Aggregation and capital allocation for portfolios of dependent risks

Aggregation and capital allocation for portfolios of dependent risks... with bivariate compound distributions Etienne Marceau, Ph.D. A.S.A. (Joint work with Hélène Cossette and Mélina Mailhot) Luminy, France École d actuariat, Université Laval April 26-30, 2010 tienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 1 / 58

1. Introduction We consider a portfolio of n possibly dependent risks X 1... X n We study the stochastic behavior of S = X 1 +... + X n We examine the risk assessment of the portfolio with risk measures We compute the allocation of the capital to each risk X 1,..., X n Basic ingredients: Multivariate model for (X 1,..., X n ) Choose a risk measure to evaluate the total capital requirement of the portfolio Numerical strategies to evaluate the risk measures Capital allocation method to quantify the contribution of each risks Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 2 / 58

1. Outline Risk measures and some examples Mixed Erlang distributions Aggregate claim amount for a portfolio of n risks Capital allocation Bivariate compound distributions Bivariate counting distributions Numerical examples Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 3 / 58

2. Risk measures de nitions Risk assessment based on risk measures Value at risk: VaR κ (X ) = inf (x 2 R, F X (x) x) Tail Value at risk (or expected shortfall): TVaR κ (X ) = 1 VaR u (X ) du 1 κ κ ) Expression of TVaR κ (X ): h i E X 1 fx >VaRκ (X )g + VaR u (X ) (F X (VaR κ (X )) κ) Z 1 1 κ If X is continuous (with an eventual mass at zero): ) F X (VaR κ (X )) κ = 0 and h i TVaR κ (X ) = E X 1 fx >VaRκ (X )g 1 κ Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 4 / 58

2. Risk measures Gamma distribution Suppose that X Gamma (α, β) with p.d.f. and c.d.f. with x > 0. Expression of TVaR κ (X ): TVaR κ (X ) = E f X (x) = h (x; α, β) = βα Γ(α) x α F X (x) = H (x; α, β) = h i X 1 fx >VaRκ (X )g 1 κ Z x 0 1 e βx h (y; α, β) dy = E [X ] H (VaR κ (X ) ; α + 1, β) 1 κ Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 5 / 58

2. Risk measures Compound distribution Let the r.v. X have a compound distribution with X = M k=1 B k, M > 1 0, M = 0 r.v. M 2 N = number of claims r.v. B k represents the amount of the kth claim B 1, B 2,... form a sequence of i.i.d. positive r.v. s (B k B) and independent of M Expression of F X (x): F X (x) = Pr (M = 0) + k=1 Pr (M = k) Pr (B 1 +... + B k x) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 6 / 58

2. Risk measures Compound distribution Expression of E X 1 fx >bg :... = = k=0 Pr (M = k) E X 1 fx >bg jm = k Pr (M = k) E (B 1 +... + B k ) 1 fb1 +...+B k >bg k=1 Expression of TVaR κ (X ):... = k=1 Pr (M = k) E 1 κ + VaR κ (X ) (F X (VaR κ (X )) κ) 1 κ h i (B 1 +... + B k ) 1 fb1 +...+B k >VaR κ (X )g Assume that the claims B 1, B 2,... are continuous (with an eventual mass at 0) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 7 / 58

2. Risk measures Compound distribution The distribution of X has a probability mass at 0 and with a continuous part for x > 0. If κ < Pr (M = 0) then VaR κ (X ) = 0. Moreover, when κ > Pr (M = 0), it implies that VaR κ (X ) > 0 so that F X (VaR κ (X )) = κ. ) Expression for TVaR κ (X ): h i k=1 Pr (M = k) E (B 1 +... + B k ) 1 fb1 +...+B k >VaR κ (X )g 1 κ Only interesting when B 1 +... + B k belongs to a family of distributions closed under convolution (ex: Gamma distribution, mixed Erlang distribution) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 8 / 58

