On the Calculation of the Solvency Capital Requirement Based on Nested Simulations (and some extensions)

Transcription

1 2015 ASTIN and AFIR/ERM Colloquium Sydney, NSW August 26, 2015 The Bob Alting von Geusau Memorial Prize On the Calculation of the Solvency Capital Requirement Based on Nested Simulations (and some extensions) Daniel Bauer, 1 Andreas Reuss, 2 and Daniela Singer 2 Georgia State University 1 and Institute for Finance and Actuarial Science (ifa) 2

2 Page 2 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer!!! So!when!does! Solvency!II!come! into!effect?! In!two!years! Dani!and!Andy!in!2011!

3 Page 2 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer!!! So!when!does! Solvency!II!come! into!effect?! In!two!years!!!!! So!when!does! Solvency!II!come! into!effect?! In!two!years! Dani!and!Andy!in!2011! Dani!and!Andy!in!2014! I Insurance companies appear to have difficulty with the implementation of Solvency II, particularly with the calculation of the Solvency Capital Requirement (SCR) [Pillar 1] I Difficulties in understanding the underlying details and struggle with numerical implementation, particularly in internal models

4 Page 3 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Formalization of SCR Calculation and Application of Advanced Simulation Techniques ON THE CALCULATION OF THE SOLVENCY CAPITAL REQUIREMENT BASED ON NESTED SIMULATIONS * BY DANIEL BAUER, ANDREAS REUSS AND DANIELA SINGER I Introduce framework for calculation of SCR (precise formulas) ABSTRACT Within the European Union, risk-based funding requirements for insurance companies are currently being revised as part of the Solvency II project. However, many life insurers struggle with the implementation, which to a large extent appears to be due to a lack of know-how regarding both, stochastic modeling and efficient techniques for the numerical implementation. The current paper addresses these problems by providing a mathematical framework for the derivation of the required risk capital and by reviewing different alternatives for the numerical implementation based on nested simulations. In particular, we seek to provide guidance for practitioners by illustrating and comparing the different techniques based on numerical experiments. KEYWORDS Solvency II, Value-at-Risk, nested simulations, screening procedures. 1. INTRODUCTION Within the European Union, risk-based funding requirements for insurance companies are currently being revised as part of the Solvency II project. One key aspect of the new regulatory framework is the determination of the required risk capital for a one-year time horizon, i.e. the amount of capital the company must hold against unforeseen losses during the following year. In particular, the regulation allows for a company-specific calculation based * Parts of this paper are taken from an earlier paper called Solvency II and Nested Simulations a Least-Squares Monte Carlo Approach and from the third author s doctoral dissertation (cf. Bergmann (2011)). The authors are grateful for helpful comments from an anonymous referee and seminar participants at the 2009 ARIA meeting, the 2009 CMA Workshop on Insurance Mathematics and Longevity Risk, the 2010 International Congress of Actuaries, Georgia State University, Humboldt University of Berlin, Ulm University, and the University of Duisburg-Essen. All remaining errors are ours. Astin Bulletin 42(2), doi: /AST by Astin Bulletin. All rights reserved.

5 Page 3 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Formalization of SCR Calculation and Application of Advanced Simulation Techniques ON THE CALCULATION OF THE SOLVENCY CAPITAL REQUIREMENT BASED ON NESTED SIMULATIONS * BY DANIEL BAUER, ANDREAS REUSS AND DANIELA SINGER ABSTRACT Within the European Union, risk-based funding requirements for insurance companies are currently being revised as part of the Solvency II project. However, many life insurers struggle with the implementation, which to a large extent appears to be due to a lack of know-how regarding both, stochastic modeling and efficient techniques for the numerical implementation. The current paper addresses these problems by providing a mathematical framework for the derivation of the required risk capital and by reviewing different alternatives for the numerical implementation based on nested simulations. In particular, we seek to provide guidance for practitioners by illustrating and comparing the different techniques based on numerical experiments. KEYWORDS Solvency II, Value-at-Risk, nested simulations, screening procedures. 1. INTRODUCTION Within the European Union, risk-based funding requirements for insurance companies are currently being revised as part of the Solvency II project. One key aspect of the new regulatory framework is the determination of the required risk capital for a one-year time horizon, i.e. the amount of capital the company must hold against unforeseen losses during the following year. In particular, the regulation allows for a company-specific calculation based I Introduce framework for calculation of SCR (precise formulas) I We survey, extend, and adapt different advanced techniques for the calculation of the SCR based on nested simulations. Specifically: I I I I Optimal allocation of numerical resources within simulation Confidence intervals for the SCR Screening procedures Variance reduction and resampling techniques (jackknife) * Parts of this paper are taken from an earlier paper called Solvency II and Nested Simulations a Least-Squares Monte Carlo Approach and from the third author s doctoral dissertation (cf. Bergmann (2011)). The authors are grateful for helpful comments from an anonymous referee and seminar participants at the 2009 ARIA meeting, the 2009 CMA Workshop on Insurance Mathematics and Longevity Risk, the 2010 International Congress of Actuaries, Georgia State University, Humboldt University of Berlin, Ulm University, and the University of Duisburg-Essen. All remaining errors are ours. Astin Bulletin 42(2), doi: /AST by Astin Bulletin. All rights reserved.

