An introduction to computational risk management of equity linked insurance

Ax expected present value of a life insurance that pays1 immediately upon the death of a life-agex ax expected present value of an annuity continuously payable at the rate of1 per period

An Introduction

to Computational Risk Management

of Equity-Linked

Insurance

CHAPMAN & HALL/CRC

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Chapman-and-HallCRC-Financial-Mathematics-Series/book-series/CHFINANCMTH

An Introduction

to Computational Risk Management

of Equity-Linked

Insurance

Runhuan Feng

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1.1 Fundamental principles of traditional insurance 1

1.1.1 Time value of money 1

1.1.2 Law of large numbers 3

1.1.3 Equivalence premium principle 5

1.1.4 Central limit theorem 8

1.1.5 Portfolio percentile premium principle 8

1.2 Variable annuities 9

1.2.1 Mechanics of deferred variable annuity 10

1.2.2 Resets, roll-ups and ratchets 15

1.2.3 Guaranteed minimum maturity benefit 17

1.2.4 Guaranteed minimum accumulation benefit 18

1.2.5 Guaranteed minimum death benefit 20

1.2.6 Guaranteed minimum withdrawal benefit 21

1.2.7 Guaranteed lifetime withdrawal benefit 24

1.2.8 Mechanics of immediate variable annuity 27

1.2.9 Modeling of immediate variable annuity 29

1.2.10 Single premium vs flexible premium annuities 31

1.3 Equity-indexed annuities 32

1.3.1 Point-to-point option 33

1.3.2 Cliquet option 34

1.3.3 High-water mark option 34

1.4 Fundamental principles of equity-linked insurance 35

1.5 Bibliographic notes 36

1.6 Exercises 37

vii

2 Elementary Stochastic Calculus 39

2.1 Probability space 39

2.2 Random variable 45

2.3 Expectation 49

2.3.1 Discrete random variable 54

2.3.2 Continuous random variable 55

2.4 Stochastic process and sample path 56

2.5 Conditional expectation 64

2.6 Martingale vs Markov processes 69

2.7 Scaled random walks 72

2.8 Brownian motion 76

2.9 Stochastic integral 80

2.10 Itˆo formula 88

2.11 Stochastic differential equation 92

2.12 Applications to equity-linked insurance 93

2.12.1 Stochastic equity returns 93

2.12.2 Guaranteed withdrawal benefits 96

2.12.2.1 Laplace transform of ruin time 98

2.12.2.2 Present value of fee incomes up to ruin 99

2.12.3 Stochastic interest rates 101

2.12.3.1 Vasicek model 101

2.12.3.2 Cox-Ingersoll-Ross model 102

2.13 Bibliographic notes 102

2.14 Exercises 103

3 Monte Carlo Simulations of Investment Guarantees 111 3.1 Simulating continuous random variables 111

3.1.1 Inverse transformation method 112

3.1.2 Rejection method 113

3.2 Simulating discrete random variables 115

3.2.1 Bisection method 116

3.2.2 Narrow bucket method 118

3.3 Simulating continuous-time stochastic processes 120

3.3.1 Exact joint distribution 120

3.3.1.1 Brownian motion 120

3.3.1.2 Geometric Brownian motion 120

3.3.1.3 Vasicek process 121

3.3.2 Euler discretization 121

3.3.2.1 Euler method 122

3.3.2.2 Milstein method 122

3.4 Economic scenario generator 124

3.5 Bibliographic notes 125

3.6 Exercises 125

Contents ix

4.1 No-arbitrage pricing 127

4.2 Discrete time pricing: binomial tree 131

4.2.1 Pricing by replicating portfolio 131

4.2.2 Representation by conditional expectation 137

4.3 Dynamics of self-financing portfolio 138

4.4 Continuous time pricing: Black-Scholes model 140

4.4.1 Pricing by replicating portfolio 141

4.4.2 Representation by conditional expectation 143

4.5 Risk-neutral pricing 144

4.5.1 Path-independent derivatives 144

4.5.2 Path-dependent derivatives 146

4.6 No-arbitrage costs of equity-indexed annuities 148

4.6.1 Point-to-point index crediting option 148

4.6.2 Cliquet index crediting option 149

4.6.3 High-water mark index crediting option 149

4.7 No-arbitrage costs of variable annuity guaranteed benefits 151

4.7.1 Guaranteed minimum maturity benefit 151

4.7.2 Guaranteed minimum accumulation benefit 153