2. Risk measures Compound distribution with gamma claims Suppose that claim amount B Gamma (α, β). Expression of F X : Expression of E F X (x) = Pr (M = 0) + Expression of TVaR κ (X ): k=1 Pr (M = k) H (x; αk, β) h i (B 1 +... + B k ) 1 fb1 +...+B k >VaR κ (X )g :... = kα β H (VaR κ (X ) ; αk + 1, β) TVaR κ (X ) = 1 1 κ Pr (M = k) kα β H (VaR κ (X ) ; αk + 1, β) k=1 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 9 / 58

3. Mixed Erlang Distribution De nition Consider a r.v. Y de ned by Y = K k=1 C k, K > 1 0, K = 0 C k Exp (β) K is a discrete r.v. with m.p.f. f K (k) = Pr (K = k) = f K, k 2 N C 1 +... + C k has an Erlang distribution (i.e. gamma distribution with α = k) Expression of F Y : F Y (x) = f K (0) + k=1 f K (k) H (x; k, β) Distribution of Y : mixed Erlang distribution (MixErlang (f K ; β)) Often it is assumed that f K (0) = Pr (K = 0) = 0 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 10 / 58

3. Mixed Erlang Distribution De nition Large class of distributions Special cases: Exponential distribution, Gamma distribution Finite mixtures of exponentials Generalized Erlang distribution Etc. See Willmot & Lin (2010) Expression of TVaR κ (Y ): TVaR κ (Y ) = 1 1 κ f K (k) k β H (VaR κ (Y ) ; k + 1, β) k=1 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 11 / 58

3. Mixed Erlang Distribution De nition Important: The probability distribution of any positive r.v. can be approximated by a mixed Erlang distribution Theorem (Tijms, 1994). De ne for h > 0 Then F h (x) = for any continuity point x (F B (kh) F B ((k 1) h)) H x; k, β = 1 h k=1 lim F h (x)! F B (x) h!0 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 12 / 58

3. Mixed Erlang Distribution Compound distribution Let X be de ned by M X = j=1 B j, M > 1 0, M = 0 r.v. M 2 N = number of claims r.v. B k MixErlang (f K ; β) Then, X MixErlang (f M ; β) i.e. X = M j=1 C k, M > 1 0, M = 0 r.v. M is de ned by M = M j=1 K j, M > 1 0, M = 0 K 1, K 2,...: i.i.d. r.v s where K j K C 1, C 2,...: i.i.d. r.v s where C j C Exp (β) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 13 / 58

3. Mixed Erlang Distribution Compound distribution It implies that F X (x) = f M (0) + k=1 f M (k) H (x; k, β) TVaR κ (X ) = 1 1 κ f M (k) k β H (VaR κ (X ) ; k + 1, β) k=1 Computation of f M (k) with a recursive aggregation method (see e.g. Klugman et al (2008) and Sundt and Vernic (2009)) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 14 / 58

3. Mixed Erlang Distribution Sum of independent r.v. s Let S = X 1 +... + X n X 1,...X n are independent and X i MixErlang (f Ki, β) Then, S MixErlang (f K1 +...+K n, β) Computation of f K1 +...+f Kn (k) with a recursive aggregation method Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 15 / 58

3. Aggregate claim amount for a portfolio of n risks We consider a portfolio of n risks over a xed period Total claim amount (or loss) for risk i: X i (n = 1, 2,..., n) Aggregate claim amount for the portfolio: E [S] = n i=1 E [X i ] S = X 1 +... + X n Var (S) = n i=1 Var (X i ) + n i=1 n j=1,j6=i Cov (X i, X j ) Traditional assumption for (X 1,..., X n ): independence (until mid-nineties) Independence assumption may be inappropriate in several situations Since mid-nineties, several papers propose and study multivariate models in risk theory In this talk, we x n = 2 and consider bivariate compound distributions for (X 1, X 2 ) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 16 / 58