6 Page 3 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Formalization of SCR Calculation and Application of Advanced Simulation Techniques ON THE CALCULATION OF THE SOLVENCY CAPITAL REQUIREMENT BASED ON NESTED SIMULATIONS * BY DANIEL BAUER, ANDREAS REUSS AND DANIELA SINGER ABSTRACT Within the European Union, risk-based funding requirements for insurance companies are currently being revised as part of the Solvency II project. However, many life insurers struggle with the implementation, which to a large extent appears to be due to a lack of know-how regarding both, stochastic modeling and efficient techniques for the numerical implementation. The current paper addresses these problems by providing a mathematical framework for the derivation of the required risk capital and by reviewing different alternatives for the numerical implementation based on nested simulations. In particular, we seek to provide guidance for practitioners by illustrating and comparing the different techniques based on numerical experiments. KEYWORDS Solvency II, Value-at-Risk, nested simulations, screening procedures. 1. INTRODUCTION Within the European Union, risk-based funding requirements for insurance companies are currently being revised as part of the Solvency II project. One key aspect of the new regulatory framework is the determination of the required risk capital for a one-year time horizon, i.e. the amount of capital the company must hold against unforeseen losses during the following year. In particular, the regulation allows for a company-specific calculation based * Parts of this paper are taken from an earlier paper called Solvency II and Nested Simulations a Least-Squares Monte Carlo Approach and from the third author s doctoral dissertation (cf. Bergmann (2011)). The authors are grateful for helpful comments from an anonymous referee and seminar participants at the 2009 ARIA meeting, the 2009 CMA Workshop on Insurance Mathematics and Longevity Risk, the 2010 International Congress of Actuaries, Georgia State University, Humboldt University of Berlin, Ulm University, and the University of Duisburg-Essen. All remaining errors are ours. I Introduce framework for calculation of SCR (precise formulas) I We survey, extend, and adapt different advanced techniques for the calculation of the SCR based on nested simulations. Specifically: I I I I Optimal allocation of numerical resources within simulation Confidence intervals for the SCR Screening procedures Variance reduction and resampling techniques (jackknife) I Illustrate drawbacks and advantages of the different approaches based on numerical experiments in simple participating contract framework Astin Bulletin 42(2), doi: /AST by Astin Bulletin. All rights reserved.

7 Page 4 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Application of Least-Squares Monte Carlo Techniques to SCR Estimation I To address the key issue in view of the Solvency II and Nested Simulations a numerical feasibility the nested Least-Squares Monte Carlo Approach simulation structure we propose Least-Squares Monte Carlo approach (LSMC) Daniel Bauer J. Mack Robinson College of Business, Georgia State University, 35 Broad Street, Atlanta, GA 30303, USA phone: , fax: DBauer@gsu.edu Daniela Bergmann Institute of Insurance, Ulm University, Helmholtzstraße 18, Ulm, Germany phone: , fax: daniela.bergmann@uni-ulm.de Andreas Reuss Institute for Finance and Actuarial Sciences, Helmholtzstraße 22, Ulm, Germany phone: , fax: a.reuss@ifa-ulm.de I I I Familiar from pricing American Options (Longstaff & Schwartz, 2001) Idea is to approximate conditional expected value in risk measure using "basis functions" that can be estimated from paths using OLS Experiments in simple framework encouraging First version: October This version: October Corresponding author.

8 Page 4 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Application of Least-Squares Monte Carlo Techniques to SCR Estimation I To address the key issue in view of the Solvency II and Nested Simulations a numerical feasibility the nested Least-Squares Monte Carlo Approach simulation structure we propose Least-Squares Monte Carlo approach (LSMC) Daniel Bauer J. Mack Robinson College of Business, Georgia State University, 35 Broad Street, Atlanta, GA 30303, USA phone: , fax: DBauer@gsu.edu Daniela Bergmann Institute of Insurance, Ulm University, Helmholtzstraße 18, Ulm, Germany phone: , fax: daniela.bergmann@uni-ulm.de Andreas Reuss Institute for Finance and Actuarial Sciences, Helmholtzstraße 22, Ulm, Germany phone: , fax: a.reuss@ifa-ulm.de First version: October This version: October I Familiar from pricing American Options (Longstaff & Schwartz, 2001) I Idea is to approximate conditional expected value in risk measure using "basis functions" that can be estimated from paths using OLS I Experiments in simple framework encouraging I Popular in practice and some follow-up academic work I Barrie & Hibbert, 2011; Milliman, 2013 I Olivier s (prize-winning) presentation yesterday Corresponding author.

9 Page 4 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Application of Least-Squares Monte Carlo Techniques to SCR Estimation I To address the key issue in view of the Solvency II and Nested Simulations a numerical feasibility the nested Least-Squares Monte Carlo Approach simulation structure we propose Least-Squares Monte Carlo approach (LSMC) Daniel Bauer J. Mack Robinson College of Business, Georgia State University, 35 Broad Street, Atlanta, GA 30303, USA phone: , fax: DBauer@gsu.edu Daniela Bergmann Institute of Insurance, Ulm University, Helmholtzstraße 18, Ulm, Germany phone: , fax: daniela.bergmann@uni-ulm.de Andreas Reuss Institute for Finance and Actuarial Sciences, Helmholtzstraße 22, Ulm, Germany phone: , fax: a.reuss@ifa-ulm.de First version: October This version: October Corresponding author. I Familiar from pricing American Options (Longstaff & Schwartz, 2001) I Idea is to approximate conditional expected value in risk measure using "basis functions" that can be estimated from paths using OLS I Experiments in simple framework encouraging I Popular in practice and some follow-up academic work I Barrie & Hibbert, 2011; Milliman, 2013 I Olivier s (prize-winning) presentation yesterday I However, many open questions...

