Premium Distribution and Market Competitiveness Under Rate Regulation

Transcription

1 Premium Ditribution and Maret Competitivene Under Rate Regulation April 2018

2 2 Premium Ditribution and Maret Competitivene Under Rate Regulation AUTHOR Zia Rehman, Ph.D., FCAS SPONSOR Society of Actuarie Caveat and Diclaimer The opinion expreed and concluion reached by the author are their own and do not repreent any official poition or opinion of the Society of Actuarie or it member. The Society of Actuarie mae no repreentation or warranty to the accuracy of the information Copyright 2018 by the Society of Actuarie. All right reerved.

3 3 CONTENTS Section 1: Introduction... 5 Section 2: Foundation Section 3: Cae Study Section 4: Perfectly Competitive Maret Section 4.1 Company Optimizing Function Section 4.2 Company Behavior Section 4.3 Company Premium Mix Section 5: Regulated Maret Section 5.1 Company Premium Mix Section 5.2 Deciion to Exit a Line from a State Section 5.3 Cae Study Continued (Company Premium Mix) Section 5.4 Statewide Premium Mix Section 5.5 Cae Study Continued (Pre-Regulation) Section 5.6 Predicting Statewide Regulatory Impact on Premium Mix Section 5.7 Cae Study Continued (Pot-Regulation) Section 5.8 Maret Competitivene Section 5.9 Cae Study Continued (Pre-Regulation Maret Competitivene) Section 5.10 Cae Study Dicuion Section 6: Model Limitation and Dicuion Appendix A: Determining Capped Expected Profit Appendix B: Etimation of Profit Covariance Matrix Data Triangle Model Potulate Etimation of Profit Covariance Matrix Appendix C: Linear Algebra Reult on Average Matrice Reference About The Society of Actuarie... 55

4 4 Thi paper concern optimal new buine written premium mix at the firm and tate level under varying competitive condition. We addre three common problem a well a a new reult in linear algebra, hown in an appendix. The firt i to determine a company optimal new buine written premium mix for a given geographical area uch a a tate. The approach we preent incorporate the idea of mean-variance optimization, an important attribute of competitive firm. We develop an optimizing function that maximize return on equity. The econd problem concern predicting tatewide product mixe under varying rate regulation cenario involving profit cap. Knowing the effect of propoed rate regulation will help regulator encourage adequate capacity in particular line of buine in their repective tate. The reult alo help a company now when to exit a line of buine in a given tate. The third problem, alo a regulator problem, involve meauring and teting a maret competitivene for a given line of buine. Generally peaing, competitive maret require le rate regulation. Our fourth theorem provide neceary and ufficient condition for maret competitivene. We ue hypothetical data to demontrate the uefulne of our reult, but, in practice, one can eaily generate them with widely available company level and, where appropriate, indutry-level data. The preented olution lin to company reerve and can be updated along with reerve parameter a new data come into play. The method we preent fit well with current annual (or more frequently occurring) reerve review and rate filing an advantage for both companie and regulator. The cope of the paper i international, and we ue the United State a a bae example to mae our point. The model applie to any line of buine where there i ri tranfer and hence triangulation of data i poible. Thi include property liability, individual and group life, individual and group health, diability and accidental inurance.

5 5 Section 1: Introduction The manager of any inurance company operating in any country now about at leat ome of the complexitie preented by tate inurance regulation. Thee include difference in rating law and regulation and, thu, competitive environment among the different tate and acro line of inurance. For example, auto inurance rate regulation in Michigan differ from thoe in Illinoi, and homeowner rate regulation differ from commercial property owner rate regulation in Texa. Rate regulation affect a company ability to cover loe, pay expene and ufficiently compenate capital provider for the ri that they aume. Thu, trating regulation hould and will influence an inurance company managerial deciion regarding whether to write a line of buine in a given geographic region and, if o, how much to write and at what rate. When regulator et rate cap too low, companie may reduce their writing in a tate or chooe to withdraw altogether from a line of buine in a tate maret. Either way, maret diruption occur, and in ome cae, rate may actually go up in the long term the oppoite of what wa hoped for with rate uppreion (ee, for example, Regan, Tennyon and Wei 2008). Farmer Inurance Group, one of Texa larget homeowner inurance companie, topped writing new homeowner inurance in Augut of 2002 following argument with tate regulator over policy pricing, reuming writing everal month later on a limited bai (Aociated Pre 2003). The hort-run effect wa fewer maret for homeowner when eeing coverage. More recently, State Farm announced in 2009 that they planned to leave Florida homeowner inurance maret (Simpon 2009), ubequently triing a deal with the tate to remain, but hedding 125,000 policie (Patel 2011). Many of State Farm policyholder hifted to Citizen Property Inurance, Florida tate homeowner inurer of lat reort (Patel 2011). Some point to rate uppreion a a primary reaon that Citizen eventually grew to inure approximately 1.5 million policie in 2005 before a plan for depopulating it to a more reaonable ize wa adopted (Patel 2011; Vinon 2015). Citizen Property Inurance Corporation preent it own problem becaue it i largely underfunded (Vinon 2015). The cae of Citizen how what can happen to maret tructure when the combination of public maret capital and rate uppreion crowd out private maret ri-bearing capital.

