Pricing and Capital Allocation for Multiline Insurance Firms With Finite Assets in an Imperfect Market
PPricing and Capital Allocation for Multiline InsuranceFirms With Finite Assets in an Imperfect Market
John A. Major Stephen J. MildenhallCreated 2020-08-27 12:24:17.937134
Abstract
We analyze multiline pricing and capital allocation in equilibrium no-arbitragemarkets. Existing theories often assume a perfect complete market, but when pricingis linear, there is no diversification benefit from risk pooling and therefore no role forinsurance companies. Instead of a perfect market, we assume a non-additive distortionpricing functional and the principle of equal priority of payments in default. Underthese assumptions, we derive a canonical allocation of premium and margin, withproperties that merit the name the natural allocation. The natural allocation givesnon-negative margins to all independent lines for default-free insurance but can exhibitnegative margins for low-risk lines under limited liability. We introduce novel conditionalexpectation measures of relative risk within a portfolio and use them to derive simple,intuitively appealing expressions for risk margins and capital allocations. We give aunique capital allocation consistent with our law invariant pricing functional. Suchallocations produce returns that vary by line, in contrast to many other approaches.Our model provides a bridge between the theoretical perspective that there should beno compensation for bearing diversifiable risk and the empirical observation that morerisky lines fetch higher margins relative to subjective expected values.JEL Codes: G22, G10
The complete perfect market paradigm dominates financial models of insurance pricingdeveloped since the 1970s. These models imply the existence of an additive valuationrule, meaning there is no benefit from diversification and no role for insurer intermediaries.Moreover, they typically provide no compensation for bearing diversifiable risk. Milestones inthis approach include the use of discounted cash flows, Myers and Cohn (1987) and Cummins(1990), and options pricing Doherty and Garven (1986) and Cummins (1988). Phillips,Cummins, and Allen (1998) and Myers and Read Jr. (2001) considered multiple line pricing,allowing for ex ante default rules, which Sherris (2006) and Ibragimov, Jaffee, and Walden(2010) extended to ex post rules. From Phillips’s contribution forward, there is a focus onwhat happens in default states to explain premiums by line. Aside from non-diversifiable risk1 a r X i v : . [ q -f i n . R M ] A ug nd tail-driven default, these models do not reflect the overall distribution of losses, or whatwe will call the shape of risk .Holding capital in an intermediary insurance company introduces frictional costs, whichCummins (2000) explains are primarily driven by agency conflict, taxes, and regulation. Whywould insureds incur the extra expense of buying through an insurer in a perfect market?Ibragimov, Jaffee, and Walden (2010) assume insureds do not have direct access to ultimatecapital providers or that there are additional costs to do so. This somewhat unsatisfactoryexplanation was anticipated by Grundl and Schmeiser (2007), who pointed out that pricingdoes not depend on allocation in a perfect market. Bauer and Zanjani (2013) consider capitalallocation in the context of allocating frictional costs.The fundamental theorem of asset pricing states that in a perfect market the existence ofan additive valuation rule is essentially equivalent to no arbitrage, Ross (1978), Dybvig andRoss (1989), Delbaen and Schachermayer (1994). The thought that insurance valuation rulesmust be additive has exerted a considerable influence on theory and practice, Borch (1982),Venter (1991). But while indubitably competitive, the insurance market is neither perfectnor complete.Imperfect or incomplete pricing paradigms isolate different failures of the perfect completemodel but tend to have very similar implications. They result in non-additive pricingfunctionals that are often additive on comonotonic risks, and that can be expressed as aworst-case expectation over a set of probability distribution outcomes. Wang (1996) andWang, Young, and Panjer (1997) apply a non-additive functional using distorted probabilitiesto insurance pricing, leveraging diverse theoretical underpinnings including Huber (1981),Schmeidler (1986), Schmeidler (1989), Yaari (1987), and Denneberg (1994). Within thistheory, distortion risk measures (DRM) occur repeatedly and in many guises. Kusuoka (2001)and Acerbi (2002) characterize DRMs as coherent, law invariant, and comonotonic additivefunctionals.DRMs are easy to apply in practice and have many appealing properties. However, they are notadditive and include transaction costs, via an implied bid-ask spread, Castagnoli, Maccheroni,and Marinacci (2004). We must ask whether they are consistent with arbitrage-free pricing.Fortunately, the answer is yes. The presence of transaction costs render apparent arbitrageopportunities impractical and so a non-additive pricing rule can still be arbitrage-free.Results from Chateauneuf, Kast, and Lapied (1996), De Waegenaere (2000), and especiallyCastagnoli, Maccheroni, and Marinacci (2002) and De Waegenaere, Kast, and Lapied (2003)produce general equilibrium models that allow for non-additive prices, and show that DRMpricing—in the equivalent guise as a Choquet integral—is consistent with general equilibrium.As a result, it is legitimate to use DRMs to price risk transfer. They return a premiumthat charges for the shape of risk, even diversifiable risk. DRMs can be applied both tonon-intermediated, direct-to-investor pricing and insurer intermediated pricing, where insurerssell the risk to investors.Why is it reasonable to charge for the shape of risk, especially when it is diversifiable? Thereare two related reasons: ambiguity aversion and winner’s curse.2nsurance pricing is a horse lottery, not a roulette lottery; it relies on subjective probabilitiesAnscombe and Aumann (1963). Losses are ambiguous and investors are ambiguity averse.Zhang (2002) and Klibanoff, Marinacci, and Mukerji (2005) describe ambiguity relevant toinsurance pricing. The latter paper has been applied in an insurance context by Robert andTherond (2014), Dietz and Walker (2017) and Jiang, Escobar-Anel, and Ren (2020). Epsteinand Schneider (2008) can be applied equally to underwriters who often weight bad news moreheavily than good. Premium ambiguity is confounded with risk because more risky classesof business tend to have more ambiguous premiums. The failure of the terrorism insurancemarket is a case in point.The second reason is the winner’s curse: the winning bid is biased low when there are multiplequotes, Thaler (1988). The insurance market is very competitive and is characterized bybig-data, predictive modeling pricing with heterogeneous insureds. Competing classificationplans exacerbate winner’s curse. As a result, a margin over subjective expected loss, evenwhen subjective probabilities are unbiased, is justified. Winner’s curse will be correlatedto ambiguity: greater ambiguity will result in wider quote dispersion. The winner’s cursemargin will appear in quotes but will not appear ex post in results. The winner’s curse isdistinct from adverse selection, D’Arcy and Doherty (1990)In this paper, we consider applying a DRM pricing functional to individual policies. We dothis in two steps. First, we consider pricing without insurers and associated frictional costs.In this world, insureds contract directly with investor capital providers. The capital providerscan be risk-neutral, but they are ambiguity averse. Uncertain subjective probabilities drivethe market; there are no objective probabilities. Investors may be willing to write a roulettelottery at cost, but they are unwilling to write a horse lottery at cost because they areambiguity averse and because they know their winning quotes will be biased below subjectiveexpectation. The DRM incorporates a margin to allow for ambiguity aversion and helpcorrect for the winner’s curse.Non-additive DRMs create a benefit to independent insureds who pool their risks. Estimationrisk, process risk, and entropy are all lower for the pool than for the individual risks, resultingin less ambiguous subjective probabilities. Pool pricing is closer to the subjective expectedvalue, lowering the average premium for pool participants. These conclusions do not hold if theinsureds are not independent, for example if the risk is driven by catastrophe events. Boonen,Tsanakas, and Wüthrich (2017) and Mildenhall (2017) discuss the important differencesbetween the independent and dependent cases.Pool members still have the problem of allocating any gains from diversification. We showthere is a unique way to do this consistent with the value of insurance cash flows and anassumption of equal priority in default, but relying on no other inputs. We call our methodthe natural allocation. Our approach is similar to Ibragimov, Jaffee, and Walden (2010). Wemodel fair value to insureds rather than marginal costs to the pool. The pool is a transparent,contractual pass-through.Our method identifies the conditional expectation of policy losses given total pool loss asa controlling function, which we call κ . It is exquisitely sensitive to relative risk within