Loading Factors and Equilibria in Insurance Markets

Transcription

1 Loading Factors and Equiibria in Insurance Markets Yoram Eden, * Eiakim Katz, ** and Jacob Rosenberg *** Abstract: Tis paper examines te effect of introducing positive oading factors into insurance premia, in insurance markets consisting of groups of individuas in different risk categories It is sown tat tat suc oading factors may ave far-reacing effects on insurance market equiibria T INTRODUCTION e existence and nature of equiibria reaced in insurance markets consisting of groups of individuas in different risk categories as attracted muc attention in te iterature 1 At te same time, tere as been interest in te iterature in te functiona forms of te oading factor component in insurance premia 2 Surprisingy, te reation between insurance oading factors and te type of equiibria in insurance markets as not been examined in te iterature It is te purpose of tis paper sow tat oading factors may ave far-reacing effects on insurance market equiibria Te point is made by constructing a simpified mode of an insurance market Te equiibrium in tis market is ten examined bot in te absence and in te presence of a simpe and very commony used type of oading factor Specificay, it is assumed tat tere are two types of riskaverse consumers ig and ow-risk and tat tese consumers can eiter fuy insure teir risk or remain fuy exposed to it Insurance coverage is indivisibe Te market is competitive Under tese assumptions and in te absence of oading factors, te market wi be caracterized * Te Scoo of Business Administration, Te Coege of Management-Academic Studies, Israe ** Department of Economics, Nortern Iinois University, DeKab, IL *** Te Coege of Management Academic Studies, Israe 52 Journa of Insurance Issues, 2003, 26, 1, pp Copyrigt 2003 by te Western Risk and Insurance Association A rigts reserved

2 LOADING FACTORS AND EQUILIBRIA 53 by one of two equiibria Te first is a pooing equiibrium, in wic a consumers buy insurance Because te price of insurance is competitivey determined and equa for bot types of individuas, tis type of equiibrium impies tat te ow-risk individuas are subsidizing te price paid for insurance by te ig-risk individuas Te second equiibrium is a separating equiibrium in wic te ig-risk individuas buy insurance wereas te ow-risks do not Tis occurs if te ow-risk individuas are not sufficienty risk-averse to be induced to pay for insurance wose price partiay depends on te riskiness of te ig-risk individuas In te absence of oading factors tere wi be no equiibrium in wic no insurance is bougt Once oading factors are introduced, owever, potentia market outcomes are significanty atered In addition to te two types of equiibria described above, an equiibrium in wic no insurance is bougt becomes possibe Tis occurs if te oading factor exceeds te risk premium tat te ig-risk individuas are prepared to pay Using simuations, and a specific numerica exampe, we sow tat arge groups of potentia buyers may refrain from buying insurance as a resut of sma oading factors THE MODEL Consider a popuation in wic every individua as initia weat W and utiity function U(Z), were Z is fina weat Individuas are riskaverse, ie, U (Z) > 0 and U (Z) < 0, and tey maximize expected utiity A proportion α of tis popuation consists of ig-risk () individuas, wo stand to incur a oss equa to X wit a probabiity π Te remaining individuas, wo make up 1 α of te popuation, are ow-risk (), and tese individuas stand to ose X wit a probabiity π, were π >π Individuas risks are independent of eac oter If an individua does not incur a oss, is weat remains uncanged, Z W An individua can eiter remain exposed to te risk or buy fu insurance coverage against it 3 However, wie eac individua knows is own type, tis information is not avaiabe to insurance companies, because te cost of risk cassification is proibitivey ig Insurance companies carge premia tat refect bot expected caims and positive oading Te oading covers commissions, administrative costs, and settement expenses, as we as norma profit 4 Tus, given te expected vaue of a caim, A, te premium carged is f(a) suc tat f (A) > 1 In tis paper we assume tat f(a) (1 + λ)a were λ( 0) is a constant oading factor 5,6, wose vaue is determined by competitive market forces, 7 and investigate te reation between market equiibrium and λ As mentioned above, we find tat tere are tree

