Simon Dietz and Oliver Walker Ambiguity and insurance: capital requirements and premiums

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1 Simon Dietz and Oliver Walker Ambiguity and insurance: capital requirements and premiums Article (Accepted version) (Reereed) Original citation: Dietz, Simon and Walker, Oliver (2016) Ambiguity and insurance: capital requirements and premiums. Journal o Risk and Insurance. ISSN The American Risk and Insurance Association This version available at: Available in LSE Research Online: December 2016 LSE has developed LSE Research Online so that users may access research output o the School. Copyright and Moral Rights or the papers on this site are retained by the individual authors and/or other copyright owners. Users may download and/or print one copy o any article(s) in LSE Research Online to acilitate their private study or or non-commercial research. You may not engage in urther distribution o the material or use it or any proit-making activities or any commercial gain. You may reely distribute the URL ( o the LSE Research Online website. This document is the author s inal accepted version o the journal article. There may be dierences between this version and the published version. You are advised to consult the publisher s version i you wish to cite rom it.

2 Ambiguity and insurance: capital requirements and premiums Simon Dietz and Oliver Walker November 21, 2016 Abstract Many insurance contracts are contingent on events such as hurricanes, terrorist attacks or political upheavals, whose probabilities are ambiguous. This paper oers a theory to underpin the large body o empirical evidence showing that higher premiums are charged under ambiguity. We model a (re)insurer who maximises proit subject to a survival constraint that is sensitive to the range o estimates o the probability o ruin, as well as the insurer s attitude towards this ambiguity. We characterise when one book o insurance is more ambiguous than another and general circumstances in which a more ambiguous book requires at least as large a capital holding. We subsequently derive several explicit ormulae or the price o insurance contracts under ambiguity, each o which identiies the extra ambiguity load. JEL Classiication Numbers: D81, G22. Keywords: ambiguity, ambiguity aversion, ambiguity load, capital requirement, catastrophe risk, insolvency, insurance, more ambiguous, reinsurance, ruin, uncertainty, Solvency II. ESRC Centre or Climate Change Economics and Policy and Grantham Research Institute on Climate Change and the Environment, London School o Economics and Political Science. Department o Geography and Environment, London School o Economics and Political Science. Vivid Economics Ltd, London. We are very grateul to the editor, three anonymous reerees, Pauline Barrieu and Trevor Maynard or comments on previous drats. We would like to acknowledge the inancial support o Munich Re, the UK s Economic and Social Research Council and the Grantham Foundation or the Protection o the Environment. The usual disclaimer applies. or correspondence: s.dietz@lse.ac.uk. 1

3 1 Introduction Many (re)insurance contracts are contingent on events such as hurricanes, terrorist attacks or political upheavals whose probabilities are not known with precision. Such contracts are said to be subject to ambiguity. There may be several reasons why contracts are subject to ambiguity, including a lack o historical, observational data, and the existence o competing theories, proered by competing experts and ormalised in competing orecasting models, o the causal processes governing events that determine their value. For example, ambiguity is a salient eature in the insurance o catastrophe risks such as hurricane-wind damage to property in the southeastern United States. Here, historical data on the most intense hurricanes are limited, and there are competing models o hurricane ormation (Bender et al., 2010; Knutson et al., 2008; Ranger and Niehoerster, 2012). This ambiguity is increased by the potential role o climate change in altering the requency, intensity, geographical incidence and other eatures o hurricanes. There is by now a body o evidence to show that, aced with oering a contract under ambiguity, insurers increase their premiums, limit coverage, or are unwilling to provide insurance at all. Much o the academic evidence is survey-based: actuaries and underwriters rom insurance and reinsurance companies are asked to quote prices or hypothetical contracts in which the probabilities o loss are alternatively known or unknown (Cabantous, 2007; Cabantous et al., 2011; Hogarth and Kunreuther, 1989, 1992; Kunreuther et al., 1993, 1995; Kunreuther and Michel-Kerjan, 2009). Their responses reveal that prices or contracts under ambiguity exceed prices or contracts without ambiguity and with equivalent expected losses, which is consistent with ambiguity aversion 1 and thus in line with a much larger body o evidence on decision-making, starting with Ellsberg s classic thought experiments on choices over ambiguous and unambiguous lotteries (Ellsberg, 1961). In the industry, one can ind guidance that insurers should increase their prudential margins (i.e. capital holdings) under ambiguity (e.g. Barlow et al., 1993) and below we explain how this leads to higher premiums. Yet, despite the evidence, there is seemingly little theoretical work that can explain or ormally motivate these ambiguity loadings. In this paper we seek to ill this hole by oering a ormal analysis o the connection between, on the one hand, ambiguous inormation about the perormance o a book o insurance and, on the other hand, the premium charged or a new contract. We do so via the capital held against the book: our starting point is a wellknown model o the price o insurance, according to which the objective is to maximise expected proits subject to a survival constraint (thus in the tradition o Stone, 1973), which is imposed by managerial or regulatory iat 1 We give ormal deinitions o ambiguity, ambiguity aversion and related concepts later. 2