3. Aggregate claim amount for a portfolio of n risks Risk assessment of the portfolio is made with risk measures Example of application of risk measures ρ κ (S): Economic capital Expected aggregate claim amount for the next period: E [S] Total capital requirement: ρ κ (S) where ρ κ (S) = VaR κ (S) or TVaR κ (S) Economic capital: EC κ [S] = ρ κ (S) E [S] Other examples of applications: Choice of a retention level for a stop-loss reinsurance contract Portfolio selection (strategic asset allocation) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 17 / 58

4. Capital allocation based on TVaR We consider a portfolio with n risks X 1,..., X n Top-down approach: total requirement for the whole portfolio of n risks: TVaR κ (S) Objective: Allocate TVaR κ (S) in an appropriate way to each risk We use a rule based on the TVaR (see e.g. McNeil et al. (2005)) Contribution to risk X i based on TVaR h i h i TVaR κ (X i ; S) = E X i 1 fs >VaRκ (S )g + β S E X i 1 fs =VaRκ (S )g 1 κ with β S = ( Pr (S VaRκ (S )) κ Pr (S =VaR κ (S )), if Pr (S = VaR κ (S)) > 0, 0, otherwise Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 18 / 58

4. Capital allocation based on TVaR We can show that n i=1 TVaR κ (X i ; S) = TVaR κ (S) If X i are continuous (with an evential mass at 0), then h i TVaR κ (X i ; S) = E X i 1 fs >VaRκ (S )g 1 κ Note h i n E X i 1 fs >VaRκ (S )g n TVaR κ (X i ; S) = i=1 i=1 1 κ h i = E ( n i=1 X i ) 1 fs >VaRκ (S )g h 1 κ i = E S1 fs >VaRκ (S )g = TVaR κ (S) 1 κ Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 19 / 58

4. Capital allocation based on TVaR Basic ingredients: Multivariate model for (X 1,..., X n ) Choose a risk measure to evaluate the total capital requirement of the portfolio Numerical strategies to evaluate the risk measures Capital allocation method to quantify the contribution of each risks Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 20 / 58

4. Capital allocation based on TVaR Review of litterature Panjer (2002): multivariate normal distribution; exact expression of F S, TVaR κ (S) and TVaR κ (X i ; S) Landsman & Valdez (2003) and Dhaene et al. (2008): multivariate elliptical distributions; exact expression of F S, TVaR κ (S) and TVaR κ (X i ; S) Furman & Landsman (2005): multivariate gamma distributions; exact expression of F S, TVaR κ (S) and TVaR κ (X i ; S) Chiragiev & Landsman (2007): multivariate Pareto distributions; exact expression of F S, TVaR κ (S) and TVaR κ (X i ; S) Bargès et al. (2009): multivariate FGM copula with exponential marginals; exact expression of F S, TVaR κ (S) and TVaR κ (X i ; S) multivariate distributions based on copulas; numerical method to compute F S, TVaR κ (S) and TVaR κ (X i ; S) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 21 / 58

4. Capital allocation based on TVaR In this talk, we consider a portfolio of 2 risks Multivariate model for (X 1,..., X n ): ) bivariate compound distributions Choose a risk measure to evaluate the total capital requirement of the portfolio: ) TVaR Numerical strategies to evaluate the risk measures: ) approximation of the distributions of the claim amount r.v. s by mixed Erlang distributions Capital allocation method to quantify the contribution of each risks ) rule based on TVaR Illustrations: ) numerical examples Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 22 / 58

5. Bivariate compound distributions general expressions Let (X 1, X 2 ) be a couple of r.v. s following a bivariate compound distribution where ( M i X i = j 1 =0 B i,j i, M i > 0, (i = 1, 2) 0, M i = 0 The joint m.p.f. of (M 1, M 2 ) is given by for j 1, j 2 2 N f M1,M 2 (j 1, j 2 ) = Pr (M 1 = j 1, M 2 = j 2 ), For each i, fb i,1, B i,2,,...g form a sequence of i.i.d. r.v. s. fb 1,1, B 1,2,,...g and fb 2,1, B 2,2,,...g are independent between themselves and independent of (M 1, M 2 ) Let B 1 and B 2 be continuous positive r.v. s Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 23 / 58

