Probability Transforms with Elliptical Generators

Transcription

1 Probability Transforms wit Elliptical Generators Emiliano A. Valdez Scool of Actuarial Studies Faculty of Commerce & Economics University of New Sout Wales Sydney, AUSTRALIA Tel Fax First Draft: 9 Marc 2005 Tis Draft: 1 May 2005 Abstract Tis paper introduces te notion of elliptical transformations for possible applications in constructing insurance premium principles. It is a common notion in actuarial science to transform or sometimes said, to distort te real probability measure of a risk X and ten calculate premiums based on te expectation of te transformed distribution. Elliptical transformations are a way to distort te probability distribution of a risk and it is based on te relative ratio of a density generator of a member of te class of elliptical distributions to te density generator of te Normal distribution wic is also well-known to belong to te class of elliptical distributions. Te class of elliptical distribution models consists of distributions considered symmetric and provides a generalization of te class of Normal loss distribution models. We examine te premium principle implied by tis elliptical transformation. We find tat te elliptical transforms lead us to, as special cases, te Wang premium principle, te Wang Student-t distortion principle, as well as te Esscer premium principle. In several cases, te resulting premium principle as a replicating feature of te standard deviation premium principle. A numerical example as been elaborated to illustrate te practical implementation of tis elliptical transformation. Keywords: probability transform, distortion, risk measures, premium principles, elliptical distributions, elliptical transforms. Te autor wises to acknowledge financial support from te Australian Researc Council Discovery Grant DP and te UNSW Actuarial Foundation of Te Institute of Actuaries of Australia. Te autor also extends appreciation to Prof. Dr. Jan Daene of Katolieke Universiteit Leuven for osting is visit in Belgium wile on a sabbatical leave. Please do not quote witout autors permission. Corresponding autor s address: e.valdez@unsw.edu.au. 1

2 1 Introduction Consider an insurance company tat over some well-defined time reference, faces a random loss of X on a well-defined probability space (Ω, F, Pr). We can sometimes appropriately call X te risk faced by te insurer. For tis random loss X, assume tat its distribution function is denoted by F X (x) and its corresponding tail probability by F X (x) =1 F X (x). Assuming X is a continuous random variable, as is often te case, ten te density function of X is given by f X (x) = df X(x) dx = df X(x). dx Te task of te insurer is to assign a price tag on tis risk X leading us to examine premium principles. As pointed out by Young (2004), tat loosely speaking, a premium principle is a rule for assigning a premium to an insurance risk. Te insurance premium is usually coined risk-adjusted to refer to te fact tat it already incorporates model risks as well as parameter uncertainties. As advocated in tis paper, tese model and parameter risks are often andled by distorting te real probability measure. Tis paper introduces yet anoter metod to transform distributions. A premium principle π is a mapping from a set Γ of real-valued random variables defined on (Ω, F) to te set R of real numbers. Effectively, we ave π : Γ R so tat for every X belonging to Γ, π[x] R, is te assigned premium. To no surprise, premium principles are well-studied in te actuarial literature. See for example, Goovaerts, DeVijlder and Haezendonck (1984), Kaas, van Heerwaarden and Goovaerts (1994), and more recently Young (2004). A capter on insurance premium principles is devoted in Kaas, Goovaerts, Daene and Denuit (2001). Risk-adjusted premiums are often computed based on te expectation wit respect to a transformed probability measure, say Q, suctat π[x] =E Q [X] =E [ΨX] (1) were Ψ is a positive random variable wic tecnically is also te Radon-Nikodym derivative of te Q-measure wit respect to te real probability measure, sometimes denoted by P. See Gerber and Pafumi (1998). Tis is also sometimes called, in te Finance literature, te pricing density. In several premium principles, te Ψ follows te general form defined by a relation (X; λ) Ψ = (2) E [(X; λ)] for some function of te random variable X and a parameter λ often describing some level of risk-aversion to te insurer. Tis relation in (2) can also be justified in a decision teoretic framework as discussed in Heilman (1985) and Hürlimann (2004). Following Hürlimann (2004), for example, consider a loss function L : R 2 R wic describes te loss incurred L(x, x ) for a realized outcome X = x wen te action taken by te decision maker as been x. Decisions on premiums are determined on te basis of minimizing te expectation of tis loss function. Consider te loss function defined by L(x, x )=(x; λ) (x x ) 2. (3) 2