aportfolio. We explain our allocation in reference to the relative consumption of more or less3mbiguous, and hence costly, capital layers.Risk pools have a clear role in our model: they minimize ambiguity-based risk costs. Poolsaccumulate risk to create more credible pricing signals, credible being synonymous withstable and low volatility. Stable pools will be financed more cheaply by investors than moreambiguous pools, or than single risks. A similar data-centric role for insurance pools wassuggested in Froot and O’Connell (2008).A risk pool does not have to be structured as an insurance company intermediary. Intermedi-aries emerge as pool managers look to bolster their credibility with investors by putting skinin the game and assuming risk. This provides a solid rationale for the existence of insurersversus pools managed by non-risk bearing underwriting managers. Interestingly, non-risk-bearing managers are now quite common in the property catastrophe reinsurance market.There, credibility is enhanced by reliance on independent and highly regarded third-partycatastrophe models. Such models do not exist for most lines. Generally, aggregated data isneeded to price individual policies fairly.Insurance company pools are successful if they lower the cost of pure risk transfer by anamount that more than offsets their frictional costs of holding capital. Individual policypricing still requires an allocation of frictional costs. Since frictional costs are usually regardedas a flat tax on capital, this can be done via a capital allocation. We show there is a unique“natural” way to perform the allocation, giving a complete solution to pooled risk insurancepricing under a DRM. Like the premium allocation, the capital allocation only assumesaggregate pricing by a DRM and equal priority in default. It relies on the fact that DRMsare law invariant.Although many of the results in the paper have been known, at least in principle, for manyyears, we believe it contains several noteworthy results. In particular, we highlight thefollowing contributions.• We analyze multiline pricing and capital allocation in equilibrium no-arbitrage markets.We address the allocation of risk measures from the insured’s perspective, as in Ibragimov,Jaffee, and Walden (2010), but generalize their work to pricing in imperfect andincomplete markets with non-additive functionals.• We find there is a canonical allocation of premium and margin in a market where pricesare determined by a distortion risk measure with a rule of equal priority in default. Theallocation relies on no additional assumptions and we believe it merits the name thenatural allocation. The natural allocation gives non-negative margins to all independentlines for default-free insurance but can exhibit negative margins for low-risk lines underlimited liability. The natural allocation reveals subtle interactions between margin,default, and idiosyncratic risk.• We introduce novel conditional expectation measures of relative risk within a portfolioand use them to derive simple, intuitively appealing expressions for risk margins andcapital allocations consistent with law invariant DRM pricing functionals.• We give a unique capital allocation consistent with our law invariant pricing functional.Such allocations produce returns that vary by line, in contrast to many other approaches.• We illustrate the theory with examples. The examples elucidate the interplay of loss4nd margin between lines at various levels of portfolio risk. We demonstrate how thecost of the shape of risk reflects a complex interaction between the relative consumptionof low layer, certain, high loss ratio assets, and high layer, uncertain, low loss ratioassets.• In contrast with cost allocation-based approaches, we find that margin by line is moredriven by behavior in solvent states than in default states. Our results hold even whenthere is no possibility of default.• Our model provides a bridge between the theoretical perspective that there should beno compensation for bearing diversifiable risk and the empirical observation that morerisky lines fetch higher margins relative to subjective expected values.The remainder of the paper is structured as follows. Section 2 recalls the definition andproperties of a DRM. Section 3 states and proves canonical formulas giving the naturalallocation of loss and premium by policy under a DRM, and shows margins are non-negativefor independent risks. Section 4 discusses the κ and two associated functions that controlthe natural allocation and that are informative in their own right. Section 5 derives someproperties of the natural margin allocation by policy. Section 6 extends the natural premiumallocation to a natural allocation of capital. This allows frictional costs to be allocated.Section 7 gives two examples, one simple and one more realistic. Finally, Section 8 offersconcluding remarks, discusses limitations, and makes suggestions for further research. Notation and Conventions
We consider insurance written directly by investors or intermediated by an insurance company.We distinguish, when necessary, by saying investor-written, or intermediary-written orintermediated insurance. When irrelevant, we just say the insurance is written by a provider.In either case, insurance consists of a pool of individual policies written for insureds. Policieslast one period, with premium collected at t = 0 and losses paid in full at t = 1. Line ofbusiness or business unit or other grouping can be substituted for policy. We refer to thecomponents as lines or policies, whichever is most appropriate.When insurance is supported by finite assets, the provider has limited liability.When intermediated, the intermediary is a stock insurer. At t = 0 it sells its residual valueto investors to raise equity. At time one it pays claims up to the amount of assets available.If assets are insufficient to pay claims it defaults. If there are excess assets they are returnedto investors.The terminology describing risk measures is standard, and follows Föllmer and Schied (2011).We work on a standard probability space, Svindland (2009), Appendix. It can be taken asΩ = [0 , P Lebesgue measure. The indicator function ona set A is 1 A , meaning 1 A ( x ) = 1 if x ∈ A and 1 A ( x ) = 0 otherwise.Total insured loss, or total risk, is described by a random variable X ≥ X reflects policylimits but is not limited by provider assets. X = P i X i describes the split of losses by policy. F , S , f , and q are the distribution, survival, density, and (lower) quantile functions of X .5ubscripts are used to disambiguate, e.g., S X i is the survival function of X i . X ∧ a denotesmin( X, a ) and X + = max( X, S , P , M and Q refer to expected loss, premium, margin and equity, and a refersto assets. The value of survival function S ( x ) is the loss cost of the insurance paying 1 { X>x } ,so the two uses of S are consistent. Premium equals expected loss plus margin; assets equalpremium plus equity. All these quantities are functions of assets underlying the insurance.We use the actuarial sign convention: large positive values are bad. Our concern is withquantiles q ( p ) for p near 1. Distortions are usually reversed, with g ( s ) for small s = 1 − p corresponding to bad outcomes. As far as possible we will use p in the context p close to 1 isbad and s when small s is bad.Tail value at risk is defined for 0 ≤ p < TVaR p ( X ) = 11 − p Z p q ( t ) dt. Prices exclude all expenses. The risk free interest rate is zero. These are standard simplifyingassumptions, e.g. Ibragimov, Jaffee, and Walden (2010).
We define DRMs and recall results describing their different representations. By De Wae-genaere, Kast, and Lapied (2003) DRMs are consistent with general equilibrium and so itmakes sense to consider them as pricing functionals. The DRM is interpreted as the (ask)price for an investor-written risk transfer. The rest of the paper will explore multi-policypricing implied by a DRM.
Definition 1. A distortion function is an increasing concave function g : [0 , → [0 , satisfying g (0) = 0 and g (1) = 1 .A distortion risk measure ρ g associated with a distortion g acts on a non-negative randomvariable X as ρ g ( X ) = Z ∞ g ( S ( x )) dx. (1)The simplest distortion if the identity g ( s ) = s . Then ρ g ( X ) = E [ X ] from the integration-by-parts identity Z ∞ S ( x ) dx = Z ∞ xdF ( x ) . Other well-known distortions include the proportional hazard g ( s ) = s r for 0 < r ≤ g ( s ) = Φ(Φ − ( s ) + λ ) for λ ≥
0, Wang (1995).Since g is concave g ( s ) ≥ g (0) + sg (1) = s for all 0 ≤ s ≤
1, showing ρ g adds a non-negativemargin. 6oing forward, g is a distortion and ρ is its associated distortion risk measure. We interpret ρ as a pricing functional and refer to ρ ( X ) as the price or premium for investor-writteninsurance on X . When we price intermediated insurance we need to add frictional costs ofholding capital. This is considered in section 6.2.DRMs are translation invariant, monotonic, subadditive and positive homogeneous, and hencecoherent, Acerbi (2002). In addition they are law invariant and comonotonic additive. Infact, all such functionals are DRMs. As well has having these properties, DRMs are powerfulbecause we have a complete understanding of their representation and structure, which wesummarize in the following theorem. Theorem 1.
