1 Acknowledgement Research with Zinoviy Landsman, Depart ment of - PowerPoint PPT Presentation

1 Acknowledgement • Research with Zinoviy Landsman, Depart ment of Statistics, Actuarial Re- search Center, University of Haifa, Haifa, ISRAEL • The authors gratefully acknowledge fi nancial support of the UNSW Ac- tuarial Foundation of the Institute of Actuaries of Australia, A Research Council Discovery Grant DP0345036, Caesarea Rothschild Insti- tute and Zimmerman Foundation.

2 Aims of Research • Develop a Pricing Model for a Multi-Line Insurer including friction fi ed model for risks. and default option value for speci • Implement a recently developed dependent Gamma model for lines of busi- ness. • Develop approximations and closed form expressions for implementation.

3 Introduction - Insurance Pricing • Insurance pricing PV ( losses,assuming no default )+ PV ( surplus costs ) − PV ( default option ) • Myers and Read (2001), Phillips, Cummins and Allen (1998), Zanjani (2002), Sherris (2003). • PV ( losses, assuming no default ) not impacted by capital structure or fric- fl ects the insurance loss price of risk; PV ( surplus costs ) tional costs but re fl ects capital structure - how to determine and are the frictional costs - re PV ( default option ) re fl ects how to allocate to line of business/policy?; Q probabilities not P prob- capital structure, analogy to corporate bonds, abilities, how to allocate to line of business?

4 Model for Insurer • Insurer with portfolio of n distinct insurable risks written at the beginn of the period. • Premium P i for line of business i . P i allows for the expected losses, the risk loading, the risk of claims not being met due to the insolvency insurer and any costs of capital to be allocated to policyholders. P = P n i =1 P i . premiums collected at the start of the period are i th risk denoted by • End of period claims for the L i and total claims at the L = P n i =1 L i . end of the period

5 Model for Insurer • Arbitrage-free model where there exists a probability measure Q such that the current values of the assets and liabilities are the discounted value of end of period random payments using a risk free discount rate. • There exists a risk free asset such that an investment of 1 now will return e r at the end of the period for certain.

6 Model for Insurer - Insurer Losses/Claims • L 0 i denotes the time 0 price or fair value for line of business i ignoring default option given by L 0 i = E Q h i e − r L i i = 1 , . . . , n for all L 0 = P n i =1 L 0 i . and total value of the initial liabilities by • V 0 denotes the time 0 price or fair value for the assets given by V 0 = E Q h i e − r V r is the risk-free continuous compounding rate of interest. where i = 1 , . . . , n , from the • Complete markets so that we observe L 0 i for all V 0 from an asset market. insurance market and

7 Model for Insurer - Default Option Value • Assuming equal priority for losses by line of business in the event vency actual loss payments will be L i V L V L > V (or L ≤ 1) if V L i if L ≤ V (or L > 1) or " ¶ + # µ 1 − V 1 − L i L

8 Model for Insurer - Default Option Value • Premium for line of business i allowing for default option value " " ¶ + ## µ 1 − V = E Q e − r L i P i 1 − L " ¶ + # µ 1 − V = L 0 i − e − r E Q L i L = L 0 i − D 0 i i is denoted D 0 i . where the line of business default value for line

9 Model for Insurer - Capital Structure • Denote the insurer default option value by D 0 so that " ¶ + # µ n n X X 1 − V D 0 = e − r E Q L i = D 0 i L i =1 i =1 • Capital of the insurer based on a target solvency ratio of s will be equal to C = V 0 − P = (1 + s ) L 0 − P = sL 0 + D 0 • Beginning of period assets are V 0 = (1 + s ) L 0 = sL 0 + D 0 + L 0 − D 0 = C + P

10 Model for Insurer - Liability Assumptions • Dependent Gamma model: Mathai (1982 ), Moschopoulos (1985), Alouini, Abdi, and Kaveh (2001), Furman and Landsman (2004) • ( X 0 , ...., X n , X n +1 ) are n + 2 independent gamma distributed random γ i and common rate parameter α , denoted variables with shape parameters by G ( γ i , α ) , i = 0 , ..., n + 1 . • Probability density of the X i is α γ i Γ ( γ i ) e − αX i x γ i − 1 f X i ( x i ) = , x i > 0 i

11 Model for Insurer - Liability Assumptions • End of period claims for the n lines of business and the value of the assets V are modelled as α 0 L i = α X 0 + X i , i = 1 , ..., n α 0 V = α X 0 + X n +1 • Intuition: common factor X 0 impacts the values of each line of business fl ation) claims as well as the assets (in • Each line of business and the assets have a separate independent f X i , i = 1 , ..., n for the line of business i, and impacting them denoted by X n +1 for the assets.

12 Model for Insurer - Liability Assumptions • Claims for each line of business L 1 , ..., L n are gamma distributed random G ( λ i , α ) , i = 1 , ..., n , where λ i = γ 0 + γ i variables, • Assets V are gamma distributed as G ( γ 0 + γ n +1 , α ) , since the sum of two independent gamma random variables with the same rate parameter is also gamma with shape parameter equal to the sum of the shape parameters.