4.7.3 Guaranteed minimum death benefit 154

4.7.4 Guaranteed minimum withdrawal benefit 156

4.7.4.1 Policyholder’s perspective 156

4.7.4.2 Insurer’s perspective 156

4.7.4.3 Equivalence of pricing 157

4.7.5 Guaranteed lifetime withdrawal benefit 159

4.7.5.1 Policyholder’s perspective 160

4.7.5.2 Insurer’s perspective 161

4.8 Actuarial pricing 163

4.8.1 Mechanics of profit testing 164

4.8.2 Actuarial pricing vs no-arbitrage pricing 175

4.9 Bibliographic notes 177

4.10 Exercises 178

5 Risk Management - Reserving and Capital Requirement 183 5.1 Reserve and capital 184

5.2 Risk measures 189

5.2.1 Value-at-risk 189

5.2.2 Conditional tail expectation 192

5.2.3 Coherent risk measure 193

5.2.4 Tail value-at-risk 197

5.2.5 Distortion risk measure 199

5.2.6 Comonotonicity 202

5.2.7 Statistical inference of risk measures 205

5.3 Risk aggregation 209

5.3.1 Variance-covariance approach 209

5.3.2 Model uncertainty approach 211

5.3.3 Scenario aggregation approach 216

5.3.3.1 Liability run-off approach 217

5.3.3.2 Finite horizon mark-to-market approach 219

5.4 Risk diversification 221

5.4.1 Convex ordering 221

5.4.1.1 Thickness of tail 222

5.4.1.2 Conditional expectation 225

5.4.2 Diversification and convex order 225

5.4.3 Law of large numbers for equity-linked insurance 227

5.4.3.1 Individual model vs aggregate model 227

5.4.3.2 Identical and fixed initial payments 228

5.4.3.3 Identically distributed initial payments 230

5.4.3.4 Other equity-linked insurance products 231

5.5 Risk engineering of variable annuity guaranteed benefits 231

5.6 Capital allocation 234

5.6.1 Pro-rata principle 235

5.6.2 Euler principle 238

5.7 Case study: stochastic reserving 241

5.7.1 Recursive calculation of surplus/deficiency 242

5.7.2 Average net liability 245

5.7.3 Aggregate reserve 247

5.7.4 Reserve allocation 249

5.8 Bibliographic notes 251

5.9 Exercises 252

6 Risk Management - Dynamic Hedging 257 6.1 Discrete time hedging: binomial tree 257

6.1.1 Replicating portfolio 258

6.1.2 Hedging portfolio 260

6.2 Continuous time hedging: Black-Scholes model 262

6.2.1 Replicating portfolio 262

6.2.2 Gross liability hedging portfolio 263

6.2.2.1 Continuous hedging in theory 263

6.2.2.2 Discrete hedging in practice 264

6.2.3 Net liability hedging portfolio 266

6.2.3.1 Pricing and hedging of derivatives with exogenous cash flows 266

6.2.3.2 Discrete hedging in practice 269

6.3 Greek letters hedging 273

6.4 Bibliographic notes 277

6.5 Exercises 278

Contents xi

7.1 Differential equation methods 281

7.1.1 Reduction of dimension 284

7.1.2 Laplace transform method 285

7.1.2.1 General methodology 285

7.1.2.2 Application 286

7.1.3 Finite difference method 288

7.1.3.1 General methodology 289

7.1.3.2 Application 291

7.1.4 Risk measures: guaranteed minimum withdrawal benefit 293

7.1.4.1 Value-at-risk 294

7.1.4.2 Conditional tail expectation 296

7.1.4.3 Numerical example 297

7.2 Comonotonic approximation 300

7.2.1 Tail value-at-risk of conditional expectation 300

7.2.2 Comonotonic bounds for sums of random variables 303

7.2.3 Guaranteed minimum maturity benefit 304

7.2.4 Guaranteed minimum death benefit 307

7.3 Nested stochastic modeling 312

7.3.1 Preprocessed inner loops 314

7.3.2 Least-squares Monte Carlo 317

7.3.3 Other methods 320

7.3.4 Application to guaranteed lifetime withdrawal benefit 321

7.3.4.1 Overview of nested structure 321

7.3.4.2 Outer loop: surplus calculation 325

7.3.4.3 Inner loop: risk-neutral valuation 328

7.3.4.4 Implementation 333

7.4 Bibliographic notes 343

7.5 Exercises 343

Appendix A Illustrative Model Assumptions 349 A.1 GMMB product features 349

A.2 GLWB product features 350

A.3 Life table 351

A.4 Lapsation rates 352

A.4.1 Base lapse rates 352

A.4.2 Dynamic lapse rates 352

Appendix B Big-O and Little-o 355 Appendix C Elementary Set Theory 357 C.1 Sets 357