Subject to ρ satisfying certain continuity assumptions, the following are equiva-lent.1. ρ is a law invariant, coherent, comonotonic additive risk measure.2. ρ = ρ g for a concave distortion g .3. ρ has a representation as a weighted average of TVaRs for a measure µ on [0 , ρ ( X ) = R TVaR p ( X ) µ ( dp ) .4. ρ ( X ) = max Q ∈ Q E Q [ X ] where Q is the set of (finitely) additive measures with Q ( A ) ≤ g ( P ( A )) for all measurable A .5. ρ ( X ) = max Z ∈ Z E [ XZ ] where Z is the set of positive functions on Ω satisfying R p q Z ( t ) dt ≤ g (1 − p ) , and q Z is the quantile function of Z . The theorem combines results from Föllmer and Schied (2011) (4.79, 4.80, 4.93, 4.94, 4.95),Delbaen (2000), Kusuoka (2001), and Carlier and Dana (2003). The theorem requires that ρ is continuous from above to rule out the possibility ρ = sup. In certain situations, the suprisk measure applied to an unbounded random variable can only be represented as a supover a set of test measures and not a max. Note that the roles of from above and below areswapped from Föllmer and Schied (2011) because they use the asset, negative is bad, signconvention whereas we use the actuarial, positive is bad, convention.The relationship between µ and g is given by Föllmer and Schied (2011) 4.69 and 4.70. Theelements of Z are the Radon-Nikodym derivatives of the measures in Q . This section introduces the idea of layer densities and proves that DRM premium can beallocated to policy in a natural and unique way.
Risk is often tranched into layers that are then insured and priced separately. Meyers (1996)describes layering in the context of liability increased limits factors and Culp and O’Donnell72009), Mango et al. (2013) in the context of excess of loss reinsurance.Define a layer y excess x by its payout function I ( x,x + y ] ( X ) := ( X − x ) + ∧ y . The expectedlayer loss is E [ I ( x,x + y ] ( X )] = Z x + yx ( t − x ) dF ( t ) + yS ( x + y )= Z x + yx tdF ( t ) + tS ( t ) | x + yx = Z x + yx S ( t ) dt. Based on this equation, Wang (1996) points out that S can be interpreted as the layer loss(net premium) density. Specifically, S is the layer loss density in the sense that S ( x ) = d/dx ( E [ I (0 ,x ] ( X )]) is the marginal rate of increase in expected losses in the layer at x . Weuse density in this sense to define premium, margin and equity densities, in addition to lossdensity.Clearly S ( x ) equals the expected loss to the cover 1 { X>x } . By scaling, S ( x ) dx is the close tothe expected loss for I ( x,x + dx ] when dx is very small; Bodoff (2007) calls these infinitesimallayers.Wang (1996) goes on to interpret Z x + yx g ( S ( t )) dt as the layer premium and hence g ( S ( x )) as the layer premium density. We write P ( x ) := g ( S ( x )) for the premium density.We can decompose X into a sum of thin layers. All these layers are comonotonic with oneanother and with X , resulting in an additive decomposition of ρ ( X ), since ρ is comonotonicadditive. The decomposition mirrors the definition of ρ as an integral, eq. (1).The amount of assets a available to pay claims determines the quality of insurance, andpremium and expected losses are functions of a . Premiums are well-known to be sensitiveto the insurer’s asset resources and solvency, Phillips, Cummins, and Allen (1998). Assetsmay be infinite, implying unlimited cover. When a is finite there is usually some chance ofdefault. Using the layer density view, define expected loss ¯ S and premium ¯ P functions as¯ S ( a ) := E [ X ∧ a ] = Z a S ( x ) dx (2)¯ P ( a ) := ρ ( X ∧ a ) = Z ∞ g ( S X ∧ a ( x )) dx = Z a g ( S X ( x )) dx. (3)Margin is ¯ M ( a ) := ¯ P ( a ) − ¯ S ( a ) and margin density is M ( a ) = d ¯ M ( a ) /da . Assets are fundedby premium and equity ¯ Q ( a ) := a − ¯ P ( a ). Again Q ( a ) = d ¯ Q/da = 1 − P ( a ). Together S , M ,and Q give the split of layer funding between expected loss, margin and equity. Layers up to a are, by definition, fully collateralized. Thus ρ ( X ∧ a ) is the premium for a defaultable coveron X supported by assets a , whereas ρ ( X ) is the premium for an unlimited, default-freecover. 8he layer density view is consistent with more standard approaches to pricing. If X is aBernoulli risk with Pr ( X = 1) = s and expected loss cost s , then ρ ( X ) = g ( s ) can be regardedas pricing a unit width layer with attachment probability s . In an intermediated context, thefunding constraint requires layers to be fully collateralized by premium plus equity—withoutsuch funding the insurance would not be credible since the insurer has no other source offunds.Given g we can compute insurance market statistics for each layer. The loss, premium,margin, and equity densities are s , g ( s ), g ( s ) − s and 1 − g ( s ). The layer loss ratio is s/g ( s )and ( g ( s ) − s ) / (1 − g ( s )) is the layer return on equity. These quantities are illustrated infig. 1 for a typical distortion function. The corresponding statistics for ground-up covers canbe computed by integrating densities. . . . . . . s . . . . . . g ( s ) loss = s margin = g ( s ) − s equity =1 − g ( s ) premium = g ( s ) Insurance StatisticsWang-normal 0.730
Figure 1: Relationship between distortion g and insurance market statistics as a function ofexceedance probability s .For an investor-written risk we regard the margin as compensation for ambiguity aversionand associated winner’s curse drag. Both of these effects are correlated with risk, so themargin is hard to distinguish from a risk load, but the rationale is different. Again, recall,although ρ is non-additive and appears to charge for diversifiable risk, De Waegenaere, Kast,and Lapied (2003) assures us the pricing is consistent with a general equilibrium.The layer density is distinct from models that vary the volume of each line in a homogeneousportfolio model. Our portfolio is static. By varying assets we are implicitly varying thequality of insurance. If assets are finite and the provider has limited liability we need to to determine policy-levelcash flows in default states before we can determine the fair market value of insurance. Themost common way to do this is using equal priority in default.9nder limited liability, total losses are split between provider payments and provider defaultas X = X ∧ a + ( X − a ) + . Next, actual payments X ∧ a must be allocated to each policy. X i is the amount promised to i by their insurance contract. Promises are limited by policyprovisions but are not limited by provider assets. At the policy level, equal priority impliesthe payments made to, and default borne by, policy i are split as X i = X i X ∧ aX + X i ( X − a ) + X = (payments to policy i ) + (default borne by policy i ) . Therefore the payments made to policy i are given by X i ( a ) := X i X ∧ aX = X i X ≤ aX i aX X > a. (4) X i ( a ) is the amount actually paid to policy i . It depends on a , X and X i . The dependence on X is critical. It is responsible for almost all the theoretical complexity of insurance pricing.It is worth reiterating that with this definition P i X i ( a ) = X ∧ a . Example.
Here is an example illustrating the effect of equal priority. Consider a certainloss X = 1000 and X given by a lognormal with mean 1000 and coefficient of variation 2.0.Prudence requires losses be backed by assets equal to the 0.9 quantile. On a stand-alonebasis X is backed by a = 1000 and is risk-free. X is backed by a = 2272 and the recoveryis subject to a considerable haircut, since E [ X ∧ .
3. If these risks are pooled,the pool must hold a = a + a for the same level of prudence. When X ≤ a both linesare paid in full. But when X > a , X receives 1000( a/ (1000 + X )) and X receives theremaining X ( a/ (1000 + X )). Payment to both lines is pro rated down by the same factor a/ (1000 + X )—hence the name equal priority. In the pooled case, the expected recovery to X is 967.5 and 764.8 to X . Pooling and equal priority result in a transfer of 32.5 from X to X . This example shows what can occur when a thin tailed line pools with a thick tailedone under a weak capital standard with equal priority. We shall see how pricing compensatesfor these loss payment transfers, with X paying a positive margin and X a negative one. Expected losses paid to policy i are ¯ S i ( a ) := E [ X i ( a )]. ¯ S i ( a ) can be computed, conditioningon X , as ¯ S i ( a ) = E [ E [ X i ( a ) | X ]] = E [ X i | X ≤ a ] F ( a ) + a E (cid:20) X i X | X > a (cid:21) S ( a ) . (5)Because of its importance in allocating losses, define α i ( a ) := E [ X i /X | X > a ] . (6)10he value α i ( x ) is the expected proportion of recoveries by line i in the layer at x . Since totalassets available to pay losses always equals the layer width, and the chance the layer attachesis S ( x ), it is intuitively clear α i ( x ) S ( x ) is the loss density for line i , that is, the derivative of¯ S i ( x ) with respect to x . We now show this rigorously. Proposition 1.