3 It is straigtforward to sow ten tat te resulting premium principle follows te general form (X; λ) π[x] =E E [(X; λ)] X. Many familiar premium principles lead to tis form. For example, te net premium principle can be derived wit (X; λ) =1and te variance premium principle can be derived wit (X; λ) =1+λX. Te celebrated Esscer premium principle can be derived by coosing (X; λ) =exp(λx), and te Karlsrue premium principle discussed in Hürlimann (2004) can be derived by coosing (X; λ) =X. Even te conditional tail expectation used as a risk measure can be derived wit (X; λ) =I X>FX 1 (q) were I( ) is an indicator function and FX 1 (q) is te quantile of te distribution of X at a pre-determined probability level q. In te univariate case, we ave X belonging to te class of elliptical distributions if its density can be expressed as " µx f X (x) = C # µ 2 σ g (4) σ for some so-called density generator g (wic is a function of non-negative variables), and were C is a normalizing constant. Later in te paper, te definition of elliptical distributions is expressed in terms of its caracteristic function because tis function always exist for a distribution. For te familiar Normal distribution N µ, σ 2 wit mean µ and variance σ 2, it is straigtforward to sow tat its density generator for wic we sall denote by g N in tis paper as te form g N (u) =exp( u/2). (5) In tis paper, we define elliptical transformations by considering te functional (x; λ) to ave te following form: Φ g 1 F X (x) + λ i 2 (x; λ) = Φ g 1 N F X (x) i 2 (6) were g is te density generator of some member of te class of elliptical distributions. Te family of elliptical distributions consists of symmetric distributions for wic te Student-t, te Uniform, Logistic, and Exponential Power distributions are oter familiar examples. Tis is a ric family of distributions tat allow for a greater flexibility in modelling risks tat exibit tails eavier tan tat of a Normal distribution. Tis can be especially useful for capturing insurance losses for extreme events. As we sall observe in our discussion in te paper, te probability transform defined using te ratio of density generators in (6) effectively distort te distribution by putting eavier weigts on bot tails of te distribution. In insurance premium calculations, wat tis translates to is giving more penalty on large losses but at te same time, placing some empasis on small or zero losses tereby encouraging small claims. Te result is terefore a trade-off between small and large claims redistributing te claims distribution so tat it leads to a more sensible and equitable 3

4 premium calculation. Tis trade-off is best parameterized by te parameter λ wic also effectively introduces a sift or translation of te distribution. Te rest of te paper as been organized as follows. In Section 2, we brieflydiscuss te ric class of elliptical distributions in te univariate dimension, togeter wit some important properties. Section 3 describes te notion of elliptical transformation. Several examples follow. In Section 4, we discuss te special case were te risk as a distribution tat is location-scale. Elliptical distributions are special cases of locationscale families. In Section 5, using some claims experience data, we demonstrate practical implementation of introducing premium loadings using te ideas developed in te paper. Finally, in section 6, we provide some concluding remarks togeter wit some remarks about possible direction for furter work. 2 Te Family of Elliptical Distributions Te class of elliptical loss distribution models provides a generalization of te class of normal loss models. In te following, we will describe tis class of models first in te univariate dimension, and briefly extending it to te multivariate dimension. Te class of elliptical distributions as been introduced in te statistical literature by Kelker (1970) and widely discussed in Fang, Kotz and Ng (1990). See also Landsman and Valdez (2003), Valdez and Daene (2003), and Valdez and Cerni (2003) for applications in insurance and actuarial science. Embrects, et al. (2001) also provides a fair amount of discussion of tis important class as a tool for modelling risk dependencies. 2.1 Definition of Elliptical Distributions It is widely known tat a random variable X wit a normal distribution as te caracteristic function expressed as E[exp(itX)] = exp(itµ) exp 1 2 t2 σ 2, (7) were µ and σ 2 are respectively, te mean and variance of te distribution. We sall use te notation X N µ, σ 2. Te class of elliptical distributions is a natural extension to te class of normal distributions. Definition 1 Te random variable X is said to ave an elliptical distribution wit parameters µ and σ 2 if its caracteristic function can be expressed as E [exp(itx)] = exp(itµ) ψ t 2 σ 2 (8) forsomescalarfunctionψ. If X as te elliptical distribution as defined above, we sall conveniently write X E µ, σ 2,ψ and say tat X is elliptical. Te function ψ is called te caracteristic generator of X and terefore, for te normal distribution, te caracteristic generator is clearly given by ψ(u) =exp( u/2). It is well-known tat te caracteristic function of a random variable always exists and tat tere is a one-to-one correspondence between distribution functions and caracteristic functions. Note owever tat not every function ψ can be used to construct a caracteristic function of an elliptical distribution. Obviously, tis function 4