C.2 Infimum and supremum 358

C.3 Convex function 358

Appendix D List of Special Functions 359

D.1 Gamma functions 359

D.2 Confluent hypergeometric functions 359

D.3 Owen’s T and related functions 360

Appendix E Approximation of Function by an Exponential Sum 361

Appendix F Sample Code for Differential Equation Methods 367

F.1 Distribution function of the running supremum of Brownian motion 367

F.2 Probability of loss for guaranteed minimum withdrawal benefit 368

List of Figures

1.1 Diagram of cash flows for variable annuity contracts 11

1.2 GMAB gross liability - Case 1 19

1.3 GMAB gross liability - Case 2 19

1.4 GMDB gross liability 20

1.5 GMWB gross liability 21

2.1 Random variable: a mapping from an abstract space to real line 46

2.2 Coin counting using Riemann integral and Lebesgue integral 52

2.3 Sample paths of random walk 58

2.4 Motivation of total variation 60

2.5 Plot of a function with infinite total variation 62

2.6 Visualization of random variable and its conditional expectation 66

2.7 Sample paths of scaled symmetric random walks 74

2.8 Histograms of sample from scaled random walk 75

2.9 Sample paths of stochastic integral and corresponding simple process and Brownian motion 82

2.10 A mapping from unit circle to unit interval 104

3.1 An illustration of bisection search algorithm 117

3.2 An illustration of narrow bucket algorithm 119

4.1 Two-period binomial tree model 134

4.2 Histogram of NPV profit 174

5.1 Value-at-risk and cumulative distribution function 190

5.2 VaR does not capture information in the tail 191

5.3 The support of a bivariate risk 196

5.4 Visualization of an integral of value-at-risk 197

5.5 Visual illustration of the identity (5.12) 201

5.6 Two methods of estimating VaR 206

5.7 Upper and lower bounds of distribution function and value-at-risk of the sum ofn shifted exponentials 216

5.8 Network representation of dependence 217

5.9 Logic flow of the liability run-off approach 218

5.10 Logic flow of the finite horizon mark-to-market approach 220

5.11 Comparison on thickness of tail 222

5.12 Random variables with the same finite support and mean 224

xiii

5.13 Risk management techniques embedded in equity-linked insurance 232

6.1 Sample path of equity index value 265

6.2 A sample path of gross liability hedging portfolio 266

6.3 Effectiveness of hedging portfolios with various rebalancing frequencies 271

6.4 Comparison of gross and net liability hedging strategies 272

7.1 Pattern of mesh points in a finite difference scheme 292

7.2 An illustration of nested simulations 313

7.3 Effectiveness of hedging with deltas calculated from formula without reduction of dimension 342

7.4 Effectiveness of hedging with deltas calculated from fromula with reduction of dimension 342

E.1 Approximation of mortality density by an exponential sum 365

List of Tables

1.1 GLWB with a step-up option 25

2.1 Cost of coverage per person 65

3.1 Distribution function of a future lifetime random variable 117

3.2 Bucket table 120

4.1 No-arbitrage argument to determine forward price 130

4.2 Profit testing - section 1 - projected account values 167

4.3 Profit testing - section 2 - projected cash flows from general and separate accounts 169

4.4 Profit testing - section 2 - projected cash flows with general account 171 4.5 Profit testing - section 3 - projected distributable earnings 172

5.1 Illustrative balance sheet 184

5.2 RBC formula worksheet 187

5.3 Weighting table for 50 scenario set 208

5.4 Reserve valuation method - projected values before decrements 242

5.5 Reserve valuation method - projected values after decrements 243

5.6 Reserve valuation method - projected income statement 244

6.1 Simulation of replicating portfolio 259

6.2 Simulation of hedging portfolio 261

7.1 Finite difference approximation off′(0) with step size of 0.1 290

7.2 Finite difference approximation off′(0) with step size of 0.01 290

7.3 Finite difference approximation off′′(0) 291

7.4 Loss distribution of GMWB net liability 300

7.5 Risk measures for the GMMB net liability 310

7.6 Risk measures for the GMDB net liability withδ = 0 311

7.7 Risk measures for the GMDB net liability withδ = 0.06, σ = 0.3 312 7.8 Preprocessed inner loop estimators 315

7.9 Exact values of GMMB no-arbitrage cost for different parameters 316 7.10 Estimation of GMMB no-arbitrage cost using preprocessed inner loop method 316

