金融と保険の融合について

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1 ARTalternative risk transfer ART EVTextreme value theory ARTEVT

2 Bancassurance alternative risk transferart ART Paul - Choudhury Allfinanz

3 extreme value theoryevtevt EVT EVT ART EVT Alternative Risk TransferART ART ART ART ART ARTalternative risk transfer alternativerisk transfe ARTsigma

4 risk retention captive ART ART ART Schanz Schanz ART Alternative Solutions ART CogloneRe

5 Alternative Risk Absorbers ART Alternative Sales Channels ART holistic covers integrated risk managementbalance sheet protection Alternative Sales Channels

6 Swiss Re New Markets ; ; ;; ;; ;; ;; ;; ;; ;; ;; ;;

7 β = λ =

8 D S 1 D ={S 1 12,000 }

9 ILS insurance linked securitiesils ABS MBSILS ARTILS ILS catastrophe risk CAT

10 ILS special purpose vehicle/company ILS ILS ILS

11 PCS PCS ILS ILS Canter, Cole and Sandor

12 time-dependent probability Toyota Motor Credit ILSSchmock 1998Schmock1998Winterthur

13 Swiss Re New Markets abc Chicago Board of TradeISOInsurance Services Office ISO PCSProperty Claims Service 1996 Bermuda Commodities Exchange Guy Carpenter Catastrophe IndexGCCI Guy CarpenterIndexCo

14 PCS PCS Chicago Mercantile Exchange, CME degree days ILS Catastrophe Risk ExchangeCATEX CATEX CATEX OTC Considine CME OTC HDDheating degree-dayscddcooling degree-days CDDdegree days Considine ILSOTC ILSILS

15 Christensen1999 PCS Considine ILS ILS ILS sigma ILS ILS sigma ILS

16 dynamic financial analysis ART ART risk theory collective risk theory

17 Rolski, Schmidli, Schmidt and Teugels Embrechts, Frey and Furrer BühlmannGerber

18 LundbergCramér N ={N (t)} t intensityλ > 0 homogeneous Poisson process N (0) = 0 a.s. N 0 s < t N (t) N (s)s {N (u), u s} N 0 s < t N (t) N (s) s t s λt s Lévy N (t) t k A k A k = inf { t 0 : N (t) k} k 1 k T k T 1 = A 1, T k = A k A k 1, k = 2,3,... N (t) T k / λ k X k F X F (0 ) = 0 t S (t) N(t ) (t )= X k k=1 N = {N(t)} t, t 0 almost surely

19 S (t) compound Poisson process G t (x) P (S (t ) x) = P (N(t )= n)f n* (x), x, t 0 n=0 P (A) A I(t ) t I(t )= ct c t = u 0 U (t ) U (t ) = u + ct S (t ), t 0 T ψ (u, T ) = P (U(t )< 0, 0 t T ) T = Ψ(u) τ (T ) = inf { t : 0 t T, U(t )< 0} τ (T ) = c λµ> 0 u lim Ψ(u) = 0 λµ c net profit F n* (x) F nf n x n F 0* (x) x F 0* (x) x

20 c = (1+ρ) λµ ρsafety loading ρ = c/ λµ 1 Ψ(u) ρ 1 Ψ(u) = (1+ρ) n F n* I (u), u 0 1+ρ n=0 F I F (x) =1 F(x ) 1 x F I (x ) = F (y)dy µ 0 u = 0 Ψ(0) =1/(1+ρ) = λµ /c Ψ(u) Ψ ( u) = 1 ρ exp 1 + ρ µ ( 1+ ρ ) u Ψ(u) v > 0 c λ = exp ( νy ) F ( y ) dy = 0 m X ( ν ) 1 ν m X (v) F v > 0 Lundberg exponent

21 v > 0 u 0 Ψ (u) exp( vu) xexp (vx)f (x)dx< 0 lim exp ( νu ) Ψ ( u ) = u c λµ λm ( ν ) c X F I 2 1 F I (x) lim = 2 x 1 F I (x) 1 Ψ (u) F I (u), u p F I ε > 0 lim exp(εx) x F I (x) = F F I Ψ (u) Ψ(u,T ) diffusion approximation Grandell1991Appendix A. 4

22 λ>0 A(t )A(0) = 0 N(t )N(t ) N(s) A(t ) A(s) N(t ) inhomogeneous Poisson processa(t ) intensity measure mean value functiona(t ) t A(t) = α (s)ds 0 α(s) α(s) s α(s) s s µα(s) t I(t ) t 0 I(t ) = (1+ ρ)µα (s)ds = (1+ ρ)µa (t) N(t ) U(t ) =u + I (t ) S (t )= u + (1+ ρ)µa (t) X k k=1 A 1 (t ) = sup (s A (s) t ) N (t ) U (t ) U (A -1 (t )) = u + (1+ ρ)µt X k k=1 N (t ) hazard function