4 out o concern or ensuring solvency or avoiding a downgrading o credit. An example o such a constraint, imposed by regulation, is the European Union s new Solvency II Directive (where it is called a Solvency Capital Requirement). Our twist is that the capital held is sensitive to the range o estimates o the probability o ruin and to the insurer s attitude towards ambiguity in this sense. Based on recent contributions to the theory o decision-making under ambiguity, we characterise circumstances in which one book o insurance is more ambiguous than another, and establish general conditions under which more ambiguous books entail higher capital holdings under our capitalsetting rule. We then use the rule to derive pricing ormulae or ambiguous contracts in a way that isolates the additional ambiguity load, distinct rom the more amiliar risk load. We examine the properties o the ambiguity load under dierent assumptions about the insurer s inormation: it is shown to depend on the ambiguity o the contract being priced, as well as the insurer s ambiguity aversion. It also depends on the relationship between the ambiguity o the new contract and the ambiguity o the pre-existing book, while under some circumstances it can interact with the coventional risk load. We hope that these pricing ormulae, or urther extensions and reinements o them, may prove practically useul in the industry: one o the consequences o the lack o existing theory is that the practice o loading contract prices under ambiguity does not appear to have been codiied and may oten use back-o-the-envelope calculations and heuristics (Hogarth and Kunreuther, 1992). Our paper is a complement to recent work on how ambiguity, and ambiguity aversion, on the part o would-be policyholders aects the characteristics o optimal insurance contracts (Alary et al., 2013; Gollier, 2014). In this work, the insurer is taken to be ambiguity-neutral, whereas our insurer is ambiguity-averse. Our paper is also related to recent work on ambiguity aversion and robust control that has taken a similar approach, but applied it to dierent problems. Notable examples include Garlappi et al. (2007) on portolio selection, and Zhu (2011) on catastrophe-risk securities. Finally, our paper oers an alternative approach to previous work in the literature on insurance that has also considered ambiguity under the auspices o model uncertainty (e.g. Cairns, 2000). This work also eatures ambiguity-neutral insurers, because it is assumed that they reduce compound lotteries à la probabilistic sophistication (Machina and Schmeidler, 1992; Epstein, 1999). The rest o the paper is organised as ollows. Section 2 presents the decision problem ormally. Section 3 considers the relationship between how ambiguous a book o insurance is and how much capital the insurer must hold, drawing on elements o Jewitt and Mukerji s (2012) characterisation o the more ambiguous relation. Section 4 then derives explicit pricing ormulae 3

5 or insurance contracts under ambiguity. Finally, Section 5 concludes with a discussion o the descriptive and normative appeal o our capital-setting rule, and some interpretation o our results. 2 The insurer s decision problem We take the point o view o an insurer who aces uncertainty over the perormance o its book and wishes to maximise expected proits subject to a survival constraint. The classic treatment o this problem characterises the insurer s uncertainty using a single probability measure over the space o events determining the book s return (e.g. Kreps, 1990). Under this account the insurer may control the likelihood o insolvency/ruin by choosing a capital holding, since the likelihood o ruin is then simply the probability that the book s losses are not covered by the capital. As we explained in the Introduction, there are, however, important cases where the insurer s inormation does not take the orm o a single probability measure over a space o relevant scenarios. In such cases, it may entertain a multiplicity o possible measures over the space o payo-relevant events and not be certain which o them correctly quantiies the uncertainty it aces. Such an insurer is said to ace ambiguity. We model this kind o insurer s inormation as ollows. There is a metric space, S, known as the state space, that consists o all o the possible states o the world that are relevant to the perormance o an insurance book, with the Borel σ-algebra on S denoted B. A book is then a B-measurable mapping rom S to R. We denote the ull set o books by F and, where F, interpret (s) = x as the statement that i s turns out to be the true state o the world, book will return the monetary quantity x. A book is thus identical to a Savage act (in the sense o Savage, 1954) and, in terms o the insurer s capital setting and pricing, both the pre-existing insurance portolio and the new contract to be potentially added to this portolio may be regarded as books. In the classic account o this problem, the insurer is assumed to possess a single probability measure on B, representing its inormation about payorelevant events. We, however, wish to allow or cases where the insurer aces ambiguity and thereore endow it with a set o measures on B, Π, encompassing all probability measures it believes might characterise its uncertainty correctly. We reer to Π as the set o models, and require Π to be compact and convex. Where the insurer s book depends, or example, on weather events, Π may consist o a set o seasonal orecasts, one o which is assumed to be correct insoar as it accurately measures the likelihood o any member o B. Where B Π is a Borel σ-algebra on Π, let ν be the probability measure on B Π to represent the insurer s belies about which o the models in Π is 4

6 correct. We require supp(ν) = Π. Using B R or the Borel σ-algebra on R, or any F we can deine the probability measure P on B R as ollows: P (E) = ˆ Π ( ) π 1 (E) dν or any E B R. In words, given the insurer s belies about Π, P (E) gives the probability the insurer places on its book paying out some amount in E. Throughout this paper, we adopt the convention o using P (y) or P ({x : x < y}): the probability, given belies ν over the measures Π, that pays out less than y. Let us take as our starting point a amiliar model o insurance pricing based on maximising expected proit subject to a survival constraint, in the tradition o Stone (1973). 2 In this model, there is an insurer who, given any book F, sets its capital holding, Z, as ollows: Z = min{x : P ( x) θ} (1) That is, Z is the smallest holding such that the probability o losses exceeding it is no more than some benchmark level θ (we take it or granted here and throughout the paper that (1) well deines Z ). Given how we deine P (.), one can alternatively think o x as the Value at Risk o book with respect to the conidence level 1 θ. The requirement that the insurer holds Z may be interpreted as a managerial or regulatory constraint with the magnitude o θ representing the conservatism o the regime responsible or it. The insurer thus ocuses on the single probability as measured by P o its book paying out less than its capital holding. We extend this ramework to allow the capital holding to depend on both the range o models Π and the insurer s attitude to ambiguity about the risk o ruin. Speciically, our insurer sets Z according to { [ ] [ ] } Z = min x : ˆα max P π ( x) + (1 ˆα) min P π ( x) θ π Π π Π where ˆα [0, 1], and the measure P π on B R is deined as P π (E) = π ( 1 (E) ). In contrast to (1), (2) requires the insurer to consider the dispersion in Π. 3 2 Ignoring, however, his stability constraint on the volatility o the ratio o losses to expenses. 3 Notice that (2) does not require a probability measure ν on B Π, rather it suices to know the models that yield respectively the maximum and minimum probability o the book paying out less than the insurer s capital holding. However, to say in Section 3 that one book is more or less ambiguous than another, we do require such a probability measure. We also require it to explicitly identiy the ambiguity load in the premium price, which is the purpose o Section 4. (2) 5