5 ψ sould fulfill te requirement tat ψ(0) = 1. A necessary and sufficient condition for te function ψ to be a caracteristic generator of an ellipticial distribution can be seen in Teorem 2.2 of Fang, Kotz and Ng (1990). It is also interesting to note tat te class of elliptical distributions consists mainly of te class of symmetric distributions wic include well-known distributions like normal and Student-t. Te moments of X E µ, σ 2,ψ do not necessarily exist. However, it can be sown tat if te mean, E(X), exists, ten it will be given by E(X) =µ (9) andiftevariance,var(x), exists, ten it will be given by Var(X) = 2ψ 0 (0) σ 2, (10) were ψ 0 denotes te first derivative of te caracteristic generator. A necessary condition for te variance to exist is ψ 0 (0) <, (11) see Cambanis, Huang and Simmons (1981). In te case were µ =0and σ 2 =1, we ave wat we call a sperical distribution and te random variable X is replaced by a standard random variable Z. Tat is, we ave Z E(0, 1,ψ) and te notation S(ψ) for E(0, 1,ψ) is more typically used and tus, we write Z S(ψ). It is clear tat te caracteristic function of Z as te form E[exp(itZ)] = ψ t 2 for any real number t. Also, if we consider any random variable X satisfying X d = µ + σz, for some real number µ, some positive real number σ and some sperical random variable Z S(ψ), ten it can be sown tat X E µ, σ 2,ψ. Similarly, for any elliptical random variable X E µ, σ 2,ψ,wecanalwaysdefine te random variable Z = X µ σ wic is clearly sperical. 2.2 Densities of Elliptical Distributions An elliptically distributed random variable X E µ, σ 2,ψ does not necessarily possess a density function f X (x). In te case of a normal random variable X N µ, σ 2, its density is well-known to be " f X (x) = 1 exp 1 µ # x µ 2. (12) σ 2 σ 5

6 For elliptical distributions, one can prove tat if X E µ, σ 2,ψ as a density, it will ave te form " µx f X (x) = C # µ 2 σ g (13) σ for some non-negative function g( ) satisfying te condition 0 < Z 0 z 1/2 g(z) dz < (14) and a normalizing constant C given by Z C = z 1/2 g(z) dz 1. (15) 0 Also, te opposite statement olds: Any non-negative function g( ) satisfying te condition (14) can be used to define a one-dimensional density " µx # C µ 2 σ g σ of an elliptical distribution, wit C given by (15). Te function g( ) is called te density generator. One sometimes writes X E µ, σ 2,g for te one-dimensional elliptical distributions generated from te function g( ). A detailed proof of tese resultsfortecaseofn-dimension, using sperical transformations of rectangular coordinates, can be found in Landsman and Valdez (2003). From (12), one immediately finds tat te density generators and te corresponding normalizing constants of te normal random variable X N µ, σ 2 are given by g(z) =exp( z/2) and C = 1, respectively. Table 1 Some known elliptical distributions wit teir density generators Family Density generators g(u) Bessel g(u) =(u/b) a/2 K a (u/b) 1/2i, a> 1/2,b>0 were K a ( ) is te modified Bessel function of te 3rd kind Caucy g(u) =(1+u) 1 Exponential Power g(u) =exp[ r (u) s ], r, s > 0 Laplace g(u) =exp( u ) Logistic g(u) = Normal Student-t g(u) = exp( u) [1 + exp( u)] 2 g N (u) =exp( u/2) ³ 1+ u m (m+1)/2, m>0 an integer 6

7 As a special case, we find tat from te results concerning elliptical distributions, if a sperical random variable Z S(ψ) possess a density f Z (z), tenitwillave te form f Z (z) =Cg z 2, were te density generator g satisfies te condition (14) and te normalizing constant C satisfies (15). Furtermore, te opposite also olds: any non-negative function g( ) satisfying te condition (14) can be used to define a one-dimensional density Cg z 2 of a sperical distribution wit te normalizing constant C satisfying (15). One often writes S(g) for te sperical distribution generated from te density generator g( ). 3 Elliptical Transforms Suppose g Z (u) is te density generator of a univariate sperical random variable Z E (0, 1,g Z ), also written as Z S (g Z ). Weavesubscriptedtedensity generator to empasize te corresponding sperical random variable Z. Nowwe define te ratio of density generators as follows. Definition 2 Let X be a random variable wit tail probability function F X ( ). We define te ratio of density generators g Z and g N to be te random variable Φ g 1 Z F X (X) + λ i 2 gz (X; λ) = Φ g 1 N F X (X) i 2 (16) for some non-negative parameter λ 0 and were Φ ( ) is te cumulative distribution function of a standard Normal random variable and g N is te density generator of a Normal distribution. Notice tat in te definition we assume we ave a random variable X Γ, belonging to te set of all risks, wose tail probability function is given by F X ( ). As we sall observe later in te examples tat follow, tis non-negative parameter can be interpreted as a premium per unit of volatility (or risk). Te following teorem gives an expression for te expectation of te ratio of density generators. Teorem 1 Let X be a random variable wit tail probability function F X ( ) and let te cumulative distribution function of a standard Normal be denoted by Φ ( ). Te expectation of te ratio of density generators g Z and g N can be expressed as E [ gz (X; λ)] = 1 Z g Z z 2 dz. (17) Proof. First assuming te density of X is denoted by f X ( ), ten we can write te expectation as Z Φ g 1 Z F X (x) + λ i 2 E[ gz (X; λ)] = Φ g 1 N F X (x) i 2 f X(x) dx. 7