Expected losses to policy i under equal priority, when total losses are supportedby assets a , is given by ¯ S i ( a ) = E [ X i ( a )] = Z a α i ( x ) S ( x ) dx (7) and so the policy loss density at x is S i ( x ) := α i ( x ) S ( x ) .Proof. By the definition of conditional expectation, α i ( a ) S ( a ) = E [( X i /X )1 X>a ]. Condi-tioning on X , using the tower property, and taking out the functions of X on the rightshows α i ( a ) S ( a ) = E [ E [( X i /X )1 X>a | X ]] = Z ∞ a E [ X i | X = x ] f ( x ) x dx and therefore dda ( α i ( a ) S ( a )) = − E [ X i | X = a ] f ( a ) a . (8)Now we can use integration by parts to compute Z a α i ( x ) S ( x ) dx = xα i ( x ) S ( x ) (cid:12)(cid:12)(cid:12)(cid:12) a + Z a x E [ X i | X = x ] f ( x ) x dx = aα i ( a ) S ( a ) + E [ X i | X ≤ a ] F ( a )= ¯ S i ( a )by eq. (5). Therefore the policy i loss density in the asset layer at a , i.e. the derivative ofeq. (5) with respect to a , is S i ( a ) = α i ( a ) S ( a ) as required.To recap, eq. (7) gives a direct analog to eq. (2) for policy i losses. Note that S i is not the survival function of X i ( a ) nor of X i . Equation (7) is surprising because it gives adecomposition of S through the convolution of random variables: X P i X i X ∧ a P i X i ( a ) E [ X ] P i E [ X i ] E [ X ∧ a ] P i E [ X i ( a )] Z S P i Z S X i Z a S P i Z a α i SS P S X i , S P α i S. = === = == == = = = .4 Premiums at Different Asset Levels Premium under ρ is given by eq. (3). We can interpret g ( S ( a )) as the portfolio premiumdensity in the layer at a . We now consider the premium and premium density for each policy.Using integration by parts we can express the price of an unlimited cover on X as ρ ( X ) = Z ∞ g ( S ( x )) dx = Z ∞ xg ( S ( x )) f ( x ) dx = E [ Xg ( S ( X )))] . (9)It is important that this integral is over all x ≥ xg ( S ( x )) | a term disappears.Equation (9) makes sense because a concave distortion is continuous on (0 ,
1] and can have atmost countably infinitely many points where it is not differentiable (it has a kink). In totalthese points have measure zero, Borwein and Vanderwerff (2010), and we can ignore them inthe integral. For more details see Dhaene et al. (2012).By Equation (9), and the properties of a distortion function, g ( S ( X )) is the Radon-Nikodymderivative of a measure Q with ρ ( X ) = E Q [ X ]. In fact, E Q [ Y ] = E [ Y g ( S ( X ))] for all randomvariables Y . In general, any non-negative function Z (measure Q ) with E [ Z ] = 1 and ρ ( X ) = E [ XZ ] (= E Q [ X ]) is called a contact function (subgradient) for ρ at X , see Shapiro,Dentcheva, and Ruszczyński (2009). Thus g ( S ( X )) is a contact function for ρ at X . Thename subgradient comes from the fact that ρ ( X + Y ) ≥ E Q [ X + Y ] = ρ ( X ) + E Q [ Y ], bytheorem 1. The set of subgradients is called the subdifferential of ρ at X . If there is a uniquesubgradient then ρ is differentiable. Delbaen (2000) Theorem 17 shows that subgradients arecontact functions.We can interpret g ( S ( X )) as a state price density specific to the X , suggesting that E [ X i g ( S ( X ))] gives the value of the cash flows to policy i . This motivates the followingdefinition. Definition 2.
For X = P i X i with Q ∈ Q so that ρ ( X ) = E Q [ X ] , the natural allocationpremium to policy X j as part of the portfolio X is E Q [ X j ] . It is denoted ρ X ( X j ) . The natural allocation premium is a standard approach, appearing in Delbaen (2000), Venter,Major, and Kreps (2006) and Tsanakas and Barnett (2003) for example. It has many desirableproperties. Delbaen shows it is a fair allocation in the sense of fuzzy games and that it has adirectional derivative, marginal interpretation when ρ is differentiable. It is consistent withJouini and Kallal (2001) and Campi, Jouini, and Porte (2013), which show the rational priceof X in a market with frictions must be computed by state prices that are anti-comonotonic X . In our application the signs are reversed: g ( S ( X )) and X are comonotonic.The choice g ( S ( X )) is economically meaningful because it weights the largest outcomes of X the most, which is appropriate from a social, regulatory and investor perspective. It isalso the only choice of weights that works for all levels of assets. Since investors stand readyto write any layer at the price determined by g , their solution must work for all a .However, there are two technical issues with the proposed natural allocation. First, unlikeprior works, we are allocating the premium for X ∧ a , not X , a problem also considered inMajor (2018). And second, Q may not be unique. In general, uniqueness fails at capped12ariables like X ∧ a . Both issues are surmountable for a DRM, resulting in a unique, welldefined natural allocation. For a non-comonotonic additive risk measure this is not the case.It is helpful to define the premium, risk adjusted, analog of the α i as β i ( a ) := E Q [( X i /X ) | X > a ] . (10) β i ( x ) is the value of the recoveries paid to line i by a policy paying 1 in states { X > a } ,i.e. an allocation of the premium for 1 X>a . By the properties of conditional expectations, wehave β i ( a ) = E [( X i /X ) Z | X > a ] E [ Z | X > a ] . (11)The denominator equals Q ( X > a ) / P ( X > a ). Remember that while E Q [ X ] = E [ XZ ], forconditional expectations E Q [ X | F ] = E [ XZ | F ] / E [ Z | F ], see Föllmer and Schied (2011),Proposition A.12.To compute α i and β i we use a third function, κ i ( x ) := E [ X i | X = x ] , (12)the conditional expectation of loss by policy, given the total loss. It is an important fact thatthe risk adjusted version of κ is unchanged because DRMs are law invariant. With thesepreliminaries we can state the main theorem of this section. Theorem 2.
Let Q ∈ Q be the measure with Radon-Nikodym derivative Z = g ( S X ( X )) .1. E [ X i | X = x ] = E Q [ X i | X = x ] .2. β i can be computed from κ i as β i ( a ) = 1 Q ( X > a ) Z ∞ a κ i ( x ) x g ( S ( x )) f ( x ) dx. (13)
3. The natural allocation premium for policy i under equal priority when total losses aresupported by assets a , ¯ P i ( a ) := ρ X ∧ a ( X i ( a )) , is given by ¯ P i ( a ) = E Q [ X i | X ≤ a ](1 − g ( S ( a ))) + a E Q [ X i /X | X > a ] g ( S ( a )) (14)= E [ X i Z | X ≤ a ](1 − S ( a )) + a E [( X i /X ) Z | X > a ] S ( a ) . (15)
4. The policy i premium density is P i ( a ) = β i ( a ) g ( S ( a )) . (16)The Theorem offers two contributions. First, it shows we can replace E Q [ X i | X ] with E [ X i | X ], which enables explicit calculation. There is no risk adjusted version of κ i .Intuitively, a law invariant risk measure cannot change probabilities within an event definedby X : if it did then it would be distinguishing between events on information other than S ( X ) whereas law invariance says this is all that can matter. And second, it identifies the13remium density eq. (16), giving an allocation of eq. (3) and a premium analog of eq. (7).It provides a clear and illuminating way to visualize risk by collapsing a multidimensionalproblem to one dimension, see fig. 2. Part (4) provides a direct premium analog of eq. (5)and part (5) an analog of eq. (7). Equation (14) is the same as Tsanakas and Barnett (2003)eq. (19), although their derivation is in the context of a homogeneous portfolio whereas ourportfolio is static. Proof.
