An insurance company is considered as an intermediary between policyholders and the capital market. By applying the traditional and the generalized version of the capital asset pricing model, a class of premium principles can be derived. This class is fully compatible with Bühlmann's economic premium principle. Moreover, insurance premiums can be directly related to risk premiums on the stock exchang

Müller, Heinz H.

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ARTICLESECONOMIC PREMIUM PRINCIPLES IN INSURANCE ANDTHE CAPITAL ASSET PRICING MODELBY HEINZ H. MULLERUniversity of ZurichABSTRACTAn insurance company is considered as an intermediary between pohcyholdersand the capital market. By applying the traditional and the generalized versionof the capital asset pricing model, a class of premium principles can be derived.This class is fully compatible with Buhlmann's economic premium principle.Moreover, insurance premiums can be directly related to risk premiums on thestock exchange.KEYWORDSCapital asset pricing model; insurance premiums; price equilibrium; Neumannutility.1. INTRODUCTIONPremium calculation is one of the main objectives of risk theory. The extensiveand sophisticated literature on this topic is summarized in the recent book byGOOVAERTS, DE VYLDER and HAEZENDONCK (1984).On the other hand, equilibrium conditions on capital markets are a centralissue of the theory of corporate finance. Based on mean-variance analysis,SHARPE (1964) and LINTNER (1965) derived the capital asset pricing model(CAPM). The CAPM-formula is an equilibrium condition which relates riskpremiums to the covariances between the returns on the market portfolio and thecorresponding assets. Later, the CAPM relationship was generalized to the caseof risk evaluation by Neumann utilities (see e.g. MERTON (1982), pp. 614-618).There are only a few recent papers in which elements of risk theory have beencombined with models of corporate finance and economic theory. BUHLMANN(1980, 1984) and LIENHARD (1986) derived a class of premium principles byapplying a general equilibrium approach to the insurance market. BORCH'Sanalysis (1986) of the insurance market is based on a slightly generalized versionof the CAPM-relationship. KAHANE (1979) stressed the importance of invest-ment income on premiums and made a first step in order to apply the traditionalcapital asset pricing model to premium calculation.The aim of the present work is to unify the ideas of the articles mentionedabove by means of a simple model. In accordance with corporate finance, an in-surance company is considered as an intermediary between policyholders and theASTIN B U L L E T I N Vol .17 , No .2available at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,142 MULLERcapital market. Hence, a share of an insurance company is a combination ofclaims on the assets representing the reserves and the liabilities stemming fromthe insurance contracts issued by the company (see also KAHANE (1979), BORCH(1986)). By application of the traditional and the generalized versions of theCAPM-formula, insurance premiums can be related to risk premiums on stocks.The resulting class of premium principles corresponds exactly to the class ofpremium principles derived by BUHLMANN (1980) and LIENHARD (1986). Hence,at least in this respect, risk theory and the theory of corporate finance lead toperfectly compatible conclusions. Our main result, however, consists in a rela-tionship between insurance premiums and risk premiums on stocks which couldalso be useful for empirical research.2. THE CAPITAL ASSET PRICING MODEL (CAPM)2.1. The Sharpe-Lintner ModelIn the Sharpe-Lintner model there is one risk-free asset (h = 0) and there are nrisky assets (h = 1,...,«). The rates of return are given by:the deterministic rate of interest Ro for the risk-free asset h = 0;the stochastic rates of return Ru ...,/?„ for the risky assets h = 1,...,«.m investors are characterized by:their initial endowments represented by (n + l)-bundles of assets,their preferences with respect to final wealth, which are assumed to be compat-ible with mean-variance analysis.Furthermore, markets are assumed to be competitive. Short selling is possible,there are no transaction costs and in particular the risk-free asset can be borrow-ed and lent at the same rate of interest. For this model, SHARPE (1964) andLINTNER (1965) have shown that under mild regularity assumptions therates of return Ro,Ri, ...,Rn must satisfy the following equilibrium conditions(CAPM-relationship):(1) E(RH)-RO = ^ ^ § - ^ [ E ( R M ) - R O ] (h=l,...,n),where RM denotes the stochastic rate of return on the market portfolio. Themarket portfolio is defined as