10 Page 5 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Convergence Proofs/Orders for Least-Squares Monte Carlo Approach and Analysis of Optimal Basis Functions A Least-Squares Monte Carlo Approach to the Calculation of Capital Requirements Daniel Bauer & Hongjun Ha Department of Risk Management and Insurance. Georgia State University 35 Broad Street. Atlanta, GA USA July 2015 I Develop detailed theory around LSMC Abstract The calculation of capital requirements for financial institutions usually entails a reevaluation of the company s assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. Relying on a well-known method for pricing non-european derivatives, the current paper proposes and analyzes a novel approach to this computational problem based on least-squares regression and Monte Carlo simulations. We show convergence of the algorithm, we analyze the resulting estimate for practically important risk measures, and we derive optimal basis functions based on singular value decomposition of compact operator. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs, particularly when relying on the optimal basis functions. Keywords: Loss distribution, least-square Monte Carlo, Value-at-Risk, compact operator, singular value decomposition, variable annuities. Corresponding author. Phone: +1-(404) Fax: +1-(405) addresses: bauer@gsu.edu (D. Bauer); hha6@gsu.edu (H. Ha).

11 Page 5 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Convergence Proofs/Orders for Least-Squares Monte Carlo Approach and Analysis of Optimal Basis Functions A Least-Squares Monte Carlo Approach to the Calculation of Capital Requirements Daniel Bauer & Hongjun Ha Department of Risk Management and Insurance. Georgia State University 35 Broad Street. Atlanta, GA USA July 2015 Abstract The calculation of capital requirements for financial institutions usually entails a reevaluation of the company s assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. Relying on a well-known method for pricing non-european derivatives, the current paper proposes and analyzes a novel approach to this computational problem based on least-squares regression and Monte Carlo simulations. We show convergence of the algorithm, we analyze the resulting estimate for practically important risk measures, and we derive optimal basis functions based on singular value decomposition of compact operator. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs, particularly when relying on the optimal basis functions. Keywords: Loss distribution, least-square Monte Carlo, Value-at-Risk, compact operator, singular value decomposition, variable annuities. I Develop detailed theory around LSMC I Prove convergence and discuss issues (number of simulations vs. basis functions) as well as discuss potential biases Corresponding author. Phone: +1-(404) Fax: +1-(405) addresses: bauer@gsu.edu (D. Bauer); hha6@gsu.edu (H. Ha).

12 Page 5 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Convergence Proofs/Orders for Least-Squares Monte Carlo Approach and Analysis of Optimal Basis Functions A Least-Squares Monte Carlo Approach to the Calculation of Capital Requirements Daniel Bauer & Hongjun Ha Department of Risk Management and Insurance. Georgia State University 35 Broad Street. Atlanta, GA USA July 2015 Abstract The calculation of capital requirements for financial institutions usually entails a reevaluation of the company s assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. Relying on a well-known method for pricing non-european derivatives, the current paper proposes and analyzes a novel approach to this computational problem based on least-squares regression and Monte Carlo simulations. We show convergence of the algorithm, we analyze the resulting estimate for practically important risk measures, and we derive optimal basis functions based on singular value decomposition of compact operator. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs, particularly when relying on the optimal basis functions. Keywords: Loss distribution, least-square Monte Carlo, Value-at-Risk, compact operator, singular value decomposition, variable annuities. I Develop detailed theory around LSMC I Prove convergence and discuss issues (number of simulations vs. basis functions) as well as discuss potential biases I Discuss choice of basis functions I Optimal basis functions defined as basis functions that minimize difference between valuation and approximation (operator!) I Result: Singular functions of valuation operator optimal Corresponding author. Phone: +1-(404) Fax: +1-(405) addresses: bauer@gsu.edu (D. Bauer); hha6@gsu.edu (H. Ha).

13 Page 5 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Convergence Proofs/Orders for Least-Squares Monte Carlo Approach and Analysis of Optimal Basis Functions A Least-Squares Monte Carlo Approach to the Calculation of Capital Requirements Daniel Bauer & Hongjun Ha Department of Risk Management and Insurance. Georgia State University 35 Broad Street. Atlanta, GA USA July 2015 Abstract The calculation of capital requirements for financial institutions usually entails a reevaluation of the company s assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. Relying on a well-known method for pricing non-european derivatives, the current paper proposes and analyzes a novel approach to this computational problem based on least-squares regression and Monte Carlo simulations. We show convergence of the algorithm, we analyze the resulting estimate for practically important risk measures, and we derive optimal basis functions based on singular value decomposition of compact operator. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs, particularly when relying on the optimal basis functions. Keywords: Loss distribution, least-square Monte Carlo, Value-at-Risk, compact operator, singular value decomposition, variable annuities. I Develop detailed theory around LSMC I Prove convergence and discuss issues (number of simulations vs. basis functions) as well as discuss potential biases I Discuss choice of basis functions I Optimal basis functions defined as basis functions that minimize difference between valuation and approximation (operator!) I Result: Singular functions of valuation operator optimal I Numerical studies in the context of Guaranteed Annuitization Options in line with theory Corresponding author. Phone: +1-(404) Fax: +1-(405) addresses: bauer@gsu.edu (D. Bauer); hha6@gsu.edu (H. Ha).

