Single-Year and Multi-Year Insurance Policies in a Competitive Market

Transcription

1 Single-Year and Mlti-Year Insrance Policies in a Competitive Market Pal R. Kleindorfer INSEAD, France Howard Knrether The Wharton School University of Pennsylvania Chieh O-Yang City University of Hong Kong People s Repblic of China April 01 Working Paper # Forthcoming in Jornal of Risk and Uncertainty Risk Management and Decision Processes Center The Wharton School, University of Pennsylvania 3730 Walnt Street, Jon Hntsman Hall, Site 500 Philadelphia, PA, USA Phone: Fax:

2 CITATION AND REPRODUCTION This docment appears as Working Paper of the Wharton Risk Management and Decision Processes Center, The Wharton School of the University of Pennsylvania. Comments are welcome and may be directed to the athors. The views expressed in this paper are those of the athors and pblication does not imply their endorsement by the Wharton Risk Center and the University of Pennsylvania. This paper may be reprodced for personal and classroom se. Any other reprodction is not permitted withot written permission of the athors. THE WHARTON RISK MANAGEMENT AND DECISION PROCESSES CENTER Established in 1984, the Wharton Risk Management and Decision Processes Center develops and promotes effective corporate and pblic policies for low probability events with potentially catastrophic conseqences throgh the integration of risk assessment, and risk perception with risk management strategies. Natral disasters, technological hazards, and national and international secrity isses (e.g., terrorism risk insrance markets, protection of critical infrastrctre, global secrity) are among the extreme events that are the focs of the Center s research. The Risk Center s netrality allows it to ndertake large scale projects in conjnction with other researchers and organizations in the pblic and private sectors. Bilding on the disciplines of economics, decision sciences, finance, insrance, marketing and psychology, the Center spports and ndertakes field and experimental stdies of risk and ncertainty to better nderstand how individals and organizations make choices nder conditions of risk and ncertainty. Risk Center research also investigates the effectiveness of strategies sch as risk commnication, information sharing, incentive systems, insrance, reglation and pblic private collaborations at a national and international scale. From these findings, the Wharton Risk Center s research team over 50 faclty, fellows and doctoral stdents is able to design new approaches to enable individals and organizations to make better decisions regarding risk nder varios reglatory and market conditions. The Center is also concerned with training leading decision makers. It actively engages mltiple viewpoints, inclding top level representatives from indstry, government, international organizations, interest grops and academics throgh its research and policy pblications, and throgh sponsored seminars, rondtables and forms. More information is available at

3 Single-Year and Mlti-Year Insrance Policies in a Competitive Market Pal R. Kleindorfer 1, Howard Knrether and Chieh O-Yang 3 4 April ABSTRACT This paper examines the demand and spply of annal and mlti-year insrance contracts with respect to protection against a catastrophic risk, sch as a hrricane or earthqake, in a competitive market. Hoseholds are identical with respect to their exposre to the hazard bt with different degrees of risk aversion. They determine whether or not to prchase an annal or mlti-year contract so as to maximize their expected tility. Insrers who offer annal policies can cancel policies at the end of each year and change the premim in the following year. Mlti-year insrance has a fixed annal price for each year and no cancellations are permitted at the end of any given year. The competitive eqilibrim consists of a set of prices for a single year and mltiyear policy that segments homeowners who are not very risk averse into the noninsrance category. Consmers who are not very risk averse decide not to prchase insrance. More risk averse individals demand either single-year or mlti-year policies depending on the premims charged by insrers and their degree of risk aversion. 1 INSEAD, France The Wharton School, University of Pennsylvania 3 City University of Hong Kong, Hong Kong 4 We thank Ioannis Siskos for his assistance in ptting this paper together and in ndertaking the detailed analyses and programming of the nmerical reslts presented. Special thanks to Keith Crocker, Dwight Jaffee, Krishna Kaliannen and Kip Viscsi for very helpfl comments on an earlier draft of the paper. Partial fnding for this research was provided by the Wharton Risk Management and Decision Processes Center Extreme Events Project and INSEAD; National Science Fondation (SES ); the Center for Climate and Energy Decision Making throgh a cooperative agreement between the National Science Fondation and Carnegie Mellon University (SES ); the Center for Risk and Economic Analysis of Terrorism Events (CREATE) at the University of Sothern California; CRED at Colmbia University; 1

4 1. Introdction Insrance policies for property insrance are normally issed as annal contracts even thogh they are not preclded from offering coverage for a longer time period. If the risk of a loss is stable over time, then the insrer shold be willing to offer a mlti-year policy with a fixed annal premim reflecting risk. The variance of the losses to the insrer over time wold be redced in mch the same way that an increase in the nmber of insred individals redces the variance. In this sense, offering a policy for more than one year is another form of risk diversification. The insrer who offers a mlti-year policy at a fixed premim per year is restricted by not being able to either raise the premim and/or cancel policies if it sffers a very large loss that redces its srpls significantly. This featre is similar to garanteed renewable policy with locked premims for health insrance (Paly, Knrether, and Hirth, 1995, Frick, 1998, Paly, Knrether, Menzel, and Hirth, 011). On the other hand, the administrative cost of marketing a policy is lower for a mlti-year policy than for annal contracts that have to be renewed each year. A mlti-year policy is attractive to a risk averse consmer becase the premim is stable over time. Frthermore, the consmer knows that it will not have to incr search costs in finding another insrer if its policy is canceled that has occrred historically when major catastrophes case insrers to adjst their nderwriting criteria. This paper examines from a theoretical perspective the relative attractiveness of mlti-year policies and annal policies to insrers and to hoseholds facing a given risk in a competitive insrance market. We are particlarly interested in prsing this line of research becase individals are likely to find mlti-year property insrance attractive relative to annal policies. Adding fixed-price mlti-year property insrance policies to the men of contracts offered by insrers shold also lead to an increase in the demand for coverage. Empirical evidence spporting these points comes from a web-based controlled experiment played for real money where individals had the option of prchasing 1-year or -year insrance contracts to protect their property against damage from a hrricane With respect to the 1-year policy, the price wold increase in year if a hrricane occrred in year 1 to reflect an pdating of the probability of a disaster occrring in the following year. The premim for a -year contract was kept stable over time; however, the insrer charged a higher premim than for an annal policy offered in year 1 becase it cold not increase its premim in year if a hrricane occrred in the previos year. The data reveal that when insrers offer individals 1-year policies at an actarially fair rate, approximately 6% of the sbjects prchase insrance. When the same 1-year contract is offered along with a -year contract that has a loading cost of 5%, the demand for insrance increases to 73%---a statistically significant difference. Over 60% of those demand insrance preferred the -year policy even thogh its expected cost was 5% higher than bying two annal policies (Knrether and Michel-Kerjan, 01). In practice, insrers do offer mlti-year contracts for life insrance where the losses are normally independent of each other. Term-life policies are typically offered with premims locked in for five to ten years; byers can choose whether they want to pay extra for sch garantees over annal contracts knowing that they may drop coverage at