6 6 In their defene, determining what rate hould be charged for a given line of inurance in a particular juridiction i not an eay ta for regulator. Although they poe the guiding principle that rate hould be adequate, not exceive and not unfairly dicriminatory, the challenge of implementing them with the information provided by inurance companie in rate filing can be ignificant. Conidering what rate eem reaonable for a given line of inurance in a tate maret preent a vexing challenge, not jut for regulator, but for actuarie and inurance company manager too. In thi paper, we preent ome tool for doing jut that. In the ection that follow, we introduce the foundation for the ret of the paper, including an equation for profit, ey definition and cae tudy. The third ection focue on our baic reult on optimization when perfectly competitive maret exit and expected profit vary freely. There we pecify a company optimizing function and how neceary but not ufficient condition for maret efficiency at the company level. The fourth ection concern regulated maret, and it i here that our major contribution to the literature begin. Employing a contrained optimization problem, we enlit a reult by Gotoh (2001) to find optimal product mix weight uing eigenvector, uing Theorem 2 to decribe the circumtance. The reult help in undertanding when a company hould exit a line of buine in a particular tate. Next, we turn our attention to the cae of optimal premium mix in tatewide maret. The third and fourth theorem culminate in the major contribution of thi paper. Our third theorem, which how how the tatewide premium mix can be found from company-level information even if firm are not mean-variance optimizer, hould prove epecially ueful to tate inurance regulator charged with reviewing company profit filing. The development i made feaible by proving a new reult in linear algebra that i hown in Appendix C. Our fourth and lat theorem give regulator a mathematical way to meaure competitivene of inurance maret in their repective tate even when at leat one firm i not a mean-variance optimizer. Thi ha not been done before in the literature and i a major contribution of thi paper. In ummary, the mathematical derivation preented in the paper are not an exact or minor variation of meanvariance portfolio theory; rather, we preent new mathematical reult and an inurance etting that i applicable to regulator. The novel mathematical reult and concept will be made obviou in the paper in bold italicized font.

7 7 Although we ue hypothetical data to demontrate the uefulne of our reult, in practice, one can eaily generate them with widely available company-level and, where appropriate, indutry-level data aggregated for a particular juridiction. Hence, the model prove very practical in implementation. Alo, the preented olution lin to company reerve and can be updated along with reerve parameter a new data come into play, adding another ource of uefulne to the model. The method we preent fit well with current annual (or more frequently occurring) reerve review and rate filing, an advantage for both companie and regulator. Before proceeding, undertand that our intent i not to invetigate the competitive tructure of the U.S. inurance indutry. Other reearcher, including Cummin and Xie (2013), Choi and Wei (2005), Cummin, Wei and Zi (1999), Tomb and Hoyt (1994), Mayer and Smith (1988), and King (1975), to name a few, have one already done o uing data aggregated by firm acro the juridiction in which they operate. Critically, their method do not let them determine which line() of buine a company hould expand (or contract) in a given tate or territory. Our method allow for companie to do jut that under different rating environment. Our paper i not about determining underwriting profit proviion by line, and thee are aumed to be nown for companie. Thee underwriting profit proviion are determined a a reult of a proce that i part cience, part art and regulation. See Myer and Read (2001) for one uch approach baed on capital allocation. Taylor (1987) found that contant unit expene rate lead to optimal premium rate of ubtantial negative profitability, and the adjutment to reflect marginal expene properly can caue very ignificant change to thee low premium rate. Rothchild and Stiglitz (1992) dicued the equilibrium in competitive inurance maret with imperfect information. They focued on ale offer, which conit of both a price and a quantity, a particular amount of inurance that the individual can buy at that price. What more, fully revealed information for an individual can mae everyone better off. Paul and Haberman (2005) built the optimal control model for general inurance pricing. For two demand function, an optimal premium trategy i well defined and mooth for certain parameter choice, epecially for a linear demand function that thee trategie yield the optimal dynamic premium if the maret average premium i lognormal ditributed. Taylor

8 8 (2006) paid attention to what individual inurer were attempting to achieve in following the maret and found that optimal trategie do not follow what might be thought the obviou rule. The optimal trategie depend on variou factor, including the predict time, price elaticity of demand and rate of return. They alo found when the current coverage maret rate lie below the brea-even rate, return to ubtantial profitability in the very near future may be poible. Paul (2007) analyzed the pricing problem with two form of contraint: a bounded premium and a olvency requirement. A lower bound i placed on the premium then an analytic olution can be found, but for olvency contraint, we can get numerical reult only uing control parameter. Taylor (2008) built the dynamic model for inurance maret, which include 11 eential parameter with phyical interpretation, ome of which can be ued a regulatory control. But thee regulatory control need to be applied with great caution let they induce preerve effect. Pantelou, Athanaio and Eudoia (2013) conidered the volume of buine, average maret premium, the company premium, which i a control function, and a linear tochatic diturbance when tudying a company expected to drop part of the maret. In thi model, the optimal premium trategy can be defined analytically and endogenouly by maximizing the total expected linear dicounted utility of the wealth over a finite time horizon. Pantelou, Athanaio and Paalidou (2015) built a dicrete-time tochatic dynamic programming model to connect a company optimal trategy with maret competition, which i available for both negative and poitive effect on the volume of buine depending on the company reputation for non-life inurance pricing. When the company ha a very great reputation, the company i very flexible to chooe any premium it wihe. Pantelou, Athanaio and Eudoia (2017) introduced the quadratic utility function into a dicrete-time tochatic nonlinear premiumreerve model to optimize the reerve in a competitive inurance maret. Beide the company reerve, for the very firt time, the derived optimal premium in a competitive maret environment i alo dependent on the brea-even premium, the expectation of the maret average premium a it did in the linear model, the income inurance elaticity of demand and other factor. Our paper concern the optimal new buine written premium mix at the firm and tate level under varying competitive condition. Our whole dicuion i around the mean-variance optimizer, and four theorem are derived. We propoe an extenion of mean-variance optimization to the cae of an inurer or inurer eeing to optimize the mix of premium acro variou line of buine.