14 Page 6 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II Nested Simulations Design of Nested Simulations Approach Optimal Allocation of Computational Budget Confidence Intervals for the SCR Screening Procedures Variance Reduction Least-Squares Monte Carlo Approach Conclusion

15 Page 7 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II Background on Solvency II Nested Simulations Design of Nested Simulations Approach Optimal Allocation of Computational Budget Confidence Intervals for the SCR Screening Procedures Variance Reduction Least-Squares Monte Carlo Approach Conclusion

16 Page 8 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II Solvency II Capital Requirement in a Nutshell! Consideration over a one-year horizon now: t = 0 / then: t = 1 Solvency intuition An insurer is solvent if P ([MV t=1] > [MV t=1]) apple.5%

17 Page 8 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II Solvency II Capital Requirement in a Nutshell! Consideration over a one-year horizon now: t = 0 / then: t = 1 Solvency intuition An insurer is solvent if P ([MV t=1] > [MV t=1]) apple.5% I Ignoring all the complications, Available Capital: [AC] t =[MV Assets] t [MV Liabilities] t, t 2{0, 1} ) An insurer is solvent if (i one-year risk-free t = 0) 0.5% P ([AC] 1 < 0) =P > 0 = P B [AC]1 1 + [AC] 0 [AC] i {z } =:L 1 > [AC] 0 C A

18 Page 8 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II Solvency II Capital Requirement in a Nutshell! Consideration over a one-year horizon now: t = 0 / then: t = 1 Solvency intuition An insurer is solvent if P ([MV t=1] > [MV t=1]) apple.5% I Ignoring all the complications, Available Capital: [AC] t =[MV Assets] t [MV Liabilities] t, t 2{0, 1} ) An insurer is solvent if (i one-year risk-free t = 0) 0.5% P ([AC] 1 < 0) =P > 0 = P B [AC]1 1 + [AC] 0 [AC] i {z } =:L 1 > [AC] 0 C A

19 Page 9 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II So the company is considered fine if there is more available capital at time zero [AC] 0 than the Value at Risk (V@R) of L at the 99.5% level I Caveat: The Definition of L depends on [AC]0

20 Page 9 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II So the company is considered fine if there is more available capital at time zero [AC] 0 than the Value at Risk (V@R) of L at the 99.5% level I Caveat: The Definition of L depends on [AC]0... let s call the [AC] 0 on the right hand side the Solvency Capital Requirement (SCR) then that s approximately what [AC] 0 should be Solvency Capital Requirement SCR = argmin x {P(L > x) apple 0.5%} = 99.5% (L)

21 Page 9 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II So the company is considered fine if there is more available capital at time zero [AC] 0 than the Value at Risk (V@R) of L at the 99.5% level I Caveat: The Definition of L depends on [AC]0... let s call the [AC] 0 on the right hand side the Solvency Capital Requirement (SCR) then that s approximately what [AC] 0 should be Solvency Capital Requirement... we have SCR = argmin x {P(L > x) apple 0.5%} = 99.5% (L) SCR = argmin x {P( [AC] 1 > (1 + i) (x [AC] 0 ) apple 0.5%},... we only need to determine the 99.5%-quantile of [AC] 1, where [AC] 1 =[MV Assets] 1 E Q [Disc. Fut. Policyholder CFs F 1 ] is an F 1 random variable, i.e. we need to assess its distribution

22 Page 9 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Background on Solvency II So the company is considered fine if there is more available capital at time zero [AC] 0 than the Value at Risk (V@R) of L at the 99.5% level I Caveat: The Definition of L depends on [AC]0... let s call the [AC] 0 on the right hand side the Solvency Capital Requirement (SCR) then that s approximately what [AC] 0 should be Solvency Capital Requirement... we have SCR = argmin x {P(L > x) apple 0.5%} = 99.5% (L) SCR = argmin x {P( [AC] 1 > (1 + i) (x [AC] 0 ) apple 0.5%},... we only need to determine the 99.5%-quantile of [AC] 1, where [AC] 1 =[MV Assets] 1 E Q [Disc. Fut. Policyholder CFs F 1 ] is an F 1 random variable, i.e. we need to assess its distribution

23 Page 10 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Background on Solvency II Nested Simulations Design of Nested Simulations Approach Optimal Allocation of Computational Budget Confidence Intervals for the SCR Screening Procedures Variance Reduction Least-Squares Monte Carlo Approach Conclusion

24 Page 11 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Model Framework I d-dimensional Markov process Y =(Y t ) t2[0,t ] drives risks (assets & liabilities) I Q risk-neutral measure with respect to Numéraire B t = exp{ R t 0 r s ds} I Cash-flows from insurance business: X =(X 1,...,X T ) where X t = f t (Y s, s 2 [0, t]) (App.: Limit focus to market risk non-financial risk factors could be incorporated by appropriate choice of Y, f, and Q. Default put option should not be considered.)

25 Page 11 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Model Framework I d-dimensional Markov process Y =(Y t ) t2[0,t ] drives risks (assets & liabilities) I Q risk-neutral measure with respect to Numéraire B t = exp{ R t 0 r s ds} I Cash-flows from insurance business: X =(X 1,...,X T ) where X t = f t (Y s, s 2 [0, t]) (App.: Limit focus to market risk non-financial risk factors could be incorporated by appropriate choice of Y, f, and Q. Default put option should not be considered.) h PT i I [AC] 1 =[MVA] 1 E Q X R t t=2 t1 Y rs ds s, s 2 [0, 1] e

