this end, we deﬁne a conditional mone- t=T t=T tary risk measure process (ρt,T )t=S and an acceptance set process (Ct,T )t=S . We give the following deﬁnition of strong time-consistency.

Deﬁnition 3.5 (dynamic programming principle) . A conditional monetary risk measure process is said to be strongly time- ∞ consistent if for all processes X ∈ Rt,T and all t = S, ..., T − 1:

ρt,T (X) = ρt,T (X1[t,θ) − ρθ,T (X)1[θ,∞)), (3.5) where θ is a ﬁxed (Ft)-stopping time.

(3.5) has an natural interpretation. A conditional monetary risk measure process is said to be strongly time-consistent if it assigns the same value to the process irrespective of whether it is calculated directly or in two steps backwards in time. In addition, Cheridito et al. [21] showed that this deﬁnition is equivalent to the condition Ct,T = Ct,θ + Cθ,T . So, strong time- consistency can also be characterized in terms of the acceptance set process t=T (Ct,T )t=S . The following proposition is an immediate consequence of (3.5).

t=T Proposition 3.1 Any conditional monetary risk measure process (ρt,T )t=S that satisﬁes the recursive relation:

ρt,T (X) = ρt,T (X1{t} − ρt+1,T (X)1[t+1,∞)), (3.6) is strongly time-consistent for all t = S, ..., T − 1. Proof. The proof is based on the principle of (backward) induction. Let ∞ X ∈ Rt,T and Y := X1[t,θ)−ρθ,T (X)1[θ,∞). If t = T , then trivially ρt,T (X) = ρt,T (Y ). Assume now that ρt+1,T (X) = ρt+1,T (X1[t+1,ξ) − ρξ,T (X)1[ξ,∞)), where ξ is a ﬁxed (Ft)-stopping time. By assumption, cash invariance and (3.6), we have that

ρt,T (Y ) = ρt,T (−1{θ=t}ρt,T (X)1[t,∞) + 1{θ≥t+1}Y )

= 1{θ=t}ρt,T (X) + 1{θ≥t+1}ρt,T (Y )

= 1{θ=t}ρt,T (X) + 1{θ≥t+1}ρt,T (Y 1{t} − ρt+1,T (Y )1[t+1,∞))

= 1{θ=t}ρt,T (X) + 1{θ≥t+1}ρt,T (X1{t} − ρt+1,T (X)1[t+1,∞))

= ρt,T (X).

We will provide necessary and sufficient conditions for strong time-consistency in terms of the representation functionals. To this end, we define a concatenation operation for adapted increasing processes. The main results are finally summarized in two theorems.

1 θ Deﬁnition 3.6 Let a, b ∈ A+,A ∈ Fθ. Then the concatenation a ⊗A b is deﬁned as follows:

 c n o at on {t < θ} ∪ A ∪ h1, bi = 0 θ  θ,∞ (a⊗ b)t := . A h1,aiθ,∞ n o  aθ−1 + (bt − bθ−1) on {t ≥ θ} ∩ A ∩ h1, bi > 0 h1,biθ,∞ θ,∞

+ θ A subset D of A1 is said to be stable under concatenation if a ⊗A b ∈ D for all a, b ∈ D, every (Ft)-stopping time θ, and A ∈ Fθ. The concepts of multiplicative stability of probability measures, stability under concatenation and strong time-consistency are closely related, see Cheridito et al. [21]. The following two theorems provide necessary and suﬃcient conditions for strong time-consistency.

Theorem 3.2 (Cheridito et al. [21]) . t=T Let T ∈ N ∪ {∞} and (ρt,T )t=S a strong time-consistent conditional convex ∞ monetary risk measure process that can be represented by (X ∈ Rt,T ) n o ρt,T (X) = sup h−X, ait,T − (αmin(a))t,T , (3.7) a∈Qt,T for all t = S, ..., T . Then it follows:

θ (αmin(a))τ,T = inf (αmin(a ⊗Ω b))τ,T + E[(αmin(a))θ,T |Fτ ], (3.8) b∈Qθ,T for every pair of (Ft)-stopping times τ and θ. Proof. By deﬁnition, (α (a)) = sup h−Y, ai . The reasoning min τ,T Y ∈Cτ,T τ,T goes now as follows:

(αmin(a))τ,T = sup h−Y, aiτ,T + sup h−Y, aiτ,T Y ∈Cτ,θ Y ∈Cθ,T

= sup h−Y, aiτ,T + E[ sup h−Y, aiτ,T |Fτ ] Y ∈Cτ,θ Y ∈Cθ,T

= sup h−Y, aiτ,T + E[(αmin(a))θ,T |Fτ ]. (3.9) Y ∈Cτ,θ

We also have for all a ∈ Qτ,T and b ∈ Qτ,T :

θ D θ E (αmin(a ⊗Ω b))τ,T = sup −Y, a ⊗Ω b Y ∈Cτ,T τ,T D θ E D θ E = sup −Y, a ⊗Ω b + sup −Y, a ⊗Ω b Y ∈Cτ,θ τ,T Y ∈Cθ,T τ,T

= sup h−Y, aiτ,T + E[ sup h−Y, biθ,T h1, aiθ,T |Fτ ] Y ∈Cτ,θ Y ∈Cθ,T

= sup h−Y, aiτ,T + E[(αmin(b))θ,T h1, aiθ,T |Fτ ]. Y ∈Cτ,θ

Robust representation result (3.7) implies that ρθ,T (X) can be characterized as follows: n o ρθ,T (X) = sup h−X, aiθ,T − (αmin(a))θ,T . a∈Qθ,T

By the regularity condition, infb∈Qθ,T (αmin(b))θ,T = 0. This yields

θ inf (αmin(a ⊗Ω b))τ,T = sup h−Y, aiτ,T . (3.10) b∈Q θ,T Y ∈Cτ,θ

(3.10) combined with (3.9) proves (3.8). Theorem 3.3 (Cheridito et al. [21]) . t=T Let T ∈ N ∪ {∞} and (ρt,T )t=S a conditional convex monetary risk measure ∞ process that can be represented by (X ∈ Rt,T ) n o ρt,T (X) = sup h−X, ait,T − αt,T (a) , a∈Qt,T for all t = S, ..., T . t=T If at least one of the following two conditions is satisﬁed, then (ρt,T )t=S is strongly time-consistent. The conditions are (for all t = S, ..., T − 1 and all a ∈ Qt,T ): θ ext αt,T (a) = inf αt,T (a ⊗Ω b) + E[αθ,T (a)|Ft]. (3.11) b∈Qθ,T t+1 ext αt,T (a) = inf αt,T (a ⊗Ω b) + E[αt+1,T (a)|Ft]. (3.12) b∈Qt+1,T

ext The deﬁnition of αθ,T (·) is given below.  n o 0 on h1, ai = 0 ext  θ,T αθ,T (a) := → n o ,  h1, aiθ,T αθ,T (aθ) on h1, aiθ,T > 0 where  n o →  1[θ,∞) on h1, aiθ,T = 0 a := . θ a n o  1[θ,∞) on h1, ai > 0 h1,aiθ,T θ,T Proof. See Cheridito et al. [21].

3.4 Conditional Risk Measures on L∞ This section brieﬂy reviews the theory of conditional risk measures on L∞. ∞ Let Y ∈ Rt,T and X := 0[t,T ) + Y 1{T }. In fact, X can be regarded as an ∞ ∞ element of L (FT ). A conditional risk measure ρt,T (X) on L (FT ) is a ∞ ∞ real-valued map ρ : L (FT ) → L (Ft). 1 We introduce a set of probability measures St := Q ∈ M Q = P on Ft which are absolutely continuous with respect to P and are identical to P on the sub-σ-algebra Ft.

Under a mild continuity assumption, ρt,T (X) admits a robust representation of the following form, see for instance Detlefsen & Scandolo [30]:

ρt,T (X) = sup {EQ[−X|Ft] − (αmin(Q))t,T } , Q∈St where the minimal penalty function is given by

(αmin(Q))t,T := sup EQ[−Y |Ft] for Q ∈ St. Y ∈Cρ

We give the following definition of strong time-consistency. Definition 3.7 A set of probability measures K is called strongly time- ∞ consistent if for any pair of payoffs X,Y ∈ L (FT ), we have that ρt+1,T (X) ≤ ρt+1,T (Y ) implies ρt,T (X) ≤ ρt,T (Y ) for all t = S, ..., T − 1. This definition induces another characterization of time-consistency, i.e. if ρt+1,T (X) = ρt+1,T (Y ), then ρt,T (X) = ρt,T (Y ). A strongly time-consistent conditional monetary risk measure can then be represented in terms of the following recursive relation:

ρt,T (X) = ρt,T (−ρt+1,T (X)), (3.13) for all t = S, ..., T − 1.

Proof. By cash invariance, ρt+1,T (−ρt+1,T (X)) = ρt+1,T (X). (3.13) follows now by strong time-consistency. t=T Delbaen [28] proved that strong time-consistency of (ρt,T )t=S is equivalent to multiplicative stability of its set of probability measures. The concept of multiplicative stability is deﬁned below.

Definition 3.8 We say that a set of probability measures K is multiplica- tively stable if for elements Q0 ∈ K, Q ∈ Ke (Ke consists of the members 0 dQ0 of K that are equivalent to ) with associate martingales Z = [ |Ft] P t E dP dQ and Zt = E[ d |Ft], and for each stopping time τ, the element L defined as P 0 L = Z0 for t ≤ τ and L = Zτ Zt for t ≥ τ is a martingale that defines an t t t Zτ element in K.

4 Some Dynamic Measures of Risk

This section studies three different, but closely related, risk measures. We start our examination with a classical risk measure: the entropic risk measure. The entropic risk measure is one of the most appealing convex risk measures due to its explicit characterization. This measure is investigated in Section 4.1. If the agent’s preferences are represented by an exponential utility function, then the entropic risk measure emerges as a negative certainty equivalent payoff. In this setting, the agent has a unique prior probability distribution and a utility function such that decisions are made so as to maximize expected utility. However, empirical evidence shows that expected utility theory violates certain consistency conditions (as motivated by the Ellsberg Paradox). Moreover, agents seem to have problems with specifying prior probability distributions for actual statistical inference (i.e. preferences are not probabilistically sophisticated). The literature has introduced different generalizations of (subjective) expected utility theory. Schmeidler [65] [66] introduced nonadditive probabilities and Gilboa & Schmeidler [43] investigated multiple priors. In Section 4.1.1, we apply the latter (in a dynamic sense) to the entropic risk measure. The agent considers here many reference measures P which are equivalent to P. In Section 4.2, we introduce an entropic risk measure for random processes. The dynamic fluctuation of a random cash balance is clearly important in risk measurement, and therefore it is not sufficient to consider only total amounts of cash accumulated at some terminal date T . Jobert & Rogers [47] defined an entropic risk measure on finite trees. We give a more general characterization of the entropic risk measure for random processes.

4.1 The Entropic Risk Measure: Definition and Properties We consider the entropic risk measure and its properties. The entropic risk measure is a convex risk measure which is related to the exponential utility function u(x) = 1 − e−γx. In the special case where risk preferences are characterized by an exponential utility function, ambiguity fully disappears. This measure admits a representation in terms of an expectation under P. We first define the relative entropy19 (or Kullback-Leibler divergence) of a probability measure Q with respect to a reference probability measure P.