5. Bivariate compound distributions Expression of F S (x): f M1,M 2 (0, 0) + + + m 1 =1 m 2 =1 m 1 =1 f M1,M 2 (m 1, 0) F B1,1 +...+B 1,m1 (x) f M1,M 2 (0, m 2 ) F B2,1 +...+B 2,m2 (x) m 2 =1 f M1,M 2 (m 1, m 2 ) F B1,1 +...+B 1,m1 +B 2,1 +...+B 2,m2 (x) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 24 / 58

5. Bivariate compound distributions Expression of TVaR κ (S): 1 h 1 κ f M1,M 2 (m 1, 0) E (B 1,1 +... + B 1,m1 ) 1 f(b1,1 +...+B 1,m1 )>VaR m 1 =1 + 1 h 1 κ f M1,M 2 (0, m 2 ) E (B 2,1 +... + B 2,m2 ) 1 f(b2,1 +...+B 2,m2 )>VaR m 2 =1 + 1 1 κ m 1 =1 m 2 =1 f M1,M 2 (m 1, m 2 ) E h (B 1,1 +... + B 1,m1 + B 2,1 +... + B 2,m Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 25 / 58

5. Bivariate compound distributions Expression of TVaR κ (X 1 ; S): 1 h 1 κ f M1,M 2 (m 1, 0) E (B 1,1 +... + B 1,m1 ) 1 f(b1,1 +...+B 1,m1 )>VaR m 1 =1 + 1 1 κ m 1 =1 m 2 =1 f M1,M 2 (m 1, m 2 ) E h (B 1,1 +... + B 1,m1 ) 1 f(b1,1 +...+B 1,m Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 26 / 58

5. Bivariate compound distributions gamma claims Suppose that the claim amount B i Gamma (α i, β) for i = 1, 2. Expression of F S : f M1,M 2 (0, 0) + + + m 2 =1 m 1 =1 m 1 =1 f M1,M 2 (m 1, 0) H (x; m 1 α 1, β) f M1,M 2 (0, m 2 ) H (x; m 2 α 2, β) m 2 =1 f M1,M 2 (m 1, m 2 ) H (x; m 1 α 1 + m 2 α 2, β). Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 27 / 58

5. Bivariate compound distributions gamma claims Expression of TVaR κ (S): 1 1 κ + 1 1 κ m 1 =0 m 1 =0 m 2 =0 m 2 =0 Expression of TVaR κ (X 1 ; S): 1 1 κ m 1 =0 m 2 =0 f M1,M 2 (m 1, m 2 ) m 1α 1 β H (VaR κ (S) ; α 1 m 1 + α 2 m 2 + 1, β f M1,M 2 (m 1, m 2 ) m 2α 2 β H (VaR κ (S) ; α 1 m 1 + α 2 m 2 + 1, β f M1,M 2 (m 1, m 2 ) m 1α 1 β H (VaR κ (S) ; α 1 m 1 + α 2 m 2 + 1, β Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 28 / 58

5. Bivariate compound distributions mixed Erlang claims Suppose that the claim amount B i MixErlang (f Ki ; β) for i = 1, 2 We can express (X 1, X 2 ) as ( X i = M i j i =0 C i,j i, Mi > 0 0, Mi, = 0 where the r.v. s M1 and M 2 are de ned by ( M i = where K i,ji K i for i = 1, 2 Joint m..p.f. of (M 1, M 2 ) for j 1, j 2 2 N. C i,ji Exp (β) for all i and j i M i j i =0 K i,j i, M i > 0 0, M i = 0 f M 1,M 2 (j 1, j 2 ) = Pr (M 1 = j 1, M 2 = j 2 ), Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 29 / 58,