26 Page 11 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Model Framework I d-dimensional Markov process Y =(Y t ) t2[0,t ] drives risks (assets & liabilities) I Q risk-neutral measure with respect to Numéraire B t = exp{ R t 0 r s ds} I Cash-flows from insurance business: X =(X 1,...,X T ) where X t = f t (Y s, s 2 [0, t]) (App.: Limit focus to market risk non-financial risk factors could be incorporated by appropriate choice of Y, f, and Q. Default put option should not be considered.) h PT i I [AC] 1 =[MVA] 1 E Q X R t t=2 t1 Y rs ds s, s 2 [0, 1] e I Typically: For evaluation of insurance contracts, it is sufficient to know the values of certain state variables/accounts D (e.g. death benefit account, net survivor value,...) to determine the value of a contract! (possibly high, but) finite dimensional Markov state space " TX # [MVL] 1 = E Q X t t=2 e R Y t 1, D 1 1 rs ds (generalizes to company level)

27 Page 11 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Model Framework I d-dimensional Markov process Y =(Y t ) t2[0,t ] drives risks (assets & liabilities) I Q risk-neutral measure with respect to Numéraire B t = exp{ R t 0 r s ds} I Cash-flows from insurance business: X =(X 1,...,X T ) where X t = f t (Y s, s 2 [0, t]) (App.: Limit focus to market risk non-financial risk factors could be incorporated by appropriate choice of Y, f, and Q. Default put option should not be considered.) h PT i I [AC] 1 =[MVA] 1 E Q X R t t=2 t1 Y rs ds s, s 2 [0, 1] e I Typically: For evaluation of insurance contracts, it is sufficient to know the values of certain state variables/accounts D (e.g. death benefit account, net survivor value,...) to determine the value of a contract! (possibly high, but) finite dimensional Markov state space " TX # [MVL] 1 = E Q X t t=2 e R Y t 1, D 1 1 rs ds (generalizes to company level)

28 Page 12 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Nested Simulations Approach P RF 1 Y (1), D (1) Q Y (i), D (i) Y (N), D (N) t=0 t=1 t=t Simulate N first-year paths "under P" Simulate K 1 paths "under Q" starting in Y1 i, Di 1 to determine [AC] 1 N K 1 paths! Determine quantile via empirical distribution function

29 Page 12 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Nested Simulations Approach P RF 1 Y (1), D (1) Q Y (i), D (i) Y (N), D (N) t=0 t=1 t=t Simulate N first-year paths "under P" Simulate K 1 paths "under Q" starting in Y1 i, Di 1 to determine [AC] 1 N K 1 paths! Determine quantile via empirical distribution function I Gordy & Juneja (2010,ManSci): Diversification of error in inner step when estimating portfolio V@R! Not necessarily true in insurance context different contracts evaluated based on same (asset & liability) scenarios

30 Page 13 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Example: Participating contract model from Bauer et al. (2006, IME) (Must-Case) I Term-fix insurance, interest rate guarantee (cliquet-style), participation scheme (can serve as model for company, Kling et al. (2007,IME)) I Underlying portfolio S t. Generalized Black-Scholes model for (S t, r t ) with Ornstein-Uhlenbeck interest rates (Vasicek model) I Guaranteed rate g, accounting par. y, participation rate : I Ai+1 = A + i (S i+1 /S i ), I Li+1 = L i (1 + g)+ y A i+1 A + i gl i +, I di+1 =(1 ) y (A i+1 A + i ) 1 { y(a i+1 A + i )>g L i } +(y (A i+1 A + i ) gl i ) + 1 { y(a i+1 A + i )appleg L i } I ci+1 =(L i+1 A i+1 ) + (no defaults!) I A + i+1 = A i+1 d i+1 + c i+1! D i = L i, x i = A+ L i i, Y L i = S i, r i, R i i 0 rs ds ) DM: X i = 0, i < T, X T = L T ) IDM: X 0 = 0, X i = d i c i, i = 0,...,T 1, X T =(A T L T ) c T = R T c T, (Details: Bauer, Kling, Kiesel & Russ (2006, IME) and Zaglauer & Bauer (2008, IME))

31 Page 14 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Nested Simulations Bias in Nested Simulations (K 0 = 250, 000; N = 100, 000) ,000-4,000-2, ,000 4,000 6,000 Loss (L) K 1 =1 K 1 =10 K 1 =1, 000 K 1 SCR [AC] 0 /SCR 1 1, % 5 1, % 10 1, % 100 1, % 1,000 1, %! Choice of K 1 significantly affects SCR!

32 Page 15 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Background on Solvency II Nested Simulations Design of Nested Simulations Approach Optimal Allocation of Computational Budget Confidence Intervals for the SCR Screening Procedures Variance Reduction Least-Squares Monte Carlo Approach Conclusion

33 Page 16 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Bias (adapted from Gordy & Juneja (2010, ManSci)) [MVA] 1 [MVL] 1 L(Y 1, D 1 ) = [AC] 0, 1 + i [MVA] 1 L(Y 1, D 1 ) = [AC] g (K 0) 0 1 K 1 Pk P t 1 + i X (k) t (Y B 1 (t) (k) 1, D 1 ) and L (b99.5% Nc+1) the (b99.5% Nc + 1)-st order statistics of Then, under some regularity conditions: MSE = Var[ L b99.5% Nc+1) ]+BIAS, i BIAS = E h L(b99.5% Nc+1) SCR h h f (u) E L L Y1, D 1 = where f is density of L ii L = u u=scr f (SCR) {z } = K 1 f (SCR) I > 0 ) Bias positive! L(Y (i) 1, D(i) 1 ) +O N (1/N)+...