8 Using te transformation u = F X (x) so tat du = f X (x) dx, tenweave Z Φ 1 g 1 Z (u)+λ i 2 E[ gz (X; λ)] = du. 0 g N (Φ 1 (u)) 2i Wit anoter transformation w = Φ 1 (u) so tat du = 1 g N w 2 dw, wetenave Z g Z (w + λ) 2i 1 E[ gz (X; λ)] = g N (w 2 g N w 2 dw. ) Applying one last transformation z = w + λ, we get te desired result. Following te above Teorem, it is clear tat for a sperical Z S (g Z ) wit a normalizing constant C Z, we ave te following result (wic will later be useful): 1 Z C Z = g Z z 2 dz 1 = 1 E[gZ (X; λ)]. We now give some examples of ratios of density generators. Example 3.1: Normal-to-Normal Generators If g Z is also te density generator of a Normal distribution, ten we ave Φ g 1 N F X (X) + λ i 2 gn (X; λ) = Φ g 1 N F X (X) i 2 =exp λ Φ 1 F X (X) λ = e λ2 /2 exp λφ 1 F X (X). (18) It is easy to see tat in tis case, te expectation of (18) is equal to 1. application of Teorem 1, for example, leads us to tis. Direct Example 3.2: Student-t-to-Normal Generators If g Z is te density generator of a Student-t distribution wit m degrees of freedom, ten we ave 2 Φ g 1 Z F X (X) + λ i 2 gz (X; λ) = Φ g 1 N F X (X) i 2 = 1+ 1 m Φ 1 F X (X) + λ i 2 (m+1)/2 exp 12 Φ 1 F X (X) i 2. (19) Applying Teorem 1 to solve for te expectation, we ave E[ gz (X; λ)] = = Z 1 Z 1 1 correction made: 1-May correction made: 1-May g Z z 2 dz µ 1+ 1 m z2 (m+1)/2 dz. 8

9 Applying transformations, it can be sown tat tis expectation leads us to r Z m 1 r µ E[ gz (X; λ)] = z (m/2) 1 (1 z) (1/2) 1 m m dz = 0 B 2, 1 2 r r m Γ(m/2) Γ(1/2) m = Γ((m +1)/2) = Γ(m/2) 2 Γ((m +1)/2), were B(, ) is te Beta function and Γ( ) is te Gamma function. Here, we also note tat we applied te fact tat Γ(1/2) = π. Now interestingly, te case were m =1 leads us to te Caucy-to-Normal generators and in wic case, we would ave r r 1 Γ(1/2) Γ(1/2) π E[ gz (X; λ)] = = Γ(1) 2. It can also be sown tat in te limiting case were m, tis leads us back to te Normal-to-Normal density generators. Example 3.3: Exponential Power-to-Normal Generators Assuming now tat g Z is te density generator of an Exponential-Power distribution (see Table 1), ten we ave Φ g 1 Z F X (X) + λ i 2 gz (X; λ) = Φ g 1 N F X (X) i 2 = n exp r Φ 1 F X (X) + λ 2s 1 2 Φ 1 F X (X) io 2. (20) Applying Teorem 1 to solve for te expectation, we ave E[ gz (X; λ)] = = = Z 1 Z 1 g Z z 2 dz e rz2s dz Z 2 e rz2s dz 0 were te tird line follows because of symmetry. Let us derive an explicit form for tis expectation. First, consider te transformation u = z 2s so tat du =2sz 2s 1 dz = 2su (2s 1)/2s dz. Tis leads us to E[ gz (X; λ)] = 1 Z s u ( 2s) 1 1 e ru du. 0 Next ten, consider te transformation z = ru so tat dz = rdu. Weave Z 1 E[ gz (X; λ)] = s r1 (1/2s) z ( 2s) 1 1 e z dz 0 µ 1 1 = s r1 (1/2s) Γ, 2s were Γ( ) is te usual Gamma function. Te case were r =1/2 and s =1leads us to te Normal distribution case. 9