5 any time. Policyholders are then certain what their life insrance premims will be over the next five or ten years, regardless of what happens to their health or the overall mortality rate of their insrer s portfolio. Hendel and Lizzeri (003) examine 150 term life insrance contracts, some of which have fixed premims for 5, 10 or 0 years while others are 1-year renewable policies. They show that on average the extra prepayment of premims to protect consmers against being reclassified into a higher risk category for a fixed period of time is more costly over the total period of coverage than a series of annal term policies that can be renewed bt where premims may flctate from year to year. Still, people by those mlti-year fixed-price life insrance policies, indicating that they view the stability of premims as an important attribte in their tility fnction and are willing to pay more for it. An important difference between property and life coverage is that insrers are concerned abot catastrophic losses to property de to natral disasters sch as hrricanes and earthqakes. They ths have to reserve capital in to protect themselves against these extreme events. There is an opportnity cost to them of being forced to keep this capital in relatively liqid form rather than investing the money in secrities that earn a higher expected retrn. On the demand side, consmers may want to prchase mlti-year property insrance policies is to avoid the search costs of looking for another insrer shold their annal policy be canceled. After the 004 and 005 hrricane seasons many insrers did not renew coverage for a significant nmber of homeowners in the Glf Coast (Klein 007). While most of those residents were able to find coverage with other insrers, they typically had to pay a higher price than prior to these disasters and they were reqired to have a higher dedctible (Vitelo, 007). Others obtained coverage from state insrance pools, which grew significantly after the 004 and 005 hrricane seasons (Grace and Klein, 009). Mlti-year property insrance contracts can also improve social welfare by increasing the nmber of individals whose homes are insred prior to a disaster. They will receive claim payments to restore property damaged from a hrricane or flood and have less need for federal disaster assistance. The cost to the general taxpayer is ths redced. In this regard, the nmber of ninsred homes facing losses from natral disasters is significant. The Department of Hosing and Urban Development (HUD) revealed that 41% of damaged homes from the 005 hrricanes were ninsred or nderinsred. Of the 60,196 owner-occpied homes with severe wind damage from these hrricanes, 3,000 did not have insrance against wind loss (U.S. Government Accontability Office, 007). With respect to risks from flooding, Kriesel and Landry (004) and Dixon et al. (006) fond that only abot half of the homes in high-risk areas had flood insrance. In California, despite the well-recognized risk of earthqakes, only 1% of homeowners in the state had coverage against earthqake damage at the end of 010 (California Department of Insrance, 011). A principal reason for this lack of coverage is that many residents prchase coverage only after a disaster occrs and then allow their annal policy to lapse. Flood insrance in the U.S. provided by the federally rn National Flood Insrance Program (NFIP) since its 3

6 creation in 1968 highlights this point. Brown and Hoyt (000) analyzed data from the NFIP between 1983 and 1993 fond that flood insrance prchases were highly correlated with flood losses in the previos year. A recent analysis of all new policies issed by the NFIP from Janary 1, 001 to 31 December 009 revealed that the median length of time before these policies lapsed is 3 to 4 years. On average, only 74% of new policies were still in force 1 year after they were prchased; after 5 years, only 36% were still in place. The lapse rate is still high after correcting for migration and does not vary mch across flood zones (Michel-Kerjan et. al. in press). The paper is organized as follows. Section models the mlti-period insrance problem and develops conditions for determining a competitive eqilibrim. Section 3 analyzes the case where the loss distribtion facing each homeowner is Gassian and where homeowner s risk preferences are of the CARA form sing an exponential tility fnction. Section 4 provides illstrative examples based on the Gassian distribtion and CARA tility fnction to show how the demand for annal and mlti-year insrance contracts is impacted by changes in the costs affecting the price of insrance and the correlation between risks. Section 5 smmarizes the paper and sggests directions for ftre research.. Modeling Mlti-period Insrance We consider an insrance market operating over two periods to cover a set of hoseholds exposed to a common catastrophic risk sch as earthqake or hrricane risk. The insrance market is assmed to be competitive with free entry and exit, bt sbject to solvency reglation. Risk bearing capital is obtained from premim income and reinsrance. The price of reinsrance in period 1 is known, bt the price in period is ncertain, and is specified by a binary random variable with a specified increase or decrease relative to the price in period 1 which depends on whether Two types of prodcts, single-period policies and mlti-period policies, compete for consmer demand for insrance. Homeowners can prchase either no insrance, singleyear policies in one or both periods, or a mlti-year policy prchased at the beginning of the first period and covering both periods. The competitive insrance price is determined so that it covers expected losses, marketing and operating expenses, pls the cost of risk capital necessary to provide protection against insolvency, where the level of the reqired capital is determined exogenosly by the insrance reglator. We assme that hoseholds are identical in terms of their exposre to the hazard, bt with some correlation in the losses, sch as wold be the case if a natral disaster damaged a nmber of insred homes. Denote the set of potential homeowners in the market by A. Homeowners are assmed to be risk averse with a A being a scalar index of risk aversion and with two period separable risk preferences given by V(x 1, x, a) = U x 1, a + U x, a (1) where U(x, a) is concave increasing in x, and x 1 and x are monetary otcomes in periods 1 and. Note that for simplicity we neglect disconting of tility in period. While A represents the set of potential homeowners, the actal distribtion of homeowner risk aversion will be specified by the conting fnction defined for any a A by 4