34 Page 17 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Proposition Under the computational budget constraint =c K 0 + N K 1, for an optimal triplet that minimizes the mean square error (MSE), we have asymptotically for large K 1 : N (1 ) K , and 0 f (SCR) p (1 ) p p K c 1 K1 K 0 r 0 K 1 f (SCR) N K1 2c. I needs to be estimated in pilot simulation I Use regression on Var and finite difference approximation I We find (K0 = 1, 500, 000; N = 320, 000; K 1 = 300) approx. optimal I Bias can be reduced via jackknife

35 Page 18 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach 1,280 1,270 10%/90% quantile 25%/75% quantile median 1,260 ]SCR 1,250 1,240 1,230 1,220 N =160, 000 K1 =600 N =320, 000 K1 =300 N =640, 000 K1 =150 N =1, 500, 000 K1 =64 N K 1 Mean Empirical Estimated Estimated Corrected ( ] SCR) Variance Bias MSE Mean 160, , , , , , , , , ,500, , ,246.3

36 Page 19 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach 1,280 1,270 10%/90% quantile 25%/75% quantile median 1,260 ]SCR 1,250 1,240 1,230 1,220 N =160, 000 K1 =600 N =320, 000 K1 =300 N =640, 000 K1 =150 N =1, 500, 000 K1 =64 N K 1 Mean Empirical Estimated Estimated Corrected ( ] SCR) Variance Bias MSE Mean 160, , , , , , , , , ,500, , ,246.3

37 Page 20 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach 1,280 1,270 10%/90% quantile 25%/75% quantile median 1,260 ]SCR 1,250 1,240 1,230 1,220 N =160, 000 K1 =600 N =320, 000 K1 =300 N =640, 000 K1 =150 N =1, 500, 000 K1 =64 JACKKNIFE: Jackknife Estimator Nested Simulations Estimator N K 1 Mean Empirical Mean Empirical Corrected ( ] SCR + ) Variance ( ] SCR) Variance Mean 160, , , , , , , , , , , , ,500, , , ,246.3

38 Page 21 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Confidence Intervals for the SCR I Idea: (cf. Lan, Nelson, and Staum (2007, IEEE)) I {L (j) applescr} Bernoulli(99.5%) ) P N j=1 I {L (j) applescr} Binomial(N, 99.5%) PN ) P L (n) > SCR = P j=1 I {L (j) applescr} < n = P n 1 N j=0 j.95 j.05 N j! Determine r and s such that: P (L r apple SCR apple L s )= P s 1 j=r N j.95 j.05 N j 1 OUT I Problem: Error resulting from inner simulation not considered. I Idea: Instead of order statistics of L, for upper and lower bound consider r th and s th order statistics of L (j) ± t K0 1,1 AC 0 2 pˆ0 ± t K1 1,1 " K0 2 ˆ(j) 1 (1+i) p K 1, where " =(1 AC1 ) 1/N, IN = AC0 + AC1 AC0 AC1, IN + OUT = TOT. I Then these present a confidence interval for the level TOT.

39 Page 22 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach In Example: Confidence level TOT = 90% 1,700 1,600 1,500 1,400 1,300 1,200 ]SCR LB UB 60% 50% 40% 30% 20% 1,100 10% 1,000 50, , , , , ,000 0% 50, , , , , ,000 N N I Same budget as before ) CI VERY WIDE. Even for approx. optimal allocation

40 Page 23 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Screening Basic idea I Run inner procedure twice (based on same outer samples), first based on fewer (K 1,1 ) inner simulations then based on more (K 1,2 ) inner simulations. I However, based on the first step, "screen" out samples that are very unlikely to be in the tail, i.e. only run inner simulations for samples that have not been screened out in first step. I Approach: (cf. Lan, Nelson, and Staum (2010, OperRes) Only keep scenarios in following set 8 8 s < < I = : n :# : j : L (n) < L (j) t f (n,j),1 9 = 9 = (1 + i) 2 K 1,1, ; < N 1 r + 1 ; (ˆ(j) 1 )2 +(ˆ(n) 1 )2 where = SCREEN (N 1 r+1)(r 1) and f (n,j) (degrees of freedom) determined by Welch-Satterthwaite equation. I To reduce number of comparisons, pre-screening based on "maxima" t-quantile and "maximal" stdev can be applied

41 Page 24 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Variance Reduction Antithetic Variates N K 1, K 0 Mean Emp. Est. Est. Corrected K 1 (SCR, Var Bias MSE Mean SCR) with AV 1,070, ,000 1, ,246.0 with AV 310, ,000 1, ,245.7 w/o AV 320, ,500,000 1, ,246.4 IDM with AV 1,375, ,000 1, ,246.0 IDM with AV 115, ,000 1, ,245.7 IDM w/o AV 105, ,000 1, ,246.0 I Even more efficient when combined with screening point estimate very accurate and CI about 3% of SCR! I While IDM estimator generally performs worse, with antithetic variates it may improve

42 Page 24 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Variance Reduction Antithetic Variates N K 1, K 0 Mean Emp. Est. Est. Corrected K 1 (SCR, Var Bias MSE Mean SCR) with AV 1,070, ,000 1, ,246.0 with AV 310, ,000 1, ,245.7 w/o AV 320, ,500,000 1, ,246.4 IDM with AV 1,375, ,000 1, ,246.0 IDM with AV 115, ,000 1, ,245.7 IDM w/o AV 105, ,000 1, ,246.0 I Even more efficient when combined with screening point estimate very accurate and CI about 3% of SCR! I While IDM estimator generally performs worse, with antithetic variates it may improve