23 A 1 operational time scale λ λ mixed Poisson process P (L>0)=1 FN (t )L N N οl (N (Lt)) t 0 s < t N (t ) N (s) k { P ( N () t N () s = k ) = L ( t s)} exp { L ( t s)} df() L k! 0 L f (x) 1 f (x) α ν x ν 1 e αx Γ (ν) Γ (ν) = x ν 1 e x dx 0 N Pólya proces N (t ) doubly stochastic Poisson process λ λ Credit Suisse Financial ProductsCreditRisk + Credit Suisse Financial Products

24 Λ(t) N (t )Λ(t) N = N ο Λ {N(Λ(t))}t Cox process Λ (t) λ (t) t Λ(t) = λs (s)ds 0 λ (t)intensity process s < t N (t ) N (s) ( () ( ) ) ( ) k t 1 t P N t N s = k = E λ u du exp λ ( u ) du k! s s λ (t) = LL renewal proces N (T K ) K 1 T 2 T 3 GNT 1 ordinary renewal process G /λ T 1 G 0 G 0 (x) = λ G (s) ds stationary renewal process x 0 Λ(t) Λ(0) = t > 0Λ(t)< random measure hazard process Rolski, Schmidli, Schmidt and Teugels

25 Delbaen and Haezendonck premium calculation principles S c H c = H(S ) H c = E [S ]net-premium principle expectation principle c = E [S ]+δe [S ] variance principle standard deviation principle c = E [S ]+δvar[s ] c = E [S ]+δ Var[S ] semi-variance principle c = E [S ]+δe [{(S E [S ]) + } 2 ] u (x) u (x) = E [u (x+c S )] u (x) =(1 exp ( δx))/δ Embrechts, Frey and Furrer

26 exponential principle log E [exp(δ S)] c = δ VaR quantile principle c = F (1 δ) S F δ F (y) = inf{x R :F (x) y} δ Esscher principle E [X exp(δ X)] c = E [exp(δ X)] Bühlmann n ΩFPω Ω i X i (ω) ω i Y i (ω) ϕ Price[Y i ] Y i (ω)ϕ (ω)dp (ω)= E [Y i ϕ] ω ω ΩΣ n i=1 Y i (ω)= 0 Y risk exchange i u i i ω Ω u Xi ( ω ) + Yi ( ω ) ω Ω Y i ( ω ) ϕ( ω ) dp( ω ) dp ( ω ) ϕ, Yu i n Z = X i i=1 iy i (ω)

27 e δ Z(ω) ϕ (ω) = E [e δ Z ] X E [Xe δ Z ] E [e δ Z ] X Z X [ Xe δ Z ] δ X [ e δ ( Z E E Xe ] E = X ) δ X E [ Xe ] E [ e = δ ( Z X )] [ Xe = δ X E[ δ Z e ] E [ e δ Xeδ ( Z X ) ] E [ e δ X ] E [ e δ ( Z X )] E [ e X ] δ ] Esscher transform P Q e hx dq(x) = dp(x) E [e hx ] Q S Bachelier S t

28 ds t = µ S t dt + σ S t dw t W t no-arbitrage viable predictable self-financingφ t V t (φ ) 0 t TV 0 (φ ) = 0 V T (φ ) 0P (V T (φ ) >0)>0 complete replicable P P P previsiblelamberton and Lapeyre1996 Harrison and Pliskaadmissible Harrison and Pliska1981attainable PQP(A)> Q(A)> A F

29 P PP P TOPIX TOPIX

30 jump-diffusion model stochastic volatility model superreplicationsuperhedging quantile hedging Föllmer and LeukertVaR Sekine Föllmer and Leukert efficient hedge Schweizer

31 Delbaen and HaezendonckSondermann Delbaen and Haezendonck T t X t t X T X t p t t S t = p t +X t S t Q Meister Gerber and Shiu Bühlmann, Delbaen, Embrechts and Shiryaev Credit Suisse Financial Products CreditRisk + SBC UBSACRAActuarial Credit Risk Accounting

32 extreme value theoryevt EVT EVT EVT EVT a F nx 1, X 2,, X n M n = max (X 1, X 2,, X n ) x P (M n x ) = P (X 1 x, X 2 x,, X n x ) {F(x )} n right endpoint x F sup{x R :F (x) < 1} Embrechts, Resnick and SamorodnitskyEVT Embrechts, Klüppelberg and Mikosch Research Conference on Risk Measurement and Systemic RiskLuncheon Address FRB Work that characterizes the statistical distribution of extreme events would be useful iidindependent and identically distributed