Definition 4.1 Given two probability measures Q and P such that Q P, the relative entropy of Q with respect to P is given by: dQ dQ dQ H(Q|P) = EQ[log ] = E[ log ], dP dP dP where dQ is a Radon-Nikodym derivative. If the absolute continuity condi- dP tion is not satisfied, then H(Q|P) is defined to be ∞. The relative entropy provides a notion of distance from a probability measure Q to a reference probability measure P. The non-negative function H(Q|P) is minimal for the measure that is closest to P. Therefore, it is a well- 1 suited candidate for a penalty function. In particular, we take γ H(Q|P) for α(Q), where γ > 0 is some coefficient to account for risk aversion when risk preferences are characterized by an exponential utility function. Otherwise,

19The concept of relative entropy, or entropy, has several meanings and is used in many contexts. Entropy is most often associated with ‘energy dispersal’. The word entropy was introduced by Rudolf Clausius in 1862 and means literally ‘in turning’, which is derived from the two Greek words en (in) and trope (turning). γ can be viewed as measuring the degree of distrust the agent puts in P. The static entropic risk measure is given by e 1 ρ (X) = sup EQ[−X] − H[Q|P] . (4.1) Q∈M1 γ

e ∞ In this setting, ρ is a collection of mappings from L (FT ) → R. The entropic risk measure in closed-form is easily deduced from the variational principle for the relative entropy (see also for instance F¨ollmer& Schied [38]).

Lemma 4.1 (Variational principle for the relative entropy) . For any probability measure Q,

1 1 −γX H(Q|P) = sup (EQ[−X] − log E[e ]). (4.2) γ X∈L∞ γ

The supremum is attained by letting X = − 1 log dQ if Q . γ dP P

Proof. Assume H(Q|P) < ∞. Deﬁne a measure QX by dQ e−γX X = . dP E[e−γX ]

The transformation P → QX is usually called the Esscher transformation (see Gerber [42]). Consider now the following identity: 1 dQ 1 dQ dQ log = [log + log X ]. γ dP γ dQX dP Taking expectations on both sides with respect to Q yields 1 1 1 H(Q| ) = H(Q|Q ) + [−X] − log [e−γX ]. γ P γ X E γ E

1 1 1 −γX Since γ H(Q|QX ) ≥ 0, we conclude that γ H(Q|P) ≥ EQ[−X]− γ log E[e ]. To show the converse, take Xn = nIA, where A is such that Q[A] > 0 and P[A] = 0. Hence, 1 [−X ] − log [e−γXn ] = n · Q[A] → ∞ = H(Q| ). EQ n γ E P

Assume now that Q . Then X = − 1 log dQ satisﬁes P γ dP 1 1 H(Q| ) = [−X] − log [e−γX ]. γ P EQ γ E 1 dQ Let Xn = (−n) ∨ (− log ) ∧ (n). It follows that γ dP

1 dQ −γXn γ log E[e ] → E[e γ dP ] = 1, n o by computing the expectation [e−γXn ] over the sets − 1 dQ < 1 and E γ dP n o − 1 dQ ≥ 1 . We get the desired result by making use of dominated con- γ dP vergence of the former and monotone convergence of the latter. Finally by Fatou’s Lemma, dQ lim inf EQ[−Xn] = lim inf E[− Xn] n↑∞ n↑∞ dP 1 dQ dQ ≥ E[ log ] γ dP dP 1 = H(Q| ). γ P

1 −γXn 1 Concluding, limn↑∞ inf(EQ[−Xn] − γ log E[e ]) ≥ γ H(Q|P). The corresponding risk measure takes thus the form of: 1 ρe(X) = log [e−γX ]. γ E

Within the expected utility framework, ρe(X) can be interpreted as a negative certainty equivalent payoff −XCE. In fact, it is the guaranteed amount of money at which an agent is indifferent between accepting this guaranteed amount of money and the uncertain payoff X. Detlefsen & Scandolo [30] introduced the notion of a conditional entropic ∞ ∞ risk measure, i.e. a family of mappings from L (FT ) → L (Ft). Naturally, we define 1 ρe (X) = log [e−γX |F ]. t,T γ E t A risk measure which satisfies a supermartingale property (i.e. as time evolves, the information about X increases, which in turn lowers the per- ceived riskiness of X) and a recursive relation is given by the dynamic entropic risk measure. The dynamic entropic risk measure is a family of successive conditional entropic risk measures.

Theorem 4.1 The dynamic entropic risk measure is strongly time-consistent. Proof. It suﬃces to show (3.13). So, we get

In what follows, we assume that the dynamic entropic risk measure is t=T adapted to the discrete-time ﬁltration (Ft)t=S generated by a random walk process. We are interested in convergence of this measure to its continuous- time counterpart, i.e. the dynamic entropic risk measure adapted to the augmented continuous-time ﬁltration (Ft)t∈[S,T ] generated by Brownian motion W (i.e (Ft)t∈[S,T ] is generated by σ(Ws, s ≤ t) and the P-null sets of F). To this end, we introduce a sequence of random variables Xn, which e,n n t=T converges to X in probability as n → ∞. We identify (ρt,T (X ))t=S with a dynamic entropic risk measure in discrete-time and (ρt,T (X))t∈[S,T ] with its continuous-time counterpart.

n 20 Theorem 4.2 Let (X )n∈N be a sequence of random variables, where each n ∞ n ∞ X ∈ L (FT ). Assume that there exists some X ∈ L (FT ) such that Xn → X in probability. Then,

e,n n e sup ρt,T (X ) − ρt,T (X) → 0 in probability as n → ∞. (4.3) S≤t≤T

In order to proof (4.3), we need the notion of weak convergence of ﬁltrations.

n t=T Definition 4.2 A sequence of filtrations (Ft )t=S converges weakly to a fil- n w tration (Ft)t∈[S,T ] (in short, F → F) if and only if, for all A ∈ FT , the n t=T sequence of processes (E[1A|Ft ])t=S converges in probability under the Sko- rokhod J1-topology to the process (E[1A|Ft])t∈[S,T ]. Moreover, if this limit is continuous, then the convergence holds uniformly in t.

20 n ∞ n Since our discrete-time filtration F has finitely many atoms, the spaces L (FT ) and 2 n p n L (FT ) coincide. In fact, all L (FT )-spaces coincide. Proof of theorem. By assumption, Xn → X in probability as n → ∞. Co- n w n n quet et al. [25] showed that F → F if and only if E[X |F ] → E[X|F] in probability under the Skorokhod J1-topology for all integrable random variables X measurable with respect to FT . So, we apply the weak convergence of filtrations to conclude that

−Xn n −X sup E[e |Ft ] − E[e |Ft] → 0 in probability as n → ∞. S≤t≤T

To show (4.3), we remark that e−γXn is uniformly bounded away from zero, and log e−γXn is thus uniformly bounded away from −∞. Concluding, e,n n e supS≤t≤T |ρt,T (X ) − ρt,T (X)| → 0 in probability as n → ∞. It is fine to extent the set of time instances at which ρe is evaluated (this is especially relevant for numerical exercises). In fact, the continuous-time dynamic entropic risk measure is the natural analogue of its discrete-time counterpart. Moreover, as shown by Barrieu & El Karoui [8], the continuous- time dynamic entropic risk measure emerges in the context of backward stochastic differential equations (BSDEs). They stressed its representation in terms of a suitable BSDE. We first recall the definition of a BSDE21.

Definition 4.3 Let W = (Wt)t∈[S,T ] be a d-dimensional Brownian motion 2 defined on (Ω, F, (Ft)t∈[S,T ], P). Further, let X ∈ L (FT ) be a terminal d condition and g a P ⊗B(R)⊗B(R )-measurable coefficient, where P denotes d the predictable σ-algebra and B(R) and B(R ) are the Borel σ-algebras on d R and R , respectively. An adapted solution for the BSDE associated with (g, X) is a pair of progressively measurable processes (Yt,Zt)t≤T , with values d in R × R such that (S ≤ t ≤ T ): Z T 2 2 E[sup |Yt| ] < ∞, E[ |Zs| ds] < ∞, t≤T S

Z T Z T Yt = X + g(s, Ys,Zs)ds − ZsdWs. t t The uniqueness and existence of a solution is formulated in terms of conditions on the coeﬃcient or driver g. Generally, one of the following conditions has to be fulﬁlled: uniformly Lipschitz continuous with respect to (Yt,Zt),

21BSDEs were initiated by Bismut [9], and later studied and developed by Pardoux & Peng [58]. linear growth in (Yt,Zt) or linear growth in Yt and quadratic growth in Zt. See for instance Barrieu & El Karoui [8] and El Karoui et al. [33] (and references therein) for a more rigorous treatment on BSDEs. Under the assumption of Brownian ﬁltration, it is well-known that the dynamic entropic risk measure arises as a solution to a BSDE. This result is summarized in Theorem 4.3.

e Theorem 4.3 The dynamic entropic risk measure (ρt,T (X))t∈[S,T ] is a so- γ 2 lution of the following BSDE with quadratic coeﬃcient g(t, Zt) = 2 ||Zt|| ∞ and terminal bounded condition −X ∈ L (FT ). γ −dρe (X) = ||Z ||2dt − Z dW , ρe (X) = −X, (4.4) t,T 2 t t t T,T with γ > 0 and (Wt)t∈[S,T ] a d-dimensional Brownian motion.

Proof. Deﬁne Mt(X) := E[exp(−γX)|Ft]. Notice that Mt is a positive and bounded continuous martingale, and so by the multiplicative decomposition theorem we get dMt = γMt · ZtdWt, where Zt is a d-dimensional square integrable process. Itˆo’sLemma applied 1 to the function γ log Mt yields 1 1 1 1 1 −d log Mt = 2 d[Mt,Mt] − dMt, γ 2γ Mt γ Mt

2 where d[Mt,Mt] = (γMt · Zt) dt is the quadratic variation process of Mt. The g-coefficient associated with the dynamic entropic risk measure is of the quadratic growth type, i.e.: γ g(t, Z ) = ||Z ||2 t 2 t 2 ≤ K(1 + ||Zt|| ) dP × dt a.s., for some constant K. Under the additional condition | ∂g(t,Zt) | ≤ Kˆ (1+||Z ||) d ×dt a.s. for some ∂Zt t P constant Kˆ , Kobylanski [50] proved that there exists a unique solution of (4.4) such that Y is bounded. 4.1.1 Modeling Ambiguity In Section 4.1, the relative entropy is taken as a measure of distance from a reference measure P to a probability measure Q. Solving problem (4.1) yields the entropic risk measure in closed-form. γ can be viewed as measuring the degree of distrust the agent puts in the reference measure P. In this case, the agent has one and only one reference measure P. However, it could be that he considers many reference measures P ∼ P. Laeven & Stadje [52] introduced the notion of a dynamic entropy coherent risk measure. They provide the following definition. Definition 4.4 (Laeven & Stadje [52]) . ∞ We call a family of mappings ρ : L (FT ) → L(Ft), γ-entropy coherent if there exists a convex closed set Se ⊂ M1 such that

e 1 −γX ρ˜t,T (X) = sup log EP [e |Ft]. P ∈Se γ This deﬁnition is, in fact, a special version of the atemporal multiple-priors model of Gilboa & Schmeidler [43]; or recursive multiple-priors model of Chen & Epstein [18] and Epstein & Schneider [35]. Further, it can be shown (see Laeven & Stadje [52]) that an entropy coherent risk measure is strongly time-consistent if and only if Se is supposed to be multiplicative stable. We now introduce some notation. d A density generator is an R -valued predictable process θ = (θt) satisfying Novikov’s condition, i.e.: Z T 1 2 E[exp ||θs|| ds ] < ∞. 2 S

θ This condition ensures that the process (zt ) is a P-martingale, where θ θ θ dzt = −zt θtdWt, z0 = 1, Z t Z t θ 1 2 zt =: E(θ · W )t = exp − ||θs|| ds − θsdWs . 2 S S 22 Let θ = (θt) be a continuous martingale in BMO , then the process E(θ · W )t is a uniformly integrable martingale. It follows that θ generates a probability measure, i.e.: θ dP θ = zT . dP 22Bounded Mean Oscillation, see John & Nirenberg [48]. Delbaen [28] related multiplicative stability of Se to the existence of a closed convex set-valued mapping Θ which is predictable23.