43 Page 24 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Variance Reduction Antithetic Variates N K 1, K 0 Mean Emp. Est. Est. Corrected K 1 (SCR, Var Bias MSE Mean SCR) with AV 1,070, ,000 1, ,246.0 with AV 310, ,000 1, ,245.7 w/o AV 320, ,500,000 1, ,246.4 IDM with AV 1,375, ,000 1, ,246.0 IDM with AV 115, ,000 1, ,245.7 IDM w/o AV 105, ,000 1, ,246.0 I Even more efficient when combined with screening point estimate very accurate and CI about 3% of SCR! I While IDM estimator generally performs worse, with antithetic variates it may improve

44 Page 25 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Soo... I GOOD NEWS: I Simulation design matters! Length of confidence interval was decreased by more than a factor of 14! I Ongoing work on adaptive screening procedures" may even improve performance

45 Page 25 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Soo... I GOOD NEWS: I Simulation design matters! Length of confidence interval was decreased by more than a factor of 14! I Ongoing work on adaptive screening procedures" may even improve performance I NOT SO GOOD NEWS: I Our example contract is very simple, and we rely on scenarios, but still the length of the confidence interval is 3% of the SCR! Questionable if even these advanced approaches will present practicable results I Indication that the rather conservative confidence intervals are "useless" (?)

46 Page 25 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Design of Nested Simulations Approach Soo... I GOOD NEWS: I Simulation design matters! Length of confidence interval was decreased by more than a factor of 14! I Ongoing work on adaptive screening procedures" may even improve performance I NOT SO GOOD NEWS: I Our example contract is very simple, and we rely on scenarios, but still the length of the confidence interval is 3% of the SCR! Questionable if even these advanced approaches will present practicable results I Indication that the rather conservative confidence intervals are "useless" (?) I Remember the problem: we only need to determine the 99.5%-quantile of [AC] 1, where [AC] 1 =[MV Assets] 1 E Q [ Disc. Fut. Policyholder CFs F 1 ] is an F 1 random variable, i.e. we need to assess its distribution

47 Page 26 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Background on Solvency II Nested Simulations Design of Nested Simulations Approach Optimal Allocation of Computational Budget Confidence Intervals for the SCR Screening Procedures Variance Reduction Least-Squares Monte Carlo Approach Conclusion

48 Page 27 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Least-Squares Monte Carlo! Approach popular for valuation of non-european options (fast!) Longstaff & Schwartz (2001,RFS), Clément, Lamberton & Protter (2002,Fin&Stoch)! Idea: Use 2 approximations 1. Continuation value replaced by finite linear combination of certain basis functions 2. Use MC simulations and LS regression to approximate linear comb. in 1 ) Opt. stopping rule by comparing exerc. val. to "synthetic" contin. val.

49 Page 27 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Least-Squares Monte Carlo! Approach popular for valuation of non-european options (fast!) Longstaff & Schwartz (2001,RFS), Clément, Lamberton & Protter (2002,Fin&Stoch)! Idea: Use 2 approximations 1. Continuation value replaced by finite linear combination of certain basis functions 2. Use MC simulations and LS regression to approximate linear comb. in 1 ) Opt. stopping rule by comparing exerc. val. to "synthetic" contin. val. Idea here: I Simulate N paths (first year P, then Q) I For i = 1,...,N, determine ( TX Z ) T h i PV1 i := exp r s(! i ) ds X t (! i )=E Q PV1 i F 1 +" i 1 {z } t=2 =h(y i 1,Di 1 ) 1 Replace h by finite linear combination of certain basis functions, say h 2 Use MC simulations and LS regression to approximate linear comb. in 1 ) Use given emp. cdf of ([MVA] 1 h) to determine quantile

50 Page 28 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Example (cont.): Average of 150 runs with N = 320, 000, K 0 = 1, 500, 000 # Regression Function Mean (SCR) 1 ˆ (N) 0 +ˆ (N) 1 A ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L 1 +ˆ (N) 6 x ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L 1 +ˆ (N) 6 x 1 +ˆ (N) 7 A 1 e r ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L 1 +ˆ (N) 6 x 1 +ˆ (N) 7 A 1 e r1 +ˆ (N) 8 L 1 e r ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L 1 +ˆ (N) 6 x 1 +ˆ (N) 7 A 1 e r1 +ˆ (N) 8 L 1 e r1 +ˆ (N) 9 e A1/ I Result from Nested Sims (benchmark): 1, (takes days!) now in < 10 min!

51 Page 28 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Example (cont.): Average of 150 runs with N = 320, 000, K 0 = 1, 500, 000 # Regression Function Mean (SCR) 1 ˆ (N) 0 +ˆ (N) 1 A ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L 1 +ˆ (N) 6 x ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L 1 +ˆ (N) 6 x 1 +ˆ (N) 7 A 1 e r ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L 1 +ˆ (N) 6 x 1 +ˆ (N) 7 A 1 e r1 +ˆ (N) 8 L 1 e r ˆ (N) 0 +ˆ (N) 1 A 1 +ˆ (N) 2 A 2 1 +ˆ (N) 3 r 1 +ˆ (N) 4 r1 2 +ˆ (N) 5 L 1 +ˆ (N) 6 x 1 +ˆ (N) 7 A 1 e r1 +ˆ (N) 8 L 1 e r1 +ˆ (N) 9 e A1/ I Result from Nested Sims (benchmark): 1, (takes days!) now in < 10 min! I HOWEVER: Influence of basis function quite pronounced. Used knowledge from Nested Sims approach for derivation of regression fct. cheating! Functional analytic bias. Hard to asses...