7 G a = Nmber of Homeowners with risk aversion less than or eqal to "a". To illstrate, sppose the nmber of homeowners is 100 of which 40 have a = 0.1 and 60 have a=0.. Then, G(a)=0 for a < 0.1, G(a)=40 for 0.1 a < 0., G(a) = 100 for a 0.. We make the following assmptions concerning the hazard and the policies offered in the insrance market to homeowners. A1. Only fll insrance is offered and each hosehold a A faces the same annal/period risk X (a) of loss, distribted according to the generic cdf F 0 (x, μ, σ, ρ), with mean E X (a) = μ and variance VAR X (a) = σ, where ρ 0 is a scalar index which is intended to measre the extent to which the loss distribtion for a book of bsiness (BoB) made p of properties from the set A is fat-tailed. The loss distribtion is assmed to be identical for both periods, and statistically independent between the periods. The impact of the index ρ will be specified below; it may be thoght of as an index of the cost of reinsrance cover for a BoB made p of properties from the set A. We se ρ to model the impact of correlated losses on standard reinsrance pricing models with constant or increasing loading factors. The reinsrance will be an Excess of Loss (XoL) contract with fixed pper and lower limits designated as L 1 n, and L n respectively. A. Firms offering single-year (SY) policies may cancel any policy at the end of the first period in response to increased cost of risk capital that leads them to want to redce their BoB. Homeowners are aware of this possibility and assme that there is a probability q (0, 1) of cancellation, with an ensing transaction cost of τ 0 to search for new coverage alternatives for period if their policy is canceled..1 Demand for Single-Year (SY) and Mlti-Year (MY) Policies Homeowners face the choice of prchasing either two single-year policies or a mltipolicy to cover the risks they face over the two period horizon of interest. Of corse, they may also elect to by no insrance (NI) in one or both periods. At the beginning of period one, homeowners mst either choose an MY policy covering both periods, or they mst plan on some other seqence of SY policies or NI decisions. In doing so, we assme that homeowners have complete information on the prices they will face. A3: Homeowners know the prices for all policies in all states of the world at the beginning of period 1. For the MY policy, the price per period price P M is constant over both periods. For the SY policy, price in the first period is denoted P S1, and the state-contingent price in the second period is denoted P w S, where w d, reflects the state of the world in terms of reinsrance/capital costs with probability of state d = φ [0,1] and probability of state = 1-φ. The conseqences of these alternative states for the insrer are described below, bt their general import is that the cost of reinsrance in period will decrease in state d and increase in state relative to period 1. At the beginning of period 1, there are three alternatives available to a homeowner: 5

8 i) Prchase an SY policy for period 1, at price P S1, possibly facing cancellation of this policy at the end of period 1, with probability q and with reslting transactions costs τ. ii) Prchase an MY policy at price per period of P M covering both periods 1 and. iii) Prchase No Insrance (NI). At the beginning of period, after the state of the world w d, is known, a homeowner faces the following choices (see Figre 1): i) If the homeowner chose either an SY policy or NI in period 1, then the homeowner can either choose NI or prchase an SY policy for period at price P w S where the price can either decrease to P d S or increase to P S depending on reinsrance capital costs. ii) The homeowner who chose an MY policy in period 1 can contine to be covered nder this MY policy or can cancel it with a cancellation fee ψ 0. The cancellation fee is set by the insrer so that it breaks even at the end of period. If there were no cancellation costs associated with an MY policy, then all homeowners will want to switch to an SY policy if the following two conditions hold: the price of an SY policy decreased in period so it was less than P M ; and the price differential between the MY and SY policy is greater than the transaction cost τ of prchasing a new policy. In this case, the MY insrer wold only offer coverage in the state of the world w =. As we show in the Appendix, P M is less than the MY insrer s average cost at w =, so that the MY insrer wold sffer a loss in the process. It will have priced its policy nder the assmption that it wold recover its costs from revenes generated in both periods, bt cold not recop these costs (some of which will be snk in period 1) becase its policyholders abandoned ship at the end of period 1. Hence, for MY insrance to be viable, the cancellation fee ψ imposed by the MY insrer has to satisfy: ψ P M -P d s -τ. 5 In this case, all homeowners wold maintain their MY policy in period. For or benchmark analysis in this paper, we assme, per A3, that homeowners are perfectly informed abot prices and the probability distribtion on the states of the world w {d, }, i.e., they know φ. We assme that their beliefs abot the cancellation probability q are fixed and independent of the actal BoB changes by insrers. 6 Figre 1 shows the relevant choices for a homeowner in period. 5 Or nmerical examples show for typical vales of insrance cost that the lower bond P M -P d s -τ on the cancelation fee ψ will be only a small percentage of the MY premim. Indeed, the lower bond may even be negative if the MY insrer has significant marketing cost advantages and/or if τ is sfficiently large. 6 Of corse, q cold be adjsted in rational expectation fashion so that it reflected the eqilibrim otcomes of SY insrers between periods 1 and. 6