Theorem 4.4 (Delbaen [28]) . ρ˜e is strongly time-consistent if and only if Se is multiplicative stable. Fur- thermore, Se being multiplicative stable is equivalent to the existence of a closed convex set-valued mapping Θ which is predictable such that   θ is predictable e   S = E(θ · W )T θt(ω) ∈ Θt(ω)

 E(θ · W ) is a positive uniformly integrable martingale.  As a result, the dynamic entropy coherent risk measure can be written as follows:

e 1 −γX 1 θ −γX ρ˜t,T (X) = sup log EP [e |Ft] = sup log E[zT · e |Ft]. e P ∈S γ θt∈Θt γ Laeven & Stadje [52] proved thatρ ˜e is γ-entropy coherent if and only if its γ 2 g-coefficient takes the form of: g(t, Zt) = 2 ||Zt|| +g ¯(t, Zt), whereg ¯(t, ·) is sublinear for every t = S, ..., T . They showed that this function can be represented by g¯(t, Zt) = sup Zt · µt, µt∈−At where At = (ω, µ) (t, ω, µ) ∈ A and A is a predictable, essentially bounded set. Chen & Epstein [18] considered, in a setting where consumers maximize 24 (a generalization of) stochastic differential utility , the functiong ¯(t, Zt) = supθt∈Θt Zt · θt, where Θ is the predictable set as introduced before. d Let us now investigate a specific choice for Θ. Fix κ ∈ R and consider the following predictable and essentially bounded set Θ25, i.e:

n d o Θt(·) = θt ∈ R |θi,t| ≤ κi for all i . Optimization yields

∗ g¯(t, Zt) = sup Zt · θt = Zt · θt = ||Zt|| · κ, θt∈Θt

23The set-valued mapping is said to be predictable if the graph of Θ belongs to the d product σ-algebra P ⊗ B(R ). 24 h T i e R They considered the objective function Vt = minP ∈S EP t f(cs,Vs)ds Ft , where V is a utility function and c denotes consumption. 25 d A more ﬂexible form of Θ is given by Θt(ω) = θ ∈ R ||wt(ω)θ||ν ≤ κt(ω) , where wt(·) is a ‘weighting’ process. where ( i |Zt | i ∗ κi · i if Zt 6= 0 θ = Zt i,t i 0 if Zt = 0. θ θ In the special case where κ = 0, then zT ≡ 1 and P = P, which means that the set of prior views contains only one element (i.e. the economic agent is ambiguity neutral). Moreover, the agent’s preferences over probability measures are not represented by a speciﬁc utility function. In fact, the predictable processes θ ∈ Θ are simply truncated.

This model is called κ-ignorance in Chen & Epstein [18]. The term ||Zt|| · κ is here interpreted as modeling ambiguity aversion. So, the dynamic entropy coherent risk measure entangles risk attitude towards uncertainty (covered by the ambiguity aversion coeﬃcient κ) and towards risk (covered by the risk aversion coeﬃcient γ). The corresponding BSDE is given by γ −dρ˜e (X) = [ ||Z ||2 + ||Z || · κ]dt − Z dW , ρ˜e (X) = −X. (4.5) t,T 2 t t t t T,T

4.2 The Entropic Risk Measure for Random Processes The entropic risk measure is a convex risk measure which is defined on L∞. We now introduce the conditional entropic risk measure for bounded e ∞ ∞ discrete-time random processes, i.e. ρt,T : Rt,T → L (Ft). The entropic risk measure for random processes on finite trees is investigated in Jobert & Rogers [47]. We first define the entropy of a density process a ∈ Qt,T .

Deﬁnition 4.5 The entropy Ht,T (a) of a ∈ Qt,T is given by T X Ht,T (a) = E[ ∆as log((T + 1 − t)∆as)], s=t for all t = S, ..., T . The dynamic entropic risk measure can then be deduced from the variational principal for the entropy. Lemma 4.2 (Variational principle for the entropy) . For any density process a ∈ Qt,T , ( T !) 1 1 1 X −γXs Ht,T (a) = sup h−X, ai − log E[e |Ft] . γ ∞ γ T + 1 − t X∈Rt,T s=t 1 The supremum is attained by letting Xs = − γ log((T + 1 − t)∆as). Proof. Deﬁne a density process aX by

−γXs X e (T + 1 − t)∆as = . −1 PT −γXs (T + 1 − t) s=t E[e |Ft]

X PT X Please note that a qualiﬁes as a density process, i.e. E[ s=t ∆as ] = 1. Consider now the following identity:

1 1 ∆as X log((T + 1 − t)∆as) = [log X + log((T + 1 − t)∆as )]. γ γ ∆as Or, equivalently,

T ! 1 1 ∆as 1 1 X log((T +1−t)∆a ) = log −X − log [e−γXs |F ] . γ s γ ∆aX s γ T + 1 − t E t s s=t Taking expectations on both sides with respect to the probability measure a · P yields 1 H (a) = γ t,T T T ! 1 X ∆as 1 1 X [ ∆a log ] + h−X, ai − log [e−γXs |F ] . γ E s ∆aX γ T + 1 − t E t s=t s s=t

1 PT ∆as 1 Since γ E[ s=t ∆as log ∆aX ] ≥ 0, we conclude that γ Ht,T (a) ≥ h−X, ai − s 1 1 PT −γXs γ log T +1−t s=t E[e |Ft] . We now show the converse. First note 1 that X = − γ log((T + 1 − t)∆as) satisﬁes

T ! 1 1 1 X H (a) = h−X, ai − log [e−γXs |F ] . γ t,T γ T + 1 − t E t s=t

n 1 Let Xs = (−n) ∨ (− γ log((T + 1 − t)∆as)) ∧ (n). It follows that

T T 1 X n 1 X [e−γXs |F ] → [elog((T +1−t)∆as)|F ] = 1, T + 1 − t E t T + 1 − t E t s=t s=t

n n o −γXs 1 by computing E[e |Ft] over the disjoint sets − γ (T + 1 − t)∆as < 1 n 1 o and − γ (T + 1 − t)∆as ≥ 1 . We get the desired result by making use of dominated convergence of the former and monotone convergence of the latter. Finally by Fatou’s Lemma,

T 1 1 X n Ht,T (a) = E[ ∆as log((T + 1 − t)∆as)] ≤ lim inf h−X , ai . γ γ n↑∞ s=t

This result completes the proof. The corresponding risk measure takes thus the form of:

T ! 1 1 X ρe (X) = log [e−γXs |F ] . (4.6) t,T γ T + 1 − t E t s=t

Theorem 4.5 The dynamic entropic risk measure for random processes is strongly time-consistent.

Proof. It suﬃces to show (3.6).

e e ρt,T (X1{t} − ρt+1,T (X)1[t+1,∞)) = 1 1 T − t γρe (X) log [e−γXt |F ] + [e t+1,T |F ] . γ T + 1 − tE t T + 1 − tE t Substitution yields 1 1 −γX T − t log 1 PT [e−γXs |F ] log [e t |F ] + [e ( T −t s=t+1 E t+1 )|F ] . γ T + 1 − tE t T + 1 − tE t

This ﬁnally reduces to

T ! 1 1 1 X log [e−γXt |F ] + [e−γXs |F ] = γ T + 1 − tE t T + 1 − t E t s=t+1

T ! 1 1 X log [e−γXs |F ] = ρe (X). γ T + 1 − t E t t,T s=t We obtain convergence of the dynamic entropic risk measure for random processes to its continuous-time counterpart when taking the limit n → ∞. This result is summarized in the following theorem. n Theorem 4.6 Let (X )n∈N be a sequence of bounded adapted stochastic n ∞ ∞ processes, where each X ∈ Rt,T . Assume that there exists some X ∈ Rt,T such that Xn → X in probability. Then,

e,n n e sup ρt,T (X ) − ρt,T (X) → 0 in probability as n → ∞. (4.7) S≤t≤T

Proof. By Deﬁnition 4.2,

T T n 1 X −γX n 1 X −γXs sup [e s |F ] − [e |Ft] T + 1 − t E t T + 1 − t E S≤t≤T s=t s=t → 0 in probability as n → ∞. e,n n Note that ρt,T (X ) is uniformly bounded away from −∞, and so (4.7) follows. The dynamic entropic risk measure (4.6) assigns a uniform weight to each −γXs E[e |Ft]. It is, however, possible to vary the weights attached to each of these conditional expectations. The accompanying dynamic risk measure is strongly time-consistent if and only if we assume a mild condition on the weight scheme. Deﬁnition 4.6 Deﬁne the following dynamic risk measure:

T ! 1 X ρ (X) = log ωt [e−γXs |F ] , (4.8) t,T γ sE t s=t

t −γXs where ωs is a weight attached to E[e |Ft], which is determined at time t.

t s=T t Theorem 4.7 Let (ωs)s=t be a weight scheme, i.e. ωs ≥ 0 for every s and PT t s=t ωs = 1. The dynamic risk measure (4.8) is strongly time-consistent if and only if t t+1 ωs ωs = t (4.9) 1 − ωt for every s ≥ t + 1.

Proof. It suﬃces to show (3.6).

ρt,T (X1{t} − ρt+1,T (X)1[t+1,∞)) =

T ! 1 X log ωt [e−γXt |F ] + ωt [eγρt+1,T (X)|F ] . γ tE t sE t s=t+1 Substitution yields

T ! 1 X log PT ωt+1 [e−γXs |F ] |F ] log ωt [e−γXt |F ] + ωt [e ( s=t+1 s E t+1 ) t . γ tE t sE s=t+1

This reduces to

T T ! 1 X X log ωt [e−γXt |F ] + ωt ωt+1 [e−γXs |F ] = γ tE t s s E t s=t+1 s=t+1

T ! 1 X log ωt [e−γXt |F ] + (1 − ωt) ωt+1 [e−γXs |F ] . γ tE t t s E t s=t+1 By condition (4.9), this is equal to

T ! 1 X log ωt [e−γXt |F ] + ωt [e−γXs |F ] = γ tE t sE t s=t+1

T ! 1 X log ωt [e−γXs |F ] = ρ (X). γ sE t t,T s=t

So, ρt,T is strongly time-consistent if and only if (4.9) has been satisﬁed. ωt t+1 s1 ωs1 Condition (4.9) implies that the relative proportions (i.e. ωt = t+1 for all s2 ωs2 t and every pair of time instances s1 and s2) are the same at any time.

5 A Numerical Exercise: A Pension Fund Setting

We now turn to dynamic risk measurement in a pension fund setting. The nominal funding level26 is commonly used as a supervisory monitoring ratio. For instance, Dutch authorities impose the following restriction on the N nominal funding level Ft+1:

N N P[Ft+1 < 1] ≤ 2.5% ⇒ P[St+1 < 0] ≤ 2.5%,

N N where St+1 is the pension fund surplus ratio, i.e. Ft+1 − 1.