5. Bivariate compound distributions mixed Erlang claims Expression of f M 1,M 2 (j 1, j 2 ): j 1 = 1, 2,..., j 2 = 1, 2,... : m 1 =1 m 2 =1 j 1 = j 2 = 0 : f M 1,M 2 f M1,M 2 (m 1, m 2 ) f K1,1 +...+K 1,m1 (j 1 ) f K2,1 +...+K 2,m2 (j 2 ) (0, 0) = f M1,M 2 (0, 0) + + + m 1 =1 m 2 =1 m 1 =1 f M1,M 2 (m 1, 0) f K1,1 +...+K 1,m1 (0) f M1,M 2 (0, m 2 ) f K2,1 +...+K 2,m2 (0) m 2 =1 f M1,M 2 (m 1, m 2 ) f K1,1 +...+K 1,m1 (0) f K2,1 +...+K 2,m2 (0) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 30 / 58

5. Bivariate compound distributions mixed Erlang claims j 1 = 1, 2,..., j 2 = 0 : = f M 1,M 2 (j 1, 0) m 1 =1 + m 1 =1 f M1,M 2 (m 1, 0) f K1,1 +...+K 1,m1 (j 1 ) m 2 =1 j 1 = 0, j 2 = 1, 2,... : = f M 1,M 2 (0, j 2) m 2 =1 + m 1 =1 f M1,M 2 (m 1, m 2 ) f K1,1 +...+K 1,m1 (j 1 ) f K2,1 +...+K 2,m2 (0) f M1,M 2 (0, m 2 ) f K2,1 +...+K 2,m2 (j 2 ) m 2 =1 f M1,M 2 (m 1, m 2 ) f K1,1 +...+K 1,m1 (j 1 ) f K2,1 +...+K 2,m2 (j 2 ) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 31 / 58

5. Bivariate compound distributions mixed Erlang claims Expression of F S (x): f M 1,M2 + m 1 =1 + m 2 =1 + m 1 =1 m 2 =1 (0, 0) f M 1,M 2 (m 1, 0) H (x; m 1, β) f M 1,M 2 (0, m 2) H (x; m 2, β) f M 1,M 2 (m 1, m 2 ) H (x; m 1 + m 2, β) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 32 / 58

5. Bivariate compound distributions mixed Erlang claims Expression of TVaR κ (S): 1 1 κ + 1 1 κ m 1 =1 m 1 =1 m 2 =1 f M 1,M2 (m 1, m 2 ) m 1 + m 2 H (b; m 1 + m 2 + 1, β) β f M 1,M 2 (m 1, 0) m 1 β H (b; m 1 + 1, β) + 1 1 κ f M 1,M (0, m 2 2) m 2 m 2 =1 β H (b; m 2 + 1, β). Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 33 / 58

5. Bivariate compound distributions mixed Erlang claims Expression of TVaR κ (X 1 ; S): 1 1 κ m 1 =1 m 2 =1 f M 1,M 2 (m 1, m 2 ) m 1 β H (b; m 1 + m 2 + 1, β) + 1 1 κ f M 1,M (m 2 1, 0) m 1 m 1 =1 β H (b; m 1 + 1, β) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 34 / 58

6. Bivariate Poisson distribution Bivariate m.p.f.: f M1,M 2 (k 1, k 2 ) = e λ 1 λ 2 +α 0 min(k 1,k 2 ) j=0 for k 1, k 2 2 N and with M i Poisson (λ i ), i = 1, 2 (see e.g. Johnson et al (1997)) α j 0 (λ 1 α 0 ) k 1 j (λ 2 α 0 ) k 2 j, j! (k 1 j)! (k 2 j)! Values of the joint m.p.f. of (M 1, M 2 ), f M 1,M 2 (j 1, j 2 ) = Pr (M 1 = j 1, M 2 = j 2), for j 1, j 2 2 N can be are computed with the algorithm of Theorem 3.2 of Hesselager (1996) (see eq s 3.4, 3.5 and 3.6). Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 35 / 58