9 Figre 1: Homeowner s Choice Problem in Period Given or assmption that homeowners will not cancel MY policies in period, the state-dependent insrance decision I (a, w) of homeowner a A, w {d, } at the beginning of period will be the following: D-i) D-ii) I a, w = MY if I 1 a = MY, with reslting period tility of U(-P M,a) I a, w = SY w if I 1 a MY and U(-P w S,a) U(CE(NI, a), a) with reslting period tility of U(-P w S, a) D-iii) I a,w = NI if I 1 a MY and U(-P w S,a) < U(CE(NI, a), a) with reslting period tility of U(CE(NI, a), a) where the certainty eqivalents (CEs) of the varios policies offered, MY and SY w, w {d, }, are CE MY = -P M and CE SY w = -P w S. The CE NI, a is characterized by U(CE(NI, a), a) = E{U(-X (a), a)}.7 The above three conditions can be interpreted in the following manner. A homeowner will have an MY policy in period only if he prchased an MY policy in period 1 (D-i). A homeowner will prchase an SY policy in period if he did not prchase an MY policy in period 1 and is sfficiently risk averse so that the tility of having fll insrance is greater than the expected tility of having no insrance in period. (D-ii). He will be in state NI if the expected tility of having no insrance in period exceeds the tility of SY(D-iii). Demand in period [D z, w ] for the policy options Z = {MY, SY w, NI} follows directly from the above. Let z, a, w = 1 if I a, w = z and z, a, w =0 otherwise, where z {MY, SY w, NI}. Then, D(z,w) A (z,a,w)dg(a), z {MY, SY w, NI}. () Eqation () jst allocates homeowners in period to an MY, SY or NI policy as a fnction of their degree of risk aversion and whether the reinsrance/capital costs are in state d or. 7 For convenience, we sppress explicit dependence of certainty eqivalents on initial wealth levels W(a). 7

10 Trning to period 1, recall that an SY insrer can cancel a policy at the end of period 1, and that homeowners believe this will occr with probability q and reslt in transactions costs τ. Then the certainty eqivalent CE(SY 1,a) of a first-period SY policy is characterized by U CE SY 1, a, a = qu - P S1 + τ, a +(1-q)U(-P S1, a). (3) Eqation (3) indicates that higher vales of q redces the attractiveness of an SY policy becase the homeowner is more likely to have her policy canceled and will have to incr a search cost τ to find an insrer who will be willing to sell her a policy in period. The optimal period 1 decision (note that we ignore disconting) is then determined throgh dynamic programming as follows. First, define the expected tilities V 1 (z) associated with choosing each of the options z Z 1 = {MY, SY 1, NI} in period 1 and the optimal state-dependent choice following this in period. Then V 1 MY, a = U -P M, a (4) V 1 SY 1, a = U(CE(SY 1, a), a) + φmax[u(-p d S, a),u(ce NI, a, a)] + 1-φ Max[U(-P S, a), U(CE(NI, a), a)] (5) V 1 NI, a = U CE NI, a, a + φmax[u(-p d S, a),u(ce NI, a, a)] + 1-φ Max[U(-P S, a), U(CE(NI, a), a)] (6) where, in view of the possibility of cancellation of the SY policy at the end of period 1, the period 1 expected tility of choosing an SY policy is given by (3). Eqation (4) states that a homeowner who prchases MY insrance in period 1 will contine to be insred by the same policy in period. Eqation (5) states that a homeowner who prchases SY insrance in period 1 incrs the immediate cost of the premim P S1 and, with probability q, may incr an additional transactions cost τ if the policy is canceled at the end of period 1, as reflected in the CE of SY 1 given in (3). This same homeowner has the option of bying a second SY policy in period or NI, and will choose the best of these two options in each state of the world w {d, }. Eqation (6) states that a homeowner choosing NI in period 1 can choose prchase an SY policy in period or remain ninsred (NI) and will choose the best of these options in each state of the world w {d, }. The optimal first-period choice for homeowner a A is then given by I 1 a = ArgMax z [V 1 z, a z Z 1 = MY, SY 1, NI ]. (7) The demand in period 1[D 1 z for the policy options Z 1 = {MY, SY 1, NI} follow directly from the above. Let 1 z, a = 1 if I 1 a = z and 1 z, a = 0 otherwise, where z {MY, SY, NI}. Then, 8

11 D(z) 1 1 A (z,a)dg(a), z {MY, SY, NI}. The above characterization of demand for MY and SY policies is general. Final demands for these policies in both periods depend on the distribtion of homeowner risk preferences as reflected in their degree of risk aversion characterized by G(a). It also of corse depends on the loss distribtion X. In section 3 below we consider the special case where per period losses X are normally distribted and the risk preferences of homeowners are of the CARA form, U x,a = -e -ax. (8). Spply and Pricing of Insrance We assme a competitive insrance market which consists of two types of firms, those offering SY policies and those offering MY policies. Firms offering SY policies can adjst the size of their BoB at the end of period 1 in response to changes in the cost of reinsrance (i.e. in response to the realized state of the world w {d, }). In a competitive eqilibrim, the size of the insrer s BoB is determined by the minimm of the average cost crve for the insrer. SY insrers will therefore cancel some policies at the end of period 1 in the state of the world in which reinsrance rates increase from period 1 to period and will expand their BoB when reinsrance rates decrease. MY insrers do not have this option and mst provide coverage in both periods to all homeowners to whom they issed policies in period 1. Reinsrance costs are assmed to depend on the non-negative scalar index ρ > 0, which may be thoght of as an index of the fat-tailed natre of the distribtion of a BoB of size n from the set A (see A1). A more specific model for reinsrance pricing is discssed in section 4. The following assmption smmarizes the relationship of the reglated solvency risk level and reinsrance costs for insrers. A4. Insrers are reqired to satisfy a reglatory solvency constraint. 8 It reqires for each period their combined premim revene pls reinsrance be sfficiently large to pay all claims with a probability of at least 1 - γ *. This reglatory solvency constraint is similar to a safety first model that insrers often tilize to determine the optimal BoB and pricing decisions. It was first proposed by Roy (195), examined in the context of the theory of the firm and profit maximization by Day, Aigner and Smith (1971) and applied to insrance by Stone (1973a and 1973b). Following the series of natral disasters that occrred at the end of the 1980s and in the 1990s, many insrers focsed on the solvency constraint to determine the maximm amont of catastrophe coverage they were willing to provide. More specifically they were concerned that their aggregate exposre to a particlar risk not exceed a certain level. Today rating agencies, sch as A.M. Best, focs on insrers exposre to catastrophic losses as one element in determining credit ratings, another reason for insrers to focs on the solvency constraint (Knrether and Michel-Kerjan 011). 8 We do not analyze the costs of insolvency here, bt rather assme that any insolvencies are paid for by an independent mechanism that does not affect the spply and demand decisions modeled here. 9