7.11 Least-squares parameter estimation 319

7.12 Estimation of GMMB no-arbitrage cost using LSMC 319

xv

7.13 Delta calculation based on the formula without reduction of

7.17 Accuracy of delta estimation on the grid using PDE and LSMC 338

7.18 Accuracy of delta estimation off the grid using PDE and LSMC 339

7.19 Accuracy of delta estimation off the grid using the preprocessed

inner loop method 340

7.20 Errors in the neighborhood of a point of large error 340

7.21 Comparison of CTE (best efforts) with various techniques 341

Ax expected present value of a life insurance that pays1 immediately upon the

death of a life-agex

ax expected present value of an annuity continuously payable at the rate of1

per period for life time of a life-agex

ax:t m expected present value of an annuity continuously payable at the rate of1

per period lasting fort periods computed with the force of interest m perperiod

tpx probability that a life-agex survives t years

tqx probability that a life-agex dies within t years

µx+t force of mortality, intensity with which death occurs at agex + t, µx+t=

−( dtpx/ dt)/tpx

Tx future lifetime of a life-agex

Kx curtate future lifetime of a life-agex, Kx= ⌊Tx⌋

V, σ2 variance

FX cumulative distribution function of random variableX

FX survival function of random variableX

φ standard normal density function

Φ standard normal cumulative distribution function

[·, ·] cross/quadratic variation

h·i total variation

⌊x⌋ largest integer smaller than or equal tox

⌈x⌉ smallest integer larger than or equal tox

While the concept of risk management may be as old as our civilization, the tive modeling of complex systems of interacting risks is a fairly recent development

quantita-in fquantita-inancial and quantita-insurance quantita-industries

Most traditional insurance products were developed to protect against particulartypes of risks For example, life insurance and annuities provide coverages for mor-tality and longevity risks, whereas property and casualty insurance compensates forlosses from human hazards and natural disasters However, over the past decades,there have been tremendous innovation and development in the insurance industrywith increasingly sophisticated coverages As the particular focus of this book, theintroduction of equity-linked insurance brings revolutionary changes to the market,exposing life insurers not only to traditional mortality and longevity risks but also

to unprecedented financial risks As the industry moves into a new territory of aging many intertwined financial and insurance risks, non-traditional problems andchallenges arise, presenting great opportunities for technology development.Today’s computational power and technology make it possible for the life insur-ance industry to develop highly sophisticated models, which were impossible just adecade ago Nonetheless, as more industrial practices and regulations move towardsdependence on stochastic models, the demand for computational power continues

man-to grow While the industry continues man-to rely heavily on hardware innovations, ing to make brute force methods faster and more palatable, we are approaching acrossroads about how to proceed There are many practitioners’ publications on suchissues, most of which, however, lack fine-level technical details due to the protection

try-of their proprietary interests This book is intended not only to provide a resource forstudents and entry-level professionals to understand the fundamentals of industrialmodeling practice, but also to give a glimpse of academic research on “software”methodologies for modeling and computational efficiency

Computational risk managementin the title of this book refers to a collection ofcomputational models and techniques drawn from various fields for the purpose ofquantifying and managing risks emerging from equity-linked insurance The bookcontains a fair amount of computational techniques well adopted in the insuranceindustry and their underpinning theoretical foundations However, this book is by

no means a comprehensive review of literature on the subject matter Readers areencouraged to explore the topics further with references in the bibliographical notes

at the end of each chapter

As a guiding principle of presentation in this introductory book, we only intend toillustrate fundamental structure of practical models without regressing to accountingdetails For example, we would only touch on reserves at a high level with a brief

xix

discussion of distinctions between statutory reserve, generally accepted accountingprinciple (GAAP) reserve, or tax reserve, etc None of the examples or models in thebook would require any accounting experience.

