Moment-based approximation with finite mixed Erlang distributions

Transcription

1 Moment-based approximation with finite mixed Erlang distributions Hélène Cossette with D. Landriault, E. Marceau, K. Moutanabbir 49th Actuarial Research Conference (ARC) 2014 University of California Santa Barbara Université Laval July 15th 2014 Hélène Cossette (Université Laval) July 15th / 48

2 Acknowledgements For this research, the authors would like to gratefully acknowledge the financial support received from The Actuarial Foundation ( TAF ), The Casualty Actuarial Society ( CAS ) and The Society of Actuaries ( SOA ). This project was funded by Individual Grants Competition under the name of "Mixed Erlang Moment-Based Approximation: Applications in Actuarial Science and Risk Management". Hélène Cossette (Université Laval) July 15th / 48

3 0. Outline Introduction Definitions and Notations Mixed Erlang distribution Moment-based approximation methods Approximation method based on finite mixed Erlang distributions: 2 contexts Numerical examples Hélène Cossette (Université Laval) July 15th / 48

4 1. Introduction We consider a positive continuous rv S for which we know the first m moments. Applications in actuarial science and quantitative risk management: S = aggregate claim amount for a portfolio of insurance risks S = aggregate claim amount for a line of business S = aggregate losses for a portfolio of investment risks (e.g. credit risks) Main objective: evaluate cdf of S i.e. F S Impossible or very diffi cult to find F S analytically Possible to use aggregation methods: Methods based on recursive numerical methods Methods based on MC simulation May be very time-consuming to find numerically F S A moment-based approximation can be used Hélène Cossette (Université Laval) July 15th / 48

5 2. Definitions and notations S : rv with cdf F S j th raw moments : µ j (S) = E [ S j ], j N + Risk measure VaR VaR κ (S) = FS 1 (κ), for κ (0, 1), where FS 1 (u) = inf {x R : F S (x) u} Risk measure TVaR TVaR κ (S) = 1 1 κ 1 κ VaR u (S) du for κ (0, 1) TVaR κ (S) = E [S 1 {S >VaRκ (S )}]+VaR κ (S )(F S (VaR κ (S )) κ) 1 κ If the rv S is continuous, TVaR κ (S) = E [S 1 {S >VaRκ (S )}] 1 κ Stop-loss premium : π S (b) = E [max (S b; 0)] = E [ S 1 {S >b} ] bfs (b) Hélène Cossette (Université Laval) July 15th / 48

6 3. Mixed Erlang distribution Attractive features of this class of distributions: studied by e.g. Willmot & Woo (2007), Lee & Lin (2010), and Willmot & Lin (2010) Illustrate the versatility of this distribution to model claim amounts Illustrate the faisability to obtain closed-form expressions for various quantities of interest in risk theory. Provide several non trivial examples of distributions which belong to the class of mixed Erlang distributions Provide a detailed procedure to express e.g. mixtures of exponentials, generalized Erlang distributions in terms of mixed Erlang distributions Closed under various operations such as convolutions, Esscher transformations (risk aggregation and ruin problems). Risk measures VaR, TVaR and stop-loss premium are easily obtained. Hélène Cossette (Université Laval) July 15th / 48

7 3. Mixed Erlang distribution Tijms (1994): class of mixed Erlang distributions is dense in the set of all continuous and positive distributions any nonnegative continuous distribution may be approximated by an Erlang mixture to any given accuracy Theorem. Let F be the cdf of a positive rv. For any given h > 0, define the cdf F h by ( F h (x) = (F (jh) F ((j 1) h)) H x; j, 1 ), x 0, h j=1 where H (x; n, β) is the Erlang cdf. Then, for any continuity point x of F, lim h 0 F h (x) F (x). Moment-based approximation based on class of mixed Erlang dist. Hélène Cossette (Université Laval) July 15th / 48

8 3. Mixed Erlang distribution W : mixed Erlang rv with common rate parameter β Cdf of W : F W (y) = l ζ k H(x; k, β), l finite or infinite k=1 k 1 H(x; k, β) = 1 e βx (βx ) i i! : cdf of Erlang rv of order k i=0 ζ k : non-negative mass probability associated to the kth Erlang distribution in the mixture k=1 ζ k = 1 Hélène Cossette (Université Laval) July 15th / 48