12 For simplicity, we assme that insrers with a BoB of size n meet their solvency constraint by prchasing XoL reinsrance with limits L 1 n, L n, with L 1 n = nμ (the expected loss cost for a BoB of size n) and L n set to meet the solvency constraint. Consider an insrer with BoB = {a 1, a,, a n } and the associated total loss distribtion n X n = i=1 X (a i ), with cdf F L; X n. Then, L n is characterized by 9 γ * =1-F L (n);x n = Prob{X n >L (n)}. Figre illstrates the above assmption on reinsrance attachment points and the solvency constraint. Reinsrance contracts are on a per period basis, corresponding also to the solvency constraints that are reqired to be flfilled in each period. Figre : Illstrating reinsrance attachment points nder assmption A4 The costs to insrers of providing coverage encompass administrative and selling expenses, loss costs and reinsrance costs, and depend on the size of the BoB (n). Formally, the expected total costs per period for SY and MY policies are given as: C SY n;r,ζ =C 0 n +C m n +μn+c s (n;r, μ, σ, ρ) (9) C MY n;r,ζ =C 0 n +νc m n +μn+c s (n;r, μ, σ, ρ) (10) 9 In the insrance literatre, the negative cdf 1-F L;X n is referred to as the exceedance probability (EP) crve. It is fndamental in reinsrance calclations since expected losses between any two attachment points can be calclated as the area nder the EP crve between these attachment points. See Grossi and Knrether (005) for details. 10

13 where the vector of parameter vales is given by ζ = (μ, σ, ρ, γ *, φ, q, τ) and where the elements of total expected cost are: C 0 n represents administrative and selling expenses; C m n represents marketing expenses; μn are total expected losses; and C s (n;r, μ, σ, ρ) represents reinsrance cost. Note that, with the exception of marketing costs, all of the elements of total expected cost are identical for SY and MY insrers. Concerning marketing costs, it is assmed that ν (0, 1] so that these per period marketing costs are likely to be less for an MY insrer since its policyholders in the second period are locked in as a reslt of first period choices. Ths, if SY and MY insrers were to choose the same BoB n in both periods, and if v = 1, then both insrers wold have identical total expected cost. However, SY insrers can adjst their BoB from period to period in response to changes in reinsrance costs, while the MY insrer is constrained to offer the same BoB in both periods. Ths, SY and MY insrers will in general face different total expected costs becase of potential marketing cost advantages for the MY insrer and potential flexibility advantages of the SY insrer in responding to changing reinsrance costs. Reinsrance costs are net of expected payoffs from reinsrance contracts, which are all of the XoL variety. In other words, the reinsrance cost reflects the additional premim above the expected loss paid by the insrer to the reinsrer for protection against a portion of the loss between the attachment points of the XoL contract. We assme that the average nderwriting costs [C 0 (n)/n] are convex and decreasing as n increases to reflect the spreading of fixed costs over more policies. The average marketing costs [C m n /n] are convex and increasing in n, representing the higher marginal cost of marketing as the insrance territory increases. Losses to the insrer have a mean μ and standard deviation σ. Average reinsrance costs [C s (n;r, μ, σ, ρ)/n] are convex in n and are dependent on whether the reinsrer is in state d or so that r = r w, w d,. These costs are also assmed to be increasing in ρ (the fat-tail index) reflecting the need for the reinsrer to hold higher reserves de to an increased probability of experiencing large losses. The Appendix, Part 1, specifies the derivation of the average costs and the eqilibrim conditions in a competitive market. Competitive eqilibrim in both the SY and MY markets occrs where insrers of each type select a BoB that minimizes their average cost, C SY n;r,ζ /n, C MY n;r,ζ /n, with price given by the minimm of the respective average cost crve. The assmptions concerning the elements of average costs discssed in the Appendix, Part 1, assre the existence of the eqilibrim for both SY and MY markets. These assmptions also imply a nmber of intitively appealing reslts for the comparative statics of eqilibrim prices and BoBs for both MY and SY insrers. For example, since reinsrance costs in period increase () or decrease (d) relative to period 1, depending on the state of the world, w d,, eqilibrim prices in the SY market satisfy: P d S < P S1 < P d S and the corresponding optimal BoBs satisfy: n S > n S1 > n S. Also, d the average costs of the MY insrer satisfy: AC M < P M < AC M so that the MY insrer has positive profits in state w = d and losses in state w = In particlar, as arged earlier, the cancelation fee needs to satisfy: ψ PM -P s d -τ in order to assre that the MY insrer will have viable operations in both the profitable state of the world w = d as well as the nprofitable state of the world w =. 11