The idea of writing a book on this subject matter was inspired by my coauthor

Dr Jan Vecer, who have always multitasked many creative projects of his own Part

of this book has grown out of lecture notes for a one-semester topics course theauthor taught a few times at the University of Illinois at Urbana-Champaign It is notassumed that the reader has any substantial knowledge of life insurance or advancedmathematics beyond the level of calculus sequence and a first course in probabilityand statistics The book aims to be self-contained and should be accessible to most

of upper level undergraduate students from a quantitative background However, as

a reviewer of this book wittily quoted the Greek mathematician Euclid, “there is noroyal road to geometry” It is difficult to cover theoretical underpinnings of somesophisticated modeling without advanced machinery Therefore, the book is writtenwith the aim to strike a delicate balance between theory and practice The author isreceptive to hearing of any criticism and suggestion

Many students helped through the production of this book by pointing out typosand errors in the earlier versions of the lecture notes and demanding more examples.Special thanks should go to Haoen Cui, Longhao Jin, Fan Yang, who helped createmany graphs and examples in the book, and Justin Ho, Chongda Liu, Saumil Padhyafor correcting typos and valuable suggestions which improved the presentation of thebook

During the development of this text, the author received an endowed ship from the State Farm Companies Foundation and research grants from the Society

professor-of Actuaries and the Actuarial Foundation Any opinions expressed in this materialare those of the author and do not reflect the views of the State Farm Companies,the Society of Actuaries or the Actuarial Foundation The author is very grateful fortheir generous support

The author would also like to thank the Society of Actuaries for permission to

adapt materials in Chapter 7 from the research report Nested Stochastic Modeling for Insurance Companies(copyright © 2016 by Society of Actuaries)

Runhuan Feng, PhD, FSA, CERAUrbana and Champaign, Illinoisrfeng@illinois.edu

Modeling of Equity-linked Insurance

1.1 Fundamental principles of traditional insurance

Life insurance and annuities are contracts of financial arrangement under which aninsurer agrees to provide policyholders or their beneficiaries at times of adversitywith benefits in exchange for premiums For examples, a life insurance can make amuch needed lump sum benefit payment to a family to cover their sudden loss ofincome due to the death of a family member An annuity can provide living expensesfor an insured retiree who outlasts his or her lifetime savings Given that the industry

is in the business of long-term financial services, the design, production and agement of insurance are as much an art as a science However, as the intent of thisbook is to provide a thorough introduction and overview of fundamentals of equity-linked insurance, we tend to focus on the scientific and theoretical underpinning ofcomputational models and techniques used in the literature and in practice

man-While there is probably no “theory of everything” in insurance mathematics,there are certainly fundamental principles that form the bases of computational mod-els used in this field, which shall be reviewed in this chapter

1.1.1 Time value of money

One dollar today is not the same as one dollar in the future The entire State ofAlaska was purchased by the United States from Russia in 1827 at the price of7.2million dollars In today’s dollars, the same amount is probably only enough to buy

a penthouse in downtown Chicago The sharp contrast of purchasing power is best

explained by the concept of time value of money in the classical theory of interest.

An elementary tool to quantify the time value of money is the accumulationfunction, usually denoted bya(t), as a function of time t It describes the accumu-lated value at timet of an original investment of one dollar at time 0 By definition,

a(0) = 1 The one dollar is called principal An investor makes the investment of

principal in expectation of financial returns The amount by which the accumulated

value of the investment exceeds the principal is called interest Here we assume that

the term of investment has no bearing on the underlying interest rate Under thisassumption, for any long-term risk-free investment, it is sensible in the market todevelop an investment growth mechanism so that the accumulation satisfies the fol-

1

lowing equality at all times

which means the amount of interest earned by an initial deposit of one dollar over

t + s periods is equal to the amount of interest earned if the deposit of one dollar iswithdrawn at timet and then immediately deposited back for an additional s periods.The violation of such a simple property creates problems for both the lender and theborrower On one hand, if a bank provides a savings account that offers interest ratessuch thata(t + s) < a(t)a(s), then it is to the advantage of investors to withdrawmoney and re-deposit instantly since it accrues more interest in such a way Thefaster one can withdraw and re-deposit, the more gains Hence in theory any rationalinvestor would want to maximize the proceeds by continuously opening and closingaccounts, which induces infinite amount of transaction costs for the bank On theother hand, if the bank provides a savings account where interests are calculated insuch a way thata(t + s) > a(t)a(s), then a new investor who brings in the money

in amount ofa(t) at time t would never earn at any time in the future as much as

an old investor whose account equally values at a(t) at time t Knowing it is tothe disadvantage of new investors to invest any time after the account is offered, norational new investor would open a new account and the bank would soon run out ofthe business due to the inability to attract new clients

It can be shown mathematically*that the only continuous solution to (1.1) withthe conditiona(0) = 1 must be

a(t) = ert, t > 0,for some constantr, known as the force of interest, which is the intensity with whichinterest is operating on each one dollar invested att In theory of interest, this type

of accumulation is known as the investment with compound interest.