26The relative value of the pension fund assets and nominal liabilities, expressed as a percentage figure. Although most pension supervision rules are formulated in terms of the 27 nominal liabilities, we define the surplus ratio St as the pension fund assets minus the sum of all real liabilities divided by the sum of all real liabilities. This has a few advantages. First, we do not have to worry about the impact of future indexation on the risk assessment, i.e. liabilities are modeled as if they are fully indexed to price inflation. This abstraction seems reasonable, because the main objective of the European Central Bank is to maintain low inflation. The euro area can be seen as a low-inflation environment with an annual average rate of inflation of 0.3 % in 2009 (source: Eurostat). The pension fund surplus is then viewed solely as a solvency buffer. In fact, the indexation buffer is taken to be the difference between the nominal funding level and the real funding level. Second, the participants’ main concern is the purchasing power of the pension plan. So, it is natural to consider the surplus ratio in terms of real liabilities. The risk measure is then the smallest amount of money that would have to be put aside to make next period’s surplus ratio acceptable. A real funding level of 70 % (which may nowadays well happen) would certainly not be ‘acceptable’. It follows then from our risk assessment that trustees should take action to raise capital buffers. In the current Dutch situation, pension funds need to have a solvency buffer on top of the level of nominal liabilities. The solvency buffer is determined in such a way that the probability of underfunding in one year from now is less than 2.5%. The indexation is conditional on the pension fund’s financial position. The pension fund surplus can, in fact, be divided into a solvency buffer (i.e. to secure participants against potential losses) and an indexation buffer (i.e. to provide a hedge against inflation). It would be better to base the solvency stress test on the real liabilities (or nominal liabilities plus participants’ expectations of future indexation) and to determine the solvency buffer by means of a convex measure of risk in order to take into account model risk. The surplus can then be viewed solely as a solvency buffer. Figure 1 depicts the current Dutch situation and two proposals.

27In The Netherlands, there is currently much discussion about the role of the supervisor and the revision of the supervision rules. In particular, an adviser committee - a committee headed by Prof. dr. Frijns - of the government argues that the Financial Assessment Framework for Dutch pension funds must be aimed at the real funding level. Figure 1: Current Situation vs. Proposal (a) and (b)

In what follows, we setup two specific models of the surplus ratio. In the first model, we only take care of one time-independent source of uncertainty. The model is very stylized and is used to illustrate ideas. The second model is more elaborate and explicitly describes the dynamics of the pension fund assets and real liabilities28. Given our model of the surplus ratio, we try to measure its risk dynamically. For a suitable calibration of the parameters, we compute the risk - which can also be viewed as a valuation - of the final surplus ratio and the surplus ratio process. Throughout, we consider a time horizon of one year (simulation run length) and 5000 trajectories of Brow- nian motion (number of replications). Figures are gathered in Appendix 7.3.

5.1 One Source of Uncertainty The first case is a very stylized pension fund model. The uncertainty is fully driven by a one-dimensional Brownian motion, which represents e.g. stock market movements. The pension fund exposure to this Brownian motion is summarized in the constant parameter coefficient σ. This coefficient can be viewed as an abstract number that captures uncertainty in a very simple way. As a result, we assume that other sources of uncertainty - such as interest rate risk and currency risk - are fully hedged, absent or reinsured. In

28Future benefit liabilities are discounted at the real interest rate earned by high-quality corporate bonds. addition, the pension fund is stationary (i.e. inflow and outflow of participants guarantee a stable composition of participants, so that there is neither a trend in rights accrual nor a trend in asset accumulation) and pension fund trustees are not allowed to change policy over the evaluation period. The latter can be motivated by the fact that action is generally taken after the evaluation period. The model is given by

dFt = σFtdWt,FS = c, where

Ft = real funding level at time t σ = standard deviation of the real funding level Wt = one-dimensional standard Brownian motion at time t.

By Itô’sLemma, the dynamics dSt are given by dFt, so the surplus ratio process (St)t∈[S,T ] is described by (Ft)t∈[S,T ] − 1[S,T ]. The parameter value σ strongly depends on the pension fund characteristics. If, for instance, interest rate risk is hedged away29, then the pension fund invests its wealth in assets or derivatives that provide a hedge against interest rate risk. The equity exposure is then relatively small. From the perspective of optimal risk sharing, σ should also reflect participants’ risk preferences (i.e. a green pension fund should feature a larger value of σ than a gray pension fund)30. We somewhat arbitrarily take σ = 10% and σ = 15%. In a pension fund context, there is yet little known about the parameter value γ. As noticed before, the dynamic entropic risk measure is linked to the exponential utility function. γ is here the Arrow-Pratt measure of absolute risk aversion (Arrow [2] [3] and Pratt [60]). Information about people’s risk aversion coefficient can be obtained in different ways such as confronting agents with hypothetical questions or inference from economic data. Most studies (see for instance Arrow [3], Kehoe [49] and Kydland & Prescott [51]) conclude that the coefficient of relative risk aversion (i.e. γ · X) should be

29This can be carried out by choosing a specific asset mix. 30According to life cycle models, young generations should take more equity exposure than old generations. In particular, the fraction of total wealth (i.e. the sum of human capital and financial capital) invested in the financial market should be constant over the life course (see e.g. Merton [55]). between one and two, but certainly less than ten. Studies concerning the coefficient of absolute risk aversion indicate that this value is very small, i.e. between zero and four-hundredth (see Raskin & Cochrane [61] for an overview of commonly used risk aversion coefficients). Those studies try to measure people’s risk aversion coefficient with respect to wealth, income or consumption. However, our primary concern is not individual consumption, but the surplus ratio. To get an idea of a correct value for γ, we consider the following experiment. Suppose that the pension fund’s risk preferences are characterized by an exponential utility function with γ = 4. As stated before, the entropic risk measure can now be interpreted as a negative cer- 1 tainty equivalent payoff. If, for instance, S1 = +10% w.p. 2 or S1 = −10% 1 e w.p. 2 , then ρ (S1) ≈ 2%. This simple example illustrates that γ may well be many orders of magnitude larger than those values that we found in the literature. Furthermore, it is not clear how participants’ risk preferences relate to the pension fund’s risk preferences. Blackburn & Ukhov [10] prove that an economy consisting of risk seeking agents can lead to an aggregate economy that is risk averse. A nice topic for future research would be how to deal with aggregate preferences in an economic setting31. Alternatively, one can target γ in such a way that the risk assessment matches current regulatory requirements. In sum, it is quite troublesome to take a value for γ that is convenient for our modeling purposes.

Risk Measurement of the Final Surplus Ratio

2 32 Although the ﬁnal surplus ratio ST is an element of L (FT ) , we describe its dynamics by a simple diﬀusion process. The dynamic entropic risk measure e (ρt,T (X))t∈[S,T ] is given by the solution of a suitable BSDE. In order to compute a risk-adjustment factor (i.e. the process Zt in (4.4)), we need the dynamics of de−γSt , i.e.: 1 de−γSt = −γe−γSt dS + γ2e−γSt d[S ,S ]. t 2 t t

1 −γST The entropic risk measure at time T is given by γ log E[e |FT ] = −ST . As expressed in (4.4), the remaining entropic risk measures are obtained γ 2 by means of a BSDE with g-coeﬃcient 2 ||Zt|| . This BSDE is Markovian and its g-coeﬃcient grows less than quadratically in Zt. Appendix 7.2.1 and

31Possible settings can be pension funds, insurance companies, whole economies et cetera. 32In our numerical application, all Lp-spaces coincide. 7.2.3 describe a numerical procedure to implement such a one-dimensional BSDE33. t=1 Figure 2 and Figure 3 plot the expected risk measure process E[ρt,1(S1)t=0] as a function of γ. Alternatively, one can compute and plot the expected t=1 risk measure process based on a rolling window, i.e. E[ρt,1(S1+t)t=0]. As can be observed, the expected risk measure process satisﬁes a supermartingale property34. Further, the risk measure process is strongly time-consistent - it can be computed directly or in two steps backwards in time - which is certainly a desirable property. On the contrary, Value at Risk is not time-consistent as showed by Cheridito & Stadje [23]. A few things can be concluded from these ﬁgures.

• If σ is relatively large (i.e. upper surface), then γ becomes more important. To put the matter more clearly, if a pension fund is exposed to a large risk, then the risk assessment becomes very sensitive to γ.

• For γ = 8 and σ = 15%, the underfunded and fully funded pension fund have to keep a capital buffer of 34% and 9%, respectively. This observation is intuitive, because the underfunded pension fund is exposed to a relatively small risk (i.e. ≈ σ · 70%). In practice, the risk exposure of the new funding level (e.g. 70% + capital buffer) must be reduced or the capital buffer must be raised in order to still satisfy the regulatory requirements.

In Figure 4, we plot a particular trajectory of the surplus ratio process (i.e. bluish surface). This trajectory is obviously independent of γ. The associated risk measure process is the yellowish curvature. As can be seen, the process follows the same pattern as the surplus ratio process, but in opposite direction. If incoming information is ‘positive news’ (e.g. stock markets are doing well), then the risk assessment is revised downwards.

Value at Risk

The current Dutch regulatory requirement is based on the 2.5%-quantile of next year’s nominal funding level. We argue that the risk measurement must be aimed at the real funding level. Figure 5 plots the expected risk measure

n 33 + 1 −γSn n n − In the context of Appendix 7.2.1, Yn = γ log E[e |1 , ..., n−1, 1] and Yn = n 1 −γSn n n + − γ log E[e |1 , ..., n−1, −1]. So that both Yn and Yn are Fn−1-measurable. 34 E[ρt+1,T (ST )] ≤ ρt,T (ST ). process based on the negative of the quantile function of S1. The quantile function at time t is given by

−1 2 F (q) = inf S1 ∈ L (Ft) q ≤ F (S1) , where F is the cumulative distribution function of S1. The negative of the quantile function35 can then be interpreted as the amount of money that would have to be put aside to make S1 acceptable. Also, the quantile function is not very robust, which can be concluded from the ﬁgure.

Modeling Ambiguity

As said earlier, the entropic risk measure arises as a negative certainty equivalent payoff in the expected utility framework. Within this framework (i.e. economic agents’ feature exponential utility), there is no ambiguity concerning P. We now consider a more flexible risk measure that allows for risk aversion as well as ambiguity aversion. Its related BSDE is given by (4.5). Here, we have to specify γ and κ. Ambiguity aversion means that the agent considers many reference measure P ∼ P. Stated differently, the agent is not quite confident about the model specification dFt = σFtdWt. By Girsanov Theorem, we know that

dFt = −σθtFtdt + σFtdW˜ t, where W˜ t is a standard Brownian motion under P and θ = (θt) is a predictable process, which is deﬁned in Section 4.1.1. For θ = κ, we get dFt = −σκFtdt + σFtdW˜ t. −κσ The worst expected value of F1 is then given by e . Table 1 summa- rizes the relation between the ambiguity aversion coeﬃcient κ and the worst expected value of F1 (under some worst case measure P ).

35The quantile function satisﬁes cash invariance. κ Worst expected funding level F1 under worst case measure P 0 E[F1] = 100% 0.5 EP [F1] = 92.8% 1 EP [F1] = 86.1% 1.5 EP [F1] = 79.9% 2 EP [F1] = 74.1% 2.5 EP [F1] = 68.7% 3 EP [F1] = 63.8% 3.5 EP [F1] = 59.2% 4 EP [F1] = 54.9% Table 1: Ambiguity and Worst Case Expectations with σ = 15%

Figure 6 plots the expected entropy coherent risk measure process as a function of κ. The upper (lower) surface corresponds with γ = 8 (γ = 1). The risk measure process is very sensitive to κ and very insensitive to γ. So, ambiguity aversion is more important to the agent than risk aversion. More research is needed to investigate the relation between risk and uncertainty.