14 The model proposed here sggests a nmber of trade-offs on both the demand and the spply side in evalating the srvival and efficiency of MY vs SY policies in competitive eqilibrim. On the demand side, MY policies offer a constant price over both periods and therefore are attractive to risk-averse homeowners in avoiding intertemporal price volatility. MY policies also protect homeowners from the transactions costs of policy cancellation (represented by the parameters q, τ above) associated with changes in the eqilibrim BoB for SY insrers between periods 1 and that reslt from changes in the cost of capital and reinsrance. On the spply side, there may be marketing and policy service cost savings associated with MY policies. However, MY policies expose the insrer to increased risk of reinsrance cost volatility, since the MY insrer cannot adjst his BoB in response to the realized state of the world w d, in contrast to the SY insrer. The ltimate otcome in terms of eqilibrim prices and demands for MY vs SY policies is an empirical matter depending on which of these trade-offs dominates and on the risk preferences of homeowners. The reslts in the next sections illstrate this for the case of Gassian loss distribtions and CARA preferences. 3. The Case of Gassian Losses and CARA Preferences This section describes the case where X, the generic loss distribtion facing each homeowner, is Gassian N(μ, σ ) and where homeowner risk preferences are of the CARA form: U x, a = -e -ax, where a > 0, i.e A =, the positive reals. We frther assme that there is a non-negative correlation ρ 0, 1 between the loss distribtions facing two homeowners X (a) and X (b), identical for all a, b A (a b). Ths, an insrer with a BoB of n homes wold face an annal loss distribtion with mean nμ and variance σ [n+n n-1 ρ]. These assmptions allow certainty eqivalents (CEs) for the demand eqations to be compted easily. Reinsrance costs for this case can be also readily compted for Excess of Loss (XoL) contracts with fixed limits L 1 n, L n, and linearly increasing loading factors. These costs can be shown to satisfy all of the assmptions detailed in the Appendix Part 1, inclding increasing costs as ρ increases. Eqilibrim prices nder competition are determined by (T1) and (T) as shown in the Appendix. These determine, for any given cost fnctions and reinsrance market conditions, the price vector {P M, P S1, P d S, P S }. This price vector and the parameter vector ζ = (μ, σ, ρ, γ *, φ, q, τ) are assmed to be common knowledge for both insrers and homeowners. We calclate demand for each homeowner a A from the demand eqations ()-(8), obtaining market demand by aggregation over G(a). Note that, given the Gassian and CARA assmptions, the Certainty Eqivalent (CE) of No Insrance (NI) for homeowner a A in either period 1 or is given by: CE(NI, a) = (μ+ aσ ). Ths, conditions D-i) D-iii) translate into the following decision rles for determining period demand for homeowner a A: DN-i) I a, w = MY if I 1 a = MY, with reslting period tility of U(-P M,a) 1

15 DN-ii) If I 1 a MY, then I a,w = NI or SY whichever yields the maximm expected tility with a reslting period tility of Max{U -P w S, a, U CE NI, a, a }. Period 1 demand for a A is then specified by (4)-(6) which are smmarized as follows: D1N-i) D1N-ii) I 1 a = MY if V 1 MY, a = U -P M, a Max{V 1 SY, a, V 1 NI, a }; Else I 1 a = SY or NI, whichever gives the highest period 1 expected tility V 1 NI, a, V 1 SY, a as given in (5) (6), noting from (7) that the CE of prchasing SY insrance in period 1 is CE(SY 1, a) = - 1 a log L(a, ζ) eap S1, with L(a, ζ) = qe aτ +(1-q). The conseqences of the above rles for period choices are smmarized as follows. A homeowner who chose an MY policy in period 1 will contine it in period. A homeowner who chose either NI or SY policy in period 1 will choose the option, NI or SY that has a higher expected tility. This gives rise to two critical ctoff vales of homeowner risk preferences, a d and a, which are the ct off vales for NI vs SY choices in period given the state of the world w {d, } so that: U CE NI, a, a = Max{U -P w w S, a,u CE NI, a, a } for a < a U -P w S, a = Max{U -P w S, a,u CE NI, a, a } for a a w. The comparisons in (11) are eqivalent to determining when P w S > or μ+ 1 aσ, where the ctoff vales are therefore given by: a w = (P w S -μ), w {, d}. Since, as noted above, P d S < P S, it follows that a d < a. The ctoff vales in Period indicate that consmers who are more risk averse will want to prchase an SY policy in period 1. Frthermore if the price of insrance is lower becase the insrer is in the d rather than state, consmers who are less risk averse will still be willing to prchase insrance (i.e. a d < a ). Given the period choice, the optimal choice in period 1 is specified by D1N-i) - D1N-ii). To characterize this soltion, define the critical threshold risk aversion a (P S1 ) A as the soltion to V 1 NI, a V 1 SY 1, a. In the Appendix Part, a (P S1 ) is shown to exist and be niqe. As we will see below, homeowners with risk aversion a < a (P S1 ) will prefer NI to SY in period 1 and homeowners with risk aversion a > a (P S1 ) will prefer SY to NI in period 1 (if these were the only choices available). From the definition of V 1 NI, a, V 1 SY 1, a (see (5) and (6), we see that period payoffs from choosing NI or SY in period 1 are identical. so that a (P S1 ) is the soltion to U(CE(NI, a), a) = U(CE(SY 1, a), a). This is eqivalent to CE(NI 1, a) = CE(SY 1, a). For the CARA/Gassian this yields the following identity characterizing a (P S1 ): σ (11) 13

16 a (P S1 ) : μ+ 1 aσ = 1 a log L(a, ζ) eap S1. (1) Similarly, define the fnctions H 1 a, NI-MY, respectively H 1 a, SY-MY, as the vale of P M for which V 1 NI, a =V 1 MY,a, respectively V 1 SY 1, a =V 1 MY, a. From (4)-(6), these fnctions are characterized by: H 1 a, NI-MY : U -P M,a = U CE NI, a, a + φmax[u(-p d S, a),u(ce NI, a, a)] + 1-φ Max[U(-P S, a),u(ce(ni, a), a)]; (13) H 1 a, SY-MY : U -P M,a = U CE SY 1, a, a + φmax[u(-p d S, a),u(ce NI, a, a)] + 1-φ Max[U(-P S, a),u(ce(ni, a), a)]. (14) In the Appendix, Part, we show that there is a niqe soltion P M to (13)-(14) for every a A so that these fnctions are well defined. We also show that both fnctions are increasing in a A. Intitively, this reflects the fact that as homeowners become more risk averse, they are prepared to pay a higher premim P M for an MY contract. They prefer this mlti-period contract over the alternatives for the following reasons: Avoiding the risks of sffering a large loss from not being insred (NI) Having their policy cancelled or paying a higher premim in period if they prchased an SY policy in period 1. Using the above fnctions (1)-(14), we can smmarize the soltion to the two-period dynamic programming problem characterizing the optimal choice for homeowner a A in period 1 as shown in Figre 3. This figre shows for arbitrary bt fixed vales of the prices P S1, P d S, P S, the optimal period 1 demand response for homeowners as P M varies. Of corse, P M is also fixed at its eqilibrim vale (see (T) in the Appendix), bt one can think of this figre as illstrating alternative demand otcomes as the MY eqilibrim price P M varies, driven by the nderlying costs in (10). 14