10 Consider again g Z as te density generator of a univariate sperical random variable Z S(g Z ).Nowwedefine wat we meant by probability transformation using tese elliptical density generators. Tese transformations we will, for simplicity, call elliptical transformations. Definition 3 Let X be a random variable wit tail probability function F X ( ) and wose density function exists and is equal to f X ( ). Wedefine te transformed random variable, denoted by X, to be one wit a (transformed) density function given by Φ g 1 Z F X (X) + λ i 2 f X (x) =C Φ g 1 N F X (X) i 2 f X(x) (21) for all x in te domain or range of X and were C is a normalizing constant. By recognizing tat gz (X; λ) = g Z (Φ 1 (F X (X))+λ) 2i g N (Φ 1 (F X (X))) 2i, te ratio of density generators, we simply note tat te normalizing constant is C = 1 E[ gz (X; λ)] for wic can be easily evaluated from Teorem 1. We can, as a matter of fact, find an expression for te distribution function of te (transformed) random variable X. First, notice tat Z Z Φ g 1 Z F X (v) + λ i 2 F X (x) = f X (v) dv = C Φ x x g 1 N F X (v) i 2 f X(v) dv. Applying te transformation u = F X (v) so tat du = f X (v) dv, tis yields Z Φ F X (x) g 1 Z (u)+λ i 2 F X (x) =C du 0 g N (Φ 1 (u)) 2i and ten applying te transformation w = Φ 1 (u) so tat du = 1 g N w 2 dw, we ave F X (x) = = Z C Φ 1 (F X (x)) Z Φ 1 (F X (x))+λ g Z (w + λ) 2i g N (w 2 ) C Z g Z z 2 dz g N w 2 dw = F Z Φ 1 F X (x) + λ (22) were F Z ( ) is te distribution function of a sperical random variable wit density generator g Z. In sort, we ave Z S(g Z ). 10

11 In an independent work done by Landsman (2004), e also proposed to use a transformation of densities of elliptical random variables using a ratio of te density generator evaluated as i g ((x µ) /σ) 2 2λx g ((x µ) /σ) 2i. Landsman (2004) sowed tat tis transformation is a generalization of te Esscer transform for elliptical distributions. He furter demonstrated tat tis generalizes te variance premium principle applied again to elliptical distributions and e appropriately called tese transformations elliptical tilting. Tere are several differences between Landsman s elliptical tilting wit tose proposed in tis paper. First, unlike Landsman (2004), we take te ratio of two different density generators, one of wic is always te Normal density generator, and te oter is an arbitrarily cosen one. In effect, we are transforming te distribution by taking te relative weigts of te densities of an arbitrarily cosen elliptical random variable to te Normal random variable. Te arbitrary selection allows te decision maker (wic in tis case is te insurer) some flexibility. Landsman cooses te density generator of te random variable being tilted wic is always an elliptical random variable. Second, wile bot Landsman and tis paper allows translation of te distribution by introducing a sift parameter λ, te manner in wic te translation is accomplised differ in bot respects. Landsman introduces te sift using ((x µ) /σ) 2 2λx wile we, in tis paper, introduce te sift via Φ 1 F X (x) + λ. Tird, Landsman limited its applications to exponential tilting of elliptical random variables wile ours does not necessarily ave suc limitations. Lastly, in te case of a Normal distribution, Landsman generalizes te variance premium principle wile we generalize te standard deviation premium principle. Te standard deviation is more commonly utilized as a measure of te level of riskiness of a portfolio. See, for example, Markowitz (1952) and Merton (1990). Definition 4 Let X be a random variable wit tail probability function F X ( ) and wose density function exists and is equal to f X ( ). LetX be te transformed random variable of X according to te elliptical transformation defined in (21). Ten te expectation of X is defined to be te premium principle implied by te elliptical transformation: π[x] =E(X gz (X; λ) )=E E[ gz (X; λ)] X. Observe tat from (22), we can also derive te premium (or expectation of te transformed distribution) using π[x] =E(X )= Z 0 F Z Φ 1 F X (x) + λ dx + Z 0 F Z Φ 1 F X (x) + λ dx wic reduces to just R 0 F Z Φ 1 F X (x) + λ dx for random variables wit nonnegative support. We produce Figures 1 to 4 to elp us visualize te transformation. All figures in tis paper are attaced as appendix following te list of references. For Figures 11