7 The weight actor ˆα plays a role akin to the ambiguity attitude parameter α in the well known α-maxmin expected utility (α-meu) representation o choice under ambiguity o Ghirardato et al. (2004). In particular, we show in the next section that a more ambiguous book, which can be deined in terms o the preerences under ambiguity o an α-meu decision-maker inter alia, incurs a higher capital holding Z i and only i the weight actor ˆα 0.5. Thereore ˆα indexes the insurer s aversion to ambiguity about the risk o ruin. Notice that i ˆα = 1 then (2) simpliies to Z = min{x : max π Π P π ( x) θ}, which encodes a concern or robustness in a way analogous to the decision rule in Hansen and Sargent (2008), which itsel has been shown to be equivalent to Gilboa and Schmeidler s (1989) axiomatically ounded maxmin expected utility representation (Hansen and Sargent, 2001). The capital holding aects the premium charged on a new contract, added to the existing portolio, in the ollowing way. An insurer endowed with book who agrees to an additional contract c, itsel a book, ends up with book = + c. We deine the addition operation over F pointwise that is, or, c F, + c = where (s) = (s) + c(s) or all s and note that F is closed under addition i.e. i, c F, + c F. As a result o signing c, it needs to increase its capital holding by Z Z, and i c is competitively priced, then the underwriter s expected proit rom the contract cannot exceed the opportunity cost o this incremental capital holding. Thus, using y or the opportunity cost o capital, i c is competitively priced it must be that: µ c = y ( ) Z Z (3) The expected return on c, µ c, is equal to the price the insurer charges the counterparty to c, p c, less the expected loss on c (including administrative costs), L c, and we say an insurer is competitive 4 whenever it always sets p c such that µ c satisies (3): p c = L c + y ( ) Z Z (4) The purpose o Section 4 is to expand the pricing ormula (4) so that the ambiguity load the uplit on the premium due to ambiguity about the new contract and the existing book can be isolated under various distributional assumptions. 3 Ambiguity and the capital holding Jewitt and Mukerji (2012) provide various choice-based accounts o what it is 4 Note this our approach does not require the assumption o perect competition. y may be interpreted as a managerial target rate o return rather than opportunity cost, in which case it could be consistent with a monopolistic or oligopolistic insurance industry. 6

8 or one act book o insurance in our context to be more ambiguous than another. We ocus on one o these accounts, according to which book is more ambiguous than g whenever any ambiguity-neutral agent is indierent between the two books, any ambiguity-averse agent preers g to, and any ambiguity-seeking agent preers to g. Note that under this deinition what it takes or to be more ambiguous than g depends on what it means or an agent to be ambiguity-averse, -seeking, or -neutral, in keeping with revealed-preerence traditions. To characterise ambiguity attitude we primarily use Ghirardato, Maccheroni and Marinacci s (2004, GMM) axiomatically based α-meu representation o choice under ambiguity, according to which preerences, given by the relation over F, are such that, or any bounded, g F: g α min π Π α min π Π S u ((s)) dπ + (1 α) max π Π u (g(s)) dπ + (1 α) max π Π S S u ((s)) dπ S u (g(s)) dπ (5) where α [0, 1] is an index o ambiguity attitude, u is a continuous, nondecreasing unction representing risk attitude in the usual way, and Π, which is compact and convex, represents the belies revealed by. Where A and B belong to the class o α-meu preerences with common belies Π, A is more (less) ambiguity averse than B i and only i α A ( )α B, and u A and u B are equal up to an aine transormation (GMM, Prop. 12). Furthermore i Π is centrally symmetric, in that there exists a probability measure π known as the centre o Π such that π Π i and only i π (π π ) Π (Jewitt and Mukerji, 2012), then the preerence is ambiguity neutral i and only i α = 0.5. It ollows that, or this centrally symmetric Π, is ambiguity averse (seeking) i and only i α > (<)0.5. In the extreme case where α = 1, the preerence is equivalent to maxmin expected utility in Gilboa and Schmeidler (1989). α-meu is not the only way to characterise choice in a manner that distinguishes the decision-maker s belies and preerences towards ambiguity. An alternative is Klibano, Marinacci and Mukerji s (2005, KMM) smooth representation o choice under ambiguity, and indeed Jewitt and Mukerji (2012) apply their deinitions o more ambiguous to both the GMM and KMM representations. We will ocus on the α-meu representation here, since it is very similar in structure to our capital-setting rule (2), which in turn allows or the derivation o particularly tractable expressions or the price o contracts aected by ambiguity, developed in Section 4. Note that (5) does not constrain the preerences o agents over unbounded books. This means that i we were to deine more ambiguous in terms o the choices o all ambiguity-averse, -neutral, and -seeking agents with preerences consistent with GMM s representation, we would never be able to describe one unbounded book as being more or less ambiguous than 7