33 x F x F = 0 x < x P(M F n x ) n 1 x x F M n M n x F a.s., n (M n d n ) / c n EVT n S n ( = X 1 + X 2 + +X n )S n n S n /n n X n

34 Marcinkiewicz Zygmund p (0, 2) a S n an 0, a.s. n n 1/p E [ X p ] < a 0 a = µ p (0, 1 ) p [ 1, 2 ) E [ X] p an n 1/p p=2 ( S n nµ )/(σ n ) stable distribution F Xn S n S n = d n 1/α X+ γ n αγ n FG α characteristic exponentx c n n 1/α α α αα domain of attraction S n a n d G α, n b n d = Feller

35 a n Rb n >0FG α F DA(G α )F DA(G α ) α F DA(G ) F DA(G )EX 2 σ S n µ n d Φ, n nσ ΦEX 2 a n R Lx S n a n d G α, n n 1/α L(n) EVT Embrechts, Klüppelberg and Mikosch EVTResnick d fxslowly varyinglim f(tx) / f x (x)= t a n Embrechts, Klüppelberg and Mikosch

36 M n Hc n >0 d n R M n d n d H, n c n H extreme value distributions 0, x 0 Fréchet Φ α (x)= α > 0 exp ( x α ), x> 0 exp( ( x ) α ), x 0 Weibull Ψ α (x)= α > 0 1, x> 0 Gumbel Λ (x ) = exp{ exp( x )}, x R c n normalizing constantd n centering constant norming constant generalized extreme value distributiongev H ξ; µσ µσh ξ 1 ξ H ( x ) x µ ξ; µ, σ = exp 1 + ξ, x R σ + µσξ ξ = 0 ξ0 H ; / ( x ) x µ = exp, x R µ, σ exp 0 σ ξ > 0 α =1/ξ, µ=1, σ =1/α Φ α ξ = 0 µ =0, σ=1 Λ (50) ξ < 0 α = 1/ξ, µ=1, σ = 1/α Ψ α FGa >0b R x G ( x )=F { ( x b )/a} location parameterscale parameter

37 x GEV H maximum domain of attractionf MDAH F MDAψ α x F Φ α Λ Ψα

38 EVT regularly varying f(x) t f (tx) lim = t α f (x) x αf R α RV α α = t α t t α F MDAΨ α F MDAΨ α x F < F (x F 1 /x) R α d n = x F c n = x F F (1 1 /n) F (t) quantile function F (t) inf{x R :F (x) t}, 0<t<1 F [] x F =1 < t lim x F ( xf 1 / ( tx )) 1/ ( tx ) 1 = = t F ( xf 1/ ( x )) 1 / x F MDAΨ 1 d n d n x F c n F (1 1 /n) n c n x F x F F MDAΦ α t =

39 F MDAΦ α F (x) R α d n = 0 c n = F (1 1 /n) F (x) = 1 (a /x) b ab lim x F ( tx ) F ( x ) { a/ ( tx )} = lim = t b x ( a/ x ) b F MDA b b c n F (1 1 /n) c n d n = 0 c n n F MDAΛ F MDAΛz x F z x x F F( x ) = c ( x ) exp c (x) g(x) lim c (x) = c 1 >0 lim g(x) =1 x x F x x F a (x) lim a (x) = 0 x x F d n = F (1 1 /n)c n = a(d n ) z x g () t dt a() t cgaa x F ( ) F () t a x F = dt, ( x ) x x < x F F

40 d n c n a (x) a( x )= x 1 2π 1 2π x t exp ( k 2 / 2 ) dk 2 exp ( k / 2 ) dk ϕ ( x ) dt = x + Φ ( x ) ϕ Φ c n d n c n M n c n Λd n n= n c n d n c n d n n

41 F(x)=P(M n x) F(x)=P(M n x) M 5 EVD M 10 EVD x x F(x)=P(M n x) F(x)=P(M n x) M 50 EVD M 100 EVD x x u x F F X u x F F (x +u ) F (u ) F u (x) P (X u x X>u) =, u x x F 1 F (u ) u X excess distribution function e (u) = E (X u X>u) mean excess function