Risk Measurement of the Surplus Ratio Process

We now turn to dynamic risk measurement of the surplus ratio process. In this case, the surplus ratio can be viewed as a cash balance process. The dynamic ﬂuctuation of the surplus ratio also plays a major role. We use the following dynamic risk measure, see Section 4.2:

T ! 1 1 X ρe (X) = log [e−γSs |F ] . t,T γ T + 1 − t E t s=t This measure values the surplus ratio at each point in time when new information is prevailing. By way of construction, the volatility of Ss increases as time evolves. In addition, the agent attaches a uniform weight to each time instance. Alternatively, one can vary the weights attached to each point in time (i.e. some periods of time are considered to be more important than s 1 −γSs other periods of time). From (4.4), Yt = γ log E[e |Ft] can be computed s γY −γSs by means of a suitable BSDE. So, e t is given by E[e |Ft]. The value of the dynamic risk measure for random processes is then obtained by taking s 1 γYt the natural logarithm (scaled by γ ) of the average value of e . Figure 7 plots the expected risk measure process as a function of γ. The expected risk measure process is smaller than the expected risk measure process of the final surplus ratio only. This result is straightforward, because the risk measure also takes care of intermediate values of the surplus ratio36. Moreover, the measurement is more stable compared to only measuring the final surplus ratio when new information is gathered. This can be observed from e.g. the standard deviation of the risk measurement. From the perspective of risk management, a stable assessment would probably be preferred to an unstable assessment. Finally, we plot the risk measure process of a particular trajectory of the surplus ratio, see Figure 8. If we measure the final surplus ratio only, then the risk measure process is shifted upwards. This is caused by the fact that the dynamic fluctuation of the surplus ratio is less volatile than the fluctuation of the final surplus ratio (if the other way around would be the case, then we would observe a downward shift).

5.2 A More Realistic Model

We now explicitly model the market value of the pension fund assets At and the market value of the pension fund real liabilities Lt. The pension fund invests its wealth in risky assets (e.g. equities) and zero-coupon bonds. There are two sources of uncertainty present, i.e. one source of uncertainty associated with stock market movements and another source of uncertainty associated with fluctuations in the short-term real interest rate. Our model differs from other standard pension fund models (see for instance Cairns [15]). Those other models are suited for complex benefit and pay-out struc- tures, rather than managing interest rate risk. Please note that changes in the interest rate affect both the value of the pension fund assets and liabilities. The value of the assets is affected because of the fact that bond prices move inversely to interest rates and the value of the liabilities is affected because of the fair value principle (IAS 19).

Modeling Assumptions

In what follows, an interest rate is defined to be the real return earned on a short-term Treasury Bill. We have the following three modeling assumptions regrading the term structure of interest rates. First, the expected returns on bonds in excess of the short-term interest rate are time-invariant and are set 36In other words, the final surplus ratio fluctuates more heavily than intermediate values 1 PT −γXs −γXT of the surplus ratio (i.e. T +1−t s=t E[e |Ft] ≤ E[e |Ft]). equal to zero. So, the holding period return on bonds is just the short-term interest rate. We call this the expectations hypothesis of the term structure, because the long-term interest rates reflect the market’s expectations of future short-term interest rates (Fisher [36]). Second, the optimal forecast of future short-term interest rates can be obtained by conditioning on past and current short-term interest rates only. Third, the short-term interest rates follow a first-order autoregressive model with homoskedastic normally distributed errors (i.e. AR(1)). The latter assumption is satisfied by e.g. the Vasicek model of the term structure (Vasicek [67]). Given these assumptions, we will be able to compute the interest rate sensitivity of the pension fund assets and liabilities. The model is described by (the initial conditions will be specified later on):

dAt = [((1 − ω)rt + ωµ)At + Pt − Bt]dt + ωσ1AtdW1,t − drtAAt, dLt = [(1 + rt + it)Lt + p − b]dt − drtLLt, drt = a(b − rt)dt + σ2dW2,t, where

At = market value of assets at time t rt = holding period return on bond portfolio µ = expected return on equities ω = proportion of assets invested in equities (no dynamic allocation) Pt = actual premium rate Bt = actual beneﬁt outgo rate σ1 = standard deviation of equity returns A = (1 − ω) · B, with B the interest rate sensitivity of the bond portfolio Lt = market value of liabilities at time t it = additional indexation rate at time t p = cost based premium rate b = target beneﬁt outgo rate L = interest rate sensitivity of liabilities rt = short-term interest rate at time t σ2 = standard deviation of the short-term interest rate Wt = standard Brownian motion at time t.

The change in the pension fund assets is given by the return earned on the assets, plus the actual premium rate, minus the actual benefit outgo rate. The actual premium rate and the actual benefit outgo rate are policy instruments and do not have to match the cost based premium rate and the target benefit outgo rate. Here, however, we assume that Pt = p = Bt = b. The change in the pension fund real liabilities is given by the cost of capital (i.e. real interest rate), plus the indexation rate, plus the cost based premium rate (i.e. new accrued liabilities attributable to one year of additional service of active participants), minus the target benefit outgo rate (i.e. liabilities which are reserved for pension payments). The indexation rate is also a policy instrument (often conditional on the pension fund’s financial position). We assume that the (additional) indexation rate is constant. The correlation ρ between equity risk and interest rate risk is assumed to be zero. It is still an open question whether or not equity prices and interest rates move in tandem. Empirical evidence shows that this relation depends on economic conditions, i.e. recession or boom. The Dutch regulator beliefs that the correlation is 0.5 in bad scenarios, but this number is certainly not a best estimate. We take independent sources of uncertainty, so that the interest rate sensitivity of equities is zero. The dependence structure can easily p 2 be modeled by introducing the dynamics dW2,t = ρdW1,t + 1 − ρ dW3,t, where W1 and W3 are independent standard Brownian motions. The dynamics of the surplus ratio S = At − 1 can be expressed as follows. t Lt

1 At At 1 dSt = dAt − 2 dLt + 3 d[Lt,Lt] − 2 d[At,Lt]. Lt Lt Lt Lt Substitution yields (general case)

2 2 1 dSt = [ω(µ − rt) − it + σ2(L − LA)]Ftdt + (Pt − Bt) dt Lt At − (p − b) 2 dt + ωσ1FtdW1,t − drt(A − L)Ft. Lt

Model Calibration The short-term interest rate dynamics are described by the Vasicek model with long-term mean level b and speed of reversion coeﬃcient a. This model is based on an AR(1)-model, i.e.:

rt+1 = c + γ · (rt − c) + ηt+1, where η is white noise with mean zero and standard deviation σ2. By the expectations hypothesis of the term structure, the n-year interest rate can be expressed in terms of the short-term interest rate, i.e. ‘n-year interest rate’ = an · rt, where 1 1 − γn a = . n n 1 − γ This formula will be used to determine the interest rate sensitivity of a zero-coupon bond and the pension fund liabilities. Rewriting the AR(1)-model yields the Ornstein-Uhlenbeck stochastic process (Ornstein & Uhlenbeck [57]). In particular,

drt = [c(1 − γ) + (γ − 1)rt]dt + ηt+1

= a(b − rt)dt + ηt+1, with a · b = c(1 − γ) and a = 1 − γ. Chan et al. [17] estimated c and γ − 1 for the annualized one-month U.S. Treasury Bill yield. They performed a regression based on an estimation window from June 1964 to December 1989 and found estimates of 0.0866 for c and 0.8821 for γ. The interest rate sensitivity of a ten-year zero-coupon bond is then a10 · 10 = 6.062. So if the short-term interest rate goes up with 100 basis points, then the value of a ten-year zero-coupon bond comes down with 6.062%. Duration analysis would suggest a decrease of 10%. The standard deviation of short-term real rates is estimated to be approx- imately 1.5% (see Ang et al. [1]). An estimate of σ2 can then be obtained by solving the following formula for σ2:

σ2 V ar(r ) = 2 (1 − e−2a), t 2 · a where V ar(rt) is the variance of the annualized short-term interest rate. Although the data is outdated, we base our calibration of a on the estimate of γ. So we implicitly assume that real and nominal rates move back to their long-term means at the same speed, which might be a strong assumption. On the contrary, b is not based on these estimates. We set b equal to 1% in order to match current real interest rates.

The parameter values for µ and σ1 are taken from the Financial Assessment Framework for Dutch pension funds.

Table 2 gives an overview of all parameter values (including L, which is determined in a somewhat arbitrary way). Parameter Value a 0.1779 b 0.01 µ 0.055 i 0.0225 σ1 0.1625 σ2 0.0164 L 7.18 AS 1 LS 1 rS 0.01

Table 2: Parameter Values

We investigate the following three pension funds.

1. Relatively standard fund: 50 % equities and 50% 10-year zero-coupon bonds.

• Average drift term: almost zero. Interest rate risk: Moderate. Equity risk: Moderate.

2. Relatively aggressive fund: 75 % equities and 25% 10-year zero-coupon bonds.

• Average drift term: Positive. Interest rate risk: High. Equity risk: High.

3. Relatively safe fund: 25 % equities and 75% 10-year zero-coupon bonds.

• Average drift term: Negative. Interest rate risk: Low. Equity risk: Low.

Risk Measurement of the Final Surplus Ratio

The risk measure process is computed by means of a BSDE as mentioned in Section 5.1 (see also Appendix 7.2.2 and 7.2.3). Figure 9 plots the expected risk measure process for three different pension funds. The ‘aggressive’ pension fund is represented by the steep curvature and yields the most conser- vative risk measurement at time zero. This result is intuitive, because the uncertainty in S1 is relatively large. So capital buffers should be sufficient to cover potential losses. On the contrary, the ‘safe’ pension fund yields the most stable measurement (flat curvature). Moreover, it turns out that the ‘aggressive’ pension fund is very sensitive to γ. In our view, γ should be taken in such a way that more risk-taking is penalized in the sense that capital buffers must be raised.

Introducing Dependence between Interest Rates and Equities

As stated before, our risk measurement is based on the assumption that there is no correlation between interest rate risk and equity risk. We now p 2 introduce the dependence structure dW2,t = ρdW1,t + 1 − ρ dW3,t, where W1 and W3 are independent standard Brownian motions. In Figure 10, we plot the expected risk measure process for ρ = 0.5 (steep curvature), ρ = 0 (middle curvature) and ρ = −0.5 (ﬂat curvature). The latter yields the most optimistic risk measurement, because equities provide a hedge against interest rate risk (i.e. if the market value of the assets comes down, then the market value of the liabilities also comes down).

Risk measurement of the Surplus Ratio Process

We also compute the entropic risk measure for random processes. This is done by applying some deterministic functions on the solutions of suitable BSDEs, see Section 5.1. A particular trajectory of the surplus ratio process and its related risk measure processes are depicted in Figure 11. As noticed in Section 5.1, the risk measure process of the surplus ratio process is more stable and more ’optimistic’ than the risk measure process of the ﬁnal surplus ratio only. We also observe that this optimism declines sharply towards the end of the evaluation period, and disappears entirely at T . 6 Final Remarks

Short summary

The axiomatization of risk measures was inspired by the Basel Accords and the Basel Committee Amendments. The Basel Committee noticed that backtesting programs typically consist of a periodic comparison of the bank’s daily VaR measures with the subsequent daily profit or loss and that the VaR approach to risk measurement is generally based on the sensitivity of a static portfolio to instantaneous price shocks. For different reasons, the VaR approach is not desirable and the literature introduced coherent, and later convex, risk measures. Section 2 described static risk measurement. However, static risk measures do not accommodate for intermediate payoffs or additional information. Section 3 investigated risk measurement in a dynamic stetting. Convexity is used to give a robust representation theorem in terms of a suitable class of penalized probability measures (or density processes). This is an implicit characterization of risk measures, which implies that risk measures are not directly applicable. In Section 4.1, the reference measure is the one which is taken most seriously, and the penalty function is proportional to the deviation from a prior view to P, measured by the relative entropy. Under the assumption of Brownian filtration, the entropic risk measure emerges as a solution to a BSDE. So, the entropic risk measure (and also the entropy coherent risk measure and entropic risk measure for random processes as investigated in Section 4.1.1 and Section 4.2, respectively) can be computed by means of BSDEs37. Section 5 finally went through an extensive numerical exercise.