52 Page 29 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Choice of Regression Function (applied) I Problem: Common criteria for variable selection (Mallows C p, AIC,...) rely on homoskedasticity I Generalization: ( =(e 1,...,e m), e i =(e i (Y 1, D 1 ),...,e i (Y N, D N )) 0, V i 1 = P M k=1 ˆ k e k (Y i 1, Di 1 )) E 1 " N X i=1 PV i 1 ) \ SMSE = # 2 NX apple 2 NX V 1 i = E 1 V1 i V 1 i + i=1 {z } i=1 =SMSE NX i=1 PV i 1 V i 1 2 N X i=1 i 1 2 tr diag 11,..., ˆi1 + 2 tr ( 0 ) 1 0 diag ˆ11,..., ˆN1 N 1

53 Page 29 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Choice of Regression Function (applied) I Problem: Common criteria for variable selection (Mallows C p, AIC,...) rely on homoskedasticity I Generalization: ( =(e 1,...,e m), e i =(e i (Y 1, D 1 ),...,e i (Y N, D N )) 0, V i 1 = P M k=1 ˆ k e k (Y i 1, Di 1 )) E 1 " N X i=1 PV i 1 ) \ SMSE = # 2 NX apple 2 NX V 1 i = E 1 V1 i V 1 i + i=1 {z } i=1 =SMSE NX i=1 PV i 1 V i 1 2 N X i=1 i 1 2 tr diag 11,..., ˆi1 + 2 tr ( 0 ) 1 0 diag ˆ11,..., ˆN1 I Problem: Need nested simulations to estimate ˆi... I Some authors propose variable selection models for heteroskedastic data (see e.g. Baek, Kraman & Ahn (2005,CommStatist)). Work well in applications, but no theoretical results available.! Idea: Use homoskedasticity assumption to derive choice based on "simple criterion" N 1

54 Page 30 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Example (cont.): Choice of Regression Function Choice via Mallow s C P : # A1 L1 x1 r1 A 2 1 A 3 1 L 2 1 L 3 1 x 2 1 x 3 1 r1 2 r1 3 A1 A1 A1 L1 x1 A1L1 A1 L 2 1 A1 A2 1 L1 L2 1 A1x1 A2 1x1 Cp of L1 x1 e r 1 e r 1 e r 1 x1 x 2 1 x1 (e r 1 ) 2 e r 1 (e r 1 ) 2 e r 1 e r 1 regr x x x x x - x x x x - x x - - x x x x x x x - x x x x x - - x - x x x x x - x x - - x - x - x - - x x x x x x x - x x x x x x x x x - - x x - x x x x x - - x - x x x - - x x - x - x x x x - - x - x x x x - x x - x - x x x x - - x - x x - x - - x - - x x x x - x x x x x - x - x x x x - - x x x - - x x x x x - - x - x x - x - - x x x x x x x x x x x - - x - x x - x - - x x x - x - - x x x x x x x x - x x x x x - x x - x - - x - - x x x x - x x x x - x x x x x - x x - x - - x x x x x x x x x x - - x - x x x x x x x x x x x x x x x x x x - - x x x x x x x x x x x x x x x x x x x x x x x x x x x - x x x x x x x x - x - x x - - x x x x x x x x x - x - - x x x x x x x x x x x - x x x x x x x x x x - x - x x x x x x x x x x x x - x x x x x x x x x x x - x x x x x x x x x x x - x x x x x x x x x x x x x x x x x x x x x - x x x x x x x x 25.0 I SCR for Regression Function (5): 1,245.9 (Average of 120 runs a ). Results for "close" choices similar. I So far results in line with "generalized" variable selection criterion! To obtain accurate results, it is important not to employ arbitrary function, but it appears sufficient to rely on roughly coherent method to determine suitable choice. I Does this hold for more complex (realistic) examples?

55 Page 31 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Develop Theory (Bauer and Ha, H. Ha s dissertation) I Experiments (mathematically) not satisfactory Questions: I Does it work (well) in generality? I Can we choose basis functions in a more coherent manner? In particular, are there basis function that work well for all" contracts?

56 Page 31 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Develop Theory (Bauer and Ha, H. Ha s dissertation) I Experiments (mathematically) not satisfactory Questions: I Does it work (well) in generality? I Can we choose basis functions in a more coherent manner? In particular, are there basis function that work well for all" contracts? I Difficulties and Solutions: 1: Change of Measure in year one makes mathematics more complicated! Consider alternative measure E 1! Then P(A) = P(A) for A 2F 1 and E P [X F 1 ]=E Q [X F 1 ]

57 Page 31 Calculation of Capital Requirement August 26, 2015 Bauer/Reuss/Singer Least-Squares Monte Carlo Approach Develop Theory (Bauer and Ha, H. Ha s dissertation) I Experiments (mathematically) not satisfactory Questions: I Does it work (well) in generality? I Can we choose basis functions in a more coherent manner? In particular, are there basis function that work well for all" contracts? I Difficulties and Solutions: 1: Change of Measure in year one makes mathematics more complicated! Consider alternative measure E 1! Then P(A) = P(A) for A 2F 1 and E P [X F 1 ]=E Q [X F 1 ] 2: If certain basis functions approximate one contract, element in (, F 1, P) well, that does not provide information for other contracts, elements in (, F 1, P)! Consider operator T with T :(f 1, f 2,...,f T ) 7! E Q h X fi (Y i, D i ) F 1 i and focus on approximating T!