6. Bivariate counting distribution based on copulas We construct a bivariate with three ingredients: a copula C (u 1, u 2 ) and two marginal distributions for counting r.v. Expression of F M1,M 2 : bivariate distribution function F M1,M 2 of (M 1, M 2 ) with marginals F M1 and F M2 for (m 1, m 2 ) 2 N + N +. F M1,M 2 (m 1, m 2 ) = C (F M1 (m 1 ), F M2 (m 2 )) Expression for f M1,M 2 (m 1, m 2 ) = Pr (M 1 = m 1, M 2 = m 2 ): f M1,M 2 (m 1, m 2 ) = F M1,M 2 (m 1, m 2 ) F M1,M 2 (m 1 1, m 2 ) F M1,M 2 (m 1, m 2 1) + F M1,M 2 (m 1 1, m 2 1), for (m 1, m 2 ) 2 N + N + and where F M1,M 2 (m 1, m 2 ) = 0 if m 1 = 0 or m 2 = 0 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 36 / 58

6. Bivariate counting distribution based on copulas See Genest and Nešlehovà (2007) for an excellent review about copulas linking discrete distributions: Dependence modelling with copulas is a valid (and even attractive) approach for constructing bivariate distributions Many stochastic dependence properties of a copula are inherited by the bivariate model Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 37 / 58

7. Basic ingredients We consider a portfolio of 2 risks Multivariate model for (X 1,..., X n ): ) bivariate compound distributions Choose a risk measure to evaluate the total capital requirement of the portfolio: ) TVaR Numerical strategies to evaluate the risk measures: ) approximation of the distributions of the claim amount r.v. s by mixed Erlang distributions Capital allocation method to quantify the contribution of each risks ) rule based on TVaR Illustrations: ) numerical examples Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 38 / 58

8. Example Bivariate compound distribution based on Frank copula Frank copula C (u 1, u 2 ) = 1 α ln 1 + (e αu 1 1) (e αu 2 1) (e α 1) Counting r.v. s: M 1 Pois (λ = 4); M 2 NBinom r = 4, q = 1 2 Claim amount r.v. s risk 1: B 1 Gamma (α = 0.5, β = 0.1) Claim amount r.v. s risk 2: B 2 Gamma (α = 0.25, β = 0.1) E [X 1 ] = 20 E [X 2 ] = 10 E [S] = 30 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 39 / 58

8. Example Bivariate compound distribution based on Frank copula Dependence parameter of Frank copula = -20 κ VaR κ (S) TVaR κ (S) TVaR κ (X 1 ; S) TVaR κ (X 2 ; S) 0.25 16.64801 36.35319 24.45543 11.89777 0.995 96.18877 109.0645 81.42774 27.63675 κ VaR κ (X 1 ) VaR κ (X 2 ) VaR κ (X 1 ) + VaR κ (X 2 ) 0.25 6.921663 1.090818 8.01248 0.995 86.4245 63.3218 149.7463 κ TVaR κ (X 1 ) TVaR κ (X 2 ) TVaR κ (X 1 ) + TVaR κ (X 2 ) 0.25 25.66058 13.24903 38.90961 0.995 99.68334 75.3916 175.0749 tienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 40 / 58

8. Example Bivariate compound distribution based on Frank copula Dependence parameter of Frank copula = 20 κ VaR κ (S) TVaR κ (S) TVaR κ (X 1 ; S) TVaR κ (X 2 ; S) 0.25 11.67007 38.20261 25.34365 12.85895 0.995 117.4703 133.5035 82.57583 50.92763 κ VaR κ (X 1 ) VaR κ (X 2 ) VaR κ (X 1 ) + VaR κ (X 2 ) 0.25 6.921663 1.090818 8.01248 0.995 86.4245 63.3218 149.7463 κ TVaR κ (X 1 ) TVaR κ (X 2 ) TVaR κ (X 1 ) + TVaR κ (X 2 ) 0.25 25.66058 13.24903 38.90961 0.995 99.68334 75.3916 175.0749 tienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 41 / 58

8. Example Bivariate compound distribution based on Frank copula Dependence parameter of Frank copula = -20 κ VaR κ (S) TVaR κ (S) TVaR κ (X 1 ; S) TVaR κ (X 2 ; S) 0.25 16.64801 36.35319 24.45543 11.89777 0.995 96.18877 109.0645 81.42774 27.63675 Dependence parameter of Frank copula = 20 κ VaR κ (S) TVaR κ (S) TVaR κ (X 1 ; S) TVaR κ (X 2 ; S) 0.25 11.67007 38.20261 25.34365 12.85895 0.995 117.4703 133.5035 82.57583 50.92763 tienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 42 / 58