Comments and Conclusions

• In a pension fund setting, the convex measures of risk (e.g. entropic risk measure) are not based on P, but on the surplus ratio itself. Model risk is thus taken into account. For the entropic risk measure, γ measures in fact the degree of distrust the pension fund puts in the probabilistic model P. Also, the discussion in Section 5 highlights that the risk assessment should be based on the real funding level (or the nominal funding level plus participants’ expectations of future indexation).

37BSDEs, and especially numerical solution schemes, are nowadays extensively studied in the literature. • The existing literature on convex risk measures has mainly focused on robust representation theorems. Much remains to be done on the computability of measures of risk and on the estimation of parameter coefficients (like γ and κ as considered in Section 4). For the class of entropic risk measures, BSDEs are a way to compute these measures of risk. • The figures presented in Appendix 7.3 are based on a specific model calibration and specification. The reader should be aware of the fact that the numerical outcomes are very sensitive to parameter choices and modeling assumptions. Alternatively, one can estimate an autoregressive model (e.g. AR(p) or ARCH(p, q)) of the surplus ratio process based on e.g. maximum likelihood estimation. This model can then be used for future predictions of the surplus ratio. • The considered risk measures are strongly time-consistent. This property would probably be too strong for solvency requirements. One may also develop other risk measures that are time-consistent in a weak sense only.

7 Appendix

7.1 Proofs We prove the main results of Section 2 and 3. The proofs are based on the works of F¨ollmer& Schied [38] and Cheridito et al. [21]. In order to proof the theorems and propositions, we need the following deﬁnition and lemmas.

Deﬁnition 7.1 A function f : X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that for all x1, x2 ∈ X:

dY (f(x1), f(x2)) ≤ K · dX (x1, x2), where dY denotes the metric space on Y and dX the metric space on X.

Lemma 7.1 In a topological vector space S, any two disjoint convex sets G and H, one of which has an interior point (i.e. there exists an open ball centered at the interior point that is contained in S), can be separated by a non-zero continuous linear functional l on S:

l(x) ≤ l(y), for all x ∈ H and all y ∈ G. Proof. See Dunford & Schwartz [32].

Lemma 7.2 Let X ∈ X . The integral Z l(X) = Xdµ, deﬁnes a one-to-one correspondence between continuous linear functionals l on X and ﬁnitely additive set functions µ ∈ ba(Ω, F).

Proof. See F¨ollmer& Schied [38].

Lemma 7.3 (Hahn-Banach Separation Theorem) . Consider a locally convex topological vector space S. For a compact convex set G and a closed convex set H, there exists a continuous linear functional l on S such that sup l(x) ≤ inf l(x). x∈G y∈H

Proof. See Dunford & Schwartz [32].

Lemma 7.4 A convex subset B of L∞ is weak* closed if for every r > 0,

∞ Br = X ∈ L ||X||∞ ≤ r

1 is closed in L (with ||X||∞ := inf {c ≥ 0|P[|X| > c] = 0}).

Proof. See F¨ollmer& Schied [38].

Lemma 7.5 Any monetary risk measure ρ is Lipschitz continuous. In particular, we have that

|ρ(X) − ρ(Y )| ≤ ||X − Y ||, where ||X − Y || = supω∈Ω |X(ω) − Y (ω)| is the supremum norm.

Proof. The following holds by deﬁnition of || · ||: X ≤ Y + ||X − Y ||. By monotonicity and cash invariance: ρ(Y +||X−Y ||) ≤ ρ(X) ⇒ ρ(Y )−ρ(X) ≤ ||X − Y ||. But then also |ρ(Y ) − ρ(X)| ≤ ||X − Y ||, since X and Y are arbitrary. Proof of Proposition 2.1. 1. (2.3) is obvious. (2.4) follows by monotonicity of ρ(X) and the deﬁ- nition of Cρ. (2.5) follows by the fact that the mapping λ → ρ(λX + (1 − λ)Y ) is Lipschitz continuous according to Lemma 7.5, and hence the set {λ ∈ [0, 1]|λX + (1 − λ)Y ∈ Cρ} is closed in [0, 1].

2. If ρ(X) is a convex risk measure, then Cρ is convex by deﬁnition of Cρ. The converse follows from Proposition 2.2.2. 3. A cone is a subset of a vector space that is closed under multiplication by positive scalars. Hence, Cρ is a cone if ρ(X) is positively homogeneous. In addition, if ρ(X) is a convex risk measure and positively homogeneous, then Cρ is clearly a convex cone. The converse follows from Proposition 2.2.3. Proof of Proposition 2.2.

1. ρC satisfies monotonicity and cash invariance by definition of ρC (cf. (2.1)) and property (2.4). It remains to show that ρC takes only finite values (remember that ρC is a mapping from X → R = (−∞, ∞)). For Y ∈ C and X ∈ X , there exists a real number m such that X + m ≥ Y (since Y is bounded). By monotonicity and cash invariance, ρC(X) − m ≤ ρ(Y ) ≤ 0 ⇒ ρC(X) ≤ m < ∞. To show that ρC(X) > −∞, take X + n ≤ 0. By monotonicity, cash invariance and property (2.3), ρC(X) > ρC(0) + n > −∞.

2. Let X,Y ∈ X and λ ∈ [0, 1]. It follows that X +ρC(X),Y +ρC(Y ) ∈ C. Convexity of C implies that λ(X + ρC(X)) + (1 − λ)(Y + ρC(Y )) ∈ C. By cash invariance,

0 ≥ ρC(λ(X + ρC(X)) + (1 − λ)(Y + ρC(Y )))

= ρC(λX + (1 − λ)Y ) − (λρC(X) + (1 − λ)ρC(Y )).

Hence, ρC(X) is convex.

3. Let X ∈ X and λ ≥ 0. It follows that X + ρC(X) ∈ C. Positive homogeneity of C implies that λ(X + ρC(X)) ∈ C. By cash invariance,

0 ≥ ρC(λ(X + ρC(X)))

= ρC(λX) − λρC(X). To prove the converse inequality, take some X + ∈ X ( > 0). Then ρC(X +) < ρC(X) and X +ρC(X +) ∈/ C. Also, λ(X +ρC(X +)) ∈/ C. It follows that λρC(X + ) < ρC(λX). So, ρC(λX) = λρC(X). In addition, if C is a convex cone, then ρC is a coherent risk measure.

4. If X ∈ C, then X ∈ Cρ (remember that C is a subset of Cρ, because {X ∈ X |ρ(X) > 0} ∩ C = ∅). It remains to show that Cρ ⊆ C (i.e. if X/∈ C, then X/∈ Cρ). Suppose that C satisﬁes property (2.5) and let X/∈ C and m > ||X||, then by assumption there exists some λ ∈ [0, 1] such that λm + (1 − λ)X/∈ C. Cash invariance implies that

ρC((1 − λ)X) > λm. By Lemma 7.5, we have that

|ρC((1 − λ)X) − ρC(X)| ≤ λ||x||.

So, ρC(X) ≥ ρC((1 − λ)X) − λ||x|| ≥ λ(m − ||x||) > 0. In sum, X ∈ C ⇒ ρC(X) ≤ 0 and X ∈/ C ⇒ ρC(X) > 0. Hence, C = {X ∈ X |ρC(X) ≤ 0} = Cρ.

Proof of Theorem 2.1. We prove that the following inequality holds for all X ∈ X :

ρ(X) ≥ sup (EQ[−X] − αmin(Q)), 1 Q∈Mf

1 and at the same time there exists some QX ∈ Mf such that for X given:

ρ(X) ≤ EQX [−X] − αmin(QX ).

To prove the ﬁrst inequality, ﬁx some ρ(X) + X ∈ Cρ. Hence,

αmin(Q) ≥ EQ[−ρ(X) − X] = −ρ(X) + EQ[−X]. 1 We will now construct some QX ∈ Mf such that the second inequality holds. We will prove it for the set {X ∈ X |ρ(X) = 0}. By cash invariance, the result also holds for any X ∈ X with ρ(X) 6= 0. In addition, assume - without loss of generality - that ρ(0) = 0. As a consequence, X is not contained in the set G = {Y ∈ X |ρ(Y ) < 0}. The convex set G contains an interior point, and we may apply Lemma 7.1. This lemma states that there exists a non-zero continuous linear functional l on X such that for X given:

l(X) ≤ inf l(Y ). Y ∈G By monotonicity, Y ≥ 0. In addition, we have for all λ ≥ 0:

l(X) ≤ l(1 + λY ) = l(1) + λl(Y ).

It follows that this is only true if l(Y ) is non-negative (by letting λ ↑ ∞). Moreover, l(1) > 0. Take some Y ∈ Cρ such that ||Y || < 1. l(Y ) can be decomposed as follows: l(Y ) = l(Y +) − l(Y −) > 0. Positivity of l(Y ) implies that l(Y +) > 0 and by construction l(1 − Y +) ≥ 0. Hence, l(1) = l(Y +) + (1 − Y +) > 0. l(Y ) By Lemma 7.2, we know that EQX [Y ] can be represented by l(1) . So,

αmin(QX ) := sup EQX [−Y ] Y ∈Cρ

≥ sup EQX [−Y ] Y ∈G⊂Cρ

= − inf EQX [Y ] Y ∈G = − inf l(Y )/l(l). Y ∈G

The set G := {X ∈ X |ρ(X) < 0} ⊂ {X ∈ X |ρ(X) ≤ 0} := Cρ. However, for any X ∈ Cρ, one can construct an interval [X,X + )( > 0) such that there exists an element Y ∈ [X,X + ) that is also contained in G. Hence, αmin(QX ) = − infY ∈G l(Y )/l(l). Finally, we have that 1 EQX [−X] − αmin(QX ) = [ inf l(Y ) − l(X)] l(1) Y ∈G ≥ 0 = ρ(X).

Concluding, ρ(X) = sup 1 ( Q[−X] − αmin(Q)). Q∈Mf E Proof of Theorem 2.2. (6) ⇒ (2). Consider the following expression:

m = sup (EQ[−X] − αmin(Q)). 1 Q∈M (Ω,F,P)

We have to show that X + m ∈ Cρ. If this holds, then we know by Theorem 2.1 that ρ(X) can be represented by m. Suppose that X + m∈ / Cρ. The ∞ 1 ∞ topological space (L , σ(L ,L )) is locally convex and Cρ is weak* closed with respect to L∞. So, we may apply Lemma 7.3 with the weak* closed convex set G = Cρ and the compact convex set H = {m + X}. In fact, there exists a continuous linear functional l on (L∞, σ(L1,L∞)) such that

inf l(Y ) > l(m + X) > −∞. Y ∈Cρ l(Y ) is a Lesbegue integral and it turns out that l(Y ) = E[YZ] for some Z ∈ L1. In addition, Z is non-negative. Note that ρ(λY ) ≤ ρ(0) for any λ, Y ≥ 0 by monotonicity. Thus,

−∞ < l(m + X) < l(λY + ρ(0)) = λl(Y ) + l(ρ(0)).

It follows that l(Y ) is non-negative (by letting λ ↑ ∞) and hence Z ≥ 0. Even stronger, [Z > 0] > 0, because l(Y ) 6= 0. Thus, dQ := Z is a valid P dP E[Z] Radon-Nikodym derivative. So, we get for αmin(Q):

αmin(Q) := sup EQ[−Y ] Y ∈Cρ

= − inf EQ[Y ] Y ∈Cρ

infY ∈C l(Y ) = − ρ . E[Z] Further,

l(m + X) EQ[X + m] = E[Z] infY ∈C l(Y ) < ρ E[Z] := −αmin(Q).