The force of interest is sometimes referred as continuously compounding interestrate For example, the accumulated value at timet of one dollar invested at time 0with a fixed annual interest rate ofr payable once per year is given by (1 + r)t If theinterest is compoundedn times per year, i.e the interest payment is divided n timesand payable every1/n-th of a year, then the accumulated value at time t is

a(t) =

1 + rn

nt

Hence the name forr continuously compounding interest rate

1.1 Fundamental principles of traditional insurance 31.1.2 Law of large numbers

The quintessence of an insurance business is the pooling of funds from a large ber of policyholders to pay for losses incurred by a few policyholders It is difficultfor individuals to absorb their own risks because potential losses can be too large

num-or too unpredictable However, an insurance company can manage the collective riskbecause the aggregated nature makes it more predictable The mathematical principle

behind the predictability of the collective risk is the law of large numbers (LLN).

Theorem 1.1(Law of large numbers) LetX1, X2, · · · , be an infinite sequence of i.i.d random variables withE(X1) = E(X2) = · · · = µ Then

Xn:= 1

We can think of random variables{X1, X2, · · · } as individual claims from aninsurance coverage on any insurable risk, whether short-term insurance, such as fire,automobile, homeowner insurances, or long-term insurance, such as life and an-nuities Despite the uncertainty as to the timing and size of any individual claim,the LLN dictates that the average claim sizeXn, representing the collective risk, isroughly the theoretical mean of any individual claim, which is a fixed amount This isprecisely why the collective risk is much more manageable, as the insurer is expected

to pay a fixed amount for each contract on average

Example 1.2 (Pure endowment) Let us consider a simple example of insurable risk

– mortality risk, which is a risk of potential losses to a life insurer resulting from theuncertainty with the timing of policyholders’ deaths Suppose that an insurer sells to

n policyholders of the same age x a pure endowment life insurance, which pays alump sum ofB dollars upon the survival of the policyholder at the end of T years Wedenote the future lifetime of thei-th policyholder at age x by Tx(i) and the survivalprobability byTpx := P(Tx(i) > T ) Following actuarial convention, we write theprobability that thei-th policyholder survives k years,

kpx:= P(Tx(i) > k)

Then the insurance claim from the i-th policyholder can be represented byBI(Tx(i) ≥ T ) where the indicator I(A) = 1 if the event A is true or 0 other-wise Even though there is uncertainty as to whether each contract incurs a claim,the LLN says that the percentage of survivors is almost certain and so is the averageclaim (survival benefit) from each contract, i.e

BI(Tx(i) ≥ T ) −→ E[BI(Tx(1)≥ T )] = BTpx, n → ∞

In other words, on average, the cost of each contract, which is the amount to cover

a claim, would be fixed Therefore, mortality risk is considered a diversifiable risk,

meaning the uncertainty can be eliminated through the diversification of underwriting

a large portfolio of homogeneous contracts

Example 1.3 (Whole life insurance) Suppose that an insurer sells ton policyholdersall at agex the same life insurance, which pays a lump sum of B dollars at the end ofyear of a policyholder’s death Earlier death benefit payments represent higher costs

to insurers due to the time value of money Assume that the overall continuously

compounding yield rate on an insurer’s assets backing up its liabilities isr per year.Then the accumulated value of one dollar investment afterT years would be erT.Assuming that the growth of an investment is proportional to its principal, in order

to pay one dollar at timeT , the insurer should invest with compound interest e−rT

dollars, which is called the present value of one dollar to be paid at timeT Therefore,the present value of death benefit to be paid at the end of the year of death is givenby

Xi:= Be−r⌈Tx(i)⌉,where⌈y⌉ is the smallest integer greater than y Or equivalently, we can rewrite it as

nqx+k:= P(Tx+k(i) < n) = P(Tx(i)< k + n|Tx(i) > k)

We often suppress the subscript on the lower left corner ifn = 1 Therefore,

P(k−1 < T(i)

x ≤ k) = P(Tx(i)< k|Tx(i) > k−1)P(Tx(i) > k−1) = k−1pxqx+k−1.While the actual present value of death benefit is uncertain for each contract, theaverage cost of the life insurance is fixed with a sufficiently large pool of policies.1

Therefore, mortality risk is considered a diversifiable risk, meaning the uncertainty

can be eliminated through the diversification of underwriting a large portfolio ofhomogeneous contracts