9 9 Firt, we pecify a company optimizing function and how neceary but not ufficient condition for maret efficiency at the company level. Next, employing a contrained optimization problem, we enlit a reult by Gotoh (2001) to find optimal product mix weight uing eigenvector, uing Theorem 2 to decribe the circumtance. Then we how how the tatewide premium mix can be found from company-level information even if firm are not mean-variance optimizer. Finally, we give regulator a practical way of determining whether inurance maret remain competitive in their repective tate even when at leat one firm i not a mean-variance optimizer. Compared with other reearch about invetigating the competitive tructure of the U.S. inurance indutry, our method allow for companie to do jut that under different rating environment. In addition, we offer a new way of etimating the profit covariance in Appendix B, and our model prove very practical in implementation. We alo poit two way in which regulator might impoe contraint on inurer in the appendix, which offer an intereting and potentially ueful extenion to the problem cited.

10 10 Section 2: Foundation To develop the paper ytematically, we will firt introduce foundational material. For reader familiar with inurance, thi material may loo elementary, but ome tatitical enhancement are highlighted. A random written premium, P, for a propective (brand new) policy year can be broen into it eential component of loe, expene, profit and invetment income offet a follow: P Lo Expene Profit ( Invetment _ Income _ Offet ) Similar to the approach taen by Robbin (2004), we add here an invetment income offet term to account for the fact that invetment income earned on premium reduce the amount required to tranfer ri. The quantity (Profit Invetment Income Offet) equate to the Underwriting Profit Proviion (UPP). Companie may how UPP charge in their rate filing. Hence, we ue the equation above. Rearranging the UPP equation to olve for profit yield: Profit = UPP + Invetment Income Offet. Random loe are the undicounted, ultimate value for a new policy year and include allocated lo adjutment expene. Expene, alo random, include company overhead, mareting cot and imilar item. Some of thee expene, uch a ale commiion, depend on the random written premium, P. We treat the profit a random, and thi i a technical enhancement becaue the literature generally treat profit a a contant. Normalizing the above, Lo Expene Profit Invetment _ Income _ Offet 1 P P P P

11 11 For a new policy year,, referred to a the permiible lo ratio (PLR) 1, i random. The expene ratio and the invetment income offet ratio are aumed fixed 2 and nown a they do not change ignificantly from year to year. Finally, the ratio Profit P Lo P i random. In ymbol for a line of buine = 1, 2,, n, 1 U e R f (1) U Lo = Random permiible lo ratio P e = Fixed expene ratio a a percent of written premium Profit R = Random underwriting profit proviion a a percent of written P premium Invetment _ Income _ Offet f = Fixed invetment income offet a a percent of P written premium. Taing expectation, we get the following totality contraint with EU u, ER : (2) 1 u e f For a given line, our dataet include hitorical ri faced by the company. In Appendix B we briefly decribe the etimation of the profit covariance matrix. Equation (3) (7) are found in thi appendix. 2 Thi i the ame a target lo ratio. Reader more familiar with the term target lo ratio can replace thi when reading the paper. 3 In actuality, however, thee ratio may not be fixed. Variou factor may caue thee ratio to vary over time. For example, expene ratio may vary becaue of change in the commiion chedule for inurance agent. Alo, the invetment income offet ratio may vary becaue of change in interet rate or the return inurer earn on their invetment. Fortunately rate filing are uually done annually, and we aume that thee change are mall during the one-year period.

12 12 Section 3: Cae Study We undertand that the information preented above may feel remote to the reader. To mae the reult eem more tangible and acceible, we preent a cae tudy developed uing hypothetical data. We begin with a company writing five line of inurance. Table 1 give rate filing information for five line in company X. 3 Table 1: 2013 Rate Filing Information Permiible Lo Ratio Expene Ratio Underwriting Profit Proviion Invetment Income Offet Profit Line % 30.0% 2.0% 5.1% 3.1% Line Line Line Line We now provide an intuitive reult (Theorem 1) on perfectly competitive maret. Although the reult itelf i not tartling, it provide a mathematical bai to prove Theorem 4, and thi mae it neceary to prove it. Second, Theorem 1 i novel in the ene that it how that perfectly competitive maret aumption lead to maximization of a certain ratio. Third, Theorem 1 et the tone to thin about the problem in thi paper. 3 Figure 1 and 2 are located in Appendix B and dicu data and etimation of profit covariance matrix.

13 13 Section 4: Perfectly Competitive Maret We aume that expected profit ER i alway given to u by line of buine. Maret force or regulation determine the value of ER. Our ultimate goal in thi paper i to find the optimal premium mix,, ubject to a certain optimizing function and contraint. We pecify the optimizing function below. w Section 4.1 Company Optimizing Function Suppoe that the company propectively write a total of p 0 (in U.S. dollar) premium for line. We can define the company-wide profit a 4 R n 1 n 1 pr n p 1 w R (8) w n p 1 p (9) From (8), n Expected company profit = w ER (10) 1 Uing (8) again, we can meaure the total portfolio ri with proportion w [0 w 1] uch that 1, n w 1 4 Thi i an abtract quantity becaue profit are charged only by line. Nonethele it i mathematically correct to define the companywide profit becaue we are imply aggregating a quantity acro all line.