This contradicts our deﬁnition of m = sup 1 ( [−X] − α (Q)). Q∈M (Ω,F,P) EQ min Thus, X + m must be an element of Cρ. (2) ⇒ (1). Obvious. (1) ⇒ (4). Suppose that ρ(X) can be represented by some penalty function 1 on M (Ω, F, P). By Lesbegue’s Dominated Convergence Theorem, limn↑∞ EQ[Xn] → EQ[X]. Hence,

ρ(X) = sup (lim EQ[−Xn] − α(Q)) 1 n↑∞ Q∈M (Ω,F,P) ≤ lim inf sup (EQ[−Xn] − α(Q)) n↑∞ 1 Q∈M (Ω,F,P) = lim inf ρ(Xn). n↑∞

(4) ⇒ (3). By monotonicity, ρ(Xn) ≤ ρ(X) if Xn & X almost surely. So, ρ(Xn) % ρ(X). (3) ⇒ (5). If the set B = {ρ ≤ c} is weak* closed, then ρ is lower semi- ∞ continuous. Let Br = X ∈ L ||X||∞ ≤ r, r > 0 . If there exists a se- 1 quence (Xn)n∈N converging to X almost everywhere in L , then the Fatou 1 property implies that X ∈ Br. So, Br is closed in L . By Lemma 7.4, B is weak* closed.

(5) ⇒ (6). Obvious. Proof of Theorem 3.1. ∞ (1) ⇒ (3). A conditional monetary risk measure ρτ,θ(X) on Rτ,θ is said to n ∞ be continuous for bounded decreasing sequences (X )n∈N in Rτ,θ if limn→∞ n ρτ,θ(X ) = ρτ,θ(X) almost surely for every bounded decreasing sequence n ∞ Xt → Xt for all t ∈ N (X ∈ Rτ,θ). In order to show that (1) implies (3), take n ∞ ∞ a bounded decreasing sequence (X )n∈N in Rτ,θ which converges to X ∈ Rτ,θ n almost surely. This convergence also implies that limn→∞ h−X , aiτ,θ = n h−X, aiτ,θ almost surely and by monotonicity, ρτ,θ(X ) is increasing in n n n. Hence, limn→∞ ρτ,θ(X ) exists almost surely and limn→∞ ρτ,θ(X ) ≤ k ρτ,θ(X) almost surely. To show the converse, take a sequence (a )k∈N in Qτ,θ such that D kE k ρτ,θ(X) = sup −X, a − (αmin(a ))τ,θ . τ,θ k∈N

n k k n By deﬁnition, −X , a τ,θ − (αmin(a ))τ,θ ≤ ρτ,θ(X ) (remember that n ρτ,θ(X ) is the supremum of the left hand side). Hence, for all k ∈ N: D kE k D n kE k −X, a − (αmin(a ))τ,θ = lim −X , a − (αmin(a ))τ,θ τ,θ n→∞ τ,θ n = lim ρτ,θ(X ) a.s. n→∞ n Concluding, ρτ,θ(X) ≤ limn→∞ ρτ,θ(X ) almost surely. (3) ⇒ (2). The proof is based on the Krein-Smulian Theorem. In particular, we refer the reader to Lemma 3.11 of Cheridito et al. [21]. ∞ (2) ⇒ (1). We prove that the following inequality holds for all X ∈ Rτ,θ: n o ρτ,θ(X) ≥ sup h−X, aiτ,θ − (αmin(a))τ,θ . a∈Qτ,θ

To prove this inequality, ﬁx some X + ρτ,θ(X) ∈ Cρ. Hence, for a ∈ Qτ,θ: (α (a)) ≥ −X − ρ (X)1 , a min τ,θ τ,θ [τ,∞) τ,θ = h−X, ai − ρ (X)1 , a τ,θ τ,θ [τ,∞) τ,θ = h−X, ai − ρ (X) 1 , a τ,θ τ,θ [τ,∞) τ,θ

= h−X, aiτ,θ − ρτ,θ(X). To prove the converse, consider the following expression: n o m = sup h−X, aiτ,θ − (αmin(a))τ,θ . (7.1) a∈Qτ,θ

Suppose that X + m1[τ,∞) ∈/ Cρ. We may apply Lemma 7.3 with the ∞ 1 σ(R , A )-closed convex set G = Cρ and the compact convex set H = 1 X + m1[τ,∞) . In fact, there exists some a ∈ (Aτ,θ)+ such that

X + m1[τ,∞), a < inf hY, ai . (7.2) Y ∈Cρ

It follows that there also exists some B ∈ Fτ with P[B] > 0 such that X + m1 , a < inf hY, ai on B. [τ,∞) τ,θ τ,θ Y ∈Cρ

Construct now the following process b ∈ Qτ,θ: a b = 1B + 1Bc 1[τ,∞). h1, aiτ,θ 38 Note that h1, aiτ,θ > 0. So, it follows by (7.2) that

hX, biτ,θ + m < inf hY, biτ,θ Y ∈Cρ

= − sup h−Y, biτ,θ Y ∈Cρ

:= −(αmin(b))τ,θ for all m.

38 n o Please note that for A = h1, aiτ,θ = 0 , 1A| hY, aiτ,θ | ≤ 1A h|Y |, aiτ,θ ≤ n o 1A||Y || h1, aiτ,θ = 0, which implies that B ⊂ h1, aiτ,θ > 0 . This contradicts (7.1), and hence X + m1[τ,∞) ∈ Cρ. As a consequence, n o ρτ,θ(X) ≤ sup h−X, aiτ,θ − (αmin(a))τ,θ , a∈Qτ,θ because ρτ,θ(X) is the smallest amount of money that would have to be added to X to make it acceptable.

7.2 Solving BSDEs: A Numerical Approach

Throughout, we assume that (Ft)t∈[S,T ] is the augmented ﬁltration generated by Brownian motion. This appendix provides a numerical solution scheme for the following forward-backward stochastic diﬀerential equation:

−dYt = g(t, Xt,Yt,Zt)dt − ZtdWt,YT = ξ, (7.3) where ξ = f(XT ) and

dXt = µ(t, Xt)dt + σ(t, Xt)dWt,XS = c.

µ(t, Xt) and σ(t, Xt) are supposed to be uniformly Lipschitz continuous with respect to Xt.

Yt satisfies the following integral equation: Z T Z T Yt = ξ + g(s, Xs,Ys,Zs)ds − ZsdWs. (7.4) t t An adapted solution of the backward stochastic differential equation (7.3) defined by g and X is a pair (Yt,Zt) of progressively measurable processes d with values in R × R . The g-coefficient only depends on the uncertainty through the state of Markov process, and more specifically through the state of diffusion process. Our application of Section 5 can be handled within this framework. In fact, the surplus ratio process satisfies the Markov property (i.e. memorylessness) and the g-coefficient depends on t and Xt through Zt. The additional process Zt may be interpreted as an risk-adjustment term and is needed to have adapted solutions. The BSDE (7.3), at least in the non-linear case, was first introduced and studied by Pardoux & Peng [58]. To find a convenient numerical solution scheme is still a challenging task. There have been several numerical methods proposed in the literature. Much has been done on the investigation of numerical schemes for BSDEs with Lipschitz continuous coefficients or other special cases. Most notable are the contributions of Bally [7], Chevance [24], Ma et al. [53] and Zhang [69]. The numerical implementation is, however, less trivial for BSDEs with coefficients of quadratic growth. A nice way to overcome problems related to coefficients of quadratic growth consists of using a concept from PDE theory (see Imkeller et al. [46]). This concept is known as the exponential Cole- Hopf transformation. Under the assumption of Lipschitz continuity of g, Appendix 7.2.1 and 7.2.2 describe a numerical solution scheme for one- and multi-dimensional BSDEs. Finally, Appendix 7.2.3 considers a numerical solution scheme for BSDEs with coefficients of quadratic growth.

7.2.1 One-dimensional Case Our method presented here is based on Peng & Xu [59]. Numerical scheme for the case d = 1. n n Suppose that a discretization (X1 , ..., Xn ) of Xt is given. n 1. Let n ∈ N. Simulate a sequence {i }i=1,...,n of Bernoulli variables such that 1 [n = 1] = [n = −1] = . P i P i 2

n √1 Pk n n n n 2. Compute Wk = n i=1 i and ∆Wk = Wk+1 − Wk . n n 3. Please remark that ξ is Fn -measurable and so there exists some func- n n n n n n n n n tion Φ such that ξ = Φ (1 , 2 , ..., n). Put Yn = ξ and Zn = 0. 4. Consider now the following equation: 1 Y n = Y n + gn (n − 1,Xn ,Y n ,Zn ) − Zn ∆W n. (7.5) n−1 n n−1 n−1 n−1 n−1 n n−1 n n + − We describe a procedure to solve for Zn−1. First, deﬁne Yn and Yn as follows:

+ n n n − n n n Yn = Φ (1 , ..., n−1, 1) and Yn = Φ (1 , ..., n−1, −1).

n Zn−1 is now the unique solution of the following set of equations: 1 1 Y n = Y + + gn (n − 1,Xn ,Y n ,Zn ) − Zn √ n−1 n n−1 n−1 n−1 n−1 n n−1 n 1 1 Y n = Y − + gn (n − 1,Xn ,Y n ,Zn ) + Zn √ . n−1 n n−1 n−1 n−1 n−1 n n−1 n √ + − n Yn −Yn This yields Zn−1 = n · 2 . n 5. Solve (7.5) for Yn−1. 6. Consider now the following equation for k = n − 2, ..., 1: 1 Y n = Y n + gn(k, Xn,Y n,Zn) − Zn∆W n. k k+1 k k k k n k k n Solve for Zk by deﬁning (same procedure as under 4):

+ n n n − n n n Yk+1 = Φ (1 , ..., k , 1) and Yk+1 = Φ (1 , ..., k , −1).

n Finally, solve for Yk .

Remark 7.1

1. W n → W as n → ∞.

2. gn(k, Xn,Y n,Zn) → g(t, X, Y, Z) as n → ∞.

3. Convergence result (provided that Y is bounded):

Z T Z T n n n (Y , Zs dWs ) → (Y, ZsdWs) as n → ∞. t t

7.2.2 Multi-dimensional Case We now present a numerical solution scheme for the multi-dimensional case d (i.e. Zt ∈ R ). Suppose that we discretize a BSDE by means of a simple Euler scheme, i.e.: 1 Y n = Y n + gn(k, Xn,Y n,Zn) − Zn∆W n,Y n = ξn. k k+1 k k k k n k k n n n n Given a pair of solutions (Yk+1,Zk+1), there is no pair of Fk -measurable n n solutions (Yk ,Zk ) that satisﬁes this equation. By taking conditional expectations, we obtain a workable scheme: 1 Y n = [Y n |F n] + gn(k, Xn,Y n,Zn) ,Y n = ξn, (7.6) k E k+1 k k k k k n n n n n Zk = n · E[Yk+1∆Wk ]. (7.7) Proof. See for instance Zhang [69] (Section 6 and Lemma 2.7).

It is a well-known fact that the pair of processes (Yt,Zt) can be expressed as a deterministic function of Xt−1 (i.e. (Yt,Zt) = (u(Xt−1), v(Xt−1)). So in order to estimate the conditional expectations in (7.6) and (7.7), we can n n n n rely on a regression of Yk+1 and Yk+1∆Wk on Xk . This non-parametric regression can be carried out in different ways. For instance, Carrière[16] use a classical kernel smoothing regression method, Bouchard et al. [12] and Bouchard & Touzi [13] consider a Malliavin approach (based on Malliavin calculus) and Breiman et al. [14] investigated classification and regression trees. We base our estimation on kernel smoothing regression with a Gaus- sian kernel function.