Example 1.4 (Immediate life annuity) Another example of insurable risk is longevity

risk, which is a risk of potential losses to an annuity writer resulting from the tainty with the amount of annuity payments Since annuity payments are guaranteedfor lifetime, typically higher life expectancy of a policyholder translates to a largerloss for an annuity writer Consider a life annuity under which the first payment oc-curs one year from now and all payments of level amountC are made on an annualbasis In any given year[k, k + 1], a payment is made as long as the policyholder is

uncer-1.1 Fundamental principles of traditional insurance 5still alive Hence its present value is given bye−r(k+1)CI(Tx(i)≥ k) Adding presentvalues of all future annuity payments yields the present value of the life annuity,

asn → ∞, provided that the infinite summation is finite

1.1.3 Equivalence premium principle

A remarkable consequence of the LLN is that an insurer can charge a fixed premium

to cover random claims A very common form of the payment for short term ance is a single premium to be paid at the inception of the contract To consider thenet cost of an insurance liability, we often do not take into account expenses andtaxes and the single premium to be determined, denoted byP , is called a net pre- mium From the viewpoint of an insurance company, the individual net liability of

insur-thei-th contract is the net present value of future liability less future income, noted byLi

Whenever there is no ambiguity, we shall suppress the subscripti for brevity As

common in the literature, we often refer to a positive net liability as a loss and the negative net liability as a profit.

Justified by the LLN, we can set the premium such that the average claims outgomatches the average premium income, i.e the average individual net liability would

be zero

which implies, in the case of a single premium, that

where we useX as the generic random variable for i.i.d {X1, · · · , Xn} This

ap-proach is known as the equivalence principle.

Example 1.5 (Pure endowment) The net premium for the pure endowment contract

is the expected present value of future survival benefit

P = e−rTBTpx,where the factore−rT represents the present value of asset investment in order toaccumulate to1 at time T

Example 1.6 (Pure endowment (continued)) Consider the survival benefit to beB =

100, the probability of survivalTpx= 0.1, the T -year discount factor e−rT = 0.9.Then the net premium is given by9 Therefore, the net liability from a single pureendowment insurance contract is given by

is easy to see that the average loss for each contract, i.e.L := (1/1000)P1000

i=1 Li

has mean0 and variance V(Li)/1000 = 0.729 The variation of profit/loss is sosmall that even the probability of having a loss bigger than2 is nearly zero To beexact, P(L > 2) = 0.0030395484 This is an example of the dramatic effect of thediversification of mortality risk

In the early days of life insurance, most contracts are issued on an annual basis

As explained by the LLN and the equivalence principle, the annual premium would

be set at the average cost of annual insurance claims As a policyholder ages, aninsurer would charge a higher premium every year in order to cover rising averagecost of claims Life insurance would become unaffordable at advanced ages when thecoverage is needed the most Therefore, the funding mechanism of merely matchingincoming and outgoing cash flows on an annual basis became an impediment to thedevelopment of long term insurance A radical development came about in the lateeighteenth century, when life actuaries developed the concept of level premium Thenew funding mechanism allows policyholders to lock in a level premium rate for anumber of years or life time These policies quickly gained popularity and became

a standard practice as level premiums were much cheaper than annual premiums forthe elder, thereby insurers can retain policies for longer periods

The innovation with level premium relies not only on the diversification effectamong a large portfolio of policyholders, but also on the redistribution of insurancecosts across different ages In other words, the sustainability of level premium isthe result of young and mid-age policyholders contributing more than their actualaverage costs to compensate for the elderly paying less than their actual averagecosts

1.1 Fundamental principles of traditional insurance 7

Example 1.7 (Whole life insurance) Consider a whole life insurance that pays a

lump sumB immediately upon a policyholder’s death Level premiums are payablecontinuously at the rate ofP per year until the policyholder’s death Note that in thiscase, both the insurance liability and premium income are life contingent, that is, theincome and outgo cash flows depend on the time of the policyholder’s death Supposethat the overall yield rate on the insurer’s assets backing up the liability is given byrper year The present value of the insurer’s net liability for thei-th contract is givenby

Li= Be−rTx(i)− P

ˆ T (i) x

0

According to the LLN, the per unit average cost of the death benefit is given by

Ax:= E(e−rTx)

