Interplay of Asymptotically Dependent Insurance Risks and Financial Risks

Transcription

1 Interplay of Asymptotically Dependent Insurance Risks and Financial Risks Zhongyi Yuan The Pennsylvania State University July 16, 2014 The 49th Actuarial Research Conference UC Santa Barbara Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 1/22

2 Outline 1 Introduction Model Risks Assumptions 2 Results 3 Concluding Remarks Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 2/22

3 Introduction Model 1 Question we consider Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 3/22

4 Introduction Model 1 Question we consider 2 Model: A discrete time model for an insurer Insurer s initial wealth: W 0 = x; wealth at time m: W m Insurer s net insurance loss within the m-th period: X m Insurer s investment return in the m-th period: R m Insurer s wealth process: 3 Comments m m W m = W m 1 R m X m = x R j j=1 i=1 X i m j=i+1 R j (1) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 3/22

5 Introduction Risks Insurance risks Losses from insurance claims One-claim-causes-ruin phenomenon (Embrechts et al. (1997), Muermann (2008), etc.) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 4/22

6 1992 Hurricane Andrew Insured loss: $16 billion More than 60 insurance companies became insolvent (Muermann (2008, NAAJ))

7 2004 Indian Ocean Earthquake and Tsunami Damaged about $15 billion Not much insurance loss due to lack of insurance coverage

8 2005 Hurricane Katrina Insured loss: $41.1 billion Damaged $108 billion

9 2011 Japan Earthquake, Tsunami and Nuclear Crisis Insured loss: $ billion

10 2012 Hurricane Sandy Insured loss: $19 billion Damage: over $68 billion

11 Introduction Risks Insurance risks Losses from insurance claims One-claim-causes-ruin phenomenon (Embrechts et al. (1997), Muermann (2008), etc.) Hedging Heavy-tailedness: assumption of regular variation Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 4/22

12 Introduction Risks Insurance risks Losses from insurance claims One-claim-causes-ruin phenomenon (Embrechts et al. (1997), Muermann (2008), etc.) Hedging Heavy-tailedness: assumption of regular variation A distribution function F is said to have a regularly varying tail with index α > 0, written as F R α, if F (xt) lim x F (x) = t α, t > 0. Examples: Pareto, t-distribution, Burr distribution Losses due to earthquakes: 0.6 < α < 1.5 Losses due to hurricanes: 1.5 < α < 2.5. (See, e.g. Ibragimov et al. (2009)) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 4/22

13 Introduction Risks Financial risks Daily log-returns for the period of 01/03/1989 to 06/30/2003. (Angelidis and Degiannakis (2005)) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 5/22

14 Introduction Risks Financial risks Daily log-returns for the period of 01/03/1989 to 06/30/2003. (Angelidis and Degiannakis (2005)) Sylized facts on returns from stocks/stock indices (Basrak et al. (2002), Rachev et al. (2005), Kelly and Jiang (2014), etc.): Regularly varying with tail index 2 < α < 4 Asymmetric Q: What are the effects of these risks on the survival of the insurer? Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 5/22

15 Introduction Risks Related studies 1 Related studies: Nyrhinen (1999) Tang and Tsitsiashvili (2003). Chen (2011) Fougères and Mercadier (2012) 2 Comments. Why asymptotic dependence? Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 6/22

16 Introduction Risks (Asymptotic/Extreme) dependence Copula of (X, Y ): C(, ) Survival copula of (X, Y ): Ĉ(, ) Asymptotic dependence Ĉ(u, u) Pr (X > FX lim = lim (1 u), Y > F Y u 0 u u 0 u (1 u)) > 0 Examples Assume asymptotic dependence for (X, 1/R) = (X, Y ) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 7/22

17 Introduction Assumptions Assumptions 1 (X 1, R 1 ), (X 2, R 2 ),... are i.i.d. copies of (X, R). 2 Suppose that F X R α, α > 0, with F X ( x) = o ( F X (x) ). Also suppose that the distribution of R has a regularly varying tail at 0 with index β > 0. 3 Suppose that there exists some function H(, ) on [0, ] 2 \{0}, such that H(t 1, t 2 ) > 0 for every (t 1, t 2 ) (0, ] 2 and Ĉ (ut 1, ut 2 ) lim = H(t 1, t 2 ) on [0, ] 2 \{0}. (2) u 0 u Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 8/22

18 Introduction Assumptions Example Let (X, Y ) have an Archimedean copula C (u, v) = ϕ 1 (ϕ (u) + ϕ (v)). (3) ϕ(u) Assume that the generator ϕ satisfies ϕ(1 tu) lim u 0 ϕ(1 u) = tθ, t > 0, (4) 0 1 u for some constant θ > 1 (Note that θ 1 if (4) holds). (See Charpentier and Segers (2009)) Example: ϕ(u) = ( ln u) θ Then (2) holds with H(t 1, t 2 ) = t 1 + t 2 ( t θ 1 + t θ 2) 1/θ > 0, (t1, t 2 ) (0, ] 2. (5) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 9/22

19 Introduction Assumptions Implications 1 Let Y = 1/R and Y i = 1/R i, i = 1, 2,.... Assumption 1 = Y is regularly varying with index β > 0. 2 X and Y are asymptotically dependent. 3 The random vector ( 1/F X (X ), 1/F Y (Y ) ) MRV 1 (multivariate regularly varying), and the vague convergence ( ( ) ) 1 1 x Pr x F X (X ), 1 v ν( ) F Y (Y ) on [0, ] 2 \{0} (6) holds with the Radon measure ν defined by ν [0, (t 1, t 2 )] c = ( 1 H, 1 ), t 1 t 2 t 1 t 2 (t 1, t 2 ) (0, ) 2. (7) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 10/22