14 14 2 n in n Var( R) Var w R w w cov( R, R ) i i 1 i1 1 (11) Inpecting the covariance matrix w w cov( R, R ), i i l variance to the total portfolio variance (component ri),, the contribution of a line l w n i 1 w cov( R, R ) i (12) The above can be verified by umming acro l and that will reult in Var( R) : n n n n w w cov( R, R ) w w cov( R, R ) VarR i i i i l1 1 l1 1 Alo ince n R 1 w R, we have n n cov( Rl, R) cov Rl, w R w cov( Ri, R ) 1 1 (13) Therefore (12) can alo be written a w cov( R, R) l l (14) The return contribution for line l i w E R ). It can be verified by umming acro l, i ( i n l1 w E( R ) R l l Define wle( Rl) Mean-variance ratio of a line = (15) w cov( R, R) l l Drawing on tandard portfolio theory, we mae the aumption that companie are mean-variance optimizer and define the optimizing function with repect to line weight wi:

15 15 ER ( ) f ( w1,.. wn ) : 2 1 in n i1 1 n w ER w w cov( R, R ) i i (16) The mean-variance optimization aumption i critical and will be carried through in the firt part of our paper. Later thi aumption i relaxed through deviance multiplier (Theorem 3). A practical rationale for companie to adopt equation (16) a the optimizing function i whenever pricing capital (Robbin 2004) i trictly monotonically increaing in, then maximizing f ( w1,, w n ) i equivalent to maximizing the return on equity (ROE). Section 4.2 Company Behavior From tandard portfolio theory (Marowitz 1952), equation (15) and (16) are et equal under condition of perfect completion, including in inurance maret: wle ( Rl) ER ( ) 2 w cov( R, R) l l The rationale i that profit will et themelve to atify the above equation. To ee thi, uppoe that the left-hand ide of the above equation exceed the righthand ide. In thi cae, the company will increae wl ince it i a mean-variance optimizer. If that i not poible due to maret force, the company will lower E(Rl), a poibility that exit if maret are perfectly competitive and profit are allowed to vary freely. In repone, other inurer will change their portfolio to let thi company increae wl. The equation repreent a ind of equilibrium when maret are perfectly competitive. Uing (13), define

16 16 cov( Rl, R) l : 2 n 1 ln n l1 1 w cov( R, R ) l w w cov( R, R ) l l (17) Therefore, E( R ) E( R) l l (18) The above loo lie the capital aet pricing model (CAPM), but the quantitie are completely different. Further from (18), n n E( R) w E( R ) E( R) w l l l l l1 l1 n l1 w 1 l l (19) At a company level, the above i a neceary (but not ufficient) condition for maret efficiency. Note that becaue of randomne in etimating, it i not poible to draw concluion about maret efficiency baed on the data of a ingle company. Nonethele equation (19) i a ueful theoretical reult. l Section 4.3 Company Premium Mix We calculate optimal weight uch that the firm i mean-variance optimized. Specifically, we wih to olve for a unique combination of weight uch that f(w1,, wn) i maximized, ER ( ) f ( w1,.. wn ) 2 2 ln f ( w,.. w ) ln E( R) ln 1 n

17 17 ln f ( w,.. w ) ln w E( R ) ln w w cov( R, R ) 1 n ln n n l l 1 l1 1 Set ln f ( w1,.. wn ) 0; 1... n w ln ER ( ) wl cov( R, Rl ) w E( R ) w w cov( R, R ) l1 n ln n l l 1 l1 1 Therefore, ER ( ) ln w cov( R, R ) w E( R ) l l l1 1 ln n l1 1 n w w cov( R, R ) l l Uing (18) and (19) we have ER ( ) E( R) w w cov( R, R ) E( R) w cov( R, R ) l l l l 1 l1 l1 ln n ln n ER ( ) ER ( ) n ln ln l l l l l1 1 l1 1 ln l1 ln n l1 1 w w cov( R, R ) w w cov( R, R ) w cov( R, R ) l l w w cov( R, R ) l l Uing (17), the right-hand ide i preciely the definition of. Thu, we conclude that the no-arbitrage argument reult in optimal firm-wide weight. We tate the reult a a theorem. THEOREM 1: Under perfectly competitive inurance maret, a neceary (but not ufficient) conequence i that the company i naturally mean-variance optimized, and thu, the buine mix of a firm i optimal.

18 18 Thi theorem provide a tarting point of our paper and addree only the ituation where perfectly competitive maret exit. Thu, we now turn our attention to regulated maret.