42 F u excess-liferesidual lifetime excess-of-loss 1 e ( u) = ( x u ) df ( x ) = F ( u) u x F x 1 F F ( x ) dx F ( u) X (F (x) = exp ( λx))e (u) =1/λ u X F u u x F F u Pickands - Balkema - de Haan generalized Pareto distributiongpd GPDGPDG ξ ; β (x) u 1 (1 + ξ x /β) 1 /ξ G ξ ; β (x) = 1 exp ( x /β) ξ 0 ξ = 0 β> 0 ξ 0 x ξ <0 x 1/ξ GPDGEV ξ= 0 ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ x x

43 β= 1 ξ GPD ξ 0 ξ< 0 GPD Pickands - Balkema - de Haan FF MDAH ξ β (u) lim sup F u (x) G ξ, β (u) ( x ) =0 x x F 0< x<x F u u F u GPD ξ β (x) ξ= 0 ξ= 0 F MDAax ξ 0ξ > 0 F (x) F( x ) = c ( x ) exp z x 1 dt a() t, z < x < z ac lim c (x ) = c (0, )lim a (x ) /x = ξ 1 x x a (x ) β (x ) ξ < 0 F (x F 1/x) a (x ) β (x ) u GPD GPD F ξ= 0 β (x ) Embrechts, Klüppelberg and MikoschA

44 u=gpd u PD u x u + x F (u + x) Fu F (u + x) = F (u ) F u(x) F (u ) u N u NN u / N u uf u (x) Fu(x)=P(X u x X>u) F 0.5 (x) GPD Fu(x)=P(X u x X>u) F 1.0 (x) GPD x x Fu(x)=P(X u x X>u) F 1.5 (x) GPD Fu(x)=P(X u x X>u) F 2.0 (x) GPD x x

45 GEVGPDξ GPD u u GEV Hill estimator HillF MDAΦ α GPD POTpeaks-over-threshold POT EVT POT TOPIX EVT VaRDanielsson and de Vries POTEmbrechts, Klüppelberg and Mikosch McNeil AR AR AR TOPIX EVT iidiidevt Embrechts, Klüppelberg and Mikosch1997McNeil and Frey

46 u mean excess plotu e( u) 1 Ν en(u) = ( X i u) + N u i =1 X i i GPDξ e( u) TOPIX u GPD

47 e(u) e(u) e(u) u u u e(u) e(u) u u n x n x n, e 2,000 x n

48 GPD u GPDPOT u ξ β u POT rξ β Y N u 1 ξ ( ξ, β ; Y ) = n ln( β ) + 1 ln 1 + β ξ i = 1 Y i Y = ( Y 1, Y 2,, Y Nu ) Y = ( TOPIX i ) u i τ = ξ / β rξ β Y ξ Ν u 1 ξ= ln( 1 τy i ) N i =1 τ 1 τ + 1 N u ˆ ξ N u i = 1 Y 1 i τy i = 0 ξ u N u GPD ξ / ξ ξ u u ξ =β =N u = Embrechts, Küppelberg and Mikosch

49 ξ ξ F (u) N u / N u F F F GPD u ξ β u u

50 x POT heteroscedasticity n x n Nx n, n/(n+1))

51 EVTiid EVT EVT EVT EVT excess-of-loss cover EVT McNeil EVTRootzén and Tajvidi windstorm insurance ARTWinterthur Schmock EVT EVT VaR Artzner, Delbaen, Eber and Heath sub-additivityhomogeneitymonotonicity risk-free condition VaR Embrechts, Klüppelberg and MikoschE X X >x p xp Artzner, Delbaen, Eber and Heath x pevt VaR anielsson and de Vries1997McNeil and Frey Phoa Cruz, Coleman and Salkin

52 MTEC ART Alternative Risk Transfer Artzner, P., F. Delbaen, J.M. Eber and D. Heath, Thinking Coherently, Risk, Vol. 10 No.11, November 1997, pp Bühlmann, H., Mathematical Methods in Risk Theory, Springer-Verlag, 1970., An Economic Premium Principle, ASTIN Bulletin 11, 1980, pp , F. Delbaen, P. Embrechts and A. Shiryaev, No-Arbitrage, Change of Measure and Conditional Esscher Transforms, mimeo, ETH Zürich, Canter, M., J. Cole and R. Sandor, Insurance Derivatives: A New Asset Class for the Capital Markets and a New Hedging Tool for the Insurance Industry, The Journal of Derivatives, Winter, 1996, pp Christensen, C. V., A New Model for Pricing catastrophe Insurance Derivatives, mimeo, University of Aarhus, Considine, G., Introduction to Weather Derivatives, Weather Derivatives Group, Aquila Energy, Credit Suisse Financial Products, CreditRisk+ - A Credit Risk Management Framework, Cruz, M., R. Coleman and G. Salkin, Modeling and Measuring Operational Risk, Journal of Risk, Fall 1998, pp Danielsson, J. and C.G. de Vries, Value-at-Risk and Extreme Returns, LSE Financial Markets Group Discussion Paper, 273, London School of Economics, Delbaen, F. and J. Haezendonck, A Martingale Approach to Premium Calculation Principles in an Arbitrage Free Market, Insurance: Mathematics and Economics 8, 1989, pp Embrechts, P., R. Frey and H. Furrer, Stochastic Processes in Insurance and Finance, mimeo, ETH Zürich, 1998., C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer-Verlag, 1997., S. Resnick and G. Samorodnitsky, Extreme Value Theory as a Risk Management Tool, mimeo, ETH Zürich, Feller, W, An Introduction to Probability Theory and Its Applications, Volume 2, John Wiley and Sons, 1966.