9 3. Mixed Erlang distribution Representation of the mixed Erlang distribution as a compound distribution with discrete primary distribution {ζ k } l k=1 and secondary exponential distribution with rate parameter β : W = M C k, k=1 C k Exp (β) (k = 1, 2,...) M = discrete r.v. with pmf f M (k) = Pr (M = k) = ζ k, k N +. j th raw moment: µ j (W ) = k=1 ζ k CV (W ) > 1 l j 1 i=0 β j (k+i) Hélène Cossette (Université Laval) July 15th / 48

10 3. Mixed Erlang distribution No general closed-form expression for Value-at-risk but easily obtained with simple numerical optimization method VaR κ (W ) = F 1 W (u) = inf {x R : F W (x) u}, κ (0, 1) Explicit expression for Tail Value-at-risk: TVaR κ (W ) = = 1 Explicit expression for Stop-loss premium: 1 VaR u (W ) du, κ (0, 1) 1 κ κ 1 k 1 κ ζ k β H (VaR κ (W ) ; k + 1, β). k=1 π W (b) = E [max (W b; 0)] ( ) 1 k = 1 κ ζ k β H (b; k + 1, β) bh (b; k, β). k=1 Hélène Cossette (Université Laval) July 15th / 48

11 4. Moment-based approximation methods One approach : approximate the unknown distribution by a mixture of known distributions. Several approximation methods motivated by Tijms theorem were proposed over the years Examples of such methods: Whitt (1982) : 3 moments and 2 moments CV (S) > 1 : ( mixtures of ) 2 exponential ( distributions ) F W (x) = p 1 1 e β 1 x + p 2 1 e β 2 x CV (S) < 1 : generalized Erlang distribution ( F W (x) = H (x; β 1,..., β r ) = r i=1 r l=1,l =i β l β l β i ) ( 1 e βi x ) Hélène Cossette (Université Laval) July 15th / 48

12 4. Moment-based approximation methods Continued... Altiok (1985) : 3 moments and 2 moments CV (S) > 1 : mixture of generalized Erlang distribution and exponential distribution F W (x) = ph (x; β 1, β 2 ) + (1 p) (1 ) e β 1 x CV (S) < 1 : generalized Erlang distribution ( F W (x) = H (x; β 1,..., β r ) = r i=1 Johnson & Taaffe (1989) : r l=1,l =i β l β l β i ) ( 1 e βi x ) 3 moments: mixture of two Erlang distributions of common order and different scale factors generalize the approximation of Whitt (1982) and Altiok (1985) (for CV (S) > 1) more than 3 moments... Hélène Cossette (Université Laval) July 15th / 48

13 4. Moment-based approximation methods Substantial body of literature on 3-moment based approximations within phase-type class of distributions Matching first 3 moments: effective to provide a reasonable approximation (see Osogami and Harchol-Balter (2006)) but does not always suffi ce. Development of more flexible moment-based approximation methods: Johnson and Taaffe (1989) Horvath and Telek (2007) Our proposed method. Hélène Cossette (Université Laval) July 15th / 48

14 5. Approx. method based on finite mixed Erlang distribution S : rv with m known moments µ 1 (S),..., µ m (S) Idea : map (approximate) F S to a subclass of distributions which belongs to the class of mixed Erlang distributions Subclass = class of finite mixed Erlang distributions with F W (y) = l ζ k H(x; k, β); l < k =1 µ j (W ) = E [ W j ] = l k =1 ζ k j 1 (k +i ) i=0 β j (j = 1, 2,..., m) Hélène Cossette (Université Laval) July 15th / 48

15 5. Approx. method based on finite mixed Erlang distribution Consider a set of first m moments (µ 1,..., µ m ) = (µ 1 (W ),µ 2 (W ),...,µ m (W )) and A l = {1, 2,..., l} ME(µ 1,..., µ m, A l ) : set of all finite mixtures of Erlang with cdf F (y) = l ζ k H(x; k, β) and first m moments (µ 1,µ 2,...,µ m ). k=1 Identification of all solutions to the problem: µ j (S) = l ζ k k=1 j 1 i=0 (k + i) β j, j = 1,..., m. Constraints: β, {ζ k } l k=1 are non-negative and l ζ k = 1. k=1 Hélène Cossette (Université Laval) July 15th / 48