a portfolio made up of all assets in the economyheld according to their market value weights.Under the model assumptions, the market portfolio can be represented as asolution of an optimization problem. Analysing the optimality conditions leadsto (1).2.2. The General Version of the Capital Asset Pricing ModelFrom the theoretical point of view, portfolio evaluation by Neumann utilities ismuch more satisfactory than mean-variance analysis.available at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,ECONOMIC PREMIUM PRINCIPLES AND CAPM 143A portfolio JC is of the formnX = (Xo,Xi,...,Xn), with Yl Xh = 1and leads to the stochastic rate of return:nR(X) = Yl XhRh-A portfolio x" is called efficient relative to a Neumann utility u:R~*R if itrepresents a solution of the optimization problemmax E{u[R(x)] }xtR"*1subject tonYi Xh-\.Under mild regularity conditions, efficient portfolios satisfy the generalizedCAPM-relationship:See e.g. MERTON (1982), Theorem 3.1. The generalized CAPM-relationship (2)can be easily derived from the optimality conditions,E \ ( R h - R 0 ) u ' [ R ( x u ) ] ) = 0 ( h = 1 , . . . , « ) .Furthermore, under regularity assumptions on the rates of return, there existclasses of Neumann utilities °U with the following property:If the preferences of investors / = 1, ...,m are given by the Neumann utilitiesu'd °U, then there exists ««€ <W such that the market portfolio is efficientrelative to UM-CASS and STIGLITZ (1970) show that the following classes have this property:(3) Vt(c)= {u\u'(w) = (l3wlc + rlrc, /3 > 0 } , c 6 ( - « , 0 )U(0 , » ) ,= {u\u'(w)=e-aw,a>0\,c=oo.These classes are also well known in risk theory. According to Borch's theoremeach of these classes leads to linear risk sharing. The union of these classes,{ M | M € * ( C ) , C € ( - O O , 0 ) U ( 0 , O O ] } ,is the HARA-class (//yperbolic Absolute Risk Aversion) which is characterizedbyu' (w) a + bwavailable at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,144 MULLER3. PREMIUM PRINCIPLES AND THE CAPITAL ASSET PRICING MODEL3.1. The Modelm investors (/= 1, ...,m) face the following investment opportunities:One risk-free asset with a deterministic rate of return Ro. This asset is assumedto be in net supply zero.n risky assets (h = 1 n) representing the non-insurance sector of theeconomy. These assets are characterized by stochastic rates of returnR\, ..., Rn-One risky asset (h = n + 1) representing the single insurance company of theeconomy. The stochastic rate of return is denoted by Rn+\ and will beanalysed below.3.1.1. The non-insurance sector. Let Wh be the market value of all outstandingshares on investment opportunity h(h = 1,..., n). Then the value of all risky non-insurance assets is given byKN := S Wh.The market portfolio for this sector of the economy is of the form(4) xN:=±-{Wu...,Wn),KMi.e. all non-insurance assets are held in proportion to their market values.For the corresponding stochastic market rate of return, one obtains(5) RN-=^~ t WiRi.A N , = I3.1.2. The insurance company. As mentioned in the introduction, the in-surance company acts as an intermediary between policyholders and the capitalmarket. In order to cover the total insurance risk X, the company receives theamount T as premium payments from the policyholders. Let us assume that theamount A7 is raised from shareholders and that the company invests its totalreserves Kr+ ir in the risk-free asset h = 0. In a perfect capital market thisassumption is not restrictive. Shareholders are able to offset any investmentpolicy of the insurance company by their own investment decisions. Then, thefinal value of the company is given byand the rate of return on its share is(O; « n + l ( A ; ) « o H .A/Shareholders can themselves borrow and lend at the risk-free rate Ro(h = 0) andavailable at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,ECONOMIC PREMIUM PRINCIPLES AND CAPM 145are fully liable for losses of the insurance company. Therefore, it does not matterwhether the shareholders lend their money to the insurance company or whetherthey invest it themselves in the risk-free asset. Hence, without influencing the riskallocation in the economy, Ki may be fixed at an arbitrary level. For our pur-poses, one can in particular assume(7) K,= -7r(1>.With this normalization, the rate of return becomes(8) Rn+1 = X-\Vand holding a share of the insurance company is nothing else than short sellinginsurance contracts. Of course, the liability condition is crucial for our nor-malization. Otherwise a share of the insurance company would be a combinationof a risk-free investment and a short position in insurance contracts.3.1.3. The market portfolio. Since the risk-free asset is in net supply zero, themarket portfolio is of the form(9) xM = \-(0,Wu...,Wn,K,), vi\thK^KN + Ki=il Wh+Ki.K h = \Owing to the normalization(7) K,= -irone obtains(10) K=KN-irand the market portfolio may be decomposed into(11) xMThe corresponding rate of return is given by(12) R»=K»-R»--*—KM — ir Ajv — ir \ i rEquation (12) relates the insurance risk X and the insurance premium ir to the1. In other words, investors/= 1,..., m receive the amount ir against the obligation to cover the riskX. Without perfect liability a lower bound on Ki would be imposed by a condition of the typeprob[(A:,+ TT)(1 +RO)- X^0}^\-E.2. Thus, /?