13 Model for Insurer - Liability Assumptions • Total claims liability at the end of the period is n X L i = nα 0 L = α X 0 + X · , i =1 where n X X · = X i i =1 γ · = P n • X · is distributed G ( γ · , α ) , with i =1 γ i . However, the total claims nα 0 liability is a sum of 2 gamma random variables, α X 0 and X · , each with ff erent rates and so the sum does not have a gamma distribution. di

14 Model for Insurer - Liability Assumptions • Furman and Landsman (2004): can represent L as a mixed gamma distri- bution with mixed shape parameter L v G ( γ 0 + γ · + ν, α ) , ν is a non negative integer random variable with probabilities where p k = Cδ k , k ≥ 0 , where k X 1 ( n − 1 n γ 0 : δ k = k − 1 γ 0 ) i δ k − i , C = k > 0 : δ 0 = 1 . n i =1

15 Model for Insurer - Liability Assumptions • For the model of claims and assets that we have assumed, the Q measure i is Gamma G ( λ i , α ) , probability density of claims for line of business i = 1 , ..., n so that α λ i Γ ( λ i ) e − αy i y λ i − 1 f L i ( y i ) = , y i > 0 , i with E Q [ L i ] = λ i α and V ar Q [ L i ] = λ i α 2

16 Model for Insurer - Liability Assumptions • Note that ³ ´ = γ 0 α 2 i 6 = j Cov L i , L j Cov ( L i , V ) = γ 0 α 2 , i = 1 , .., n and ³ ´ γ 0 ρ L i , L j = q , i, j = 1 , ..., n λ i λ j

17 Model for Insurer - Default Option Value • Default option value for each line " ¶ + # µ 1 − V = e − r E Q D 0 i L i L ⎡ ! + ⎤ ¶ Ã µ α 0 α 0 α X 0 + X n +1 = e − r E Q ⎣ ⎦ α X 0 + X i 1 − nα 0 α X 0 + X · ⎡ ! + ⎤ Ã α 0 α X 0 + X n +1 = e − r α 0 α E Q ⎣ X 0 ⎦ 1 − nα 0 α X 0 + X · ⎡ ! + ⎤ Ã α 0 α X 0 + X n +1 + e − r E Q ⎣ X i ⎦ , i = 1 , . . . , n 1 − nα 0 α X 0 + X · Y n +1 Q ∼ i =0 G ( γ i , α ) . where

18 Model for Insurer - Default Option Value • Expectation in the fi rst term (see paper for proofs) ⎡ ! + ⎤ ⎡ ! + ⎤ à à α 0 α 0 α X 0 + X n +1 α X 0 + X n +1 ⎦ = γ 0 E Q α E Q 0 ⎣ X 0 ⎣ ⎦ 1 − 1 − nα 0 nα 0 α X 0 + X · α X 0 + X · where à ! n +1 à ! α γ 0 +1 α γ i Y dQ 0 = Γ ( γ 0 + 1) e − αx 0 x γ 0 Γ ( γ i ) e − αx i x γ i − 1 α γ i dx 0 . . . dx n +1 0 i i =1

19 Model for Insurer - Default Option Value • Expectation in the second term ⎡ ! + ⎤ ⎡ ! + ⎤ Ã Ã α 0 α 0 α X 0 + X n +1 α X 0 + X n +1 ⎦ = γ i E Q α E Q i ⎣ X i ⎣ ⎦ 1 − 1 − nα 0 nα 0 α X 0 + X · α X 0 + X · where ⎛ ⎞ Ã ! n +1 ⎝ α γ j α γ i +1 Y γ j − 1 Γ ( γ i + 1) e − αx i x γ i dQ i i α γ i +1 ´ e − αx j x ⎠ dx 0 . . . dx n +1 ³ = j Γ γ j j =0 ,j 6 = i i = 1 , ..., n. Y n +1 Thus Q i ∼ G ( γ i + 1 , α ) j =0 ,,j 6 = i G ( γ i , α ) , i = 1 , ..., n.

20 Model for Insurer - Default Option Value • To evaluate the expression ⎡ ! + ⎤ Ã α 0 α X 0 + X n +1 E Q i ⎣ ⎦ , i = 1 , ..., n 1 − nα 0 α X 0 + X · • Note that under Q 0 V = α 0 α X 0 + X n +1 ∼ G ( γ 0 + 1 + γ n +1 , α ) Q i , i = 1 , ..., n and under V = α 0 α X 0 + X n +1 ∼ G ( γ 0 + γ n +1 , α )

21 Model for Insurer - Default Option Value nα 0 • Under Q 0 and Q i , i = 1 , ..., n, the distribution of α X 0 + X · will not be Gamma • Its distribution can be represented as a mixture of Gamma random v ables - Furman and Landsman (2004).

22 Model for Insurer - Default Option Value • Under Q 0 L = nα 0 α X 0 + X · ∼ G ( γ 0 + 1 + γ · + ˜ ν, α ) ν is a non-negative integer random variable de ˜ fi ned in where p k = ˜ C ˜ ˜ δ k , k ≥ 0 , k X 1 ( n − 1 δ k = k − 1 ( γ 0 + 1) ) i ˜ ˜ n γ 0+1 : ˜ k > 0 : ˜ C = δ k − i , δ 0 = 1 . n i =1

23 Model for Insurer - Default Option Value • Under Q i , i = 1 , ..., n, L = nα 0 α X 0 + X · ∼ G ( γ 0 + 1 + γ · + ν, α ) , ν is a non-negative integer random variable de fi ned previously. where