16 5. Approx. method based on finite mixed Erlang distribution ME res (µ 1,..., µ m, A l ) : (restricted) subset of ME(µ 1,..., µ m, A l ) such that at most m of the mixing weights {ζ k } l k =1 are non-zero. Propose to use it as our class of approximations. ME res (µ 1,..., µ m, A l1 ) ME res (µ 1,..., µ m, A l2 ) for l 1 l 2. Members are identified by rewriting moment expressions in matrix form Hélène Cossette (Université Laval) July 15th / 48

17 5. Approx. method based on finite mixed Erlang distribution Obtain all sets of m atoms {i k } m k=1 (i 1 < i 2 <... < i m < l) in A l = {1, 2,..., l} For a given set of atoms {i k } m k=1, µ j (S) = can be written as: A T m,mζ m = Mβ l ζ k k=1 j 1 i=0 (k+i) β j j = 1,..., m ζ T m = (ζ i 1, ζ i2,..., ζ im ), M =diag(µ 1, µ 2,..., µ m ), β T = (β, β 2,..., β m ) i 1 i 2 i m1 i 1 (i 1 + 1) i 2 (i 2 + 1) i m1 (i m1 + 1) A m1,m 2 = m 2 1 m 2 1 m 2 1 (i 1 + i) (i 2 + i) (i m1 + i) i=0 i=0 i=0 m 1 : number of Erlang terms and m 2 : number of moments Hélène Cossette (Université Laval) July 15th / 48

18 5. Approx. method based on finite mixed Erlang distribution ζ m = [ A 1 m,mm ] β under the constraint that e T ζ m = 1, with e a column vector of 1s. e T [ A 1 m,mm ] β : polynomial of degree m in β. Look for positive solutions in β to e T [ A 1 m,mm ] β = 1. Complete mixed Erlang representations via identification of mixing weights through ζ m = [ A 1 m,mm ] β. Repeat procedure for all possible sets of atoms. Hélène Cossette (Université Laval) July 15th / 48

19 5. Approx. method based on finite mixed Erlang distribution Criteria of quality among all legitimate candidates in ME res (µ 1,..., µ m, A l ): Kolmogorov-Smirnov (KS) distance KS distance for two rv s S and W (with respective cdf F S and F W ): d KS (S, W ) = sup F S (x) F W (x). x 0 Denote by F Wm,l this approximation: d KS (S, W m,l ) = inf sup F S (x) F W (x), F W ME res (µ 1,...,µ m,a l ) where W m,l is a rv with cdf F Wm,l. x 0 Hélène Cossette (Université Laval) July 15th / 48

20 6. Numerical examples Example #1: Weibull rv S F S (x) = 1 exp { (x/β) τ} for x, τ, β > 0. Parameters: τ = 1.5 and β = Γ (5/3) CV = Consider class of mixed Erlang distributions ME res (µ 1,..., µ m, A 20 ) with A 20 = {1, 2,..., 20} Hélène Cossette (Université Laval) July 15th / 48

21 6. Numerical examples Cardinalities of ME res (µ 1,..., µ m, A 20 ) : m Cardinality ME res (µ 1,..., µ m, A 20 ) Resulting mixed Erlang approximations for m = 3, 4: F W3,20 (x) = H(x; 1, ) H(x; 2, ) H(x; 4, ), F W4,20 (x) = H(x; 1, ) H(x; 2, ) H(x; 4, ) H(x; 7, ), Hélène Cossette (Université Laval) July 15th / 48

22 6. Numerical examples Kolmogorov-Smirnov distances: m d KS (S, W m,20 ) Quality of the approximation improves from 3 to 5 moments. Hélène Cossette (Université Laval) July 15th / 48