12 1 troug 4, we assume te random variable X being transformed is Uniform on te interval (0, 100). Tis coice as been made to elp us more vividly visualize te effects of te transformation. Te straigt orizontal broken line found in tese figures is te un-transformed (or original) Uniform density. In Figure 1, we consider te case were we coose Z to be te Normal distribution and we present te various transformation for different values of λ. Generally, we see tat using te Normal density generator, tis puts more weigts on te rigttail of te distribution. So if te risk are losses as in te case of insurances, tis penalizes more eavily large amount of losses. Te effect of λ introduces a sift of te distribution to te rigt for larger λ puttinginaneveneavierpenaltyonte tails. In Figures 2, 3 and 4, we consider te case were we coose Z to be te Student-t distribution. Figure 2 examines te effect of increasing te degrees of freedom (fixing λ =1) wile Figure 3 examines te effect of increasing te risk aversion parameter λ (fixing m =5). In Figure 4, we present te case were λ =0wen tere is no sift in te distribution. Here we observe equal penalty of bot ends of te tail. Te Student-t distortion as been first introduced by Wang (2004) in te pricing of catastrope bonds. As pointed out in is paper, putting more weigts to te tails of te underlying distributions is a way to reflect investor beavior and preferences tat we often observe in te market. Investors generally fear large unexpected losses, but tey also like large unexpected gains. Interestingly as special cases, we recover some of te familiar premium principles as discussed below. Tis in some sense generalizes tese premium principles. Example 3.4: Wang Premium Principle If g Z is cosen to be te density generator of a Normal distribution, ten it is immediate to get te Wang transformation: F X (x) =Φ Φ 1 F X (x) + λ. See Wang (1996) and Wang (2000). Example 3.5: Wang s Student-t Distortion Premium Principle If g Z is cosen to be te density generator of a Student-t distribution wit m degrees of freedom, ten we ave as defined in Wang (2004) F X (x) =Q Φ 1 F X (x) + λ, were, following Wang s notation, Q( ) denotes te distribution function of a Studenttwitm degrees of freedom. Wang actually as set λ =0and used F X (x) = Q Φ 1 F X (x) instead. Example 3.6: Esscer Premium Principle Because 1 Φ(x) =Φ( x), ten in te special case were X is Normally distributed say N µ, σ 2, it is rater straigtforward to see tat te elliptical transformation leads to te Esscer transform. In tis case, we ave gz (X; λ) E[ gz (X; λ)] = e λ2/2 exp λφ 1 F X (X) E e λ2 /2 exp λφ 1 F X (X) i = exp λσ X E exp λ σ X. See Esscer (1932) and also more recent articles, for example, Gerber and Siu (1994). 12

13 4 Location-Scale Families By considering te risk X to belong to families of location-scale distributions, we can furter simplify te premium principle implied by te elliptical transformation. Members of te elliptical distributions also belong to te larger family of locationscale distributions. As a matter of fact, in tis case, we are able to recover a similar concept to te standard deviation premium principle. We give te following teorem. Teorem 2 Let µ X be a random variable belonging to a location-scale family so tat x µ F X (x) =F Z for some Z = X µ wose distribution is independent of σ σ te location parameter µ and scale parameter σ. Ten te premium principle implied by te elliptical transformation can be expressed as i π[x] =µ + E Z F 1 Z (Φ(Z λ)) σ. Proof. First assuming te density of X exists and is denoted by f X ( ), tenwe can write te expectation as Z π[x] =C x Φ g 1 Z F X (x) + λ i 2 Φ g 1 N F X (x) i 2 f X (x) dx. Applying te transformation u = F X (x) so tat du = f X (x) dx, tis yields to Z Φ 1 π[x] =C F 1 g 1 Z (u)+λ i 2 X (u) du. 0 g N (Φ 1 (u)) 2i Wit anoter transformation w = Φ 1 (u) so tat du = 1 g N w 2 dw, wetenave Z π[x] =C F 1 g Z (w + λ) 2i 1 X (Φ(w)) g N (w 2 g N w 2 dw. ) Applying one last transformation z = w + λ, weget π[x] = Z F 1 X (Φ(z λ)) C Z g Z z 2 dz R were C Z = g Z z 2 i 1 dz is te normalizing constant of te sperical random variable Z. Tus, we see tat i π[x] =E Z F 1 X (Φ(Z λ)) were te expectation E Z is evaluated wit respect to te sperical random variable Z. Now, since X is location scale, we can re-write F 1 X (Φ(Z λ)) = µ + F 1 Z (Φ(Z λ)) σ and te desired result immediately follows. 13

14 Tis result gives us a way of interpreting te parameter λ, wic is actually a risk premium per unit of risk (or volatility) as measured by te standard deviation. IntecasewereX is normally distributed N µ, σ 2 and we coose Z to be te standard Normal, ten we get i E Z F 1 Z (Φ(Z λ)) = E Z (λ Z) =λ E Z (Z) =λ, so tat te resulting premium principle leads us to te familiar form of te standard deviation premium principle: π[x] =µ + λσ. Terefore, we ave λ = π[x] µ, a risk premium per unit of risk as measured σ by σ. Furtermore, it can also be sown tat by using a first-order Taylor s series expansion, we can ave te following approximation: E Z F 1 Z (Φ(Z λ)) i F 1 1 Z (Φ( λ)) = F Z Φ(λ), were Φ denotes te tail probability function of a standard normal. 5 Practical Implementation In tis section, we illustrate ow to practically implement te premium principles developed in tis paper using te empirical losses investigated in Frees and Valdez (1998). Te experience consisted of a random sample of 1,500 claims from a general insurance liability portfolio provided by te Insurance Services Office, Inc. For our purposes, we only include in tis analysis te observations from claims, for wic we denote ere as te risk X. Unfortunately, te experience data we consider ere consists of observations tat generated claims arising from an insurance portfolio. In an ordinary insurance portfolio, te case is usually one were tere is a mass of zero claims, some observations tat never generated claims. In te pricing of insurance, one would take into account te probability mass of zero claims. For te lack of available insurance claims data, we resort to tis experience data, owever, we ask te reader to exercise caution tat normally in insurance pricing, te probability mass of zero claims must be taken into account. Tis mass as been ignored in te ensuing analysis. Summary statistics can be found in Frees and Valdez (1998), but just to reiterate tese statistics, te average loss was 41,208, te median was 12,000 and te standard deviation was 102,748. In fitting for te best distribution for tis experience loss data, censoring ad to be considered because claims ave policy limits and if te actual loss exceeded te policy limit, only te policy limit was recorded. At any rate, according to Frees and Valdez (1998), te best fitting loss distribution model for te data as te Pareto form described by µ λ θ F X (x) =, for x>0. λ + x Its density function terefore as te form µ 1 θ+1 f X (x) =θλ θ. λ + x 14