8. Example 2 Bivariate Poisson compound distributions Bivariate Poisson distribution: λ 1 = 10, λ 2 = 12, α 0 = 0, 5, 10 Claim amount risk 1: B 1 Pareto (α = 3, λ = 20) ) E [B 1 ] = 10 Claim amount risk 2: B 2 LNorm (µ = 2, σ = 1) ) E [B 2 ] = 12.18249 Basic facts: E [S] = E [X 1 ] + E [X 2 ] = 246.1899 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 43 / 58

8. Example 2 Bivariate Poisson compound distributions Approximation of F B1 F B1 (x) = 1 λ λ + x α by F e B 1 F e B 1 (x) = = k=1 k=1 (F B1 (kh) F B1 ((k 1) h)) H x; k, β = 1 h λ α λ α H x; k, β = 1 λ + (k 1) h λ + kh h with h = 1 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 44 / 58

8. Example 2 Bivariate Poisson compound distributions Fit of F B1 (Pareto cdf) by F e B 1 (mixed Erlang cdf) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 45 / 58

8. Example 2 Bivariate Poisson compound distributions Fit of F B1 (Pareto cdf) by F e B 1 (mixed Erlang cdf) κ VaR k (B 1 ) (Pareto) VaR k B e1 (Mixed Erlang) 0.99 72.83178 74.4270 0.999 180.00000 181.8154 0.9999 410.88694 412.8020 0.99999 908.31777 910.2785 0.999999 1980.00000 1981.9818 0.9999999 4288.86938 4290.8610 0.99999999 9263.17765 9265.1737 tienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 46 / 58

8. Example 2 Bivariate Poisson compound distributions Comments Approximation by mixture of Erlangs is based on the result from Tijms (1994) Applied in actuarial science by e.g. Willmot & Woo (2007), Lee & Lin (2010), Willmot and Lin (2010) Applied in applied probability by e.g. Wang et al. (2005) and Panchenko & Thümmler Interesting for applications such as nite sums of r.v. s or compound sums Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 47 / 58

8. Example 2 Bivariate Poisson compound distributions Approximation of F B2 F B1 (x) = 1 λ λ + x α by F e B 2 F e B 2 (x) = ln (kh) µ F Z σ k=1 ln ((k 1) h) µ F Z H x; k σ with Z Norm (0, 1) and h = 1 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 48 / 58

8. Example 2 Bivariate Poisson compound distributions Fit of F B2 (lognormal cdf) by F e B 2 (mixed Erlang cdf) Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 49 / 58

8. Example 2 Bivariate Poisson compound distributions Fit of F B2 (lognormal cdf) by F e B 2 (mixed Erlang cdf) κ VaR k (B 2 ) (lognormal) VaR k B e2 (Mixed Erlang) 0.99 75.66743 77.35703 0.999 162.42759 164.49209 0.9999 304.60519 306.97872 0.99999 525.78416 528.42684 0.999999 856.98834 859.87261 0.9999999 1338.54380 1341.64912 0.99999999 2022.32121 2025.63147 Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 50 / 58

8. Example 2 Bivariate Poisson compound distributions α 0 VaR 0.99 (S) TVaR 0.99 (S) TVaR 0.99 (X 1 ; S) TVaR 0.99 (X 1 ; S) 0 545.5184 644.1487 5 562.8459 661.7200 10 578.8346 678.1905 tienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 51 / 58

8. Conclusion Consider portfolios with n > 2 risks Other multivariate counting distributions Etc. Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 52 / 58