23 6. Numerical examples Comparison of pdfs of W 3,20, W 4,20, W 5,20 and S. The 3-moment approximation of Johnson and Taffee (1989) is also provided. Hélène Cossette (Université Laval) July 15th / 48

24 6. Numerical examples Examine the tail fit: VaR and TVaR for the exact and approximated distributions κ VaR κ (W 3,20 ) VaR κ (W 4,20 ) VaR κ (W 5,20 ) VaR κ (S) κ TVaR κ (W 3,20 ) TVaR κ (W 4,20 ) TVaR κ (W 5,20 ) TVaR κ (S) Hélène Cossette (Université Laval) July 15th / 48

25 6. Numerical examples Example #2: Lognormal rv S S = exp (ν + σz ) where Z is a standard normal rv Consider Example 5.4 of Dufresne (2007) where ν = 0 and σ 2 = CV = Lognormal has a heavier tail than mixed Erlang: no guarantee that our mixed Erlang approximation would perform well, especially for tail risk measures. Hélène Cossette (Université Laval) July 15th / 48

26 6. Numerical examples Consider class of mixed Erlang distributions ME res (µ 1,..., µ m, A 50 ). Kolmogorov-Smirnov distances: m d KS (S, W m,50 ) KS distance increases from the 4-moment to the 5-moment approximation. Remark: F W5,50 uses Erlang-50 cdf, where 50 is the upper boundary point of A 50 : believe that a mixed Erlang approximation with a KS distance lower than could be found by increasing the value l in set A l. Hélène Cossette (Université Laval) July 15th / 48

27 6. Numerical examples Histograms of the KS distance for all the mixed Erlang distributions in ME res (µ 1,..., µ m, A 50 ), m = 3, 4, 5 KS distances (x-axis) vs counts (y-axis) Hélène Cossette (Université Laval) July 15th / 48

28 6. Numerical examples Figure: 4-moment approximations Hélène Cossette (Université Laval) July 15th / 48

29 6. Numerical examples Figure: 5-moment approximations Hélène Cossette (Université Laval) July 15th / 48

30 6. Numerical examples Overall quality of the approximations (judged by values and dispersion of KS distances) increases with number of moments matched. Comparison of the pdfs of W 3,50, W 4,50, W 5,50, and S. The 3-moment approximation of Johnson and Taaffe (1989) is also plotted. Hélène Cossette (Université Laval) July 15th / 48

31 6. Numerical examples All three mixed Erlang approximations provide an overall good fit to the exact distribution. To further examine the tail fit, specific values of VaR and TVaR for the exact and approximated distributions are provided below: κ VaR κ (W 3,50 ) VaR κ (W 4,50 ) VaR κ (W 5,50 ) VaR κ (S) κ TVaR κ (W 3,50 ) TVaR κ (W 4,50 ) TVaR κ (W 5,50 ) TVaR κ (S) Hélène Cossette (Université Laval) July 15th / 48

32 6. Numerical examples VaR and TVaR values of our mixed Erlang approximations compare reasonably well to their lognormal counterparts. Improvement is not monotone with the number of moments matched: well known that increasing the number of moments does not necessarily lead to a higher quality approximation in moment-matching techniques. Hélène Cossette (Université Laval) July 15th / 48

33 6. Numerical examples Example #3: Real data Normalized damage amounts from 30 most damaging hurricanes in United States from 1925 to 1995 (provided by Pielke and Landsea (1998) and analyzed by Brazauskas et al. (2009)). Purpose of this example: not to carry an exhaustive statistical analysis of this dataset, but provide a simple fit with a finite mixed Erlang distribution First 4 empirical moments: CV = j µ j Hélène Cossette (Université Laval) July 15th / 48

34 6. Numerical examples Perform the approximation with ME res (µ 1,..., µ m, A 30 ) for m = 3 and 4. Kolmogorov-Smirnov distances (with empirical distribution) : m d KS (S, W m,30 ) Critical value of the KS hypothesis test at a significance level of 1%: 1.63/ 30 = Do not reject both distributions as a plausible model for the dataset. Hélène Cossette (Université Laval) July 15th / 48