Part (1) follows in the same way as eq. (11).Since ρ is comonotonic additive ρ ( X ) = ρ ( X ∧ a ) + ρ (( X − a ) + ) and hence ρ ( X ) = E Q [ X ∧ a ] + E Q [( X − a ) + ] ≤ ρ ( X ∧ a ) + ρ (( X − a ) + ) = ρ ( X ). But since E Q [ X ∧ a ] ≤ ρ ( X ∧ a ) and E Q [( X − a ) + ] ≤ ρ (( X − a ) + ) it follows E Q [ X ∧ a ] = ρ ( X ∧ a ) and E Q [( X − a ) + ] = ρ (( X − a ) + ).Therefore the contact functions for X and X ∧ a are the same and it is legitimate to assume Z = g ( S ( X )) when allocating premium for X ∧ a .To prove Part (2), note that by eq. (11) β i ( a ) g ( S ( a )) = E Q [( X i /X )1 X>a ]. Conditioning on X , using the tower property, and taking out the known functions of X on the right, shows β i ( a ) g ( S ( a )) = E [ E [( X i /X ) g ( S ( X ))1 X>a | X ]]= E [( E [( X i | X ] /X ) g ( S ( X ))1 X>a ]]= Z ∞ a E [ X i | X = x ] x g ( S ( x )) f ( x ) dx. It follows from the definition of X i ( a ), eq. (4), and the fact Z is a contact function for X ∧ a that ¯ P i ( a ) = E [ X i ( a ) g ( S ( X ))]= E [ X i g ( S ( X ))1 X ≤ a ] + E [ a ( X i /X ) g ( S ( X ))1 X>a ]= E [ X i g ( S ( X )) | X ≤ a ](1 − S ( a ))+ a E [( X i /X ) g ( S ( X )) | X > a ] S ( a )= E Q [ X i | X ≤ a ](1 − g ( S ( a )))+ a E Q [( X i /X ) | X > a ] g ( S ( a )) , giving Part (3).Rearranging eq. (13) and differentiating gives dda ( β i ( a ) g ( S ( a ))) = − E [ X i | X = a ] a g ( S ( a )) f ( a ) . Now use integration by parts to compute Z a β i ( x ) g ( S ( x )) dx = xβ i ( x ) g ( S ( x )) (cid:12)(cid:12)(cid:12)(cid:12) a + Z a x E [ X i | X = x ] x g ( S ( x )) f ( x ) dx = aβ i ( a ) g ( S ( a )) + E Q [ X i | X ≤ a ](1 − g ( S ( a ))= ¯ P i ( a ) 14y eq. (14). As a result, the policy i premium density in the asset layer at a , i.e. the derivativeof ¯ P i ( a ) with respect to a , is P i ( a ) = β i ( a ) g ( S ( a )), giving Part (4).The proof writes the price of a limited liability cover as the price of default-free protectionminus the value of the default put. This is the standard starting point for allocation in aperfect competitive market taken by Phillips, Cummins, and Allen (1998), Myers and ReadJr. (2001), Sherris (2006), and Ibragimov, Jaffee, and Walden (2010). They then allocate thedefault put rather than the value of insurance payments directly.The problem that can occur when Q is not unique, but that can be circumvented when ρ is a DRM, can be illustrated as follows. Suppose ρ is given by p -TVaR. The measure Q weights the worst 1 − p proportion of outcomes of X by a factor of (1 − p ) − and ignoresthe others. Suppose a is chosen as p -VaR for a lower threshold p < p . Let X a = X ∧ a be capped insured losses and C = { X a = a } . By definition Pr ( C ) ≥ − p > − p . Pickany A ⊂ C of measure 1 − p so that ρ ( X ) = E [ X | A ]. Let ψ be a measure preservingtransformation of Ω that acts non-trivially on C but trivially off C . Then Q = Q ψ willsatisfy E Q [ X a ] = E Q [ X a ψ − ] = ρ ( X a ) but in general E Q [ X ] < ρ ( X ). The natural allocationwith respect to Q will be different from that for Q . The theorem isolates a specific Q toobtain a unique answer. The same idea applies to Q from other, non-TVaR, ρ : you canalways shuffle part of the contact function within C to generate non-unique allocations. Seesection 7.1 for an example.To recap: the premium formulas eqs. (14) and (16) have been derived assuming capital isprovided at a cost g and there is equal priority by line. They are computationally tractableand require no other assumptions. There is no need to assume the X i are independent. Theyproduce an entirely general, canonical determination of premium in the presence of sharedcostly capital. This result extends Grundl and Schmeiser (2007), who pointed out that withan additive pricing functional there is no need to allocate capital in order to price, to thesituation of a non-additive DRM pricing functional.The key formulas we have derived are summarized in table 1. In this section we explore properties of α i , β i , and κ i , see eq. (6), eq. (10), and eq. (12), andshow how they interact to determine premiums by line via the natural allocation.For a measurable h , E [ X i h ( X )] = E [ κ i ( X ) h ( X )] by the tower property. This simple observa-tion results in huge simplifications. In general, E [ X i h ( X )] requires knowing the full bivariatedistribution of X i and X . Using κ i reduces it to a one dimensional problem. This is trueeven if the X i are correlated. The κ i functions can be estimated from data using regressionand they provide an alternative way to model correlations.Despite their central role, the κ i functions are probably unfamiliar so we begin by giving15 u a n t i t y L o ss P r e m i u m C a s hfl o w X i ( a ) = X i X ∧ a X N /a M e a s u r e O b j ec t i v e , S ( x ) , f ( x ) R i s k a d j u s t e d , Q , g ( S ( x )) , g ( S ( x )) f ( x ) E x p e c t a t i o n ¯ S i ( a ) = E [ X i ( a ) ] ¯ P i ( a ) = E Q [ X i ( a ) ] = E [ X i ( a ) g ( S ( X )) ] C o nd i t i o n i n g e x p e c t a t i o n E [ X i | X ≤ a ] F ( a ) + a E [ X i / X | X > a ] S ( a ) E Q [ X i | X ≤ a ] ( − g ( S ( a ))) + a E Q [ X i / X | X > a ] g ( S ( a )) Sh a r e f un c t i o n α i ( x ) = E [ X i / X | X > x ] β i ( x ) = E Q [ X i / X | X > x ] D e r i v a t i v e o f s h a r e f un c t i o n ( α i S ) ( x ) = − E [ X i | X = x ] f ( x ) / x = − κ i ( x ) f ( x ) / x ( β i g ( S )) ( x ) = − E [ X i | X = x ] g ( S ( x )) f ( x ) / x = − κ i ( x ) g ( S ( x )) f ( x ) / x L ee i n t e g r a l e x p e c t a t i o n Z a α i ( x ) S ( x ) d x Z a β i ( x ) g ( S ( x )) d x O u t c o m e i n t e g r a l e x p e c t a t i o n Z a κ i ( x ) f ( x ) d x + a α i ( a ) S ( a ) Z a κ i ( x ) g ( S ( x )) f ( x ) d x + a β i ( a ) g ( S ( a )) S c e n a r i o i n t e g r a l e x p e c t a t i o n Z F ( a ) κ i ( q ( p )) d p + a α i ( a ) S ( a ) Z − g ( S ( a )) κ i ( q ( − g − ( − p ))) d p + a β i ( a ) g ( S ( a )) T a b l e : D i ff e r e n t w a y s o f c o m pu t i n g e x p ec t e d l o ss e s a nd t h e n a t u r a l a ll o c a t i o n . κ functions
1. If Y i are independent and identically distributed and X n = Y + · · · + Y n then E [ X m | X m + n = x ] = mx/ ( m + n ) for m ≥ , n ≥
0. This is obvious when m = 1 becausethe functions E [ Y i | X n ] are independent across i = 1 , . . . , n and sum to x . The resultfollows because conditional expectations are linear. In this case κ i ( x ) = mx/ ( m + n ) isa line through the origin.2. If X i are multivariate normal then κ i are straight lines, given by the usual least-squaresfits κ i ( x ) = E [ X i ] + cov ( X i , X ) var ( X ) ( x − E [ X ]) . This example is familiar from the securities market line and the CAPM analysis ofstock returns. If X i are iid it reduces to the previous example because the slope is 1 /n .3. If X i , i = 1 ,