17 Figre 3. Period 1 Demand for NI, SY, MY H 1 (a, NI-MY) The logic of Figre 3 is as follows. Each of the fnctions in (1)-(14) separates the set of homeowners A into two sbsets for each of the three sets of binary choices {NI, SY}, {NI, MY}, {SY, MY} into those who prefer the first of these binary choices and those who prefer the second. Ths, in evalating the two choices NI and SY in period 1, any homeowner a < a (P S1 ) prefers NI to SY and in any a > a (P S1 ) prefers SY to NI. Similarly, in evalating the two choices NI and MY in period 1, any homeowner a A sch that P M > H 1 a, NI-MY prefers NI to MY and prefers MY to NI if P M < H 1 a, NI-MY. Finally, in evalating the two choices SY and MY in period 1, any homeowner a A sch that P M > H 1 a, SY-MY prefers SY to MY and prefers MY to SY if P M < H 1 a, SY-MY. This leads to a complete determination f consmer demand for any eqilibrim price vector P = {P M, P S1, P d S, P S }. The six vales of Θ i i=1 6 in Figre 3 illstrate how one determines the optimal policy choice for homeowners in period 1. Consider the (a, P M ) pair Θ 1 Given its location relative to the three binary choice crves, homeowner a(θ 1 ) prefers MY to SY, SY to NI and MY to NI: so that this homeowner s optimal period 1 choice at price P M (Θ 1 ) is MY as shown. For the Θ pair, homeowner a(θ ) prefers SY to MY, SY to NI and MY to NI: so that this homeowner s optimal period 1 choice at price P M (Θ ) is SY as shown. For the Θ 3 pair, homeowner a(θ 3 ) prefers SY to MY, SY to NI and MY to NI: so that this homeowner s optimal period 1 choice at price P M (Θ ) is SY as shown. 15

18 For the Θ 3 pair, homeowner a(θ 3 ) prefers SY to MY, SY to NI and MY to NI: so that this homeowner s optimal period 1 choice at price P M (Θ 3 ) is SY as shown. For the Θ 4 pair, homeowner a(θ 4 ) prefers NI to MY, NI to SY and SY to MY: so that this homeowner s optimal period 1 choice at price P M (Θ 4 ) is NI. For the Θ 5 pair, homeowner a(θ 5 ) prefers NI to MY, NI to SY and MY to SY: so that this homeowner s optimal period 1 choice at price P M (Θ 5 ) is NI. For the Θ 6 pair, homeowner a(θ 6 ) prefers MY to NI, NI to SY, and MY to SY: so that this homeowner s optimal period 1 choice at price P M (Θ 6 ) is MY. All that is reqired to prodce consistent preferences is that the intersection point Θ * of H 1 a, NI-MY and H 1 a, SY-MY occr at a (P S1. This follows directly from the fact that V 1 MY, a = V 1 NI, a on H 1 a, NI-MY and V 1 MY, a = V 1 SY 1, a on H 1 a, SY-MY, so that, at the intersection Θ *, V 1 NI, a = V 1 SY 1, a. Since a (P S1 is the niqe soltion to (1), all three of the binary choice fnctions mst intersect at the same point Θ *. The logic of Figre 3 determines period 1 demand for any homeowner a A. Aggregate demand for each policy type (NI, SY, MY) is then determined by the distribtion G(a) of homeowner risk aversion in the poplation. A typical distribtion fnction G(a) is shown in Figre 3. With period 1 choices determined as above, period choices follow directly from DN-i) - DN-ii) following the logic of Figre 1. More specifically, if a homeowner boght an MY policy in period 1 then she will contine to have this policy in period. If the homeowner did not by an MY policy in period 1, then she will decide to either by an SY policy in period or be ninsred (NI) depending on on whether reinsrance costs increase () or decrease (d) relative to period 1. Ths, the CARA/Gassian case is completely solved. To smmarize, the price vector P = {P M, P S1, P d S, P S } is determined for the SY and MY markets from the conditions defining competitive eqilibrim in these markets, as characterized by (T1)-(T) in Appendix, Part 1. We define the critical vales of risk aversion a NM and a SM as the niqe soltions to: P M = H 1 a NM, NI-MY ; P M = H 1 a SM, SY-MY. We ths have the following critical vales with respect to risk aversion in determining the option in period 1 which maximizes the homeowner s expected tility when only two alternatives are available: a NM when only NI and MY are available to the homeowner. a SM when only SY and MY are available to the homeowner. a (P S1 ) when only NI and SY are available to the homeowner. Given these critical vales determining all possible binary choices, demand is then determined as follows: 16

19 i) There is a region from a = 0 to some a NI (P) = min{a (P S1 ), a NM }, sch that homeowners with a < a NI (P) all choose NI in period 1 (in period they choose the best of the two options of SY or NI depending on the state-dependent prices that obtain). ii) Defining a SY P = Max{a SM, a (P S1 )}, there is a region, possibly empty, between a (P S1 ) and a SY P, sch that homeowners with a [a (P S1 ), a SY P ] choose SY in period 1 (in period they choose the best of the two options of SY or NI depending on the state-dependent prices that obtain). iii) Homeowners with a max{a NI (P), a SM } all choose MY in period 1 (and they stick with this in period ). Total market demand is then determined by () and (8), by integrating over the distribtion G(a) of homeowner risk preferences. We ths have a complete soltion to the CARA/Gassian case. 4. Illstrative Examples for the CARA/Gassian Case This section provides some nmerical examples to illstrate the otcomes for the CARA/Gassian case derived in section 3. The elements of the expected total cost fnctions [see (9)-(10)] are specified as follows: Operating expenses C 0 n are represented as the sm of a fixed cost and a variable cost depending on the size of the BoB: C 0 n = c 0 n + K, c 0 > 0, K > 0. (15) Marketing and selling expenses are assmed to be identical in both periods for the SY insrer and are specified by: C m n = c m n s, c m > 0, s >. (16) The marketing and selling expenses for the MY insrer are assmed to be allocated eqally to both periods (althogh they are the same or lower in Period ) and are represented as a fraction ν 1 of an SY insrer for the same BoB. Total marketing and selling costs for the MY insrer with a BoB of size n eqals ν C m n. Reinsrance costs vary across periods 1 and. For period 1, they are specified in the form of a linearly increasing loading factor. This implies that reinsrance costs inclde a factor that is proportional to the expected vale of the reinsrance coverage and a second factor that increases qadratically as the size L (n) - L 1 (n) of the reinsrance tranche covered by the XoL treaty increases. Appendix, Part 3, characterizes the reinsrance costs facing the insrer. For the base case the parameters for the total cost fnction and the reinsrance costs specified in the Appendix satisfy the stylized assmptions in Table 1 for a mlti-line insrer offering catastrophe coverage: 17

