Modeling Anti-selective Lapse and Optimal Pricing in Individual and Small Group Health Insurance by Andrew Wei

Transcription

1 Modeling Anti-selective Lapse and Optimal Pricing in Individual and Small Group Health Insurance by Andrew Wei Andrew Wei, FSA, MAAA, is a vice president, senior modeler with Assurant Health, based in Milwaukee, Wis. He can be reached at or andrew.wei@assurant. com. 1. Introduction The task determining the optimal rate increase for a block individual or small group health (ISH) insurance policies presents a special challenge for actuaries. A key complicating factor is that a rate increase ten leads to anti-selective a tendency that healthy lives at a higher rate than impaired lives resulting in an adverse change in the health mix insureds. (For brevity, the term insured or life is used throughout this paper to refer to an individual, a family or a small group). In addition, a prit maximizing optimal rate increase solution needs to take into account the market price or the prevailing competitive prices in the market. In traditional actuarial models, anti selective is simply assumed and market price not explicitly considered. In this article, we present a new model in which anti selective emerges endogenously, as a result rate restriction and market competition, and an optimal pricing solution can be obtained. Many us have an intuitive understanding how anti selective occurs. A distinguishing feature the ISH market is differential rate restriction. Specifically, there are strict regulatory limits on how much renewal rates may vary within a class insureds to reflect each insured s current health status or claim risk, although renewal increases applied at the block level are generally not restricted. (These limits exist because in the absence these limits, insureds with serious long term illnesses are vulnerable to selective high renewal rate increases.) As a consequence, within a block, impaired lives renewal rates are subsidized by healthy lives. On the other hand, new business rates are not subject to the same strict limits. Insurers are allowed (or at least, with much less restriction) to medically underwrite and rate new policies. If an insured decides to switch insurers, his new policy rates will be set according to his current risk. It is then natural that when there is a large renewal rate increase, insureds would shop around; those who could find lower price alternatives in the market would likely, and those who could not would likely stay. The former are disproportionately healthy lives and the latter impaired lives. One key task in this new model is to formalize this intuition. In the following, we present the model with minimal technical details. Our focus is on the concepts, relations, and implications the model. We first describe the individual behavior an insured and then the aggregate behavior a block. (The mathematical details the model can be found in a paper by this author in the 29.1 issue the ARCH available on the SOA Web site. A quick note on notation: In the detailed paper, log transformed variables were extensively used for technical reasons. In this article, only standard variables are used for the sake readability.) 2. Individual Model Individual Lapse Behavior We first consider individual behavior arising from an insured switching from the current insurer to another insurer for a lower price in the market (We shall ignore, for simplicity, other types ). The probability price induced L due to switching for insured x is expressed as a function 2. Individual Model adjusted price P and market price M Individual Lapse Behavior We first consider individual. behavior arising from an insured sw current insurer to another insurer for a lower price in the market (We Adjusted simplicity, price other types P =P/A, ). where The P probability is price or price-induced premium rate and A is a premium adjustment fac- laps switching for insured x is expressed as a function adjusted price M tor for switching cost and product quality r differences. for switching cos So we can alternatively. write. Market price M is specific to insured x Adjusted and calculated price, a weig =P/A, as where P is, price a weighted or premium average rate and A is a p factor competitor for switching prices cost can Pand j be for product shown insured quality u x. Significantly, differences. So we can a write. Market price M is specific to insured x and c it can be shown under certain conditions that if as, a weighted average competitor prices for insu P =M, then L=1/2 and vice versa. S is an s -shaped =1/2 monotonically it can and be vi shown under increasing certain function conditions with that if properties: =M, then L =1/2 and s -shaped as monotonically, increasing function with properties: if z=1. A for as S special is a, s ith case S(z)= as for if z <1, S S(z) is =1 a if if step z z=1. >1 and A function special case for S is with S(z)= if z <1, S(z) =1 if z >1 and if z=1 (Figure 1). (Figure 1). 28 FEBRUARY 21 Health Watch