2, are compound Poisson with the same severity distribution then κ i areagain lines through the origin. Suppose X i has expected claim count λ i . Write theconditional expectation as an integral, expand the density of the compound Poissonby conditioning on the claim count, and then swap the sum and integral to see that κ ( x ) = E [ X | X + X = x ] = x E [ N ( λ ) / ( N ( λ )+ N ( λ ))] where N ( λ ) are independentPoisson with mean λ . This example generalizes the iid case. Further conditioning on acommon mixing variable extends the result to mixed Poisson frequencies where eachaggregate can have a separate or shared mixing distribution. The common severity isessential. The result means that if a line of business is defined to be a group of policiesthat shares the same severity distribution, then premiums for policies within the linewill have rates proportional to their expected claim counts.4. A theorem of Efron says that if X i are independent and have log-concave densitiesthen all κ i are non-decreasing, Saumard and Wellner (2014). The multivariate normalexample is a special case of Efron’s theorem.Denuit and Dhaene (2012) define an ex post risk sharing rule called the conditional mean riskallocation by taking κ i ( x ) to be the allocation to policy i when X = x . A series of recentpapers, see Denuit and Robert (2020) and references therein, considers the properties of theconditional mean risk allocation focusing on its use in peer-to-peer insurance and the casewhen κ i ( x ) is linear in x . α i , β i By definition α i ( x ) is the expected proportion of losses from policy i in 1 { X>x } and β i ( x ) isthe risk adjusted proportion. They are average proportions not proportions of the averages:17 i ( x ) = E [ X i /X | X > x ] = E [ X i | X > x ] / E [ X | X > x ] because of Jensen’s inequalityapplied to the convex function x /x .To better understand the shape of α i and β i we can compute their derivatives. Differentiating α i ( x ) S ( x ) = E [( X i /X )1 X>x ] and re-arranging gives α i ( x ) = α i ( x ) − κ i ( x ) x ! f ( x ) S ( x ) . (17)The results for β i are analogous. The function h ( x ) := f ( x ) /S ( x ) is called the hazardrate. If X models a lifetime, h is called the force of mortality. For thick right-skeweddistributions h is typically an eventually decreasing function. For thin tailed distributions itis typically an eventually increasing function. It is constant for the exponential distribution.The action of g is to make the right tail thicker and so to decrease the hazard rate. Since h ( x ) = − d/dx (log( S ( x ))) it follows that S ( x ) = exp (cid:18) − Z ∞ t h ( s ) ds (cid:19) . The integral is called the cumulative hazard function. From this formulation it is clear theproportional hazard g ( s ) = s r , 0 < r ≤
1, acts on the hazard function as multiplication by r ,hence justifying its name.Equation (17) shows that α i ( x ) = 0 if f ( x ) = 0 and S ( x ) close to 1, which will occur in theextreme left tail when X includes some level of near certain losses. Then α i will be flat forsmall x , while f ( x ) ≈
0. Flat behavior can also occur if α i ( x ) − κ i ( x ) /x = 0, but that is anexceptional circumstance.For thick tailed insurance distributions h ( x ) is eventually decreasing but remains strictlypositive. If κ i ( x ) /x is decreasing then α i ( x ) < α i ( x ) is the probability weightedintegral of κ i ( t ) /t over t > x , and so α i ( x ) < κ i ( x ) /x . Conversely if κ i ( x ) /x is increasing α i ( x ) > P i κ i ( x ) = x it follows that P i κ i ( x ) = 1. It is typical for the thickest tail distribution, i say, to behave like κ i ( x ) ≈ x − P j = i E [ X j ] for large x . Then κ i ( x ) = 1 and the remaining κ j ( x ) ≈ E [ X j ] are almost constant for large x . In that case κ j ( x ) /x > α j ( x ) and so α j ( x ) < α i ( x ) >
0. To have two policies with α i increasing requires a very delicate balancing ofthe thickness of their tails with κ i ( x ), growing with order x . A compound Poisson with thesame severity is an example. We now explore margin, equity, and return in total and by policy. We begin by consideringthem in total. 18y definition the average return with assets a is¯ ι ( a ) := ¯ M ( a )¯ Q ( a ) (18)where margin ¯ M and equity ¯ Q are defined in the paragraph following eq. (3).Equation (18) has important implications. It tells us the investor priced expected returnvaries with the level of assets. For most distortions return decreases with increasing capital.In contrast, the standard RAROC models use a fixed average cost of capital, regardless ofthe overall asset level, Tasche (1999). CAPM or the Fama-French three factor model areoften used to estimate the average return, with a typical range of 7 to 20 percent, Cumminsand Phillips (2005). A common question of working actuaries performing capital allocation isabout so-called excess capital, if the balance sheet contains more capital than is required byregulators, rating agencies, or managerial prudence. Our model suggests that higher layers ofcapital are cheaper, but not free, addressing this concern.The varying returns in eq. (18) may seem inconsistent with Miller Modigliani. But that saysthe cost of funding a given amount of capital is independent of how it is split between debtand equity; it does not say the average cost is constant as the amount of capital varies. The natural allocation has two desirable properties. It is always less than the stand-alonepremium, meaning it satisfies the no-undercut condition of Denault (2001), and it producesnon-negative margins for independent risks.
Proposition 2.
Let X = P ni =1 X i , X i non-negative and independent, and let g be a distortion.Then (1) the natural allocation is never greater than the stand-alone premium and (2) thenatural allocation to every X i contains a non-negative margin.Proof. It is enough to prove for n = 2 by considering X and X = X + · · · + X n .By theorem 1 we know that ρ ( X ) = E Q [ X ] where Q has Radon-Nikodym derivative g ( S X ( X )).By definition, the natural allocation is ¯ P = E [ X g ( S X ( X ))]. Therefore,¯ P = E [ X g ( S X ( X ))] ≤ sup Q ∈Q E Q [ X ] = ρ ( X )which shows Part (1), that the natural allocation is never greater than the stand-alonepremium.Let ˜ X = X + E [ X ] and ˜ X = X − E [ X ]. Then by Rothschild and Stiglitz (1970) or Machinaand Pratt (1997) ˜ X + ˜ X (cid:23) ˜ X , where (cid:23) denotes second order stochastic dominance.Bäuerle and Müller (2006) shows that DRMs respect second order stochastic dominance.Therefore ρ ( ˜ X + ˜ X ) ≥ ρ ( ˜ X ) .
19y translation invariance ρ ( ˜ X ) = ρ ( X ) + E [ X ]. Since ˜ X + ˜ X = X + X we conclude ρ ( X + X ) ≥ ρ ( X ) + E [ X ] . Combining these results we get¯ P + ¯ P = ρ ( X + X ) ≥ ρ ( X ) + E [ X ]= ⇒ ¯ P ≥ ρ ( X ) − ¯ P + E [ X ] ≥ E [ X ]as required for Part (2).Part (1) is well known. The proof if Part (2) leverages the fact ρ is translation invariant. Ifwe add X = c to X then its natural allocation is c . In a sense, this is the best case . Anynon-constant independent variable, no matter how low its variance, must slightly increaserisk. It does not make sense to grant the new variable a credit off expected loss, when wewould not do so for a constant. A credit is possible for dependent variables, however.Since ¯ P i = E [ κ i ( X ) g ( S ( X ))] we see the no-undercut condition holds if κ i ( X ) and g ( S ( X ))are comonotonic, and hence if κ i is increasing, or if κ i ( X ) and X are positively correlated(recall E [ g ( S ( X ))] = 1). Since P i κ i ( x ) = x at least one κ i , say κ i ∗ , must be increasing.Policy i ∗ is the capacity consuming line that will always have a positive margin. In this way κ differentiates relative tail thickness. We start with a corollary of the results in section 3 which gives a nicely symmetric andcomputationally tractable expression for the natural margin allocation in the case of finiteassets.
Corollary 1.
The margin density for line i at asset level a is given by M i ( a ) = β i ( a ) g ( S ( a )) − α i ( a ) S ( a ) . (19) Proof.