9. References Bargès, M., Cossette, H. & E. Marceau (2009). «TVaR-based capital allocation with copulas». Insurance : Mathematics and Economics. In press. Acerbi, C., Tasche, D., 2002. On the coherence of expected shortfall. Journal of Banking & Finance 26 (7), 1487 1503. Chiragiev, A., Landsman, Z., 2007. Multivariate pareto portfolios: Tce-based capital allocation and divided di erences. Scandinavian Actuarial Journal (4), 261 280. Klugman, S.A., Panjer, H.H. & G.E. Willmot (2008). Loss Models: From Data to Decisions (3rd edn). Wiley: New York. Feldmann, A., Whitt, W., 1998. Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Performance Evaluation 31 (3 4), 245 279. Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 53 / 58

9. References Furman, E., Landsman, Z., 2005. Risk capital decomposition for a multivariate dependent gamma portfolio. Insurance: Mathematics & Economics 37 (3),635 649. Furman, E., Landsman, Z., 2007. Economic capital allocations for non-negative portfolios of dependent risks. ASTIN BULLETIN Klugman, S.A., Panjer, H.H., Willmot, G.E., 2008. Loss models: From data to decisions, third ed. In: Wiley Series in Probability and Statistics, John Wiley & Sons Inc.,Hoboken, NJ. Landsman, Z.M., Valdez, E.A., 2003. Tail conditional expectations for elliptical distributions. North American Actuarial Journal 7 (4), 55 71. Lee, S.C.K. & X.S. Lin (2010). Modeling and evaluating insurance losses via mixtures of Erlang distributions. North American Actuarial Journal. In press. Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 54 / 58

9. References McNeil, A., Frey, R., and P. Embrechts (2005). Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton Press, Princeton. Panchenko, A. & A. Thümmler (2007). Performance Evaluation 64, 629 645. Panjer, H.H., 2002. Measurement of risk, solvency requirements and allocation of capital within nancial conglomerates. Research Report 01 15, Institute of Insurance and Pension Research, University of Waterloo. Schmock, U., 2006. Modelling dependent credit risks with extensions of creditrisk+ and application to operational risk. In: Lecture Notes. PRisMa Lab, Institute for Mathematical Methods in Economics, Vienna University of Technology. Schmock, U., Straumann, D., 1999. Allocation of risk capital and performance measurement. In: Talk at the Conference on Quantitative Methods in Finance, Sydney, Australia. Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 55 / 58

9. References Tasche, D., 1999. Risk contributions and performance measurement. Working Paper, Technische Universitt Mnchen. Tijms H.C. (1994). Stochastic Models: An Algorithmic Approach. Wiley: Chichester. Wang, JF, Zhou, H.X., Li, L. & F. Xu (2005). Accurate Long-tailed Network Tra c Approximation and Its Queueing Analysis by Hyper-Erlang Distributions, Proceedings of the IEEE Conference on Local Computer Networks 30th Anniversary. Wang, JF, H.X. Zhou, M.T. Zhou and L. Li (2006). A General model for long-tailed network tra c approximation, Journal of Supercomputing, 38(2), 155-172. Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 56 / 58

9. References Willmot, G.E. & X.S. Lin (2010). Risk modelling with the mixed Erlang distribution. Applied Stochastic Models in Business and Industry. In press. Willmot, G.E. and J.K. Woo (2007). On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal 11(2), 99 115. Étienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 57 / 58

10. Appendix Fit of F B1 (Pareto cdf) by F e B 1 (mixed Erlang cdf) κ VaR k (B 1 ) (Pareto) VaR k e B 1 (Mixed Erlang) 0.01 0.0671146 0.07391354 0.05 0.3448954 0.37938160 0.10 0.7148834 0.78514899 0.15 1.1133438 1.22080262 0.20 1.5443469 1.69055100 0.30 2.5249576 2.75398514 0.40 3.7126220 4.03305025 0.50 5.1984210 5.62130709 0.60 7.1441762 7.68446839 0.70 9.8760316 10.55540571 0.80 14.1995189 15.05324817 0.90 23.0886938 24.18916040 tienne Marceau (École d actuariat, Université Laval) Aggregation and capital allocation April 26-30, 2010 58 / 58