2 For, the fact that S is monotonically increasing implies that Modeling Anti-selective policy L =S(1)=1/2. The insured is indifferent. In this case, when L=1/2, P > M, due to the existence switching cost and a positive quality premium. The s -shape function S is intended to capture the fact that a homogeneous class FIGURE 1 insureds tends to exhibit from heterogeneous L and price P response using the to inverse price and any prediction will have a non-zero function. variance. This is If convenient we could as predict the detailed individual competitive prices for many types policies are ten not ness ten leads rate to is unre- perfectly, When with the a new busi- LAPSE FUNCTIONS zero variance, then S is reduced to a step function. The step function 1. greatly simplified models. available. Function S is derived from historical relationship L =S(1)=1/2. between The insured L and is indifferent. price P. policy In L this =S(1)=1/2. case, when The L=1/2, insured P > is M, indifferent. due to In this case, when L=1 stricted, it turns out policy.8 Let s consider how the we existence might estimate switching market cost and price a positive M the and quality existence premium premium. switching adjustment that cost factor and market a positive A. price quality premium..6 Note the relation that if L = ½ then =M or P/A =M. So if M is known and L=1/2, we The Setting s -shape Premium function S is Rates intended to The capture s -shape the fact that function a homogeneous S intended class to capture the fact that a homo can calculate S-Function A=P/M. M is initially calculated as a weighted insureds In this tends market, to exhibit insurers heterogeneous are assumed insureds average response to set tends to to price exhibit competitive M is proportional.4 and heterogeneous any prediction response to price and an prices. Step In many Function cases, we P based will may have reasonably on a the non-zero insured s variance. assume expected If that we could A is will the cost predict have same C plus individual a for non-zero all a to policies variance. cost perfectly, in If C a we with for could a all predict individual lap.2 block. Once A is known, zero variance, M can then be subsequently S is reduced to a calculated step function. zero variance, from The step then S L is and function reduced price ten to P a step leads function. to The step func margin, but there are three exceptions: 1) deviation using the inverse greatly function. simplified This models. is convenient as greatly the detailed simplified competitive models. insureds prices for in a block.. many types policies are renewal ten rate not from available. cost due Function to rate S restriction, is derived 2) from historical Let s insurers consider inability how we might to accurately estimate market forecast Let s price consider M medical and how premium we might adjustment estimate factor market A. price M and premium ad relationship 1. between L and price P. Note cost the trend, relation and that 3) if strategic L = ½ then pricing =M in or Note which P/A the =M. the relation So price if M that is known if L = ½ and then L=1/2, =M we or P/A =M. So if M is known Price (Log scale) can calculate A=P/M. M is initially calculated can as calculate a weighted A=P/M. average M is initially competitive calculated as a weighted average o Setting Premium Rates is set above or below the cost (for prit or market prices. In many cases, we may reasonably assume prices. In that many A is cases, the same we may for all reasonably policies in assume a that A is the same fo In this market, insurers block. share). Once assumed A is known, to set M price can be P subsequently based block. on calculated the Once insured s A is from known, expected M can L and be cost price subsequently C P calculated from using the inverse function. This is convenient using the as inverse the detailed function. competitive This prices is convenient for as the detailed com To illustrate, consider a policy plus with a premium margin, but rate there are three exceptions: 1) deviation renewal rate from cost due many When types the new policies business are ten not is available. unrestricted, many Function types it turns S policies is derived are from ten historical not available. Function S is derived fr P= $1, the cost switching to rate equal restriction, 1 percent 2) insurers inability to accurately forecast relationship out that market between price M L and is proportional price P. relationship medical between cost trend, L and and price 3) P. to C for premium, and a product strategic quality (which pricing can in which the price is set above or below the cost (for prit or market share). Setting all insureds Premium in Rates a block. Setting Premium Rates be either higher or lower than average competition) commanding an extra 5 percent pre- plus Excess a margin, risk but there are three exceptions: plus 1) a deviation margin, but renewal there are rate three from exceptions: cost due 1) deviation renewal r In this market, insurers are assumed to set price In this P market, based on insurers the insured s are assumed expected to set cost price C P