Using eqs. (7) and (16) we can compute margin ¯ M i ( a ) in ¯ P i ( a ) by line as¯ M i ( a ) = ¯ P i ( a ) − ¯ L i ( a )= Z a β i ( x ) g ( S ( x )) − α i ( x ) S ( x ) dx. (20)Differentiating we get the margin density for line i at a expressed in terms of α i and β i asshown.Margin in the current context is the cost of capital, thus eq. (19) is an important result. Itallows us to compute economic value by line and to assess static portfolio performance byline—one of the motivations for performing capital allocation in the first place. In many waysit is also a good place to stop. Remember these results only assume we are using a distortion20isk measure and have equal priority in default. We are in a static model, so questions ofportfolio homogeneity are irrelevant. We are not assuming X i are independent.What does eq. (19) say about by margins by line? Since g is increasing and concave P ( a ) = g ( S ( a )) ≥ S ( a ) for all a ≥
0. Thus all asset layers contain a non-negative totalmargin density. It is a different situation by line, where we can see M i ( a ) ≥ ⇐⇒ β i ( a ) g ( S ( a )) − α i ( a ) S ( a ) ≥ ⇐⇒ β i ( a ) α i ( a ) ≥ S ( a ) g ( S ( a )) . The line layer margin density is positive when β i /α i is greater than the all-lines layer lossratio. Since the loss ratio is ≤ β i ( a ) /α i ( a ) >
1. But when β i ( a ) /α i ( a ) < α and β in more detail.It is important to remember why proposition 2 does not apply: it assumes unlimited cover,whereas here a < ∞ . With finite capital there are potential transfers between lines causedby their behavior in default that overwhelm the positive margin implied by the proposition.Also note the proposition cannot be applied to X ∧ a = P i X i ( a ) because the line paymentsare no longer independent.In general we can make two predictions about margins. Prediction 1 : Lines where α i ( x ) or κ i ( x ) /x increase with x will have always have a positivemargin. Prediction 2 : A log-concave (thin tailed) line aggregated with a non-log-concave (thicktailed) line can have a negative margin, especially for lower asset layers.Prediction 1 follows because the risk adjustment puts more weight on X i /X for larger X andso β i ( x ) /α i ( x ) > > S ( x ) /g ( S ( x )). Recall the risk adjustment is comonotonic with totallosses X .A thin tailed line aggregated with thick tailed lines will have α i ( x ) decreasing with x . Now therisk adjustment will produce β i ( x ) < α i ( x ) and it is possible that β i ( x ) /α i ( x ) < S ( x ) /g ( S ( x )).In most cases, α i ( x ) approaches E [ X i ] /x and β i ( x ) /α i ( x ) increases with x , while the layerloss ratio decreases—and margin increases—and the thin line will eventually get a positivemargin. Whether or not the thin line has a positive total margin ¯ M i ( a ) > a . A negative margin is more likely for less wellcapitalized insurers, which makes sense because default states are more material and theyhave a lower overall dollar cost of capital. In the independent case, as a → ∞ proposition 2eventually guarantees positive margins for all lines.These results are reasonable. Under limited liability, if assets and liabilities are pooled thenthe thick tailed line benefits from pooling with the thin one because pooling increases theassets available to pay losses when needed. Equal priority transfers wealth from thin to thickin states of the world where thick has a bad event, c.f., the example in section 3.2. Butbecause thick dominates the total, the total losses are bad when thick is bad. The negativemargin compensates the thin-tailed line for transfers.21nother interesting situation occurs for asset levels within attritional loss layers. Mostrealistic insured loss portfolios are quite skewed and never experience very low loss ratios.For low loss layers, S ( x ) is close to 1 and the layer at x is funded almost entirely by expectedlosses; the margin and equity density components are nearly zero. Since the sum of margindensities over component lines equals the total margin density, when the total is zero itnecessarily follows that either all line margins are also zero or that some are positive andsome are negative. For the reasons noted above, thin tailed lines get the negative marginas thick tailed lines compensate them for the improved cover the thick tail lines obtain bypooling.In conclusion, the natural margin by line reflects the relative consumption of assets by layer,Mango (2005). Low layers are less ambiguous to the provider and have a lower margin relativeto expected loss. Higher layers are more ambiguous and have lower loss ratios. High risklines consume more higher layer assets and hence have a lower loss ratio. For independentlines with no default the margin is always positive. But there is a confounding effect whendefault is possible. Because more volatile lines are more likely to cause default, there is awealth transfer to them. The natural premium allocation compensates low risk policies forthis transfer, which can result in negative margins in some cases. Although eq. (19) determines margin by line, we cannot compute return by line, or allocatefrictional costs of capital, because we still lack an equity allocation, a problem we now address.
Definition 3.
The natural allocation of equity to line i is given by Q i ( a ) = β i ( a ) g ( S ( a )) − α i ( x ) S ( a ) g ( S ( a )) − S ( a ) × (1 − g ( S ( a ))) . (21)Why is this allocation natural? In total the layer return at a is ι ( a ) := M ( a ) Q ( a ) = P ( a ) − S ( a )1 − P ( a ) = g ( S ( a )) − S ( a )1 − g ( S ( a )) . We claim that for a law invariant pricing measure the layer return must be the same forall lines . Law invariance implies the risk measure is only concerned with the attachmentprobability of the layer at a , and not with the cause of loss within the layer. If return withina layer varied by line then the risk measure could not be law invariant.We can now compute capital by layer by line, by solving for the unknown equity density Q i ( a ) via ι ( a ) = M ( a ) Q ( a ) = M i ( a ) Q i ( a ) = ⇒ Q i ( a ) = M i ( a ) ι ( a ) . − g ( S ( a )) is the proportion of capital in the layer at a , eq. (21) says the allocation toline i is given by the nicely symmetric expression β i ( a ) g ( S ( a )) − α i ( x ) S ( a ) g ( S ( a )) − S ( a ) . (22)To determine total capital by line we integrate the equity density¯ Q i ( a ) := Z a Q i ( x ) dx. And finally we can determine the average return to line i at asset level a ¯ ι i ( a ) = ¯ M i ( a )¯ Q i ( a ) . (23)The average return will generally vary by line and by asset level a . Although the returnwithin each layer is the same for all lines, the margin, the proportion of capital, and theproportion attributable to each line all vary by a . Therefore average returns will vary byline and a . This is in stark contrast to the standard industry approach, which uses the samereturn for each line and implicitly all a . How these quantities vary by line is complicated.Academic approaches emphasized the possibility that returns vary by line, but struggled withparameterization, Myers and Cohn (1987).Equation (23) shows the average return by line is an M i -weighted harmonic mean of the layerreturns given by the distortion g , viz1¯ ι i ( a ) = Z a ι ( x ) M i ( x )¯ M i ( a ) dx. The harmonic mean solves the problem that the return for lower layers of assets is potentiallyinfinite (when g (1) = 0). The infinities do not matter: at lower asset layers there is little orno equity and the layer is fully funded by the loss component of premium. When so funded,there is no margin and so the infinite return gets zero weight. In this instance, the sense ofthe problem dictates that 0 × ∞ = 0: with no initial capital there is no final capital regardlessof the return. An equity allocation to policy is needed to price intermediated insurance because of thefrictional costs of holding capital in an insurance company.The price of investor-written insurance is ρ ( X ). A cat bond transaction is an example ofinvestor-written insurance. Following Myers and Read Jr. (2001) and Ibragimov, Jaffee, and23alden (2010) we model frictional costs as a tax on equity at rate δ . The density and limitedprice of intermediated insurance becomes P Ii ( a ) = P i ( a ) + δQ i ( a ) (24)¯ P Ii ( a ) = ¯ P i ( a ) + δ ¯ Q i ( a ) . (25)(26)The relative size of M i and δQ i is a topic for future research. Consider a two line example where X takes values 0, 9 and 10, and X values 0, 1 and 90,the lines are independent and X = X + X . Suppose the outcome probabilities are 1 / , / / g ( s ) = √ s . There are nine possible outcomes shown in table 2. The naturalallocation appears to depend on the ordering of the two outcomes where X = 10. If thesetwo rows are swapped the allocations are different, as shown in the last two rows of the table.Table 2: Nine possible outcomes showing ambiguous ordering for X = 10. The naturalallocation E Q [ X i ] appears to depend on the ordering of outcomes 4 and 5. The next to lastrow shows E Q with these rows swapped. Outcome X X X P S ( x ) g ( S ) Q Z E [ Z | X ] ˜ Q E E Q The last three columns of the table compute the measure ˜ Q corresponding to the unique Q | X . The ˜ Q -expected values of X and X are 6.2048488 and 45.183836, respectively. Notethese values for the natural allocation are different from the average of the the two orderingsof rows 4 and 5.Table 3 replaces X i with κ i ( x ) = E [ X i | X = x ], resulting in one row per value of X, and usestheorem 2 to compute expectations. The results are the same as using ˜ Q .24able 3: Combining outcomes 4 and 5 and working with E [ X i | X ] resolves the ambiguityand produces the natural allocation. Outcome E [ X | X ] E [ X | X ] X P S ( x ) g ( S ) Q E E Q Actuaries commonly perform this type of calculation, often with catastrophe model output.They make the simplifying assumption that X i = E [ X i | X ] when all rows are distinct.However, the ordering problem illustrated does occur in real data, especially when limits andretentions are involved. Theorem 2 shows how to rigorously resolve the ordering problem tocompute the unique natural allocation. Example 2 is based on two distributions with mean 1. X is thin tailed with a gammadistribution with coefficient of variation 0.25. X is a translated lognormal X = 0 . . X ,where X has a coefficient of variation 1 . / .