15 Te maximum likeliood parameter estimates are b λ =14, 453 and b θ =1.135, wit respective standard errors of 1,397 and In Figure 5, we provide a frequency istogram of te logaritm of te observed losses (called logloss) wit te fitted Pareto density function of te logloss. Tis figure ignores te censored observations. For simplicity, in constructing te premium principle used to illustrate, we ave considered te coice of a Student-t density generator wit m =3degrees of freedom. Tere is no empirical basis for tis coice, except te Student-t appears to be te most sensible as it allows flexibility wit te additional parameter for degrees of freedom. Tus, from Example 3.2, we can write 3 gz (X; λ) = " exp " µ µ ³ # Φ 1 λ θ 2 2 λ+x + λ 1 2 µ µ ³ θ # 2. Φ 1 λ λ+x Te purpose ere is terefore to develop risk-adjusted premiums. Te results are summarized in Figures 6 and 7. Figure 6 provides te risk-adjusted premiums for varying values of λ and it is not surprising to see tat te risk-adjusted premiums increase wit increasing λ. For te different risk-adjusted premiums implied by te different λ, we computed ten te probability of aving losses smaller tan tese riskadjusted premiums, under te original probability measure. Tis level of probability we denote ere by q, and in Figure 7, we sow te risk-adjusted premiums as a function of tis probability level. Again, tere is iger risk-adjusted premiums associated wit larger q. Notice tat te risk-adjusted premium does not always yield a larger value tan te unadjusted premium (tat is, te net premium equal to λ/ (θ 1) in te Pareto case), but because of its increasing nature, for some level λ and correspondingly for some level q, beyond wic tere will be a positive risk premium. Numerically, tese results are summarized as Table 2 for convenience. Here, te net premium level is 107,059, and at λ = (or correspondingly q =0.9108), tis premium level will equal te risk-adjusted premium. See Table 2. Table 2 Te risk-adjusted premiums and teir probability levels λ unadjusted premium E(X) risk-adjusted premium π[x] probability level q ,059 4, ,059 11, ,059 31, ,059 91, , , ,059 1,051, ,059 4,462, correction made: 1-May

16 6 Final Remarks Tis paper introduces te notion of elliptical transformation leading to a premium principle wic in some sense generalizes te familiar Wang transformation as well as te premium principle introduced by Wang (2004) using te Student-t distortion function. We also note tat in te special case of transforming random variables belonging to te location-scale families, te resulting premium principle is te standard deviation premium principle. Furtermore, we observe from te figures presented in te paper tat in some sense, te transformation introduces a eavy penalty on te extreme rigt tails of te distribution but also encourages small losses by placing relatively reasonable weigts on te extreme left tails of te distribution. How muc tis penalty is can depend upon te coice of te density generator togeter wit te parameter λ wic in a sense gives a measure of aversion to te level of risk of te insurer. Tis parameter also introduces a sift in te distribution of te risk. In te Normal distribution case, te same parameter leads to an interpretation of a risk premium per unit of volatility, or risk. We are also able to sow tat te elliptical transformation recovers many oter familiar premium principles. Te notion of elliptical transformation introduced in tis paper can also be applied as a risk measure to compute economic capital. It will be interesting ow tis risk measure may be extended to te case were tere is a need to aggregate several risks or to re-allocate te total economic capital into various constituents. Moreover, in te future, it will be an interesting work to examine some properties of tis premium principle. References [1] Cambanis, S., Huang, S., and Simons, G. (1981) On te Teory of Elliptically Contoured Distributions, Journal of Multivariate Analysis 11: [2] Embrects, P., McNeil, A., Straumann, D. (2001). Correlation and Dependence in Risk Management: Properties and Pitfalls. Risk Management: Value at Risk and Beyond, edited by Dempster, M. and Moffatt, H.K. Cambridge University Press. Also available at < [3] Esscer, F. (1932) On te Probability Function in te Collective Teory of Risk, Skandinavisk Aktuarietidskrift 15: [4] Fang, K.T., Kotz, S. and Ng, K.W. (1987) Symmetric Multivariate and Related Distributions. London: Capman & Hall. [5] Frees, E.W. and Valdez, E.A. (1998) Understanding Relationsips Using Copulas, Nort American Actuarial Journal 2: [6] Gerber, H.U. and Pafumi, G. (1998) Utility Functions: from Risk Teory to Finance, Nort American Actuarial Journal 2: [7] Gerber, H.U. and Siu, E. (1994) Option Pricing by Esscer Transforms, Transactions of te Society of Actuaries XLVI: [8] Goovaerts, M., DeVijlder, F. and Haezendonck, J. (1984) Insurance Premiums. Amsterdam: Nort Holland Publisers. 16