based on the insured When the new business rate is unrestricted, it turns out that market price M is mium. Then the premium adjustment factor A= to rate restriction, 2) insurers inability to accurately to restriction, forecast 2) medical insurers cost inability trend, and to accurately 3) forecast medical proportional to cost A central notion this model is excess risk. strategic C for all pricing insureds in which in a the block. (1+1%)*(1+5%) = The adjusted price P = price is set above strategic or below pricing the in cost which (for the prit price or is market set above or below the cost (for share). Conceptually, excess risk is the portion share). market $1/1.155 = $865. Suppose that the market price Excess risk price which is not in the actual price, because M is also $865, then the probability for this A central notion When this rate model the restrictions. new is business excess Consider rate risk. unrestricted, Conceptually, a block When it heterogeneous turns excess the out new that risk business market is the rate price portion is M unrestricted, is it turns out that market pri policy L=S(1)=1/2. The insured is indifferent. In proportional to cost C for all insureds in a block. proportional to cost C for all insureds in a block. market price which is insureds. not the Let actual x be an price, insured because with adjusted rate restrictions. premium Consider a this case, when L=1/2, P > M, due to the existence block heterogeneous Excess rate insureds. P, risk cost C, Let and x market be an insured price M, with xexcess a adjusted standard risk life premium rate, cost switching cost and a positive quality premium. C, and market price A M, with central adjusted a notion standard premium this life model with rate is excess adjusted P, cost risk. A premium Ccentral Conceptually,, and notion market rate excess this, cost model risk is is the, excess and portion risk. Conceptually, excess risk market price price M, which and xis 1 an not impaired in the actual life price, with market because adjusted price which rate pre-restrictionsmium an heterogeneous impaired life is not in the Consider actual price, a because rate restriction The s -shape function S is market intended price to capture, and block rate P 1, cost insureds. with C 1, and Let adjusted market x be an price block premium insured Mheterogeneous 1. with rate Define adjusted, cost insureds. premium, Let rate and x be, an cost insured with adjusted pre the fact that a homogeneous class market insureds price tends. Define C, excess and market excess risk price V risk for M, V x, for expressed a x, standard expressed relative life with C, relative and to adjusted market a standard to premium a price standard M, rate a life, standard, cost as life, and with adjusted premium rate to exhibit heterogeneous response to price market life, as price, and an impaired life with market adjusted price premium, and rate an impaired, cost life, and with adjusted premium rate and any prediction will have a non zero market price. Define excess risk V for market x, expressed price relative. Define to a excess standard risk life, V for as x, expressed relative to a st variance. If we could predict individual perfectly, with a zero variance, then S is reduced to a For impaired life x 1, it is easy to see that step function. The step function For impaired ten leads life to, it is easy to see that due to the subsidy received due to the subsidy received by x 1. The excess. For risk stand greatly simplified models. by. The excess risk For impaired life, it is easy.. to For see. that For impaired life standard For standard life life, the due,, to the it the is excess easy subsidy to see risk received that due to. the For s by excess. The. To risk excess risk by suppose. To. The. For illustrate, excess $1 standard risk life, the excess risk. To illustrate,. For standard suppose life = Let s consider how we might estimate market price. To illustrate, suppose =$9, =$1, =$11 and suppose =$9, =15,. then = =$1, To illustrate, = (15/1 suppose =$11 and =$9, =15, =$1, then. To illustrate, =$11 = (15/1)/ and suppose (11/9) =$9, = =$1 M and premium adjustment factor A. =15, Note then rela-tion that if L = ½ then P =M or price P/A =M. not reflected So if M is in price represents (15/1)/ (11/9) = represents the extra market =15, then 227 = (15/1)/(11/9) prese (11/9) xtr = =15, = then represents price = (15/1)/ not reflected the extra in (11/9) market the insured s = premium represents rate due th to the not insured s reflected the extra premium in the market insured s rate price premium due not to price rate reflected rate not due restriction. reflected to in rate the restriction. in the insured s premium rate due to rate restriction known and L=1/2, we can calculate A=P/M. M is insured s premium rate due to rate restriction. initially calculated as a weighted average competitive prices. In many cases, we may reasonably If M is proportional to If C, M is we proportional have an to alternative C, we have an alternative formula for excess risk assume that A is the same for all policies in a block. formula for excess risk.. Once A is known, M can be subsequently calculated Lapse Probability L (P) Anti-selective Lapse CONTINUED Note that ON PAGE 3 implies that. Now the function for an insured with excess risk V can be rewritten as Health Watch FEBRUARY 21 29