7, resulting in a coefficient of variation of 1 . X . Total assets are 12.5, corresponding to capital at a 563 year return period. Theaggregate coefficient of variation is 0.637 in total. X approximates a moderate limit bookof commercial auto and X a catastrophe exposed property book with a stable attritionalloss component. In aggregate the portfolio would be considered as volatile. The distortion g uses a Wang transform with λ = 0 . X and 1.889 for line (94.6 percent and 52.4 percentloss ratios), producing an overall 67.6 percent loss ratio, all without expenses. The profit isrealistic for a gross portfolio with these characteristics.Figure 2 illustrates the theory we have developed. We refer to the charts as ( r, c ) for row r = 1 , , , c = 1 , ,
3, starting at the top left. The horizontal axis shows theasset level in all charts except (3 ,
3) and (4 , ,
3) where itshows loss. Blue represents the thin tailed line, orange thick tailed and green total. Whenboth dashed and solid lines appear on the same plot, the solid represent risk-adjusted anddashed represent non-risk-adjusted functions. Here is the key.• (1 ,
1) shows density for X , X and X = X + X ; the two lines are independent. Bothlines have mean 1.• (1 , , X , X ) lives . The diagonallines show X = k for different k . These show that large values of X correspond to large25alues of X , with X about average.• (2 , κ i is clear from looking at (1 , κ peaks at x = 2 .
15 with maximumvalue 1.14. Thereafter it declines to 1.0. κ is monotonically increasing.• (2 , α i functions. For small x the expected proportion of losses is approximately50/50, since the means are equal. As x increases X dominates. The two functions sumto 1.• (2 , β i and the dashed lines α i from (2 , α decreases β ( x ) ≤ α ( x ). This can lead to X having a negative margin in low asset layers. X isthe opposite.• (3 , X using eq. (7)and eq. (16). For low asset layers α ( x ) S ( x ) > β ( x ) g ( S ( x )) (dashed above solid)corresponding to a negative margin. Beyond about x = 1 .
38 the lines cross and themargin is positive.• (4 , X . Since α is increasing, β ( x ) > α ( x ) for all x andso all layers get a positive margin. The solid line β gS is above the dashed α S line.• (3 , X requires a positive margin and X a negative one,reflecting the benefit the thick line receives from pooling in low layers. The overallmargin is always non-negative. Beyond x = 1 . X ’s margin is also positive.• (4 , dollar-swapping layers and then increases. It is always non-negative. The curves in thisplot are the integrals of those in (3 ,
2) from 0 to x .• (3 , − S ( x ) , x ) = ( p, q ( p )) (dashed) and premium (1 − g ( S ( x )) , x ) = ( p, q (1 − g − (1 − p )) (solid, shifted left) for each line and total. Themargin is the shaded area between the two. Each set of three lines (solid or dashed)does not add up vertically because of diversification. The same distortion g is appliedto each line to the stand-alone S X i . It is calibrated to produce a 10 percent returnoverall. On a stand-alone basis, calculating capital by line to the same return periodas total, X is priced to a 83.5 percent loss ratio and X a 51.8 percent, producing anaverage 64.0 percent, vs. 67.6 percent on a combined basis. Returns are 28.7 percentand 9.6 percent respectively, averaging 10.9 percent, vs 10 percent on a combined basis.• (4 , , i , dashed shows ( p, E [ X i | X = q ( p )]), i.e. the expectedloss recovery conditioned on total losses X = q ( p ), and solid shows ( p, E [ X i | X = q (1 − g − (1 − p ))]), i.e. the natural premium allocation (see the bottom row of table 1).Here the solid and dashed lines add up vertically: they are allocations of the total.Looking vertically above p , the shaded areas show how the total margin at that losslevel is allocated between lines. X mostly consumes assets at low layers, and the bluearea is thicker for small p , corresponding to smaller total losses. For p close to 1, largetotal losses, margin is dominated by X and in fact X gets a slight credit (dashedabove solid). The change in shape of the shaded margin area for X is particularlyevident: it shows X benefits from pooling and requires a lower overall margin. Thenatural allocation returns are 5.3, 10.6 and 10.0 percent. The overall premium tosurplus leverage is 0.308 to 1; on an allocated basis it is 0.986 and 0.223 to 1 for eachline. 26here may appear to be a contradiction between figures (3 ,
2) and (4 ,
3) but it should benoted that a particular p value in (4 ,
3) refers to different events on the dotted and solid lines.Plots (3 ,
3) and (4 ,
3) explain why the thick line requires relatively more margin (0.698 outof a total 0.728): its shape does not change when it is pooled with X . In (3 ,
3) the greenshaded area is essentially an upwards shift of the orange, and the orange areas in (3 ,
3) and(3 ,
4) are essentially the same. This means that adding X has virtually no impact on theshape of X ; it is like adding a constant, as discussed after the proof of proposition 2. Thiscan also be seen in (4 ,
3) where the blue region is almost a straight line.These shifts are illustrated in fig. 3. The left hand plot shows that the stand-alone marginarea for X shifted up by 1, the mean of X , lies almost exactly over the total margin area(orange over green). The right hand plot compares the stand-alone margin areas for each linewith the natural margin allocation. X is shifted up by 1 for clarity. Again, there is essentiallyno difference for X , especially in the expensive, large loss states where p is close to 1. X iscompletely transformed: its margin is much lower (smaller area) and it is concentrated in low p , small total loss events. X actually gets a credit at large total losses because its losseswill be close to the mean, and hence low ambiguity, whereas X ’s loss will be large and moreambiguous. We have explored how the shape of risk impacts the price of risk transfer in an imperfect,incomplete market—a holy grail for practicing actuaries. We provide a natural, assumption-free allocation of aggregate premium to policy, incorporating an allocation of capital consistentwith law invariance in order to price individual policies in the presence of frictional costs.Premium by policy is determined by the relative consumption of low and high ambiguityassets in a complex, but intuitively reasonable manner. The margin by line is driven moreby behavior in solvent states than in default states (default states are often the major focusof allocation methods). Premium is interpreted as the value of insurance cash flows undera risk-dependent state price density. This is in contrast to most other approaches thatadopt a cost allocation perspective. The natural allocation is insured-centric, rather thaninsurer-centric.The natural allocation depends on the fact that DRMs are law invariant and comonotonicadditive. It does not apply to more general convex risk measures. Notwithstanding thislimitation, it provides a useful and practical method that can be applied by an insurancecompany to understand how to share its diversification benefit between policies.Further research is needed to determine how the shapes of the X i interact to determine thenatural allocation, as well as the impact of different distortions g and capitalization standards.The underlying distortion can be calibrated to market prices. Market calibration to cat bonddata and standard intermediated insurance data would reveal how much of the cost of capitalarises from frictional costs and how much from shape of risk. The work can also be extended27 × Density
X1X2Total Log Density
X1X2Total L i n e X Bivariate Density i ( x ) = E [ X i X = x ] X1X2Total i ( x ) = E [ X i / X X > x ] X1X2 i ( x ) = E Q [ X i / X X > x ] X1X2
Line = X1 Sg ( S ) S X1 g ( S ) X1 Margin density M i ( x ) X1X2Total
Stand-alone M Line = X2 Sg ( S ) S X2 g ( S ) X2 Margin M i ( x ) X1X2Total
Natural M Figure 2: A thin tailed line combined with a thick tailed line. See text for a key to the graphs.28 .0 0.2 0.4 0.6 0.8 1.00123456
Thick line vs. TotalStand Alone Basis X + 1 , stand-aloneTotal Impact of Natural Allocation X , natural allocation X , stand-alone X + 1 , natural allocation X + 1 , stand-alone Figure 3: Impact of the natural allocation by line. Left: stand-alone margin area for thickline shifted up by mean of thin line lies almost exactly over the margin area for the total.Right: stand-alone vs. natural allocation margin areas, showing minimal impact for thickline but dramatic impact on thin. 29o dynamic portfolios and then applied to questions of optimal risk pooling under costlycapital.
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