9 another. And as we wish to do just this, we characterise more ambiguous relative to a narrower class o preerences than those consistent with GMM s representation. We thus deine P Π as the set o all preerences over F that are consistent with GMM s representation, that rank any, g F according to (5) provided all the expectations in (5) are deined, and that share belies given by Π. The ambiguity attitude o any P Π is determined by α just as in GMM s representation. We say P Π is - constrained i and only i, given the unction u and weight α associated with, α min π Π S u ((s)) dπ + (1 α) max π Π S u ((s)) dπ is deined. Let us now deine what it means or one book to be more ambiguous than another. We denote the symmetric component o using as usual. Deinition 1. For any, g F, is P Π -more ambiguous than g i: [i.] For all - and g-constrained, ambiguity-neutral P Π, g; [ii.] For any - and g-constrained A, B P Π, where A is ambiguity neutral: i B is more ambiguity averse than A, g B ; and i A is more ambiguity averse than B, B g. Where the particular coniguration o belies is unimportant or obvious rom the context, we will omit the qualiication P Π - and simply say is more ambiguous than g. We describe book as unambiguous i, or all g F, either g is more ambiguous than, or is not more ambiguous than g. can only be unambiguous i P π = P π or all π, π Π. An ambiguous book is then any book that is not unambiguous. To state our irst results we require some urther terminology. First, a Markov kernel rom (Π, B Π ) to itsel is any map (π, E) K π (E) such that K π is a probability measure on B Π. For any pair o books and g, we say K π-garbles into g whenever, or all E B R, the ollowing condition holds or all π Π: ˆ Pg π (E) = P π (E)dK π (6) The existence o a π-garbling rom to g implies that the likelihood that g pays out in E conditional on any π is a weighted average o the likelihood that pays E across all π Π. In this sense, s payo depends more sensitively on the realisation o the true probability model than g s does. Let Π be compact, convex and centrally symmetric with centre π. Then Jewitt and Mukerji characterise a Markov kernel K rom (Π, B Π ) to itsel as centre-preserving i and only i, or all E B R, ˆ P π (E) = P π (E)dK π Π Π 8

10 Whenever there is a centre-preserving π-garbling rom into g, P π (E) = Pg π (E) or all E B R. Thus a centre-preserving π-garbling is analogous to a mean-preserving spread amiliar rom the analysis o risk: just as a meanpreserving spread preserves the expected payo o a prospect, but makes this payo more sensitive to the true state o the world, a centre-preserving π-garbling preserves a book s payo-distribution at the centre, π = π, but makes the payo-distribution more strongly dependent on the true model in Π. Given this, the irst result we report rom Jewitt and Mukerji should not come as a surprise. Proposition 1. [Jewitt-Mukerji 1] For any, g F, i there is a centrepreserving π-garbling rom to g then is more ambiguous than g. It thus ollows that i there is a centre-preserving π-garbling rom into g, then any ambiguity-averse agent whose preerences belong to P Π would preer to g. The next step is to show that, given the suicient condition set out in Proposition 1, a more ambiguous book incurs a higher (lower) capital holding than its counterpart g i and only i ˆα ( )0.5. Proposition 2. Suppose Z and Z g are well deined by (2). Then i there is a centre-preserving π-garbling rom to g, Z ( )Z g i and only i ˆα ( )0.5. Proo : See Appendix. Proposition 2 shows that our capital-setting rule encodes ambiguity attitude through the parameter ˆα in a parallel manner to the parameter α in the GMM representation. It implies that, all else being equal, an ambiguityaverse insurer, represented by ˆα > 0.5, will hold a larger amount o capital against the risk o ruin o a more ambiguous book, where the deinition o more ambiguous comes rom Proposition 1. By contrast, an ambiguityneutral insurer with ˆα = 0.5 will hold neither more nor less capital, while an ambiguity-seeking insurer (ˆα < 0.5) will hold less capital. In turn, we can use this result to consider how ambiguity attitude aects the premium charged or a speciic contract: Corollary 1. Let + c = and + c =, and suppose there is a centrepreserving π-garbling rom to. Then on the assumption that L c = L c and y > 0, p c > p c or any insurer setting its capital holding according to (2), with ˆα > 0.5, and its premium price according to (4). The Corollary considers the case where the addition o a new contract c to the existing book results in a more ambiguous book, compared with the addition o an alternative new contract c, which results in book. 9

11 The most straightorward reason or this would be that c is in itsel more ambiguous than c, although the dierence between the ambiguity o the resulting books and could also stem rom ambiguity about how the payos rom the new contracts co-vary with the existing book. The assumption that L c = L c amounts to a condition that the administrative costs o c and c be the same (since both c and c have the same expected loss), while y > 0 will surely hold. The Corollary thus tells us that, when setting premiums or any pair o new contracts, an ambiguity-averse insurer will charge a higher premium or the contract that results in a more ambiguous insurance portolio, all else being equal. In Section 4 we explore circumstances in which book is more ambiguous than under various distributional assumptions. 3.1 U-Comonotonicity Proposition 1 applies to any pair o books under any set o belies, but it provides only a suicient condition or one book to be more ambiguous than another. Thus, Proposition 2 does not establish that (2) encodes ambiguity aversion over all pairs o books. However, we can use a second result rom Jewitt and Mukerji s analysis that provides suicient and necessary conditions or book to be more ambiguous than g, provided, g and Π satisy a certain condition known as U-comonotonicity in relation to each other: 5 Deinition 2. Π is U-comonotone or F F i Π can be placed in a linear order U such that or all non-decreasing bounded unctions u: ˆ ˆ π U π u((s))dπ u((s))dπ or all F S In words, Π is U-comonotone over F i all expected utility maximisers with bounded utility non-decreasing in money and a book belonging to F would agree on a single ranking o which o any pair in Π represented better news about the true probability model. This might be the case, or example, where the set F consisted o books that paid out a ixed sum in case o an extreme weather event: Π could then be ordered such that π U π i and only i π places a lower probability on the extreme weather event than π does. Indeed, this example indicates when it might be plausible to assume U-comonotonicity, namely when each o the set o acts under consideration is stochastically similar, in the sense that the realisation o a π Π has similar consequences or the likelihood o the acts in F producing good or adverse consequences. Jewitt and Mukerji give the example o a pair o bets on the S&P equities index as being stochastically similar (events), 5 Note this is a special case o a more general deinition, which can be ound in Jewitt and Mukerji (2012). S 10