3 Modeling Anti-selective FROM PAGE 29 Density If If M If is M is proportional is proportional If to M to C, is to C, proportional we C, we have we If have M have an is an alternative proportional to an C, alternative we have formula to formula an C, alternative we for have excess for excess an formula risk alternative risk risk for excess formula risk for excess risk If M is proportional to C, we have an alternative.. formula for excess is proportional to C, we have an alternative formula for excess risk. risk If If M M is is proportional to to C, C, we we have have an If an M alternative is proportional formula to for C, for. we excess have risk risk an alternative. formula for excess risk..... Lapse In addition to their differences in excess risk, these Anti-selective Anti-selective Lapse Lapse Anti-selective Lapse Anti-selective Lapse Lapse Anti-selective Lapse i-selective Lapse Note that five blocks. Now the also vary by base cost. The base cost Anti-selective Lapse Anti-selective Note Note that that Lapse Note that Note implies that implies that that implies that implies. Now. Now that the the. Now the. Now the Note that implies that. Now the e that Note Note that that implies that implies Note that for an with risk V can a block be reflects as the provider discount well as the function function that. Now function the for. an Now for. implies Now insured function an the insured the that with for with excess an excess insured risk for risk V with can V can insured be excess. Now be rewritten the rewritten risk with V as excess can as be risk rewritten V can be as rewritten as function for an insured with excess risk V can be rewritten as ction general expense an insurer. Figure 3 positions function for an insured with for for an excess an insured risk function with an V with can insured excess be rewritten with risk risk V excess V can as can for be an be risk insured rewritten V can with as be as excess rewritten risk as: V can be rewritten as these five blocks in a two dimensional map average excess risk vs. base cost. In particular, For For For, the, the, the fact For fact that that S S is S is, the For is monotonically fact that, S the increasing is fact monotonically increasing increasing implies implies increasing, that the fact that that implies S is that monotonically that, thus, thus increasing that implies S is implies monotonically has increasing that that low that excess implies increasing risk that but implies high that base cost and, in contrast, life, For, the fact that S is monotonically increasing implies that, the For fact For that S, is the, monotonically the fact fact that that S S is increasing is monotonically For, thus or or healthy or, implies thus healthy life that life s or thus s Block healthy s at at a 5 at a higher life high a or higher healthy rate s excess rate rate life at risk a higher s but rate low a In base higher addition cost. rate to In their differe, thus or healthy life s at a higher, thus or healthy than life than impaired s at life a higher. So. So the rate the above above equation is a is a formula for the for anti-selective real rate, thus, thus than impaired or or healthy life than life. So impaired the s above, thus than at life at a at impaired equation a a. higher So rate or the rate is life healthy above a rate formula. than So life equation the above s anti-selective is a equation formula a world, higher. is for. a rate Block. anti-selective formula Note Note Note 1 represents for anti-selective. Note insurer The. base Note that cost is a block r than impaired life. So the above equation is impaired life. So the above equation is a formula that for that anti-selective. is Note not is not that a formula assumed, anti-selective for anti-selective but but emerges is naturally not. good assumed, Note than than impaired life life. So. So the the above than equation within at underwriting but the the emerges model. naturally but not within good the in obtaining model. an insurer. provider discount and managing expenses, excess and block risk vs. 5 base cost. Figure 3 positi that impaired anti-selective is is a a life formula. that for So for the anti-selective is above not assumed, equation. but is Note is a emerges is Note not formula a assumed, formula naturally anti-selective but emerges within the naturally. model. Note within the model. that anti-selective is not assumed, but emerges naturally within the model. anti-selective that that anti-selective is not assumed, is but is not emerges not assumed, that naturally for but but anti-selective emerges within the naturally model. is. not within assumed, Note the the model. but that emerges anti-selective naturally within the model. and, in contrast, h represents the opposite is Aggregate not assumed, Model but emerges 