53 Föllmer, H. and P. Leukert, Efficient Hedging: Cost versus Shortfall Risk, mimeo, Humboldt- Universität, and, Quantile Hedging, Finance and Stochastics, Vol.3, No.3, 1999, pp Gerber, H, An Introduction to Mathematical Risk Theory, S. S. Huebner Foundation Monograph Series No.8, and E. Shiu, Option Pricing by Esscher Transforms, Transactions of the Society of Actuaries XLVI, 1994, pp Grandell, J., Aspects of Risk Theory, Springer-Verlag, Harrison, J. M. and S. Pliska, Martingales and Stochastic Integrals in the Theory of Continuous Trading, Stochastic Processes and their Applications 11, 1981, pp Hill, B, A Simple General approach to Inference about the Tail of a Distribution, The Annals of Statistics, Vol. 3, No. 5, 1975, pp Lamberton, D. and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall, McNeil, A., Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory, ASTIN Bulletin, Vol. 27, No. 1, 1997, pp , Extreme Value Theory for Risk Managers, working paper, ETH Zürich, and R. Frey, Estimation of Tail-Related Risk Measures for Hetero-scedastic Financial Time Series: an Extreme Value Approach, mimeo, ETH Zürich, Meister, S., Contributions to the Mathematics of Catastrophe Insurance Futures, Diplomarbeit, ETH Zürich, Paul-Choudhury, S., Getting Down to Business, Insurance Risk Special Report, Risk July 1998, pp. 1. Phoa, W., Estimating Credit Spread Risk Using Extreme Value Theory, The Journal of Portfolio Management, Spring 1999, pp Resnick, S., Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, Rootzén, H. and N. Tajvidi, Extreme Value Statistics and Wind Storm Losses: a Case Study, Scandinavian Actuarial Journal, 1, 1997: pp Rolski, T., H. Schmidli, V. Schmidt and J. Teugels, Stochastic Processes for Insurance and Finance, John Wiley & Sons, Schanz, K. U., The Convergence of Re insurance and Capital Markets - the Financial Services Industry Reinventing Itself,, Schmock, U., Estimating the Value of the WINCAT Coupons of the Winterthur Insurance Convertible Bond: a Study of the Model Risk, Discussion paper, ETH Zürich, Schweizer, M., Option Hedging for Semimartingales, Stochastic Processes and their Applications 37, 1991, pp Sekine, J., Quantile Hedging for Defaultable Securities in an Incomplete Market, mimeo, Graduate School of Engineering Science Osaka University, 1999.

54 sigma, Insurance Derivatives and Securitization: New Hedging Perspectives for the US Catastrophe Insurance Markets? No. 5, 1996, Swiss Re, Zürichhttp://www. swissre.com., Too Little Reinsurance of Natural Disasters in Many Markets, No. 7, 1997, Swiss Re, Zürich Alternative Risk Transfer ART for Corporations: A Passing Fashion or Risk Management for the 21st Century?, No.2, 1999, Swiss Re, Zürichhttp:// Sondermann, D., Reinsurance in Arbitrage-Free Markets, Insurance: Mathematics and Economics 10, 1991, pp Swiss Re New Markets, Integrated Risk Management Solutions - Beyond Traditional Reinsurance and Financial Hedging, Swiss Re Publications, 1998.

1 Tokyo Daily Rainfall (mm) Days (mm)

1 Tokyo Daily Rainfall (mm) Days (mm) ( ) r-taka@maritime.kobe-u.ac.jp 1 Tokyo Daily Rainfall (mm) 0 100 200 300 0 10000 20000 30000 40000 50000 Days (mm) 1876 1 1 2013 12 31 Tokyo, 1876 Daily Rainfall (mm) 0 50 100 150 0 100 200 300 Tokyo,