7.2.3 A Transformation Method We use a transformation method (in PDE theory known as the exponential Cole-Hopf transformation) to obtain a BSDE with a Lipschitz continuous coeﬃcient. Our main reference is Imkeller et al. [46]. The following theorem is based on the transformation P = eγY and Q = γP Z.

d Theorem 7.1 For γ ∈ R, let g :[S, T ] × R × R → R be of the form: γ g(t, Y ,Z ) = l(t, Y ) + a(t, Z ) + ||Z ||2, t t t t 2 t where l and a are measurable, l is uniformly Lipschitz continuous in Y , a is uniformly Lipschitz continuous and homogeneous in Z, and l and a are continuous in t. Then the modiﬁed BSDE

Z T Z T γξ Pt = e + G(s, Ps,Qs)ds − QsdWs, t t where the coefficient is defined by logP G(t, P ,Q ) = γP l(t, t ) + a(t, Q ) t t t γ t simplifies (7.4) and features a uniformly Lipschitz continuous coefficient G.

Proof. See Imkeller et al. [46]. 7.3 Figures

Figure 2: Expected Entropic Risk Measure Process - Funding level 100%

Figure 3: Expected Entropic Risk Measure Process - Funding level 70% Figure 4: Scenario Entropic Risk Measure Process

Figure 5: Expected Risk Measure Process based on Quantile Function Figure 6: Expected Entropy Coherent Risk Measure Process (with Ambiguity)

Figure 7: Entropic Risk Measure vs. Entropic Risk Measure for Random Processes Figure 8: Entropic Risk Measure vs. Entropic Risk Measure for Random Processes

Figure 9: Three Diﬀerent Pension Funds Figure 10: Correlation between Interest Rate Risk and Equity Risk

Figure 11: Entropic Risk Measure vs. Entropic Risk Measure for Random Processes References

[1] Ang A., G. Bekaert & M. Wei (2008). “The Term Structure of Real Rates and Ex- pected Inflation”, Journal of Finance, American Finance Association, vol. 63(2), 797-849, 04. [2] Arrow K.J. (1965). “Aspects of the Theory of Risk-Bearing”. [3] Arrow K.J. (1971). “Essays in the theory of risk-bearing”, North-Holland, Amster- dam. [4] Artzner P., F. Delbaen, J.M. Eber and D. Heath (1997). “Thinking coherently”, Risk, 10. [5] Artzner P., F. Delbaen, J.M. Eber and D. Heath (1999). “Coherent risk measures”, Mathematical Finance, 9:203-228. [6] Artzner P., F. Delbaen, J.M. Eber, D. Heath and H. Ku (2007). “Coherent multiperiod risk adjusted values and Bellman’s principle”, Annals of Operations Research, 152:5-22. [7] Bally V. (1995). “An approximation scheme for BSDEs and applications to control and nonlinear PDEs”, Preprint. [8] Barrieu P. and N. El Karoui (2009). “Pricing, hedging and optimally designing derivatives via minimization of risk measures”, In: R. Carmona (ed) Volume on Indifference Pricing, Princeton University Press. [9] Bismut J. M. (1973). “Théorieprobabiliste du contrôledes diffusions”, Memoirs of the American Mathematical Society, 176 [10] Blackburn D. W. and A. D. Ukhov (2008). “Individual vs. Aggregate Preferences: The Case of a Small Fish in a Big Pond”, Working paper series. [11] Bion-Nadal, J. (2008). “Dynamic Risk Measures: time-consistency and risk measures from BMO martingales”, Fiance and Stochastics, 12: 219-245. [12] Bouchard B., I. Ekeland and N. Touzi (2002). “On the Malliavin approach to Monte Carlo approximation of conditional expectations”, Preprint. [13] Bouchard B. and N. Touzi (2002). “Discrete-Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations”, Stochastic Processes and their Applications, 111:175-206. [14] Breiman L., J.H. Friedman, R.A. Olshen and C.J. Stone (1983). “Classification and Regression Trees”, Wadsworth. [15] Cairns A.J.G. (1996). “Continuous-Time Pension-Fund Modeling”, in Proceedings of the 6th AFIR Colloquium, Nuremberg, 1: 609-624 [16] CarrièreJ.F. (1996). “Valuation of the Early-Exercise Price for Options using Sim- ulations and Nonparametric Regression”, Insurance: mathematics and Economics, 19:19-30. [17] Chan K.C., A. Karolyi, F.A. Longstaff and A.B. Sanders (1992). “An Empirical Comparison of Alternative Models of the Short-term Interest Rates”, Journal of Finance, Volume 47, No. 3, 1209-1227. [18] Chen, Z. and L.G. Epstein (2002). “Ambiguity, risk, and asset returns in continuous time”, Econometrica 70:1403-1443. [19] Cheridito P., F. Delbaen and M. Kupper (2003). “Coherent and convex risk measures for bounded càdlàgprocesses”, Working Paper. [20] Cheridito P., F. Delbaen and M. Kupper (2005). “Coherent and convex monetary risk measures for unbounded càdlàgprocesses”, Finance and Stochastics, 9(3):369-387. [21] Cheridito P., F. Delbaen and M. Kupper (2006). “Dynamic monetary risk measures for bounded discrete-time processes”, Electronic Journal of Probability, 11:57-106. [22] Cheridito P. and M. Kupper (2006). “Composition of time-consistent dynamic monetary risk measures in discrete time”, Preprint. [23] Cheridito P. and M. Stadje (2008). “Time-inconsistency of VaR and time-consistent alternatives”, Preprint. [24] Chevance D. (1997). “Resolution numerique des ‘equations dioeerentielles stochas- tiques retrogrades”, Ph.D. thesis, Universite de Provence, Provence. [25] Coquet F., J. Méminand J. Slominski (2001). “On weak convergence of filtrations”, Séminaire de ProbabilitésXXXV, Lecture Notes in Mathematics, Springer. [26] Cvitanic J. and I. Karatzas (1999). “On dynamic measures of risk”, Finance and Stochastics, 3:451-482. [27] Delbaen F. (2002). “Coherent risk measures on general probability spaces”, Advances in Finance and Stochastics, Springer, New York. [28] Delbaen F. (2003). “The structure of m-stable sets and in particular of the set of risk neutral measures”, Preprint. [29] Deprez O. and H.U. Gerber (1985). “On convex principles of premium calculation”, Insurance: Mathematics and economics 4, 179-189. [30] Detlefsen K. and G. Scandolo (2005). “Conditional and dynamic convex risk measures”, Finance and Stochastics, 9:539-561. [31] Dufresne D. (1990). “The distribution of a perpetuity, with applications to risk theory and pension funding”, Scandinavian Actuarial Journal, 39-79. [32] Dunford N. and J. Schwartz (1958). “Linear Operators. Part I: General Theory”, Interscience Publishers, New York. [33] El Karoui N., S. Peng and M. Quenez (1997). “Backward stochastic differential equations in finance”, Mathematical Finance, 7:1-71. [34] Ellsberg D. (1961). “Risk, Ambiguity, and the Savage Axioms”, Quarterly Journal of Economics, 75(4): 643-669. [35] Epstein L.G. and M. Schneider (2003). “Recursive multiple-priors”, Journal of Eco- nomic Theory, 113:1-31. [36] Fisher I. (1896). “Appreciation and Interests”, Publications of the American Eco- nomic Association, 11, 21-29. [37] FöllmerH. and A. Schied (2002). “Convex measures of risk and trading constraints”, Finance and Stochastics, 6:429-447. [38] FöllmerH. and A. Schied (2002). Stochastic Finance - An Introduction in Discrete Time, de Gruyter, Berlin. [39] FöllmerH. and I. Penner (2006). “Convex risk measures and the dynamics of their penalty functions”, Statistics and Decisions, 24:61-96. [40] Frittelli M. and E. Rosazza Gianin (2002). “Putting order in risk measures”, Journal of Banking and Finance, 26:1473-1486. [41] Frittelli M. and E. Rosazza Gianin (2004). “Dynamic Convex Risk Measures”, In: G. Szeg. Risk Measures for the 21ste century, Chapter 12, Wiley Finance. [42] Gerber H.U. (1980). “An Introduction to Mathematical Risk Theory”, Huebner Foun- dation. [43] Gilboa I. and D. Schmeidler (1989). “Maxmin expected utility with non-unique prior”, Journal of Mathematical Economics, 18:141-153. [44] Goovaerts M.J., R. Kaas and R.J.A. Laeven (2010). “A note on additive risk measures in rank-dependent utility”, Preprint. [45] Huber P. (1981). Robust Statistics, Wiley, New York. [46] Imkeller P., J. Zhang and G. dos Reis (2010). “Results on numerics for BSDE with drivers of quadratic growth”, Contemporary Quantitative Finance (Essays in Honour of Eckhard Platen), Springer. [47] Jobert A. and L.C.G. Rogers (2008). “Valuations and dynamic convex risk measures”, Mathematical Finance, 18(1):1-22. [48] John F. and L. Nirenberg (1961). “On functions of bounded mean oscillation”, Com- munications on Pure and Applied Mathematics 14: 415-426. [49] Kehoe P.J. (1983). “Dynamics of the current account: Theoretical and empirical analysis”, Working paper, Harvard University, Cambridge, MA. [50] Kobylanski M. (2000). “Backward stochastic differential equations and partial differential equations with quadratic growth”, Annals of Probability, 28(2):558-602. [51] Kydland F.E. and E.C. Prescott (1982). “Time to build and aggregate fluctuations”, Econometrica, 50:1345-1370. [52] Laeven R. and M. Stadje (2010). “Dynamic Entropy Coherent and Entropy Convex Measures of Risk”, Preprint. [53] Ma J., P. Protter and J.M. Yong (1994). “Solving forward-backward stochastic differential equations explicitly - a four step scheme”, Probability Theory and Related Fields 98, 3:339-359. [54] Markowitz H.M. (1952). “Portfolio Selection”, Journal of Finance, 7:77-91. [55] Merton R.C. (1969). “Lifetime portfolio selection under uncertainty: the continuous time case”, Review of Economics and Statistics, 51:247-257. [56] Neumann J. von and O. Morgenstern (1944). “Theory of Games and Economic Be- havior”, Princeton University Press, Princeton, NJ. [57] Ornstein, L.S. and G.E. Uhlenbeck (1930). “On the theory of Brownian Motion”, Physical Review, 36:823-841. [58] Pardoux E. and S. Peng (1990). “Adapted solution of a backward stochastic differential equation”, Systems and Control Letters, 14:55-61. [59] Peng S. and M. Xu (2009). “ Numerical Algorithms for 1-d Backward Stochastic Differential Equations: Convergence and Simulations”, Working paper. [60] Pratt J. W. (1964). “Risk Aversion in the Small and in the Large”, Econometrica, 32:122-136. [61] Raskin R. and M.J. Cochrane (1986). “Interpretations and transformations of scale for the Arrow-Pratt absolute risk aversion coefficient: implications for generalized stochastic dominance”, Western Journal of Agricultural Economics, 11:204-210. [62] Riedel F. (2004). “Dynamic coherent risk measures”, Stochastic Processes and their Applications, 112(2):185-200. [63] Roorda B., J. M. Schumacher and J. Engwerda (2005). “Coherent acceptability measures in multiperiod models”’, Mathematical Finance, 15(4):589-612. [64] Scandolo G. (2003). “Risk measures in a Dynamic Setting”, PhD thesis. [65] Schmeidler D. (1986). “Integral representation without additivity”, Proceedings of the American Mathematical Society, Vol. 97, 255-261. [66] Schmeidler D. (1989). “Subjective probability and expected utility without additivity”, Econometrica, Vol. 57, 571-587. [67] Vasicek O. (1977). “An Equilibrium Characterization of the Term Structure”, Journal of Financial Economics, 5:177-188. [68] Wang T. (1999). “A class of dynamic risk measures”, Working Paper, University of British Columbia. [69] Zhang J. (2004). “A numerical scheme for BSDEs”, Annals of Applied Probability, 1:459-488. [70] Zitkovic G. (2002). “A filtered version of the Bipolar Theorem of Brannath and Schachermayer”, Journal of Theoretical Probability 15: 41-61.