3. Aggregate naturally Model within represents an insurer that 3. Aggregate Model 3. Aggregate Model 3. Aggregate Model discount and managing ex ggregate Model Aggregate Model 3. Aggregate the model. Model FIGURE 3 Next Next Next we we consider we consider the Next the aggregate the we aggregate consider Next behavior behavior we the consider aggregate a a block block a the block behavior aggregate policies. policies. a behavior Specifically, block Specifically, policies. a block we we want we Specifically, want want policies. to to to Specifically, we want to we want to Next we consider t we consider the aggregate behavior a block policies. determine the determine aggregate Specifically, the the aggregate behavior determine we, want,, a the block determine loss to aggregate loss loss ratio, ratio, policies. ratio, the and, and aggregate and prit prit Specifically, loss prit using using ratio,, using several and we several loss prit want representative ratio, AVERAGE representative using to and prit EXCESS blocks. using blocks. several RISK representative AND BASE blocks. COST Next Next we we consider the the aggregate behavior Next we a consider a block the policies. aggregate Specifically, behavior we we a want block want to to policies. Specifically, we several want to representative blocks. determine the aggregate 3. Aggregate, loss ratio, and Model prit using several representative blocks. BY BLOCK rmine the aggregate determine, the the loss aggregate ratio, and, prit loss loss determine ratio, using ratio, and several and the prit aggregate representative using several, blocks. loss representative ratio, and prit blocks. using several representative blocks. Next Consider Consider we consider five five five representative representative Consider the aggregate five blocks Consider representative blocks behavior that that that five vary vary vary representative in in blocks the a in the block health the health that health mix vary blocks mix mix in insureds the that insureds health vary and and mix and the cost cost health cost insureds mix and insureds cost and cost Consider five representative sider five representative blocks that vary in the health structure. mix structure. The insureds The blocks The health structure. health that and mix vary cost mix The in structure. the insureds health health mix a The a mix block block a health block insureds can can insureds can mix be be characterized in be a characterized and insureds block cost can using be using a block using characterized f(v), f(v), can f(v), a a density be a characterized density using f(v), a using density f(v), a density Consider five five representative blocks Consider that that vary policies. vary five in in the representative the Specifically, health mix mix blocks we insureds that want and vary and to cost cost determine the health the mix insureds 65 and cost structure. The health cture. The health mix insureds in a block can be distribution characterized distribution mix insureds using insured distribution insured a f(v), excess block a excess density risk distribution risk can insured risk for for be the for characterized the excess block. the block. block. insured risk Figure Figure for using excess 2 the 2 shows shows f(v), 2 block. risk shows a the for the density Figure excess the excess block. 2 risk shows risk risk Figure the excess 2 shows risk the excess risk structure. The The health mix mix insureds structure. in a a block The can can health be be mix characterized insureds using f(v), a f(v), block a a can density be characterized using f(v), a density distribution insured aggregate ribution insured excess risk for the block. Figure distributions 2 shows distributions excess, risk the excess for for these for loss these distributions for risk these ratio, five five block. five and blocks. distributions blocks. Figure prit Intuitively, these Intuitively, 2 five shows using for blocks. Blocks these several Blocks excess Intuitively, five 11 and and blocks. 1 risk 6 and 22 have have 2 Blocks Intuitively, have the the highest the 1 and highest Blocks 2 proportion have proportion 1 the and highest 2 have proportion the highest proportion distribution insured excess risk risk for for the the block. Figure insured 2 2 excess shows the risk the for excess risk risk block. Figure 2 shows the excess risk distributions for ributions for these five blocks. Intuitively, Blocks 1 and healthy 2 healthy have lives, lives, lives, highest Blocks Blocks healthy proportion 44 and and 4 lives, and 5 5 the healthy the 5 Blocks lowest the lowest lives, 4 and proportion Blocks 5 the lowest 4 and healthy healthy 5 proportion the lives, lives, lowest lives, and and proportion and healthy Block Block lives, 3 the the 3 the healthy and Block lives, 3 and the the distributions for for these these five five blocks. distributions representative these five blocks. Intuitively, Blocks for these blocks. Intuitively, Blocks 1 and 2 have the highest proportion 55 1 and 1 and five 2 have 2 blocks. have the the Intuitively, highest proportion Blocks 1 