12 in comparison with a pair o bets, where one is on the S&P and the other is on the outcome o a boxing match. The acts in this paper are books o insurance and may thereore be considered stochastically similar, especially since the principal interpretation o Π that we oer is o competing estimates o catastrophic risks, which should aect many books pay-os in a similar ashion. To state Jewitt and Mukerji s characterisation o ambiguity aversion under U-comonotonicity, we require a deinition o the comparative ambiguity o events E, E B R, in addition to the deinition o the comparative ambiguity o acts given previously in Proposition 1. For any pair o payos x and y, xey denotes the binary act that pays x i the realised state s E and y otherwise. Deinition 3. Given events E, E B R, E is a more ambiguous event than E i, or all ambiguity neutral A P, xe y A xey and x( E )y A x( E)y; or all B P, such that B is more ambiguity averse than A, xe y B xey and x( E )y B x( E)y; or all B P, such that A is more ambiguity averse than B, where x > y. xe y B xey and x( E )y B x( E)y, In the speciic context o α-meu preerences, E is a more ambiguous event than E i and only i E is a centre-preserving π-garbling o E or Π compact, convex and centrally symmetric, i.e. where acts and g in (6) are unit bets on E and E respectively. The ollowing Proposition characterises ambiguity aversion under U- comonotonicity, by establishing that events constituting adverse payos under book are more ambiguous events than the corresponding adverse payos under book g. 6 Proposition 3. [Jewitt-Mukerji 2] Suppose Π is compact, convex and centrally symmetric with centre π and is U-comonotone on {, g}. Then the ollowing three statements are equivalent: 6 The result reported here is slightly dierent to that in Jewitt and Mukerji, who deine U-comonotonicity in terms o all non-decreasing (bounded and unbounded) utility unctions but consider only bounded books. Our statement o the result encompasses all books but deines U-comonotonicity in terms o bounded utility unctions; the proo is nonetheless as in Jewitt and Mukerji with obvious modiications. 11

13 [1.] is P Π -more ambiguous than g; [2.] For each x, E x {s S : (s) x}, Ex g {s S : g(s) x} B R, E x is a more ambiguous event than Ex g ; [3.] There is a centre-preserving π-garbling rom to g, or π, π Π, π U π, the map (α, h) P απ+(1 α)π h is supermodular on [0, 1] {, g}. Speciically or 0 α < α 1, Pg απ+(1 α)π P α π+(1 α )π g P απ+(1 α)π P α π+(1 α )π The intuition behind Proposition 3 is that i is more ambiguous than g, its payo distribution is more sensitive to the realisation o the true model in Π. This result allows us to obtain the equivalent o Proposition 2. Proposition 4. I Π is U-comonotone on {, g} and is more ambiguous than g then Z ( )Z g i and only i ˆα ( )0.5. Proo : see Appendix. Proposition 4 has the same implications or the capital holdings o insurers as Proposition 2, the dierence being in the way in which any book o insurance is deined as being more ambiguous than another. An ambiguityaverse insurer with ˆα > 0.5 will hold a larger amount o capital against the risk o ruin o the more ambiguous book in the pair, all else being equal. In addition, Proposition 4 has the same implications or the premium price attached to a new contract that results in a more ambiguous insurance portolio. It will be higher than the premium charged or a new contract that results in a less ambiguous insurance portolio, deined according to Proposition 3, i and only i the insurer is ambiguity-averse with ˆα > Contract pricing under ambiguity We now examine the impact o ambiguity on the price o an individual contract given a capital holding set according to (2) and competitive pricing o premiums according to (4). We derive several pricing ormulae, which show explicitly how introducing ambiguity leads to a departure rom a benchmark pricing ormula in the absence o ambiguity, i.e. we explicitly identiy an additional ambiguity load. Our starting point is a model, inluential in the actuarial literature and in the insurance industry, which utilises inormation about the mean and variance o losses on the new contract, as well as on the existing book (Kreps, 1990). We examine our types o ambiguity, chosen on the basis o their analytical tractability and applicability to real insurance problems. The our cases dier on the distribution o model parameters assumed under the measure ν. 12

14 We only consider contract pricing, thus we ignore deductibles, co-insurance and other design options that an insurer might use to manage ambiguity. These would be interesting avenues or uture research (see Amarante et al., 2015). 4.1 Benchmark: no ambiguity As a benchmark or what ollows we review the case where the insurer s inormation is unambiguous. The set o books under consideration is F 0 F, where F 0 is deined relative to a given Π as ollows: F 0 i the density o (s) under P π on {S, B} or all π Π is parameterised by mean µ and variance σ 2. We deine the addition operation over F 0 pointwise that is, or, F 0, + = where (s) = (s) + (s) or all s and note that F 0 is closed under addition i.e. i, F 0, + F 0. It is worth emphasising that even in this ramework we assume that the insurer sets its capital holding according to rule (2) and we allow Π to be non-singleton implying that, across the class o all books, the insurer may ace some ambiguity. However, because we restrict our ocus in the benchmark case to F 0, the insurer aces no ambiguity and thereore its capital holding rule is equivalent to that in (1). We adopt this approach in order to make clearer the generalisations that ollow to richer sets o books. Where Φ is a standardised cd, given whatever assumption about unctional orm the insurer inds appropriate (e.g. normal, gamma, etc.), and z = Φ 1 (θ), the insurer s capital holding or F 0 is determined by: Given competitive pricing (3) this implies: µ c = Z = zσ µ (7) yz (1 + y) (σ σ ) Recalling that where ρ c, is the correlation coeicient or the random variables c(s) and (s), σ 2 = σ2 + σ2 c + 2σ c σ ρ c,, we have: σ σ = σ c 2σ ρ c, + σ c σ + σ And hence, where R c, := (yz/(1 + y)) (2σ ρ c, + σ c )/(σ + σ ): µ c = R c, σ c Using the general expression or the price o a premium set by a competitive insurer, we can now state Kreps s (1990) more speciic pricing result, the proo o which is immediate rom the analysis above: 13