19 19 Section 5: Regulated Maret The mathematical novelty in thi ection i due to introduction of equal column matrix C under unit um weight contraint. Thi reduce our problem to the wellnown problem of maximizing a certain ratio f( w) and will be made clear to ' wcw ' ww the reader. Section 5.1 Company Premium Mix Under regulated or le-competitive maret ituation, we cannot appeal to noarbitrage argument. Intead, we now have a contrained optimization problem. Suppoe that the expected profit for a given line are capped at becaue of rate regulation. They are of the form min( ER, ). We addre the iue of profit cap more completely in Appendix A. At thi point, we need not aume an explicit formula for capping and can continue imply by requiring that pot-regulation capped load exit. To avoid new notation, we will not introduce capped notation and aume that E(R) i capped and nown. We wih to maximize the optimizing function and olve for weight: f ( w,.. w ) 1 n 1 ln n l1 1 n ubject to the totality contraint 1. Thi i the contrained optimization problem. Let ER ( ) o that the problem can be written in matrix notation: w E( R ) w w cov( R, R ) l l n 1 w

20 20 w [ w,... w ] 1 Cov( R, R ) 1 2 l l, [,... ] 1, C...,... i [1,1...1] n n n n nn * n w We want to maximize f( w) ubject to the contraint wi 1. Then the w w n ' feaible olution are found in the et W w R wi 1, w 0. Now, 1, 1,, w 1 w1 w Cw ( w1, w2,, wn) w w1 w w2 w w n, n,, w n w n n = 1 2 w ( w w w ) w n The lat line follow from wi 1. Thu, the problem change to maximizing f( w) ' wcw ' ww ubject to the contraint wi 1. Auming that i a noningular, poitive definite matrix, the olution i facilitated by Gotoh (2001): 5 THEOREM 2: Auming that i noningular poitive definite, the maximum of f( w) n with repect to w R \{0} i given by the larget eigenvalue of the matrix 1 C and i attained by the eigenvector aociated with the larget eigenvalue of 1 C. 5 We have reduced the problem into a form that can be olved uing their theorem.

21 21 Note that the final olution i given by w, and multiplication by a contant i till reult in w a an eigenvector for but enure that the linear contraint i atified. Section 5.2 Deciion to Exit a Line from a State Sometime companie need a formal tudy 6 to decide the exit of a line from a given tate. Since both negative and poitive eigenvector are olution in Theorem 2, the poitive component are taen a the olution ince weight are poitive. However, if the ign of the component of i change, then ome weight are necearily negative, implying an exit from the tate. In thi cae the line hould be removed and weight redetermined until all component ign are the ame. In ome cae, more than one line ha an oppoite ign. In thi cae, the choice to ER remove a line could be baed on inpecting it ratio 2 with the lowet ratio removed firt. Section 5.3 Cae Study Continued (Company Premium Mix) Uing Table 1 in our cae tudy, in Table 2 we form profit matrix. C Table 2: Profit Matrix C 3.13% 3.13% 3.13% 3.13% 3.13% Inurer will conider different factor in maing their deciion a to whether to exit a line of inurance in a tate. Beyond how rate are regulated in that line or tate (a well a an inurer ene of how rate will be regulated in future year), their conideration would include the amount of un cot they would loe by exiting a maret, how their exit would affect their relationhip with inurance agent, and any economie of cope they achieve by writing multiple line of inurance.

22 Next, we how below tep to calculate covariance matrix baed on five line and the correponding triangle. In Table 3 we how calculation of covariance of line 1 and 2 uing triangle. The column 0 i the permiible lo ratio for the policy year time it written premium. It i an inerted column, and it rationale i explained in Appendix B. Table 3: Line 1 Data and Error Triangle LINE 1: DATA AND ERROR TRIANGLE pol_yr _name_ InRi InRi InRi InRi InRi InRi InRi InRi InRi InRi Log Ratio

23 23 LINE 2: DATA AND ERROR TRIANGLE pol_yr _name_ InRi InRi InRi InRi InRi InRi InRi InRi InRi InRi Log Ratio The covariance by age (uing error) i calculated uing two error triangle in Table 4. Table 4: Covariance Matrix by Age % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % um 0.07% The covariance matrix by line i then aembled imilarly for all line in Table 5. We how a ingle entry pertaining to line 1 and 2, preented a cae tudy in our paper, and other introduced line will have imilar calculation. Table 5: Covariance Matrix by Line Line 1 Line 2 Line 3 Line 4 Line 5 Line % 0.07% (um, a above) 0.24% 0.38% 0.26% Line Line Line Line

24 24 The error covariance matrix can be converted to profit covariance matrix by multiplying by the permiible lo ratio (PLR) for each line. We ip thi tep. We 1 how in Table 6 the eigenvalue and eigenvector correponding to C. 1 Table 6: Eigenvalue and Eigenvector for the Matrix C Eigenvalue E E E E-17 Eigenvector Note that four out of five eigenvalue are zero in Table 6. To undertand thi mathematically, refer to Appendix C. Next, we find the normalized eigenvector correponding to the larget 1 eigenvalue of the matrix C hown a Mix, hort for product mix, in Table 7. Thi reult correpond with Theorem 2. Table 7: Normalized Eigenvector and Derived Product Mix Line Eigenvalue Derived Mix % Total

25 25 Section 5.4 Statewide Premium Mix The theorem proved in thi ection i novel for two reaon. Firt, it hould prove epecially ueful to tate inurance regulator charged with reviewing rate filing and conidering the effect of different profit on premium mix tatewide. Second, for mathematician, thi analyi lead to a new reult in linear algebra formalized in Appendix C. We now determine the premium mix at a tatewide level. Aume that each company i mean-variance optimized and there are = 1, 2,, m companie in the tate. Each company ha a current nown total premium in dollar p and nown eigenvector ' i of the type dicued in the previou ection. The tatewide premium for all companie combined would be m 1 p '. The tatewide premium i mix i therefore m p ' m 1 i r m ' 1 i p 1 (20) r p m 1 p The above how that tatewide premium mix i a weighted average of company premium mix (eigenvector). Now uing the fact that C, 1 ' i i the eigenvector of C i i 1 ' '

