An Introduction to Computational Risk Management of Equity-Linked Insurance CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding slice of the financial sector This series aims to capture new developments and summarize what is known over the whole spectrum of this field It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners The inclusion of numerical code and concrete real-world examples is highly encouraged Series Editors M A H Dempster Centre for Financial Research Department of Pure Mathematics and Statistics University of Cambridge Dilip B Madan Robert H Smith School of Business University of Maryland Rama Cont Department of Mathematics Imperial College Stochastic Volatility Modeling Lorenzo Bergomi The Financial Mathematics of Market Liquidity From Optimal Execution to Market Making Olivier Gueant C++ for Financial Mathematics John Armstrong Model-free Hedging A Martingale Optimal Transport Viewpoint Pierre Henry-Labordere Stochastic Finance A Numeraire Approach Jan Vecer Equity-Linked Life Insurance Partial Hedging Methods Alexander Melnikov, Amir Nosrati High-Performance Computing in Finance Problems, Methods, and Solutions M A H Dempster, Juho Kanniainen, John Keane, Erik Vynckier For more information about this series please visit: https://www.crcpress.com/ Chapman-and-HallCRC-Financial-Mathematics-Series/book-series/CHFINANCMTH An Introduction to Computational Risk Management of Equity-Linked Insurance Runhuan Feng CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20180504 International Standard Book Number-13: 978-1-4987-4216-0 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To my beloved Ge, Kelsey and Kyler Contents List of Figures xiii List of Tables xv Symbols xvii Preface xix Modeling of Equity-linked Insurance 1.1 Fundamental principles of traditional insurance 1.1.1 Time value of money 1.1.2 Law of large numbers 1.1.3 Equivalence premium principle 1.1.4 Central limit theorem 1.1.5 Portfolio percentile premium principle 1.2 Variable annuities 1.2.1 Mechanics of deferred variable annuity 1.2.2 Resets, roll-ups and ratchets 1.2.3 Guaranteed minimum maturity benefit 1.2.4 Guaranteed minimum accumulation benefit 1.2.5 Guaranteed minimum death benefit 1.2.6 Guaranteed minimum withdrawal benefit 1.2.7 Guaranteed lifetime withdrawal benefit 1.2.8 Mechanics of immediate variable annuity 1.2.9 Modeling of immediate variable annuity 1.2.10 Single premium vs flexible premium annuities 1.3 Equity-indexed annuities 1.3.1 Point-to-point option 1.3.2 Cliquet option 1.3.3 High-water mark option 1.4 Fundamental principles of equity-linked insurance 1.5 Bibliographic notes 1.6 Exercises 1 8 10 15 17 18 20 21 24 27 29 31 32 33 34 34 35 36 37 vii viii Contents Elementary Stochastic Calculus 2.1 Probability space 2.2 Random variable 2.3 Expectation 2.3.1 Discrete random variable 2.3.2 Continuous random variable 2.4 Stochastic process and sample path 2.5 Conditional expectation 2.6 Martingale vs Markov processes 2.7 Scaled random walks 2.8 Brownian motion 2.9 Stochastic integral 2.10 Itˆo formula 2.11 Stochastic differential equation 2.12 Applications to equity-linked insurance 2.12.1 Stochastic equity returns 2.12.2 Guaranteed withdrawal benefits 2.12.2.1 Laplace transform of ruin time 2.12.2.2 Present value of fee incomes up to ruin 2.12.3 Stochastic interest rates 2.12.3.1 Vasicek model 2.12.3.2 Cox-Ingersoll-Ross model 2.13 Bibliographic notes 2.14 Exercises 39 39 45 49 54 55 56 64 69 72 76 80 88 92 93 93 96 98 99 101 101 102 102 103 Monte Carlo Simulations of Investment Guarantees 3.1 