and 2 have the highest proportion healthy lives, Blocks ealthy lives, Blocks 4 and 5 the lowest proportion medium healthy medium 4 lives, and proportion 5 the and medium lowest Block healthy healthy 3 proportion proportion medium lives. lives. lives. In In practice, In healthy healthy practice, we lives. lives, we can we can In and can obtain practice, obtain Block obtain lives. excess we excess In practice, risk can risk risk obtain distribution distribution we excess can obtain risk distribution excess risk distribution healthy lives, Blocks 4 and 4 and 5 the 5 the lowest healthy proportion lives, Blocks healthy 4 and 5 lives, the and lowest and Block proportion 3 the 3 the healthy lives, 5 3 the and the medium proportion ium proportion healthy lives. In practice, we can f(v) obtain f(v) f(v) by by healthy aggregating excess by aggregating lives. risk f(v) In distribution insureds by practice, insureds aggregating by by f(v) we excess by by excess can insureds aggregating obtain risk risk risk V V over by over excess V over excess insureds a a block. block. risk a block. risk distribution by V excess over a risk block. V over a block. medium proportion healthy lives. medium In Consider In practice, proportion we five we can can representative obtain healthy excess lives. risk risk In blocks practice, distribution that we can vary obtain excess risk distribution f(v) by aggregating insureds by excess risk V over a block. 45 by aggregating f(v) f(v) insureds by by aggregating by excess insureds risk V by over by excess a f(v) block. the risk by risk health aggregating V V over over mix a a block. insureds insureds by excess and risk cost V structure. over a block. The 4 health mix insureds in a block can be characterized using f(v), a density distribution insured 3 35 excess risk for the block. Figure 2 shows the excess 1% 2% 3% 4% risk distributions for these five blocks. Intuitively, Average Excess Risk V Blocks 1 and 2 have the highest proportion healthy lives, Blocks 4 and 5 the lowest propor- Now let denote a standard insured in Now a block, let denote a stand tion healthy Now lives, let denote and Block a standard 3 the insured medium in a block, the adjusted the mar adjusted premium rate for, and the market market premium price for rate x level. We and shall refer as market P as premium price level. rate Let R denote t preprice for proportion healthy lives. In practice, we can the market price for. We shall refer as obtain excess risk distribution f(v) by aggregating price level. Let R denote the percentage rate level increase. and Let M as (R) market denote price as a level. Let R denote the insureds by excess risk V over a block. percentage rate increase. Let P (R) denote P as FIGURE 2 a function rate increase R. Then P () is initial EXCESS RISK DISTRIBUTION premium rate level for x when R=. Define R =M BY BLOCK / P () -1. Then P ().(1+ R )=M Intuitively, R is the percentage rate increase needed to bring P to match M Excess Risk V Base cost For ease comparison, initial premium rate level P () is assumed to be 2 percent below the market price level for all blocks. In other words, for all blocks, if R=2%, then P = M or the premium rate level matches the market price level. So R =2% for all blocks. 3 FEBRUARY 21 Health Watch

4 Modeling Anti-selective R. Then Within () is initial a premium block, initial rate level premium for when rates R=. vary by. Then insureds benefits. Intuitively, and risk load. is the All percentage insureds rate within a block are assumed to receive the same percentage to match. rate increase (This is not restrictive because the risk initial premium load rate can level vary). Furthermore, () is assumed to the be step 2 percent function vel for all is blocks. used In for other simplicity. words, for all blocks, if R=2%, then ate level matches the market price level. So =2% for all Applying the formula L=S(P (R) / (M *V)) to each insured in a block, we calculate the aggregate emium rates vary rate by insureds the block benefits as and a function risk load. All rate increase re assumed to receive the same percentage rate increase (This is R (Figure 4). e risk load can vary). Furthermore, the step function is FIGURE 4 ula AGGREGATE to each insured LAPSE in BY a block, BLOCK we pse rate the block as a function rate increase R (Figure 4). Lapse rate 9% 8% 7% 6% 5% 4% 3% 2% 1% % 2% 4% 6% 8% 1% We make a few observations. When R < R (R =2%), the premium rate level P (R) < market ions. When price R < level ( =2%), M, and the aggregate premium rate level rate (R) = < in all ate nd aggregate = in all blocks. rate When = in R all Rblocks.,, then When P (R) M,, then and aggregate R. is These increasing blocks rate with show is R. increasing different These blocks with show R. different These blocks ate asing with rate h Blocks the highest 1 and show proportion 2, with different the highest healthy proportion lives, sensitivity healthy to price. lives, Blocks 1 ases ocks in 4 and, and 5, and with 2, Blocks with the lowest the 4 and highest proportion 5, with proportion the lowest