26 26 1 C r C r C i m 1 m m m 1 r r jm ' jm ' 1 1 i 1 1 rjj rjj j1 j1 We have the opportunity to chooe any matrice on the left-hand ide. The 1 equality i due to the additivity property of matrice. Now, C i a matrix with identical column. Note that each column i alo the eigenvector of the repective 1 C. Therefore, 1,, m r ; ' 1 i Cr ' r i r ; ' i i ' (21) The eigenvalue i the um of component of ' i. Hence from equation (21), r C r r r i i i m m 1 m m m n m r r ' r jm jm ' jm ' ' rj j i rj j rj j j1 j1 j1 Thu m r i recognized a the eigenvector of i 1 ' M r C m 1 jm 1 rj j j1 aociated with an eigenvalue of 1. The reult i tated in the theorem. To prove that remaining eigenvalue of M are zero, note that M i an equal column matrix with the column a the eigenvector. With a ingle eigenvector and 2 j m, M j and M, o that j and j 1 0. Thu, the larget eigenvalue

27 27 max max : 1, 0. Since we ee nonzero eigenvector, remaining eigenvalue j : j 0;2 j n. THEOREM 3: Auming that each company i mean-variance optimized, the tatewide premium mix m r 1 ' i can be determined from the eigenvector of the matrix r C m 1 jm 1 rj j j1 aociated with eigenvalue of 1. Alternatively, the tate premium can be obtained directly from mixe are nown. m r 1 ' i ince company premium Section 5.5 Cae Study Continued (Pre-Regulation) We preent pre- and pot-regulation reult eparately. Only one cenario will be preented where expected profit are capped for line 2 and 3 under potregulation. Oberved written premium weight r by company for a tate Y i given in Table 8. Table 8: Oberved Written Premium Weight Written Premium Weight Company 1 $10,400, % Company 2 20,000, Company 3 5,200, Company 4 28,000, Total 63,600, Uing the ame method a above with company X, next find eigenvector and company premium mix. The detail will be ipped here. In Table 9 we preent the expected profit by line and company.

28 28 Table 9: Expected Profit by Line and by Company Company 1 Company 2 Company 3 Company 4 Line % 4.96% 4.95% 5.50% Line Line Line Line Uing the weight r above we can combine the calculated company mix to obtain m tatewide premium mix r in Table 10. Thi reult connect to Theorem 3. ' 1 i Table 10: Statewide Premium Mix Company 1 Company 2 Company 3 Company 4 Computed Statewide Line % 23.39% 19.64% 29.62% 23.87% Line Line Line Line Total Section 5.6 Predicting Statewide Regulatory Impact on Premium Mix Suppoe the tate regulator want to predict the impact of profit cap on tatewide premium mix. For example, line with rate cap are expected to have a le dedicated capacity, pot-regulation. The regulator now the tatewide premium mix of each company 1,2, m before capping. They want to now the revied premium mix of the line after capping. We olve thi problem below.

29 29 DEVIANCE MATRIX: Pre-regulation, the regulator can ue Theorem 3 and obtain the eigenvector m 1 r '. However, Theorem 3 aume that each company in i the tate i mean-variance optimized. To the extent that thi i true, our theoretical premium mix will match the current maret premium mix. However, uch a mean-variance optimization aumption i unliely to hold true in practice. If Theorem 3 wa applied, we would get a premium mix that would differ from actual. Let u call thi phenomenon maret deviance. We ue a deviance matrix with real element to meaure thi phenomenon: d D dn After application of Theorem 3 we will have. Suppoe we now the current n oberved tatewide premium mix : 1. Then d i defined a 1... n 1 (writing the vector and in component form) d : The uefulne of defining d 1... n (22) in thi way will be explained later in Theorem 4. PREDICTED MATRIX (POST-REGULATION): Revie the appropriate row of the profit matrix 1 C for each company to get a capped matrix C. To avoid unneceary notation, we will not introduce any new ymbol and aume that in thi ection the matrix i capped through C. Next, ue Theorem 3 to recalculate the eigenvector m 1 r '. The predicted pot regulation premium mix i given by i

30 30 D m r 1 ' i ' m D r i ' 1 i (23) The above adjut for maret deviance a long a D remain unchanged between pre- and pot-regulation period. The deviance matrix correct for thi violation once at the pre-regulation time. If the violation itelf change, then D will alo change. Note alo that the covariance matrice remain unchanged potregulation a the regulator i intereted in capping C with component of the form r E R rather than the random variable R with = 1,, n. The ue of deviance matrix D in equation (22) i an ad hoc adjutment to reflect maret deviance, becaue factor driving the deviance come from outide the model. Hence, the matrix D m r 1 ' i will no longer um to 1 and normalization m D r ' 1 i i neceary, leading to. ' m D r i ' 1 i Section 5.7 Cae Study Continued (Pot-Regulation) Now we cap expected profit at 5% for line 2 and 3 (ee Table 11). Table 11: Profit Cap Company 1 Company 2 Company 3 Company 4 Line % 4.96% 4.95% 5.50% Line Line

