An Introduction to Computational Risk Management of Equity-Linked Insurance

Content: Modeling of Equity-linked Insurance Fundamental principles of traditional insurance Time value of money Law of large numbers Equivalence premium principle Central limit theorem Portfolio percentile premium principle Variable annuities Mechanics of deferred variable annuity Resets, roll-ups and ratchets Guaranteed minimum maturity benefit Guaranteed minimum accumulation benefit Guaranteed minimum death benefit Guaranteed minimum withdrawal benefit Guaranteed lifetime withdrawal benefit Mechanics of immediate variable annuity benefit Modeling of immediate variable annuity Single premium vs flexible premium annuities Fundamental principles of equity-linked insurance Equity-indexed annuities Point-to-point designs Cliquet designs High water mark designs Bibliographic notes Exercises Elementary Stochastic Calculus Probability space Random variable Expectation Discrete random variable Continuous random variable Stochastic process and sample path Conditional expectation Martingale versus Markov processes Scaled random walks Brownian motion Stochastic integral It^o's formula Stochastic differential equation Applications to equity-linked insurance Stochastic equity returns Guaranteed withdrawal benefitsLaplace transform of ruin time Present value of total fees up to ruin Stochastic interest rates Vasicek model Cox-Ingersoll-Ross (CIR) model Exercises Monte Carlo Simulations of Investment Guarantees Simulating continuous random variables Inverse transformation method Rejection method Simulating discrete random variables Simulating continuous-time stochastic processes Exact joint distribution Brownian motion Geometric Brownian motion Vasicek process Euler discretization Euler method Milstein method Economic scenario generator Exercises Pricing and Valuation No-arbitrage pricing Discrete time pricing: binomial tree Pricing by replicating portfolio Representation by conditional expectation Dynamics of self-financing portfolio Continuous time pricing: Black-Scholes model Pricing by replicating portfolio Representation by conditional expectation Risk-neutral pricing Path-independent derivatives Path-dependent derivatives No arbitrage costs of investment guarantees Guaranteed minimum maturity benefit Guaranteed minimum accumulation benefit Guaranteed minimum death benefit Guaranteed minimum withdrawal benefit Policyholder's perspective Insurer's perspective Equivalence of pricing Guaranteed lifetime withdrawal benefit Policyholder's perspective Insurer's perspective Actuarial pricing Mechanics of profit testing Actuarial pricing vs no-arbitrage pricing Exercises Risk Management - Reserving and Capital Requirement Reserve and capital Risk measures Value-at-Risk Conditional tail expectation Coherent risk measure Tail-value-at-risk Distortion risk measure Comonotonicity Statistical inference of risk measures Risk aggregation Variance-covariance approach Model uncertainty approach Scenario aggregation approach Liability run-o_ approach Finite horizon mark-to-market approach Risk diversification Convex ordering Thickness of tail Conditional expectation Individual model vs aggregate model Law of large numbers for equity-linked insurance Identical and fixed initial payments Identically distributed initial payments Risk engineering of variable annuity guaranteed benefitsCapital allocation Pro-rata principle Euler principle Stochastic reserving by example Exercises Risk Management - Dynamic Hedging Discrete time hedging: binomial tree Replicating portfolio Hedging portfolio Continuous time hedging: Black-Scholes model Greek letters hedging Advanced Computational Methods Differential equation methods Reduction of dimension Laplace transform method General methodology Application Finite difference method General methodology Application Application to guaranteed minimum withdrawal benefit Value-at-risk of individual net liability Conditional tail expectation of individual netliability Numerical example Comonotonic approximation Tail value-at-risk of conditional expectation Comonotonic bounds for sums of random variables Guaranteed minimum maturity benefit Application to guaranteed minimum benefit Guaranteed minimum death benefit Nested stochastic modeling Preprocessed inner loops Least-squares Monte Carlo Application to guaranteed lifetime withdrawal benefit Overview of nested structure Outer loop: surplus calculation Inner loop: risk-neutral valuation Computational techniques Exercises