3 54 EDEN ET AL possibe outcomes, depending on te vaues of α and λ In te event tat λ 0, owever, te tree outcomes coapse into two A Separating Equiibrium NO LOADING FACTOR: λ 0 A separating equiibrium is one in wic different individuas, in tis case bot te ig and ow-risks, beave differenty Tus, tere are two potentia separating equiibria In te first, ig-risk individuas insure and ow-risk individuas do not In te second, ow-risk individuas insure and ig-risk individuas do not However, as is intuitive and sown beow, ony te first of tese equiibria is viabe To see tis, note tat if ony igrisks insure, te insurance premium is P s π X Simiary, if ony ow-risks insure, te insurance premium is P s π X Ceary, an equiibrium in wic ony ow-risks insure is not viabe, since if te ow-risks are prepared to insure at a premium equa to P s, ten, a fortiori, so wi te ig-risks Given a premium equa to of insurance to ig-risk individuas is P s, te benefit B s UW ( π X) ( 1 π )UW ( ) π UW ( X) Aso, given a premium equa to P s te benefit of insurance to ow-risk individuas is B s UW ( π X) ( 1 π )UW ( ) π UW ( X) Hence, since B s B s ( π π )[ UW ( ) UW ( X) ] > 0,

4 LOADING FACTORS AND EQUILIBRIA 55 a separating equiibrium, in wic te ow-risks insure but te ig-risks do not, is not possibe Moreover, if ony te ig-risks insure, and te insurance premium is, terefore, π X, te net benefit of insurance to a igrisk individua is B s UW ( π X) ( 1 π )UW ( ) π UW ( X), wic is positive by te concavity of U Tus, ig-risks insure even if owrisks do not Note tat B s > 0 for a vaues of α A Pooing Equiibrium A pooing equiibrium is one in wic a individuas beave in te same way Tus, tere are, potentiay, two suc equiibria: first, an equiibrium in wic bot ig and ow-risk individuas purcase te insurance offered, and second, an equiibrium in wic bot ig and ow-risk individuas do not purcase te insurance offered In a pooing equiibrium in wic a individuas insure, te expected caim per poicy is given by A ( απ + ( 1 α)π ), impying tat te insurance premium is P p A ( απ + ( 1 α)π )X In deciding weter or not to purcase insurance, eac individua considers is expected utiity wit and witout insurance In te absence of insurance, te fina weat of a ow-risk individua is W wit probabiity 1 π, and W X wit probabiity π If e does buy insurance, is fina weat is W P p wit certainty In a pooing equiibrium, terefore, te net benefit of buying insurance for an individua in risk category i, (i, ) is i, were i UW ( P p ) ( 1 π i )UW ( ) π i UW ( X) i An individua in group i buys insurance if and ony if since 0 Hence, ( π π )[ UW ( ) UW ( X) ] > 0,

5 56 EDEN ET AL tis pooing equiibrium exists if and ony if UW [ ( απ + ( 1 α)π )X] ( 1 π i )( UW ( )) π UW ( X) 0 It transpires tat weter or not a pooing equiibrium exists depends on α and on te utiity function of te individuas making up te popuation 8 Note tat if α 0, ten UW [ π X] ( 1 π )UW [ ] π UW [ X] > 0, because U is concave Tus, for a sufficienty sma α, a pooing equiibrium wi exist Ceary, B p ( π, α π )XU'( W P p ) < 0 ie, an increase in α reduces An increase in te number of ig-risk individuas raises te premium carged in a pooing equiibrium Tis makes te buying of insurance ess attractive to ow-risk individuas Tere may terefore exist a vaue of α, α (defined by 0 ) suc tat a pooing equiibrium exists for α α but ony a separating equiibrium exists for α> α Of course, te vaue of α* is a function of W, X, π, and π, as we as te risk aversion of a representative individua It is usefu to iustrate tis wit a specific utiity function and a simpe numerica exampe Let te utiity function be U Z 1 r were r is a constant and equa to reative risk aversion, 1 > r 0 Let W 10, X 9, π 02, and π 01 Ten te premium carged in a pooing equiibrium in wic everyone insures is P p 09α + 09,