17 [9] Heilman, W.R. (1987) A Premium Calculation Principle for Large Risks, in Operations Researc Proceedings 1986, pp Berlin-Heidelberg: Springer Verlag. [10] Hürlimann, W.(2004) Is te Karlsrue Premium a Fair Value Premium? Blätter der Deutscen Gesellscaft für Versiceringsmatematik XXXVI(4): [11] Kaas, R., Goovaerts, M., Daene, J. and Denuit, M. (2001) Modern Actuarial Risk Teory. Neterlands: Kluwer Academic Publisers. [12] Kaas, R., van Heerwaarden, A.E. and Goovaerts, M. (1994) Ordering of Actuarial Risks. Amsterdam: Institute for Actuarial Science and Econometrics, University of Amsterdam. [13] Kelker, D. (1970). Distribution Teory of Sperical Distributions and Location- Scale Parameter Generalization. Sankya 32: [14] Landsman, Z. (2004) On te Generalization of Esscer and Variance Premiums Modified for te Elliptical Family of Distributions, Insurance: Matematics & Economics 35: [15] Landsman, Z. and Valdez, E.A. (2003) Tail Conditional Expectations for Elliptical Distributions, Nort American Actuarial Journal 7: [16] Markowitz, H. (1952) Portfolio Selection, Journal of Finance 7: [17] Merton, R.C. (1990) Continuous Time Finance. Massacusetts: Blackwell Publising Ltd. [18] Panjer, H.H. (2002) Measurement of Risk, Solvency Requirements, and Allocation of Capital witin Financial Conglomerates, Institute of Insurance and Pension Researc, University of Waterloo, Researc Report [19] Valdez, E.A. and Cerni, A. (2003) Wang s Capital Allocation Formula for Elliptically Contoured Distributions, Insurance: Matematics & Economics 33: [20] Valdez, E.A. and Daene, J. (2003) Bounds for Sums of Non-Independent Log-Elliptical Random Variables, paper presented at te Sevent International Congress on Insurance: Matematics & Economics, Lyon, France. [21] Wang, S.S. (1996) Premium Calculation by Transforming te Layer Premium Density, ASTIN Bulletin 21: [22] Wang, S.S. (2000) A Class of Distortion Operators for Pricing Financial and Insurance Risks, Journal of Risk and Insurance 67: [23] Wang, S.S. (2004) Cat Bond Pricing Using Probability Transforms, invited article, Geneva Pa-pers: Etudes et Dossiers, special issue on Insurance and te State of te Art in Cat Bond Pricing, 278: [24] Young, V.R. (2004) Premium Principles in Encyclopedia of Actuarial Science, ed. Sundt, B. and Teugels, J., New York: Jon Wiley & Sons, Ltd. 17

18 Transformed Densities lam bda=1 lam bda=2 lam bda=3 original x Figure 1: Elliptical transformation using te Normal density generator. Tis transformation effectively results in te Wang transformation sowing large penalty on te rigt tail. Transformed Densities m=1 m=5 m=10 original x Figure 2: Elliptical transformation using te Student-t density generator, varying te degrees of freedom. Tis transformation effectively penalizes bot extremes of te distribution, but more so wit rigt tail. 18

19 Transformed Densities lambda=1 lambda=3 lambda=5 original x Figure 3: Elliptical transformation still using te Student-t density generator, but varying te parameter λ. Here we ave fixed te degrees of freedom to 5. Transformed Densities x Figure 4: Elliptical transformation using te Student-t density generator. Tisistecasewereλ =0and degrees of freedom m =5. It sows tat equal penalties are imposed at extremes of te distribution. 19

20 Pareto Fit LOGLOSS Figure 5: Frequency istogram of te Logaritm of claims. Te smoot curve superimposed on te istogram is te fitted Pareto distribution. Risk-Adjusted Premiums lambda Figure 6: Te risk-adjusted premium as a function of te distribution sift parameter λ. Te broken orizontal line gives te unadjusted net premium, tat is, te expected value of te un-transformed distribution. A positive loading terefore results only wen λ is cosen so tat te smoot curve is above te orizontal line. 20

21 Risk-Adjusted Premiums probability (q) Figure 7: Te risk-adjusted premium as a function of te level of probability q. Te broken orizontal line gives te unadjusted net premium, tat is, te expected value of te un-transformed distribution. 21