20 Introduction Assumptions Implications cont d The random vector (X, Y ) follows a nonstandard MRV structure; i.e., for some Radon measure µ, the following vague convergence holds: (( ) ) X x Pr b X (x), Y v µ( ) on [0, ] 2 \{0}, (8) b Y (x)) ( ) ( ) where b X (x) = 1 F (x) and by (x) = 1 X F (x). Y See Resnick (2007). Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 11/22

21 Results 1 Introduction Model Risks Assumptions 2 Results 3 Concluding Remarks Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 12/22

22 Results Probabilities of ruin Finite-time horizon: ( ) ψ(x; n) = Pr min W m < 0 1 m n W 0 = x m m = Pr min x R j = Pr Infinite-time horizon: 1 m n max 1 m n ψ(x) = Pr m j=1 X i i=1 j=1 max 1 m< m X i R j i=1 j=i+1 i Y j > x m X i i=1 j=1 i Y j > x < 0 Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 13/22

23 Results Main result 1 Theorem 2.1 Under Assumptions 1 3 we have n i ψ(x; n) Pr Y j > x X i i=1 j=1 ( n 1 i=0 ) ( E [Y αβ/(α+β)]) i v(a) b (x), where the set A = {(t 1, t 2 ) [0, ] 2 : t 1/α 1 t 1/β 2 > 1}, the function b( ) = ( ) ( ) 1/F X 1/FY ( ), and the measure v is defined by relation (7). Note: ψ( ; n) R αβ/(α+β) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 14/22

24 Results Main result 1 cont d Theorem 2.2 In addition to Assumptions 1 3, assume that E [ Y αβ/(α+β)] < 1. Then ψ(x) 1 1 E [ v(a) Y αβ/(α+β)] b (x), On the estimation of ν. See, e.g., Resnick (2007), Nguyen and Samorodnitsky (2013). Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 15/22

25 Results Asymptotic dependence vs asymptotic independence Roughly, under (asymptotic) independence, the probability of ruin decays faster. Examples: Under corresponding conditions, ψ(x; n) C n F X (x) (Chen (2011)) ψ(x; n) C n F X (x) + D n F Y (x) (Li and Tang (2014)) Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 16/22

26 Results Main result 2 (& role of regulation) Prudent Person Investment Principle (PPIP) under Solvency II: no requirement on what insurers can invest and what they can not, but they are encouraged to reduce their investment in equities (take less investment risks) due to high capital charges, which would reduce the overall profit. If the insurer only invests a proportion π < 1 of its wealth into risky assets, and the rest earns a risk free return r f 1, then R > (1 π)r f, which would violate Assumption 2. Assumption 2 : Suppose that F X R α, α > 0, with F X ( x) = o ( F X (x) ). Also suppose that R is bounded below by some positive number r. Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 17/22

27 Results Main result 2 Theorem 2.3 Under the Assumptions 1, 2, and 3, we have ( n i n 1 ) ψ(x; n) Pr Y j > x (E [Y α ]) i r α F X (x). (9) Theorem 2.4 X i i=1 j=1 In addition to the assumptions of Theorem 2.3, assume that E [Y α ] < 1. Then 1 ψ(x) 1 E [Y α ] r α F X (x). (10) i=0 Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 18/22

28 Concluding Remarks 1 Introduction Model Risks Assumptions 2 Results 3 Concluding Remarks Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 19/22

29 Concluding Remarks Concluding Remarks Asymptotically dependent insurance risks and financial risks both play a significant role in affecting the insurer s survival. By regulating insurers investment behavior, like discouraging too risky investments, regulators can help significantly reduce their probability of ruin. Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 20/22

30 Concluding Remarks References Angelidis, T.; Degiannakis, S. Modeling risk for long and short trading positions. Journal of Risk Finance 6 (2005), no. 3, Basrak, B.; Davis, R. A.; Mikosch, T. Regular variation of GARCH processes. Stochastic Processes and Their Applications 99 (2002), no. 1, Charpentier, A.; Segers, J. Tails of multivariate Archimedean copulas. Journal of Multivariate Analysis 100 (2009), no. 7, Chen, Y. The finite-time ruin probability with dependent insurance and financial risks. Journal of Applied Probability 48 (2011), no. 4, Embrechts, P.; Klüppelberg, C.; Mikosch, T. Modelling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin, Fougères, A.; Mercadier, C. Risk measures and multivariate extensions of Breiman s theorem. Journal of Applied Probability 49 (2012), no. 2, Ibragimov, R., Jaffee, D., Walden, J.. Non-diversication traps in markets for catastrophic risk. Review of Financial Studies 22 (2009), Kelly, B.; Jiang, H. Tail Risk and Asset Prices. Review of Financial Studies 2014, to appear. Li, J.; Tang, Q. Interplay of insurance and financial risks in a discrete-time model with strongly regular variation. Bernoulli (2014), to appear. Muermann, A. Market price of insurance risk implied by catastrophe derivatives. North American Actuarial Journal 12 (2008), no. 3, Nguyen, T.; Samorodnitsky, G. Multivariate tail estimation with application to analysis of covar. Astin Bulletin 43 (2013), no. 2, Rachev, S.T., Menn, C., Fabozzi, F.J. Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing. Wiley, Hoboken, NJ, Resnick, S. I. Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. Springer, New York, Tang, Q.; Tsitsiashvili, G. Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Processes and Their Applications 108 (2003), no. 2, Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 21/22

31 Concluding Remarks Thank you! Zhongyi Yuan (Penn State University) Asymptotically Dependent Insurance/Financial Risks 22/22