35 6. Numerical examples Hélène Cossette (Université Laval) July 15th / 48

36 7. Moment-based approx. with known rate parameter Slightly different context. Distribution function F S is known to be of mixed Erlang form with known β > 0 and (µ 1, µ 2,..., µ m ). Distribution itself unknown or diffi cult to evaluate. Restrict to sets of finite mixture of Erlang distributions. Bounds on risk measures can be established. Connection with extremal points of a discrete moment-matching problem. Hélène Cossette (Université Laval) July 15th / 48

37 7. Moment-based approx. with known rate parameter F S ME(µ 1,..., µ m, β): set of all mixed Erlang dist. for l =, rate parameter β and first m moments (µ 1,..., µ m ). ME(µ 1,..., µ m, A l, β): subset of ME(µ 1,..., µ m, β) for a given l N +. ME ext (µ 1,..., µ m, A l, β): subset of ME(µ 1,..., µ m, A l, β) such that at most (m + 1) of mixing weights {ζ k } l k=1 are non-zero. Consider two approaches to derive bounds on E [φ(s)] for φ a given function (such that expectation exists): Based on discrete s-convex extremal distributions Based on moment bounds on discrete expected stop-loss transforms Hélène Cossette (Université Laval) July 15th / 48

38 8. Discrete s-convex extremal distributions D(α 1,..., α m, A l ) : all discrete dist. with support A l with first m moments α = (α 1,..., α m ). D ext (α 1,..., α m, A l ) : all discrete dist. with support A l with at most (m + 1) non-zero mass points with first m moments are α. For a given β > 0 : one-to-one correspondence between discrete classes and mixed-erlang classes Each dist. in D(α 1,..., α m, A l ) (and D ext (α 1,..., α m, A l )) corresponds to a mixed Erlang dist. in ME(µ 1,..., µ m, A l, β) (and ME ext (µ 1,..., µ m, A l, β)) (see De Vylder 1996) Allows to use theory on sets of discrete distributions e.g. in Prékopa (1990), Denuit, Lefèvre and Mesfioui (1999), Courtois et. al (2006). Hélène Cossette (Université Laval) July 15th / 48

39 8. Discrete s-convex extremal distributions Definition s-convex: Let C be a subinterval of R or a subset of N and φ a function on C. For two rv s X and Y defined on C, X is said to be smaller than Y in the s-convex sense, namely X C s cx Y, if E [φ(x )] E [φ(y )] for all s-convex functions φ. Examples of s-convex functions: φ(x) = x s+j and φ(x) = exp(cx) for c 0. K s,min and K s,max : s-extremum rv s on D(α 1,..., α m, A l ) E [φ(k s,min )] E [φ(k )] E [φ(k s,max )] for any s-convex function φ and any K D(α 1,..., α m, A l ). Hélène Cossette (Université Laval) July 15th / 48

40 8. Discrete s-convex extremal distributions General distribution forms of K s,min and K s,max are given in Prékopa (1990) and Courtois et al. (2006) K W K = C j be a mixed Erlang rv. j=1 Denuit, Lefèvre and Utev (1999) state that the s-convex order is stable under compounding. Lemma: If K A l s cx K, then W K R+ s cx W K. Can apply this Lemma to W Ks min and W Ks max : W Ks min R+ s cx W K R+ s cx W Ks max Allows to find general distribution forms of F WKs min and F WKs max For s-convex functions φ(x) = x s+j and φ(x) = exp(cx),can obtain bounds: [ ] [ ] E W s+j K s min E [W K ] E W s+j K s max E [ exp(cw Ks min ) ] E [exp(cw K )] E [exp(cw Ks max )] Hélène Cossette (Université Laval) July 15th / 48

41 9. Moment bounds on discrete expected stop-loss transforms Extrema with respect to s-convex order allows to derive bounds on E [φ(s)] for all s-convex functions φ. Approach not appropriate to derive bounds for TVaR and stop-loss premium when m 2. Use an approach (based on increasing convex order) inspired from Courtois and Denuit (2009) and Hürlimann (2002). Main idea: consider D(α 1,..., α m, A l ) for m {2, 3,...} find lower and upper bounds for E [(K k) + ] on D(α 1,..., α m, A l ) for all k A l from lower (upper) bound, derive corresponding rv K m low (K m up ) E [(K m low k) + ] E [(K k) + ] E [(K m up k) + ] on D(α 1,..., α m, A l ) for all k A l implies under the increasing convex order: K m low icx K icx K m up Hélène Cossette (Université Laval) July 15th / 48