31 31 Line Line Uing the weight r above we can combine the calculated (capped) company mix m to obtain in Table 12 the capped tatewide premium mix r. ' 1 i Table 12: Capped Statewide Premium Mix Computed Company 1 Company 2 Company 3 Company 4 Statewide Mix Line % 23.34% 21.38% 29.94% 24.15% Line Line Line Line Total Now we calculate the final maret deviance-adjuted predicted tatewide premium mix (ee Table 13). Thi mix i what hould exit if a 5% cap i enacted for line 2 and 3. Table 13: Predicted Statewide Premium Mix Computed Statewide Mix Maret Deviance Predicted Mix Line % % Line Line Line Line Total

32 32 Section 5.8 Maret Competitivene The primary purpoe of thi ection i to provide a mathematically formal way to meaure maret competitivene becaue thi i lacing in the literature. To do o, we need our fundamental reult in Theorem 1, which will be ued to prove Theorem 4. THEOREM 4: Suppoe that there i at leat one company in the indutry > 0 that i not mean-variance optimizing it portfolio for line() i 0 i n. Then a neceary and ufficient condition of perfect maret competitivene for line i exitence of d 1. i i 0 i n i 0 i n i the NECESSITY: We are given that maret are perfectly competitive. Hence, from Theorem 1, the ubet of line i 0 i n for each individual company are naturally mean-variance optimized. Thu, i i 0 i n ince the neceary condition to calculate i are identical to actual maret condition. SUFFICIENCY: We are given that i i 0 i n a well a the fact that there i at leat one company in the indutry > 0 that i not mean-variance optimizing it portfolio for line() i 0 i n i 0 competitive for line.. We need to how that maret are perfectly i n We claim that the theoretical indutry premium mix i given by component et i m r ' 1 i i iff each company i mean-variance optimized. The ufficiency of thi tatement i obviou from our dicuion that each meanvariance optimized company will lead to component et ' i i a the optimal

33 33 olution for a company. For neceity note that from equation (20) the component m et r implie that each company ha a premium mix given by i ' 1 i i the component et ' i i. But thi et maximize the objective function for the combination of company and line. Hence, if the component et m i i r ' 1 i i wa oberved, then each company i alo actually mean-variance optimized for line i 0. But uch mean-variance optimization cannot be due to a company own effort ince at leat one company i not mean-variance optimized. Thu, uch optimization i due to maret condition. To recap briefly, we have ome maret condition that lead to mean-variance optimized portfolio for all companie in the indutry. From Theorem 1, we recognize thi to be perfectly competitive maret condition. We now turn our attention to a cae tudy to demontrate how the model might be deployed. i n Section 5.9 Cae Study Continued (Pre-Regulation Maret Competitivene) Next, we calculate the deviance matrix uing actual oberved tatewide line premium and the computed tatewide mix above (ee Table 14). To calculate the oberved tatewide mix we would require an expanded Table 8 with written premium by both line and company. The expanded figure i not hown here. Table 14: Maret Deviance Computed Statewide Mix Oberved Statewide Mix Maret Deviance Line % 25.17% 1.054

34 34 Line Line Line Line Total Employing Theorem 4, we ee that line 1 and 3 have a maret deviance cloe to 1, maing them reaonably competitive. 7 Section 5.10 Cae Study Dicuion The oberved tatewide premium mix (pre-regulation) can be compared to the predicted mix (pot-regulation) to try to undertand the effect of rate cap. A a reult of rate cap, the predicted rate mix for line 2 ugget a more than 1.5% decline in writing in line 2, going from an oberved tatewide mix of 19.44% to a predicted hare of 17.91%. The difference for line 3 wa much maller and in a different direction, with an oberved hare of 18.35% adjuting lightly upward with the predicted mix of 18.66% to reflect a mall hift from line to 2 to line 3 writing. Liewie, line 1, 4 and 5 each piced up a mall hare of the offet from line 2 hift. Further, upon inpecting the maret deviance, we note that line 1 and 3 are reaonably competitive, with value cloe to 1. 7 Some ubjectivity i involved becaue maret deviance are not exactly equal to one.

35 35 Section 6: Model Limitation and Dicuion Since the model i applicable to regulator, we offer obervation to place the ignificance of rate regulation in the broader context of the full et of regulatory policie and practice. Specifically, we provide dicuion on file and ue veru prior approval rate regulation law. Firt, the empirical literature indicate that inurance maret are tructurally competitive at the national and tate level. Thi i the cae for any maret of any ignificance, uch a auto inurance, home inurance, worer compenation inurance and the lie. It i generally the concluion of academic reearcher that trict rate regulation doe not improve maret performance but can create ignificant maret ditortion. It i poible that inurer in a given tate or line may be more aggreive in competing with each other; that i, competition in a given tate or line may exceed the tandard for worable competition, at leat for a limited period of time. Hence, the degree of competitivene could till vary acro line of buine in a tate with the qualification tated above. Second, in the United State, the type of rate filing ytem (e.g., prior approval, file and ue) in a given tate and for a given line of buine, by itelf, i not necearily a good indicator of how inurer rate are regulated. In ome prior approval tate, for a given line of inurance, regulator may attempt to contrain inurer rate, wherea in other prior approval tate, for a given line of inurance, regulator do not attempt to contrain inurer rate. By the ame toen, in ome file and ue tate, regulator do not attempt to contrain inurer rate, wherea in other they do attempt to contrain inurer rate. In our model thi i not a problem becaue rate regulation i reflected only in term of the extent to which regulator might attempt to contrain inurer rate or profit, which could vary by tate and line of inurance.