Simulating continuous random variables 3.1.1 Inverse transformation method 3.1.2 Rejection method 3.2 Simulating discrete random variables 3.2.1 Bisection method 3.2.2 Narrow bucket method 3.3 Simulating continuous-time stochastic processes 3.3.1 Exact joint distribution 3.3.1.1 Brownian motion 3.3.1.2 Geometric Brownian motion 3.3.1.3 Vasicek process 3.3.2 Euler discretization 3.3.2.1 Euler method 3.3.2.2 Milstein method 3.4 Economic scenario generator 3.5 Bibliographic notes 3.6 Exercises 111 111 112 113 115 116 118 120 120 120 120 121 121 122 122 124 125 125 ix Contents Pricing and Valuation 4.1 No-arbitrage pricing 4.2 Discrete time pricing: binomial tree 4.2.1 Pricing by replicating portfolio 4.2.2 Representation by conditional expectation 4.3 Dynamics of self-financing portfolio 4.4 Continuous time pricing: Black-Scholes model 4.4.1 Pricing by replicating portfolio 4.4.2 Representation by conditional expectation 4.5 Risk-neutral pricing 4.5.1 Path-independent derivatives 4.5.2 Path-dependent derivatives 4.6 No-arbitrage costs of equity-indexed annuities 4.6.1 Point-to-point index crediting option 4.6.2 Cliquet index crediting option 4.6.3 High-water mark index crediting option 4.7 No-arbitrage costs of variable annuity guaranteed benefits 4.7.1 Guaranteed minimum maturity benefit 4.7.2 Guaranteed minimum accumulation benefit 4.7.3 Guaranteed minimum death benefit 4.7.4 Guaranteed minimum withdrawal benefit 4.7.4.1 Policyholder’s perspective 4.7.4.2 Insurer’s perspective 4.7.4.3 Equivalence of pricing 4.7.5 Guaranteed lifetime withdrawal benefit 4.7.5.1 Policyholder’s perspective 4.7.5.2 Insurer’s perspective 4.8 Actuarial pricing 4.8.1 Mechanics of profit testing 4.8.2 Actuarial pricing vs no-arbitrage pricing 4.9 Bibliographic notes 4.10 Exercises 127 127 131 131 137 138 140 141 143 144 144 146 148 148 149 149 151 151 153 154 156 156 156 157 159 160 161 163 164 175 177 178 Risk Management - Reserving and Capital Requirement 5.1 Reserve and capital 5.2 Risk measures 5.2.1 Value-at-risk 5.2.2 Conditional tail expectation 5.2.3 Coherent risk measure 5.2.4 Tail value-at-risk 5.2.5 Distortion risk measure 5.2.6 Comonotonicity 5.2.7 Statistical inference of risk measures 5.3 Risk aggregation 5.3.1 Variance-covariance approach 183 184 189 189 192 193 197 199 202 205 209 209 368 F Sample Code for Differential Equation Methods end end F.2 Probability of loss for guaranteed minimum withdrawal benefit % return approximate solution matrix s function s = val(dt, dx, dy, T, b, c, K, w, r, mw, sig, mu, m) % set up a uniform grid Nt = round(T / dt); Nx = round(b / dx); Ny = round(c / dy); % initialize solution matrix s = zeros(Nx + 1, Ny + 1, Nt + 1); % set up initial condition for i = 1:(Nx + 1) for j = 1:(Ny + 1) s(i, j, 1) = 1; end end % set up boundary condition for ‘‘time-like" variable for i = 1:(Nx + 1) for k = 1:(Nt + 1) s(i, 1, k) = 1; end end % define constants alpha = / dt; beta = dx / dy; gamma = w / dx; % update matrix solution recursively for k = 1:Nt C = diag((1:Nx) * beta * exp(- r * (T - (k - 1) * dt))); B1 = diag(((1:Nx) ˆ 2) * (sig ˆ 2) + (1:Nx) * beta * (exp(- r * (T - (k - 1) * dt))) + alpha, 0); B2 = diag(- (((1:(Nx - 1)) ˆ 2) * (sig ˆ 2) + (1:(Nx - 1)) * (mu - m) - gamma) / 2, 1); B3 = diag(- (((2:Nx) ˆ 2) * (sig ˆ 2) - (2:Nx) * (mu - m) + gamma) / 2, - 1); B = B1 + B2 + B3; F.2 Probability of loss for guaranteed minimum withdrawal benefit 369 B(Nx, Nx - 1) = - (sig ˆ 2) * (Nx ˆ 2); [L, U] = lu(B); for j = 1:(Ny) rhs3 = zeros(Nx, 1); rhs3(1, 1) = ((sig ˆ 2) - mu + m + gamma) * ((j - 1) * dy