42 9. Moment bounds on discrete expected stop-loss transforms Increasing convex order is stable under compounding: W Km low icx W K icx W Km up From Denuit et al. (2005): TVaR(W Km low ) TVaR(W K ) TVaR(W Km up ) Hélène Cossette (Université Laval) July 15th / 48

43 10. Example - Portfolio of dependent risks Portfolio of n dependent risks (common mixture model of Cossette and al. (2002)) S = X X n : aggregate claim amount with X i = B i I i. Conditional on a common mixture rv Θ with pmf p Θ, {I i } n i=1 are assumed to form a sequence of independent Bernoulli rv s with Pr (I i = 1 Θ = θ ) = 1 r i θ for r i (0, 1). B i (i = 1,..., n) are assumed to form a sequence of iid rv s, independent of {I i } 20 i=1 and Θ. B i (i = 1,..., n) : exponentially distributed with mean 1 Distribution of S : two-point mixture of a degenerate rv at 0 and a mixed Erlang with l = n and β = 1. Hélène Cossette (Université Laval) July 15th / 48

44 10. Example - Portfolio of dependent risks Parameters: n = 20 risks Θ has a logarithmic distribution with pmf p Θ (j) = (0.5) j /(j ln 2) for j = 1, 2,... constants r i [ are set such that the (unconditional) mean of I i is q i = 1 E (r i ) Θ] with q 1 =... = q 10 = 0.1 and q 11 =... = q 20 = It Perform moment-based approximation on rv Y = (S S > 0) rather than S j-th moment of Y : µ j E [ Y j ] = E[S j ] 1 F S (0) CV (Y ) = Methods of Whitt (1982) and Altiok (1985) not applicable here: constraints on CV and third moment (µ 3 µ 1 1.5µ 2 2 ) not satisfied. Method of Johnson and Taaffe (1989): r = 2, β 1 = , β 2 = and p = Hélène Cossette (Université Laval) July 15th / 48

45 10. Example - Portfolio of dependent risks First approach: discrete s-convex extremal distributions Find cdfs F WKs min and F WKs max for m = 4, 5 (s = m + 1) Consider two distributional characteristics of S : higher-order moments E [ S j ] for j = 4, 5, 6 exponential premium principle ϕ η (S) = 1 η ln E [e ηs ] for η > 0. Distributions F WKm+1 min and F WKm+1 max provide bounds to these risk measures associated to the rv S Hélène Cossette (Université Laval) July 15th / 48

46 10. Example - Portfolio of dependent risks Bounds on E [ S j ] and ϕ η (S) = 1 η ln E [ e ηs ] : j ] E [W j K5 min ] E [W j K6 min E [ S j ] ] E [W j K6 max ] E [W j K5 max θ ( ) ϕ η WK5 min ( ) ϕ η WK6 min ϕ η (S) ϕ η (W K6 max ) ϕ η (W K5 max ) Bounds get sharper as the number of moments involved increases. Hélène Cossette (Université Laval) July 15th / 48

47 10. Example - Portfolio of dependent risks Second approach: moment bounds with discrete expected stop-loss transforms Values of TVaR for W Km low and W Km up (m = 4, 5): Exact J&T TVaR κ (...) for m = 3 κ TVaR κ (S) TVaR κ (W ) W 3 low W 3 up Hélène Cossette (Université Laval) July 15th / 48

48 10. Example - Portfolio of dependent risks Exact TVaR κ (...) for m = 4 TVaR κ (...) for m = 5 κ TVaR κ (S) W K4 low W K4 up W K5 low W K5 up Inequality verified: TVaR(W Km low ) TVaR(W K ) TVaR(W Km up ) Interval estimate of TVaR κ (S) shrinks as number of moments matched increases. Hélène Cossette (Université Laval) July 15th / 48