6 LOADING FACTORS AND EQUILIBRIA 57 so tat ( 10 ( 09α + 09) ) 1 r 09( 10) 1 r 01 0 Te vaue of α tat just dissuades ig-risk individuas from buying insurance is terefore given by n( exp( 23026r) ) r α exp r Te reader can gauge te impact of risk aversion on te possibiity of a pooing equiibrium from Tabe 1 beow If an individua s reative risk aversion exceeds 0854, ten, given te above parameter vaues, a pooing equiibrium wi emerge, regardess of te vaue of α Tus, if r 0854, ten even if tere exist ony a negigibe number of ow-risk individuas, so tat te insurance premium refects ony ig-risk individuas, te ow-risk individuas wi noneteess buy insurance Tabe 1 Te reation between r and α r α As is cear from Tabe 1, as r decines, te toerance of te ow-risk individuas to α aso decines Tus, for exampe, wie for r 07, α must be ess tan 0743 to ensure a pooing equiibrium, for r 02, α must be ess tan 0156 to ensure a pooing equiibrium

7 58 EDEN ET AL A POSITIVE LOADING FACTOR: λ > 0 A Separating Equiibrium Once again tere is ony one viabe separating equiibrium ie, igrisk individuas insure and ow-risk individuas do not If ony ig-risks insure, te insurance premium is P s ( 1 + λ)π X Given a premium equa to individuas is P s, te benefit of insurance to ig-risk B s UW ( π ( 1 + λ)x) ( 1 π )UW ( ) π UW ( X) However, even if λ > 0, ten, te concavity of te utiity function ensures tat for some parameters combinations, B s > 0 A separating equiibrium can emerge 9 However, since B s π, λ XU' ( W π ( 1 + λ)x) < 0 a sufficienty arge λ wi reduce a demand for insurance to zero Te reader may gauge te reation between λ and risk aversion by considering our above functiona and numerica exampe In tis case P s ( 1 + λ) ( π X) 18( 1 + λ) so tat B s ( 10 18( 1 + λ) ) 1 r 08( 10) 1 r 02 0 Te vaue of λ tat just dissuades ig-risk individuas from buying insurance is terefore given by

8 LOADING FACTORS AND EQUILIBRIA 59 n( exp( 23026r) ) r λ exp r To give te reader a sense of te magnitudes invoved, λ is given in Tabe 2 beow, for different vaues of r 10 Tabe 2 Te reation between r and λ* r λ As is cear from Tabe 2, as r decines, te toerance of te ig-risk individuas to λ aso decines Tus, for exampe, wie for r 07, λ must be ess tan 0642 to ensure tat tis separating equiibrium is viabe, for r 01, λ must be ess tan to ensure a separating equiibrium Pooing Equiibria As te oading factor, λ, becomes arger, buying insurance is ess attractive for bot categories Tus, tere are two possibe pooing equiibria In te first, no insurance is bougt by eiter group, and in te second bot groups by insurance No insurance bougt In tis case, te parameters are suc tat even te ig-risks do not buy insurance Tus, if λ >λ*, as defined above, a pooing equiibrium emerges, in wic no insurance is bougt Bot groups buy insurance In tis case, te parameters are suc tat risk aversion is sufficienty ig to cause even ow-risks to buy insurance, despite paying a premium tat is affected by te ig-risks and despite te oading factor To ensure tat bot groups buy insurance we require tat te net benefit of te ow-risks is non-negative ie, tat UW [ ( 1 λ) ( απ + ( 1 α)π )X] ( 1 π )UW [ ] π UW [ X] 0,