15 Proposition 5. [Pricing without ambiguity] I, c F 0, then a competitive insurer with book will set p c as ollows: p c = L c + R c, σ c (8) The second element on the right-hand side is the risk load or contract c. Note that it arises solely as a consequence o the insurer s need to limit the probability o ruin to a certain level (encoded in rule (2)): without this constraint the competitive price o the contract would simply be L c. As one would expect, the risk load is increasing in the riskiness o the contract (measured by σ c ), the contract s correlation with the insurer s pre-existing book (ρ c, ), the opportunity cost o capital (y), and it is decreasing in the acceptable probability o loss (increasing in z a decreasing unction o θ). 4.2 Mean uncertain; variance known Mean uniormly distributed Our irst case o ambiguity involves considering a space o books F 1 that, given some Π and ν, satisies 7 : (1.i) or all F 1 and all π Π, (s) under P π on {S, B} has mean µπ and variance σ2 ; (1.ii) or all F 1, µ π is uniormly distributed on [a, b ] given ν on {Π, B Π }; (1.iii) F 1 is closed under addition; and (1.iv) F 0 F 1. Note that it is impossible to satisy the additivity condition without violating (1.ii) unless, or all, F 1 : µ π = a + (µπ a ) (b a ) (b a ) (9) which implies that Π is U-comonotone or F 1. In the cases examined here, a more ambiguous book thereore incurs a higher capital holding as per Propositions 3 and 4 in Section 3. To illustrate where a structure like this might apply, consider the ollowing example. Example 1. Suppose our insurer has a collection o orecasts at its disposal, all o which agree on the payo-variance o any given book, but amongst which there is disagreement over certain books payo-expectations. Speciically, there is a most pessimistic simulation, which reports the lowest mean payo or all the books or book this is a and a most optimistic simulation, which gives the highest reported mean or any book b or book. For any book, it is sure that the variance is as reported σ 2 or book and thinks the true mean must lie somewhere between these optimistic and 7 Note that F 1 may not be unique given Π and ν. This is also the case or F 2, F 3, and F 4 introduced below. 14

16 pessimistic bounds. It constructs Π and ν using two assumptions. First, the members o Π are ordered according to their pessimism so that (9) is satisied and or any c [a, b ], µ π = c or one π Π. Second, ν is set such that condition (1.ii), imposing a uniorm distribution on µ π, holds. The irst assumption may be justiied in case the insurer inds it reasonable, while the second is reasonable provided it has no evidence to suggest any value o µ π in [a, b ] is more plausible than any other, in which case the uniormity o µ π ollows rom the principle o insuicient reason. Under these assumptions, any book it considers belongs to F 1 given Π and ν. Given the decision rule (2), Z is set to reduce the probability o ruin to an acceptable level under the weighted sum o the most pessimistic and optimistic models, i.e. or any F 1 Z = zσ [ˆα a + (1 ˆα) b ] = zσ ˆαa + (ˆα 1)b Where the models are uniormly distributed this implies ( ) Z = zσ + (2ˆα 1) 3sd[µ π ] µ (10) where sd[µ π ] is the standard deviation o the random variable µπ, equal to 1/12(b a ) 2 under the uniormity assumption. We now proceed in parallel to the exposition o the previous sub-section, supposing that a competitive insurer with book F 1 accepts the urther contract c F 1 and thereby ends up with the book = + c. Using (10) and (3) as above, we obtain: Using the act that µ c = R c, σ c + 3y(2ˆα 1) 1 + y ( ) sd[µ π ] sd[µπ ] Var[µ π ] = Var[µ π ] + Var[µ π c ] + 2Cov[µ π, µ π c ] and urthermore that, given (9), sd[µ π ]sd[µπ c ] = Cov [µ π, µπ c ], we can deine A c,,1 [ ] ( 3y(2ˆα 1) 2sd[µ π ] + sd[µ π ) c ] 1 + y sd[µ π ] + sd[µ π ] so that our irst pricing result under ambiguity ollows straightorwardly: Proposition 6. [Pricing with uniorm mean] I, c F 1, then a competitive insurer with book will set p c = L c + R c, σ c + A c,,1 sd[µ π c ] 15

17 It is easy to see how Proposition 6 generalises Proposition 5. I c is unambiguous then it must belong to F 0, in which case sd[µ π c ] = 0 and so p c is set according to (8). However, i c is in F 1 \ F 0 that is to say c has ambiguous returns then p c also incorporates an ambiguity load equal to A c,,1 sd[µ π c ]. The ambiguity load is positive, provided the index o ambiguity aversion ˆα > 0.5. It is increasing in ˆα and in sd[µ π c ], the (approximate) measure o ambiguity in c 8. It is also increasing in the cost o capital, and it is increasing in the ambiguity o the pre-existing book (measured by sd[µ π ]) whenever ˆα > Mean triangularly distributed We now consider an alternative space o books, F 2, deined such that given Π and ν: (2.i) or all F 2 and all π Π, (s) under P π on {S, B} has mean µ π and variance σ2 ; (2.ii) or all F 2, µ π has a symmetric triangular distribution on [a, b ] given ν on {Π, B Π }; (2.iii) F 2 is approximately closed under addition 9 ; and (2.iv) F 0 F 2. Conditions (2.i), (2.iii), and (2.iv) mirror their counterparts in the analysis o a uniorm mean. Once again, (2.ii) and (2.iii) may only be satisied when (9) holds or all, F 2 and Π is U-comonotone or F 2. To illustrate the applicability o F 2, we extend Example 1. Example 2. Suppose the insurer rom Example 1 thinks that, or any book, values o µ π closer to the midpoint o the range [a, b ] are more probable than those urther away rom it, i.e. roughly speaking that models with more extreme orecasts o the mean loss are less likely to be correct. Provided these belies are reasonably approximated by the assumption that µ π is triangularly distributed 10 with minimum a, maximum b and mode (a + b )/2, it might proceed by again assuming the members o Π are ordered according to their pessimism and by setting ν so that (2.ii) is satisied. Given Π and ν thus constructed, every book it considers will belong to F 2. For any F 2 we have: ( ) Z = zσ + (2ˆα 1) 6sd[µ π ] µ 8 See discussions on this point in Jewitt and Mukerji (2012) and Maccheroni et al. (2010). 9 Triangular distributions are not closed under addition, so what this requires is that an insurer inds it appropriate to approximate the sum o two triangular distributions, F 2 with another triangular distribution that is a member o F We choose this distribution as (2) does not well-deine the capital holding unless µ π has a bounded support, however triangular distributions are also requently used to characterise subjective probability distributions in probability-elicitation exercises. 16