20 Table 1: Assmptions Underlying Calibration for the Base Case nμ- Re-Payot: Average Annal Loss net of Reinsrance Payot = 45% of the insrer s total cost C 0 : Operating/Underwriting Expenses (exclding marketing), inclding attritional losses and loss adjstment expenses = 5% of the insrer s total cost C m : Marketing and selling Expenses = 15 % of the insrer s total cost C s : Reinsrance premim = 15% of the insrer s total cost We assme for the base case that capital costs vary by ± 0%. The distribtion of risk aversion in the poplation of homeowners is assmed to be lognormal with mean and variance so that 0% of the homeowners choose NI at the base case eqilibrim prices. The nmber of homeowners was set at N=1000. These assmptions, together with the calibration benchmarks in Table 1, give rise to the parameters in Table for the base case corresponding to (15)-(16) and (T11)-(T17) in Appendix. Part 3. Tables 3 and 4 detail the base case otcomes. The nmber of homeowners who demand MY policies in period 1 is 580 with the remaining 0 prchasing an SY policy. Demand for SY policies in period increases to 416 if w = d since almost all of the ninsred individals in period 1 now decide to prchase coverage. When w = all the individals who were NI in period 1 will remain ninsred in period and those who boght an SY policy in period 1 will decide not to renew, so there will be no demand for SY policies in period. 18

21 Nmber of Homeowners N Table : Base Case Parameter Vales A dg(a) G(a) = NG A (a), G A (a) = lognormal(μ A,σ A ) μ, σ, ρ, φ, τ, q N = 1000 μ A = σ A = μ = 0000, σ = 6000, ρ = 0.5, φ = 0.5, τ = 100, q = 0.1 C 0 n = c 0 n + K c 0 = 150, K=65000 SY: C m n = c m n s, MY:ν C m n c m = 35, s =.01, ν = 1 L (n) C s1 (n;r 1, ζ) = λ 1-1+ξ 1 x 1-F x, n, ζ dx L 1 (n) λ 1 = 1.5, ξ 1 = L (n) C s (n;r (w), ζ) = λ (w)-1+ξ (w)x 1-F x, n, ζ dx L 1 (n) λ d = (1-δ)λ 1, ξ d = (1-δ)ξ λ = (1+δ)λ 1, ξ = (1+δ)ξ δ = 0. Table 3. Otcomes for the Base Case for the SY Insrer 1 st Period nd Period w = d nd Period w = 1) Eqilibrim prices P S1, P d S, P S 40,705 39,530 41,857 ) Size of the BoB n ) Homeowner Demand for SY policies ) Average Annal Losses 50, , ,000 5) Expected Reinsrance Payots 44,464 46,143 4,786 6) Reinsrance Premim 149,783 17, ,780 7) Operating/Underwriting Expenses 68,900 69,050 68,750 8) Marketing and selling Expenses 164,11 177, ,680 Total Expected Cost per period ( ) Total Expected Cost Colmn1+φColmn +(1-φ)Colmn3 1,058,340 1,067,314 1,046,44,115,08 19

22 Table 4. Otcomes for the Base Case for the MY Insrer 1 st Period nd Period w = d nd Period w = 1) Eqilibrim price P M 40,705 ) Size of the BoB n ) Homeowner Demand for MY policies 580 4) Average Annal Losses 50,000 5) Expected Reinsrance Payots 44,464 6) Reinsrance Premim 149, ,86 179,739 7) Operating/Underwriting Expenses 68,900 8) Marketing and selling Expenses 164,11 Total Expected Cost per period ( ) 1,058,340 1,08,383 1,088,96 Total Expected Cost Colmn1+φColmn +(1-φ)Colmn3,116,679 As the BoB for the MY insrer is constant over both periods and all states of the world, the only otcomes that change between periods1 and are the reinsrance premims, and hence the Total Expected Cost. Note that the BoB changes for the SY insrer as a fnction of the state of the world with a slightly larger (BoB when reinsrance premim decreases nder the state of the world w = d and a slightly lower BoB when reinsrance premims increases when w =. SY insrers choose the optimal BoB in each state of the world while the MY insrer mst choose a constant BoB over both periods. This gives rise to lower Total Expected Costs over both periods for the SY insrer relative to the MY insrer since or base case assmes no marketing cost advantages for the MY insrer (ν = 1). The reslts for Demand for the base case are intitive and follow the logic of Figre 3. We now consider some comparative reslts on SY vs MY policies as key parameters change. It shold be noted from Figre 3 that in general neither policy type can be expected to dominate the other for all homeowners. The eqilibrim otcomes will depend on the distribtion of homeowner risk aversion. The more risk averse the poplation of homeowners, and the more significant are marketing cost advantages of MY insrers, the greater the demand for MY policies. Similarly, the greater the perceived probability q of policy cancellation, and the greater the transactions costs τ of finding a new insrer, the greater the advantage of MY over SY policies. These general advantages of MY policies are conterbalanced by the greater flexibility of SY insrers to adjst 0