„ + ! is the rate of return on insurance contracts. Since investors hold a short position it isnot surprising that typically^ , n1 < Ro•Kholds.available at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,146 MULLERmarket rate of return RM. Applying the traditional and generalized versions ofthe CAPM-formula to RM and Rn + i = (X/ir) - 1 allows us to derive a class ofpremium principles.3.2. Premium Principles Based on the Traditional Capital Asset Pricing ModelIn this section the Sharpe-Lintner model will be applied to premium calculation.Since the Sharpe-Lintner model is based on (/*, a)-analysis, the resultingpremium principles will only depend on the first two moments of the underlyingdistributions. First of all, we shall see that the special case where X and RN areuncorrelated leads to the well-known variance principle. Later, the general casewill be cealt with, where the insurance risk X and the rate of return on the non-insurance sector RN may be correlated.3.2.1. Application of the Sharpe-Lintner model. As already mentioned, theCAPM-relationship (8) has to be applied on (12). For h = n + \, this leads to(13) E(XITT)- \-RO = (3{E(RM)-RO}, with (3 =Using (12) one obtains(14) Cov(Xlir,RM) = - KN COV(X,RN)-~ — - — Var(A').7T A J V — 7T 7T A^A' — 7T(15)N~ IT/ \ A N - 7T_ _ _ 2 ^ v _ C o v ( x ^ N )(16) 1A A 1 — 7T A ^ — 7Insertingfl = A-,y-7r KNCo\{X,RN)-Vai(X)(17)•Kinto (13) yields' A"i VarC/?^)-!-Var(A')-2A'N Cov(A',/?A')x & [£(/?A')-/?o]-[^^-l-Jor(19) [KZN Var(/?N) - KN Cov(X,RN)] [E(X) - TT(1 + Ro)]= [KN COV(X,RN) - Var(X)]KN[E(RN) - Ro].available at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,ECONOMIC PREMIUM PRINCIPLES AND CAPM 147By (12) we get( 2 0 ) ir(l+R0)-E(X) _ Cov(-X,RM)[E(RN)-RQ]KN Cov(KNRN,RM)'Obviously TT(1 + Ro) - E(X) is the net premium on the insurance risk X and[E(RN) - Ro]KN is the market risk premium on all non-insurance assets.Therefore (20) relates the premium on the insurance risk X to the market riskpremium on all non-insurance assets. This relationship allows for differentinterpretations.3.2.2. The capital asset pricing model and the variance principle. Applying thedecomposition formula (12) for the market return yieldsir(\+Ro)-E(X)= Var(X)-Cov(X,KNRN)( ' [E(RN)-R0]KN War(KNRN)-Cov(X,KNRN)'In the special case(22) Cov(X,KNRN) = 0i.e. if the risks of the insurance and the non-insurance sector are uncorrelated,(21) is reduced toir(\+R0)-E(X)^ Var(X)[E(RN)-R0]KN Var(KNRN)'In other words, the insurance risk X and the non-insurance risk KNRN areevaluated by the well-known variance principle.In general, however, (22) does not hold and the covariance terms in (21) leadto a deviation from the variance principle. If KN, Ro, E(RN), E(X), Var(A'),Var(/?'v) and Cov(A', RN) are known, the premium -K can be calculated by meansof (21). In this sense (21) provides us with a premium principle which fully reflectscapital market conditions and depends only on the first and second moments ofthe underlying distributions.3.2.3. The capital asset pricing model and general equilibrium theory.BUHLMANN (1980,1984) applied general equilibrium concepts to a risk exchangemarket. He proved the existence of equilibrium prices in the continuous case andderived, together with LiENHARD (1986), explicit price formulae for all types ofNeumann utilities belonging to the HARA-class.If Biihlmann's concept is applied to our model, the premium for a risk X isgiven by(24) £(X,RM):=E[X(p(RM)}where the price density <p is defined by(25) ^3. See also LIENHARD (1986).available at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,148 MULLERHence, one obtainsF(X R M \ = -a-E(RM)(26) e , ) = - C ° V ( 5 ' O + E(X)and (20) becomesTT(1 +RO)-E(X) _ £{X-E(X),RM\(27) [E(RN)~R0]KN e{KN[E(RN)-RN],RM}'The numerator of the right-hand side denotes the premium for the insurance riskX-E(X) under the market rate of return RM.The denominator may be interpreted as the premium for the risk of the non-insurance sector -KN[RN- E(RN)] under the market rate RM. Again thecapital asset pricing model is perfectly