18 which, or = + c and, c F 2, yields: 6y(2ˆα 1) ( ) µ c = R c, σ c + sd[µ π 1 + y ] sd[µπ ] ( Where we use A c,,2 to denote ( 6y(2ˆα 1) 2sd[µ π 1+y ) ]+sd[µ π c ] sd[µ ), π ]+sd[µπ ] this gives us: µ c = R c, σ c + A c,,2 sd[µ π c ] rom which the next pricing result is immediate: Proposition 7. [Pricing with triangular mean] I, c F 2, then a competitive insurer with book will set p c = L c + R c, σ c + A c,,2 sd[µ π c ] The result generalises Proposition 5 by incorporating an ambiguity load that is zero or c F 0 and increasing in sd[µ π c ]. Like Proposition 6 it is also increasing in the index o ambiguity aversion ˆα and in the cost o capital, while the relationship between the ambiguity load and the ambiguity o the existing book is the same as beore. However, the switch rom a uniorm distribution to a triangular distribution implies that, or any given standard deviation, the range o possible values increases, thus an ambiguity-averse reinsurer would require a greater ambiguity load. 4.3 Mean known; variance uncertain We now ocus on a space o books, F 3, deined or a given Π and ν such that: (3.i) or all F 3 and all π Π, (s) under P π on {S, B} has mean µ and variance (σ π)2 ; (3.ii) or all F 3, σ π has a uniorm distribution on [a, b ] given ν on {Π, B Π }; (3.iii) F 3 is closed under addition; and (3.iv) F 0 F 3. As in previous sections, additivity and the uniormity o σ π imply that or any, F 3 and π Π, σ π and σπ are linearly related as ollows: σ π = a + (σπ a ) (b a ) (a b ) (11) Unless F 3 = F 0, Π is not U-comonotone or F 3. We imagine this case applying to an insurer in an analogous position to that described by Example 1, except with a range o estimates o the standard deviation o losses and certainty over the mean Though note the appeal to the principle o insuicient reason to justiy the uniormity o σ π or all is weaker here. The insurer could equally invoke the principle to impose the uniormity o ( σ π ) 2, in which case the collection o books it considers could not satisy (3.ii). 17

19 Working as beore, Z is set to reduce the probability o ruin to an acceptable level under the weighted sum o the most pessimistic and optimistic models, i.e. or any F 3 Z = ˆα z b + (1 ˆα) z a µ = z [ˆαb + (1 ˆα)a ] µ [ ( )] = z E[σ π ] + (2ˆα 1) 3sd[σ π ] µ (12) and thus, where, c F 3 and = + c, (3) implies: µ c = yz ( ) 3yz(2ˆα 1) ( ) E[σ π 1 + y σπ ] + sd[σ π 1 + y ] sd[σπ ]) 2σ π ρ c +σ π c σ π +σπ Using the act that E[σ π σπ ] = E[σπ c ], we can urther decompose ) the risk load yz 1+y (E[σ π σπ ] into two terms, irst recovering the equivalent o the risk load in the absence o ambiguity, and second obtaining a term capturing how the risk load depends on ambiguity over σc π : 3yz(2ˆα 1) ( ) µ c = E[R c, σc π ] + sd[σ π 1 + y ] sd[σπ ]) 3yz(2ˆα 1) ( ) = E[σc π ]E[R c, ] + Cov[σc π, R c, ] + sd[σ π 1 + y ] sd[σπ ]) To obtain the ambiguity load, take a similar approach as [ beore, ] using the act that Var[σ π ] = Var[σπ ]+Var[σπ c ]+2sd[σ π]sd[σπ c ]corr σ π, σπ c and, given (11), that sd[σ π]sd[σπ c ] = Cov [σ π, σπ c ]. Thus, deining we have ( 3yz(2ˆα 1) 2sd[σ π ] + sd[σ π ) c ] A c,,3 = 1 + y sd[σ π] + sd[σπ ] µ c = E[σ π c ]E[R c, ] + Cov[σ π c, R c, ] + A c,,3 sd[σ π c ] which gives us our next pricing result: Proposition 8. [Pricing with uniorm standard deviation] I, c F 3, then a competitive insurer with book will set p c = L c + E[σ π c ]E[R c, ] + Cov[σ π c, R c, ] + A c,,3 sd[σ π c ] Once again, whenever c F 0, the pricing ormula above reduces to (8). In contrast to our previous results, however, introducing ambiguity aects the 18