proportion healthy lives, he slowest rate increase-induced increases. The rate s increase-induced these exhibit the steepest increases in s, in and these Blocks 4 ti-selective. in nature. and 5, with the lowest proportion healthy lives, exhibit the slowest increases. The rate increaseinduced s in these blocks are necessarily antiselective in nature. Next we calculate the aggregate loss ratio as a function rate increase for each block (Figure 5). Loss ratio 13% 12% 11% 1% 9% 8% 7% 6% 5% 4% FIGURE 5 LOSS RATIO BY BLOCK % 2% 4% 6% 8% 1% Several loss ratio patterns emerged among these blocks. When R=, the initial loss ratios differ considerably. When R < R (R =2%), loss ratio is decreasing with R in all blocks When R R, loss ratio is decreasing with R, at a slower rate, hoin Blocks 4 and 5, but is increasing with R in Blocks 1, 2, and 3. The loss ratio trajectory experiences a sudden shift at the point where s begin to rise. The various loss ratio patterns reflect the different impacts anti-selective s as well as the different cost structures in these blocks. Assessment spiral The loss ratio behavior in, incidentally, provides an illustration for the assessment spiral phenomenon. Note that if a large renewal rate increase, R=85%, is given, the loss ratio after the increase, 76 percent, is actually higher than the loss ratio before the renewal increase, 72 percent. Figure 6 shows the aggregate prit as a function rate increase for these blocks. The aggregate prit patterns mirror those loss ratio in these blocks. When R=, the initial prits vary considerably. When R < R (R =2%), prit is increasing with R in all blocks. When R R, prit is increasing with R in Blocks 4 and 5, but ho decreasing with R in Blocks 1, 2, and 3. Likewise, the different prit patterns reflect the changes in the mix insureds and the cost structures in these blocks. CONTINUED ON PAGE 32 Health Watch FEBRUARY 21 31

5 Modeling Anti-selective FROM PAGE 31 FIGURE 6 AGGREGATE PROFIT BY BLOCK as a function internal drivers and an underwriting cycle index: Prit 32 FEBRUARY 21 Health Watch Pr Prit Capacity=Policy Inforce C ( A B V ) A is the price adjustment A is the price factor adjustment described before, factor B is described a firm-specific before, cost B factor, V is average excess risk a for firm-specific a block, C is cost market factor, cost level i is or average a weighted excess average all competitors cost risk for for a standard a block, life, C is market is an underwriting cost all level cycle or a weighted index ( = market pr level Block / market 3 cost level), s average ma all competitors cost for s maa standard life, The Block formula 5 captures is an what underwriting we intuitively cycle know. index To increase ( = market prit capacity, price an insurer can employ three level basic / market strategies: cost 1) raise level), A: increase perceived quality and the cost switching; 2) reduce B: lower cost and obtain better provider discount, 3) lower V : improve underwriting The formula and risk assessment. captures what is outside we intuitively the insurer s know. control, and pr % 2% 4% 6% 8% 1% capacity fluctuates with. Due to leveraging, small changes in A, B, V, or lead to To increase prit capacity, an insurer can employ large swings in prit capacity. three basic strategies: 1) raise A: increase perceived To illustrate, suppose quality Policy and the Inforce cost =1,, switching; C = $2,, 2) reduce =1.5, B: lower A=1., V =1.4, B= 4. Optimal Pricing 1., we can calculate cost and prit obtain capacity better = $2,,. provider discount, Furthermore, 3) s lower ma a 1 percent i: change i Two distinct optimal pricing strategies any emerged. the drivers improve A, B, V underwriting, or produces and a 14 risk percent 15 assessment. percent swing is outside the insurer s s ma control, and prit capacity fluctu- all in prit First, for Blocks 4 and 5, the two blocks capacity. with the least healthy mixes insureds, the optimal strategy ates with s ma. Due to leveraging, small changes in A, is to maximize rate increase R (to the extent 6. Non-step possible within the limits the rating law). Second, all Lapse B, Effect i, or lead to large swings in prit capacity. So far, the step To illustrate, function was s ma suppose used. As Policy a result, Inforce we were =1,, able to obtain C = a simplifie for Blocks 1, 2, and 3, which have healthier mix model with easy-to-understand results. In the real world, we cannot predict individual insureds, the optimal strategy is to set R=R perfectly, $2,, =1.5, A=1., i or so a non-step function is =1.4, more realistic. B= 1., Let s we can look calculate how a prit non-step capacity function = all $2,,. might affect Furthermore, model. at an example set P = M, i.e., set the premium rate level which to illustrates the market price level. s a ma 1 percent change in any the drivers A, B, i, or Figure 7 compares the two methods calculating aggregate prit function produces a 14 percent 15 percent = market swing price in all for Block prit When the non-step function is used, the overall effect is a shift the aggregate