9 60 EDEN ET AL since te insurance premium in tis case is P p ( 1 + λ) [( απ + ( 1 α)π )]X Once again, te critica vaues of te parameters α and λ are functions of W, X, π, and π, as we as te risk aversion of a representative individua To get furter insigt, we revert to te above-specified utiity function and numerica exampe Te premium in tis case is P p 09( 1 + λ) ( 1 + α) 11 ie, tat To obtain a pooing equiibrium we terefore require tat ( W P p ) 1 r ( 1 π )( W) 1 r π ( W X) 1 r 0, [ 10 09( 1 + λ) ( 17) 1 r r ] 0 Te reader may find it of interest to examine te tradeoff between α and r in deterring te ow-risks from insuring Tabe 3 λ* for combinations of α and r α\r Te above tabe suggests tat even a sma oading factor may ave a arge impact on te type of equiibrium reaced: In order to generate a pooing equiibrium in wic a individuas buy insurance, λ must be smaer tan λ* Tus, for exampe, if 10 percent of te consumers are igrisks, insurance wi not be bougt by te oter 90 percent, for r 0, regardess of λ Moreover, even if r 03, no insurance wi be bougt by te ow-risks, uness λ < 0136 A oading factor of 15 percent, say, woud

10 LOADING FACTORS AND EQUILIBRIA 61 act to excude 90 percent of te potentia market Tis soud be compared wit te earier resut, tat in te absence of oading costs, a individuas wi insure for tese parameters vaue, if r > 0075 CONCLUSION Tis paper demonstrates tat te incorporation of oading factors into te anaysis of insurance market equiibria may yied interesting and potentiay important resuts Te resuts presented ere are based on a particuar, toug commony used, functiona form for te oading factor Tey suggest tat furter investigation of te reationsip between te type and existence of equiibria in insurance markets is in order NOTES 1 Te semina works in tis area are Rotscid and Stigitz (1976) and Wison (1977) 2 See Pitkanen (1975) and Kaane (1979) for eary discussions of tis issue For a more recent approac, see, for exampe, Tayor (1994) 3 No partia insurance is permitted in tis mode 4 See Kaane (1979) 5 See Pitkanen (1975) Goier (1996) aso refers to tis form in discussing te standard in te iterature 6 Proportiona oading factors are often used by insurance companies for severa reasons First, tis is a simpe and straigtforward pricing metod Second, it corresponds to te cost structure of te insurance company Commissions, wic are te main costs tat te oading factor must cover, are generay determined as a fraction of te premium Tird, tis pricing metod enabes te insurers to pubis a price ist of tariffs, in wic te required premium is expressed as a percentage of te insured sum Fourt, it enabes insurance companies to decentraize te underwriting decisions And fift, it avoids te need to negotiate wit te customer on eac ordinary poicy 7 Tus λ is suc tat insurance firms cover teir costs and earn norma profits 8 Remember tat a individuas ave te same utiity function 9 Tis wi occur if te risk facing te ow-risk is sufficienty ow to cause tem to reject insurance tat is (partiay) based on te risk facing te ig-risk group 10 Te vaue of r tat dissuades ig-risk individuas from buying insurance, even for λ 0, is Note tat, in genera, λ and α do not enter te insurance premium symmetricay Te fact tat tey do so ere is a coincidence owing to te parameter vaues cosen REFERENCES Goier C (1996) Optimum Insurance of Approximate Losses, Te Journa of Risk and Insurance, Vo 63, No 3, pp

11 62 EDEN ET AL Kaane Y (1979) Te Teory of Insurance Risk Premium A Re-examination in te Ligt of Recent Deveopments in Capita Market Teory Te Astin Buetin, Vo 10, No 2, pp Pikanen P (1975) Tariff Teory, Te Astin Buetin, Vo 8, part 2, pp Rotscid M, and J Stigitz (1976) Equiibrium in Competitive Insurance Markets, Quartery Journa of Economics, Vo 90, No 4, pp Tayor G (1994) Fair Premium Rating Metods and te Reation Between Tem, Te Journa of Risk and Insurance, Vo 61, No 4, pp Wison C (1977) A Mode of Insurance Markets wit Incompete Information, Journa of Economic Teory, Vo 16, pp