20 price o a contract via two additional terms rather than one. First, as in our earlier results, there is a term, A c,,3 sd[σc π ], that is increasing in the ambiguity o c, sd[σc π ], in the index o ambiguity aversion ˆα, in the cost o capital and the smaller is the acceptable probability o loss (the larger is z). The dependence o the ambiguity load on z is new and ollows immediately rom (12) it is due to the act that ambiguity in this example concerns the variance o returns, rather than mean returns. The second term, Cov[σc π, R c, ], relects the act that uncertainty over σc π leads to uncertainty over the risk load. Since Cov[σc π, R c, ] could in principle depend negatively on the ambiguity o c, the overall ambiguity load could, in contrast to the other cases examined so ar, be negative. 4.4 Mean and variance uncertain As a inal exercise, we consider an inormational structure that nests two o the cases described above: where both the mean and variance are independently uniormly distributed. Thus we consider a space o books, F 4, that satisies: (4.i) or all F 4 and all π Π, (s) under P π on {S, B} ) 2; has mean µ π (σ and variance π (4.ii) or all F4, µ π is uniormly distributed on [a, b ], σ π is uniormly distributed on [a, b ], and µπ and σπ are independent given ν on {Π, B Π }, ; (4.iii) F 4 is closed under addition; and (4.iv) F 0 F 4. Given this deinition, or any pair, F 4, µ π and σ π must satisy conditions (9) and (11) (the latter with obvious relabelling). Apart rom cases where F 4 {F 0, F 1 }, Π is not U-comonotone or F 4. Proceeding in the usual way, we have, or any F 4 : [ ( )] ( ) Z = z E[σ π ] + (2ˆα 1) 3sd[σ π ] + (2ˆα 1) 3sd[µ π ] µ So or, c F 4, a competitive insurer with book prices c such that ) µ c = yz 3yz(2ˆα 1) ) 1+y (E[σ π σπ ] + (sd[σ π 1 + y ] sd[σπ ]) ) (sd[µ π ] sd[µπ ] + 3y(2ˆα 1) 1+y It is then clear that we can progress using steps rom our analyses o F 1 and F 3 above to reach our inal pricing ormula. Proposition 9. [Pricing with independent uniorm mean and standard deviation] Where, c F 4, a competitive insurer with book will oer p c = L c + E[σ π c ]E[R c, ] + Cov[σ π c, R c, ] + A c,,1 sd[µ π c ] + A c,,3 sd[σ π c ] Thus, where books and contracts belong to F 4, the ambiguity load or any contract is the sum o a component (A c,,1 sd[µ π c ]) arising due to ambiguity in the contract s mean, a component (A c,,3 sd[σ π c ]) relecting ambiguity 19

21 in its standard deviation, and there is also the eect o ambiguity about the standard deviation o the contract on the risk load. That A c,,1 sd[µ π c ] and A c,,3 sd[σ π c ] are additive results rom our restriction that the mean and standard deviation are independent; another way o arriving at the same ormula would be to assume that the mean and standard deviation were linearly related, with higher variances corresponding to lower means. 5 Concluding Remarks The main contribution o this paper has been to establish a clear connection between ambiguity and the pricing o (re)insurance. We show, at a general level, that under our capital-setting rule increasing ambiguity leads to higher capital holdings and thus to higher costs, provided the (re)insurer is averse to ambiguity about the risk o ruin. We then show how, under a range o distributional assumptions, our capital-setting rule gives rise to particular pricing ormulae or insurance contracts, all composed o distinct risk and ambiguity loads. These pricing ormulae are testable predictions o the theory. Admittedly we have had to make relatively speciic assumptions about the probability distributions describing ambiguity. Since they must be bounded in order that our capital-setting rule is well deined, we employ uniorm and triangular distributions. However, these two distributional orms have quite strong appeal as characterisations o ambiguous belies. The uniorm distribution ollows rom the application o the principle o insuicient reason, which might oten be deemed appropriate, i or some reason (e.g. insuicient data or dependence o dierent models) the comparative perormance o dierent orecasting models cannot be evaluated. The triangular distribution is also requently used to characterise subjective probability distributions in probability-elicitation exercises. But how tenable is our assumption that the capital-setting rule takes the orm speciied in (2)? From a descriptive perspective, we have already shown that its implications or pricing decisions are consistent with the behavioural evidence in the literature. O the survey-based studies mentioned in the Introduction, Hogarth and Kunreuther (1992) is distinctive in that it provides tentative evidence rom a sample o actuaries o the decision procedures they actually ollowed. There was some evidence o the use o heuristics to load the premium, such as a simple, ad hoc multiplying coeicient on the expected value o the premium, or on the variance o the loss distribution. This is on the ace o it at odds with the mechanics o the decision process posited here. At the same time, however, there was also evidence that actuaries had in mind the eect the new contract would have on the overall risk o the insurer s ruin, as in our ramework. Indeed, the risk o ruin is known to be an 20

22 important consideration more generally when insurers set capital holdings and price contracts, especially or catastrophe risks (e.g. Kunreuther and Michel-Kerjan, 2009). Considering the rule rom a normative perspective, one could evaluate the axiomatic oundations o the similar α-meu rule as set out in Ghirardato et al. (2004). However, decision rules like this are typically motivated rom the perspective o an individual decision-maker or social planner whether it is rational or a corporate entity to ollow them remains an open question. The results in Section 4 suggest insurance contract prices should be increasing in the insurer s degree o ambiguity aversion, and in their ambiguity (as measured by the variance o their uncertain distributional parameters), provided the insurer is ambiguity averse overall. In practice this may not hold i our assumption that models are ordered by their pessimism over the uncertain parameters is violated, or in these cases increasing ambiguity in a contract may allow the insurer to hedge against the ambiguity in its pre-existing book. We do not explore this kind o inormation structure or reasons o tractability and note that, in any case, our assumption is reasonable or some classes o insurance book. For instance, models o the losses arising rom natural disasters or terrorism may be ranked according to their pessimism over the likelihood o these events. Reerences Alary, D., C. Gollier, and N. Treich (2013): The eect o ambiguity aversion on insurance and sel-protection, Economic Journal, 123, Amarante, M., M. Ghossoub, and E. Phelps (2015): Ambiguity on the insurer s side: the demand or insurance, Journal o Mathematical Economics, 58, Barlow, C., R. P. Nye, M. G. White, and H. L. Wincote (1993): Prudential Margins, in 1993 General Insurance Covention. Bender, M., T. Knutson, R. Tuleya, J. Sirutis, G. Vecchi, S. Garner, and I. Held (2010): Modeled impact o anthropogenic warming on the requency o intense Atlantic hurricanes, Science, 327, 454. Cabantous, L. (2007): Ambiguity aversion in the ield o insurance: insurers attitude to imprecise and conlicting probability estimates, Theory and Decision, 62, Cabantous, L., D. Hilton, H. Kunreuther, and E. Michel-Kerjan (2011): Is imprecise knowledge better than conlicting expertise? Evi- 21