prit Sustainable blocks level / market cost level), curve to the right. capacity. The optimal rate increase shifted from R= R to R= R 1 where R 1 is Blocks 1 and 2 belong The to formula an important captures somewhat class we intuitively higher than know. R To increase prit capacity, insurer blocks called sustainable 6. Non. But the essential characters and general results the aggreg step Lapse Effect can employ blocks, three which basic prit are strategies: characterized by switching; 2) reduce B: lower cost and So obtain far, the better step provider discount, function 3) was lower used. : As a result, function 1) remain raise A: unchanged. increase perceived quality and the cost improve underwriting The and non-step risk assessment. function we were has is similar able outside to effects the obtain insurer s on aggregate a control, simplified and and prit model loss ratio with behavior. capacity fluctuates with. Due to leveraging, easy to small understand changes results. in A, B, In, or the real lead to A high proportion healthy lives world, we large swings in prit capacity. Prit is maximized when R=R cannot predict individual perfectly, so a nonstep function is more realistic. Let s look at or P =M At optimal price, the To price-induced illustrate, suppose Policy is zero Inforce =1,, = $2,, =1.5, A=1., =1.4, B= 1., we can calculate prit capacity = an $2,,. example which Furthermore, illustrates a 1 percent how a change non-step in Note that these blocks any are theoretically drivers A, sustainable B,, or produces function a 14 might percent 15 affect percent the model. swing in prit as it is in the insurer s capacity. self-interest to keep rate increases moderate and s minimal. Setting Figure 7 compares the two methods calculating aggregate prit function for. When the non R=R amounts to giving 6. Non-step only the Lapse trend Effect increases in the long run. step function is used, the overall effect is a shift So far, the step function was used. the As aggregate a result, we prit were curve able to to obtain the right. a simplified The optimal 5. Prit Drivers model with easy-to-understand results. rate In increase the real world, shifted we from cannot R=R predict individual to R=R 1 where R 1 is perfectly, so a non-step function is more realistic. Let s look at an example Define prit capacity which as maximum illustrates how prit a non-step attained somewhat higher than R function might affect the model.. But the essential characters and general results the aggregate prit func- (at optimal price) in a block. It turns out that for sustainable blocks, prit Figure capacity 7 compares can be the expressed two methods tion calculating remain aggregate unchanged. prit function for. When the non-step function is used, the overall effect is a shift the aggregate prit curve to the right. The optimal rate increase shifted from R= to R= where is somewhat higher than. But the essential characters and general results the aggregate prit function remain unchanged. The non-step function has similar effects on aggregate and loss ratio behavior.

6 Modeling Anti-selective Prit The non step function has similar effects on aggregate and loss ratio behavior FIGURE 7 AGGREGATE PROFIT WITH NON-STEP LAPSE RESPONSE Non-Step Prit Step Prit independently: 1) make sure to track competitive prices by market; 2) for optimal pricing, is as important as cost or risk; 3) build a good model; 4) monitor different types, especially base vs. price-induced ; 5) if rates exceed a base level, then prices are probably too high; and 6) set a target rate for each class insureds. Finally, the model can be generalized. The excess risk distributions used can be easily extended to a general form. The basic framework the model also allows new businesses to be incorporated. The model structure lends itself well to computer modeling. n The model has several practical applications and yielded new insights into the behavior ISH insurance in a competative market. % 2% 4% 6% 8% 1% 7. Conclusion We developed a model in which anti selective emerges naturally as a result differential rate restriction and market competition. We applied the model to determine optimal rate increases in representative blocks with different mix insureds and cost structures. The model has several practical applications and yielded new insights into the behavior ISH insurance in a competitive market. One key insight is that for a class sustainable blocks, insurers can maximize prit while keeping rate increases moderate and s low. Sustainable blocks are good for both insurers and insureds. This could have implications for product design as well as regulation. A potential implication is that a disciplined insurer could fer contractually, at no extra cost, a renewal rate guarantee linked (within a small range) to a broad market price index, or alternatively, linked to, as a proxy, a medical cost index. It may be worth mentioning some additional practical implications or potential rules for optimal pricing. These rules, though model based, can be used Health Watch FEBRUARY 21 33