36 36 Third, other apect of tate regulatory policie and practice may alo influence inurer operation and their deciion regarding what they conider to be their optimal mix of buine acro different line of inurance. Thee apect include the regulation of olvency, underwriting and pricing, policy deign and claim ettlement practice, among other. The paper provide an extenion of mean-variance optimization to the cae of an inurer or inurer eeing to optimize the mix of premium acro variou line of buine. However, in practice, an inurer could chooe to exit a maret or reduce it premium in a given line of inurance for reaon other than achieving it optimal et of weight. Regarding applicability of the paper from a regulator tandpoint, we mae ome comment. Generally, what regulator conider to be mot important are the affordability and availability of coverage for a given line of inurance. When an inurer file for a large rate increae, epecially in a maret where rate are already high, regulator can become concerned that if they approve (or do not diapprove) uch a rate increae, it would have a ignificant and negative financial impact on conumer. At the ame time, regulator are generally aware that if they place evere contraint on inurer rate, thi could negatively affect the upply of inurance. Hence, regulator tend to balance conideration with repect to both the affordability and availability of inurance. That aid, a uggeted in the Introduction, regulator in a given ituation may not be able to predict how their deciion on rate filing will affect inurer deciion regarding how much coverage they will offer, although, in ome cae, inurer may inform regulator on the conequence of their deciion. The wor preented in thi paper could

37 37 erve a a foundation for further wor on model that could help regulator to predict the effect of their deciion. Appendix A poit two way in which regulator might impoe contraint on inurer: (1) placing a cap on their expected profit and (2) etting a cap on the maximum allowable rate increae. The firt way eem more liely to happen in practice. The econd way (i.e., a uniform cap on rate increae that would apply to all inurer) may be le conitent with reality. There may be ome ituation where regulator do impoe a uniform cap, but the more common cenario i for regulator to impoe cap on inurer rate increae that vary by inurer.

38 38 Section 7: Concluion The manager choice of what product to offer in what quantitie in a given tate, ditrict or territory i an important and highly pragmatic deciion. Mean-variance optimization provide one way of conidering how allocation uch a product choice ought to be made under riy circumtance. Mot would agree that underwriting inurance i an inherently riy buine. Finance theory ugget that return from firm activitie hould be ufficient to compenate capital provider for their ri. Inurer create return through two primary activitie: underwriting and invetment. Thee return can be incorporated into inurance pricing model ued by actuarie and regulator alie, commonly through a profit term and an invetment income offet term. We ue hypothetical data in howing ome practical reult of four theorem developed in thi paper. We begin with the unregulated maret cae. Our firt theorem tate that under perfectly competitive maret, a neceary but not ufficient conequence i that the company i naturally mean-variance optimized, and, thu, the buine mix of the firm i optimal. Companie uing the optimizing function will maximize ROE. Next, we introduce rate-regulated maret. Our econd theorem enlit a reult by Gotoh (2001) to find optimal product mix weight by firm uing eigenvector in a contrained optimization problem. The reult provide guidance in determining whether a company hould exit from a line of buine in a particular tate. Next, we turn our attention to the cae of an optimal premium mix in tatewide maret. Our third theorem, which how how the tatewide premium mix can be found from company-level information even if firm are not mean-variance optimizer, hould prove epecially ueful to tate inurance regulator charged with reviewing company profit filing. Our fourth and lat theorem give regulator a practical way of determining whether

39 39 inurance maret remain competitive in their tate, even if at leat one firm i not a mean-variance optimizer.

40 40

41 41 Appendix A: Determining Capped Expected Profit Our approach require the regulator to now capped profit. However, tatewide rate regulation for any line can tae different form, and not all of them will directly provide capped profit. We dicu the two mot common type of rate regulation: 1. Cap on expected profit in rate filing and 2. Maximum allowable rate increae. In the firt cae, the regulator mae the aumption that abent the regulatory capping, profit in the propective period would have remained unchanged becaue all companie would have obtained the required rate change. Thu, in the propective period, the profit are impacted only by regulatory capping. Thi cae i handled uing the formula given in the paper, and we explicitly conider the impact on companie when the current (i.e., propective) expected profit are capped at a certain level. The econd cae require a dicuion. A part of finding the appropriate allowable rate increae, the regulator conduct tatewide rate-level indication and determine a tatewide permiible lo ratio (PLR). With thi information, a hypothetical maximum allowable rate increae i et for further review. The impact of thi rate increae cap varie by company, with different companie affected differently. We how here how the rate increae cap convert to expected profit cap for each individual company. To illutrate, uppoe that a hypothetical rate increae cap for line 1 i et at +5%, baed on a tatewide rate-level indication of +10% and a PLR = 70%. Conider two companie with thee profile: COMPANY A, PRE-REGULATION PROFILE: PLR = 65%, expene ratio (ER) 30%, Underwriting Profit Proviion (UPP) = 5% Suppoe that company A file for an 8% rate increae and receive only 5% becaue of the cap. The reulting hortfall i 3%. The new UPP equal 3%: PLR = 65%/0.97 = 67% (loe are the ame but written premium i deficient by 3%) ER = 30% (expene ratio i percentage of written premium and thu remain unchanged) UPP = 1 67% 30% = 3% (atifie the totality contraint)