compatible with risk theory.3.3. Premium Principles Based on the Generalized Capital Asset Pricing Model3.3.1. The general case. In this section the preferences of the m investors aregiven by Neumann utilities u': R -> R,i = \,...,m. Furthermore, one assumes(A) u\...,u'"€; <W(c)(4) for some c€ ( -°o ,0)U(0 ,oo] .As we have seen in Section 2.2, this assumption implies:There exists a Neumann utility MM€ ^ ( C ) , such that the market portfolio Ris efficient relative to UM-For h = n + 1, R(x") = RM the generalized CAPM-relationship (2) leads to(28) E(Rn+l)- R0=-'7 y;pAr::,M t ; ;"^ ' ; ' [ g ^ M ) - *<>].By (12) one obtainsE(X) - ic{l + Ro)(29) [E(RN)-R0]KN- [E(X)-TT(\+R0)}E{uM'(RM)[X-E{X)]}E\uM' (RM)[RN - E(RN)]KN\ -E\uM'(RM)[X-E(X)}orTT(\+R0)-E(X) _[E(RN)-R0]KN(E{UM'{RM)[X-E(X)]}\I/E{UM'{RM)[E(RN)-RN}K*E\uM'(RM)\ j \ \ E[uM'(RM)\Now we can apply Biihlmann's concept again. Let4. W(c) is defined by (3).available at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,ECONOMIC PREMIUM PRINCIPLES AND CAPM 149be the price density and(32) c(Y,RM):=E[Y<p(RM)]be the premium for a risk Y under the market rate of return RM. Then (30) isof the formir(\+R0)-E(X) = e[X-E(X),RM\[E(RN)-R0]KN e{KN[E(RN)-RN],RM}'For the price density given by (31), a straightforward calculation leads to(a + hRM)~c(34) "V^EUa+bR*})-] Wi th 'C>°for c€ ( - oo,0) o r c 6 (0, oo), and toe~bRM(35) ^RM) = E[e-bRM] w i t h f c > 0for c= oo.The formulae (34) and (35) correspond exactly to LIENHARD'S (1986) results.This allows us to conclude that Biihlmann's premium concept for insurance risksand the generalized capital asset pricing model are compatible for the HARA-class.3.3.2. Special casesThe Traditional Capital Asset Pricing ModelFor c= — 1, the Neumann utilities are quadratic, (34) coincides with (25) and thetraditional capital asset pricing model results as a special case.The Esscher PrincipleFor c = oo the premium formula is(36) M **By assuming independence of the insurance risk X and the non-insurance riskKNRN, one obtains in analogy with BUHLMANN (1980) the Esscher principle:F(Ypbx) h4. CONCLUSIONSIn this paper the connection between capital asset pricing and some results of risktheory was analysed. For all Neumann utilities belonging to the HARA-class, itavailable at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,150 MULLERwas shown that the generalized capital asset pricing model may be interpreted asan extension of Biihlmann's economic premium principle to an economy with anon-insurance sector. In particular, it was possible to derive the variance and theEsscher principle from the CAPM-condition.Finally, our analysis led to relationships between premiums for insurance andfor non-insurance risks. These formulae could be considered as premium prin-ciples which are based on general capital market conditions.ACKNOWLEDGEMENTSI am indebted to H. Buhlmann, H. Garbers, P. Zweifel and two anonymousreferees for very helpful suggestions.REFERENCESBORCH, K. (1986) Risk theory and serendipity. Insurance: Mathematics and Economics 5, 103-112.BUHLMANN, H. (1980) An economic premium principle. ASTIN Bulletin 11, 52-60.BUHLMANN, H. (1984). The general economic premium principle. ASTIN Bulletin 14, 13-21.CASS, D. and STIGLITZ, J. E. (1970). The structure of investor preferences and asset returns, andseparability in portfolio allocation: a contribution to the pure theory of mutual funds. Journalof Economic Theory 2, 122—160.GOOVAERTS, M. J., DE VYLDER, F. and HAEZENDONCK, J. (1984) Insurance premiums. North-Holland, Amsterdam.KAHANE, Y. (1979) The theory of insurance risk premiums — a re-examination in the light of recentdevelopments in capital market theory. ASTIN Bulletin 10, 223-239.LIENHARD, M. (1986) Calculation of price equilibria for utility functions of the HARA class. ASTINBulletin 17 in press.LINTNER, J. (1965) The valuation of risk assets and the selection of risky investments in stockportfolios and capital budgets. Review of Economics and Statistics 47, 13-37.MERTON, R. C. (1982) On the microeconomic theory of investment under uncertainty. In Handbookof Mathematical Economics, Vol. II. North-Holland, Amsterdam.SHARPE, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk.Journal of Finance 19, 425—442.HEINZ MULLERInstitut fur Empirische Wirtschaftsforschung, Universitat Zurich, Kleinstrasse15, CH-8008 Zurich, Switzerland.available at https:/www.cambridge.org/core/terms. https://doi.org/10.2143/AST.17.2.2014969Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 13:56:51, subject to the Cambridge Core terms of use,

Economic Premium Principles in Insurance and the Capital Asset Pricing Model

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