Similarly, the average of a policyholder’s regular payments of1 per time unit, known

as a life annuity, is given by

1 = Ax+ rax.This is in fact an application of the simple calculus rule - integration by parts,

a profit or loss on an individual contract, the product breaks even when the life ance is sold at the net premium

insur-Example 1.8 (Deferred life annuity) In contrast with Example 1.4, the stream ofpayments under a deferred life annuity starts after a deferred period This is oftenoffered as an employee benefit to current employees, for which annuity incomes arepaid after their retirement Consider a whole life annuity payable continuously at therate of C per year and deferred for n years Suppose that the deferred annuity ispurchased with a single premiumP Then the present value of the net liability forthei-th contract is given by

Li= C

ˆ T (i)

e−rtdt − P

According to the LLN, the average cost of the deferred annuity per dollar annuitypayment is given by

P := e−rn

npxax+n

1.1.4 Central limit theorem

A second mathematical principle of great significance in insurance modeling is thecentral limit theorem (CLT) While there are many variations of the CLT, we present

a version most commonly used in actuarial practice

Theorem 1.9(Lindeberg-L´evy central limit theorem) LetX1, X2, · · · , be an nite sequence of i.i.d random variables with finite mean and variance Then

infi-Zn:= Xn− E(Xn)

qV(Xn)

where Z is a standard normal random variable.

The type of limits used in both (1.3) and (1.8) is known as convergence in tribution It means that for any givenz ∈ (−∞, ∞), the sequence of distributionfunctions{P(Zn≤ z), n = 1, 2, · · · } converges to P(Z ≤ z) as n goes to infinity.1.1.5 Portfolio percentile premium principle

dis-While net premiums determined by the equivalence principle are used for many poses in the insurance industry, a more common approach of setting premiums is

pur-based on profitability tests at the level of a business line The aggregate net liability,

denoted by L, is the sum of individual net liabilities in the whole portfolio of policies

1.2 Variable annuities 9that the aggregate net liability is negative (or the product is profitable) If L is acontinuous random variable*, then

This is often called the portfolio percentile premium principle Of course,

apply-ing such a principle requires one’s knowledge of the aggregate net liability, the act distribution of which can be difficult to obtain However, central limit theorem(CLT) tells us that L is approximately normally distributed with E(L) = nE(Li) andV(L) = nV(Li) Therefore, the premium P can be determined by

ex-P(L < 0) = P L− E(L)

pV(L) <

−E(L)pV(L)

P =

pV(X)Φ−1(α)

Policyholders contribute premiums, also known as purchase payments, into vestment accounts at the policy issue and expect to reap financial gain on the invest-ment of their payments Typically policyholders are offered a variety of investment

in-“sub-accounts”, each of which is invested in a particular equity (stock) fund with adistinct investment objective Although subject to different regulations, these “sub-accounts” are analogues of publicly available mutual funds Here is an example ofvarious investment options:

• Aggressive Allocation Portfolio (80% stocks/20% bonds)

• Balanced Portfolio (60% stocks/40% bonds)

• Conservative Allocation Portfolio (40% bonds/60% stocks)

Once selected, the performance of a sub-account is linked in proportion to the tuation of equity index or equity fund in which it invests This is known as the

fluc-equity-linking mechanism For this reason, variable annuities are often known as unit-linked productsin Europe These sub-accounts may include domestic and in-ternational common stock funds, bond funds, money market funds, and specialtyfunds (e.g real estate investment trusts, etc.)

All of policyholders’ premiums in subaccounts are typically invested and aged by third party professional vendors From an accounting point of view, these

man-assets are owned by policyholders and considered to be in separate accounts, apart from an insurer’s own investment assets, which are in general accounts Hence vari- able annuities are often referred to as segregated funds products in Canada.

Immediate variable annuitiesare those for which periodic payments to

policy-holders commence on the purchase of the annuity, whereas deferred variable nuitiesare those for which benefit payments to policyholders are deferred to somefuture date, for example, a retirement age Note, however, policyholders may elect totake their money at the future date in a lump sum, in which case the variable annuity

an-is not an annuity in the strict sense Or they may elect to annuitize their money, inwhich case a series of benefit payments occur afterwards

1.2.1 Mechanics of deferred variable annuity

Sources of investment

There are two sources of variable annuity sales in the US market The majority of

annuity sales come from qualified assets, which are funds that comply with federal

tax code retirement plans such as traditional individual retirement accounts (IRA),etc and therefore are eligible for certain tax advantages, whereas other retail sales

are referred to non-qualified assets which come from an after-tax source such as

a savings account, the sale of real estate, etc Sometimes variable annuity designsspecify different investment guarantees for qualified and non-qualified assets.Investment guarantees

In the early days, variable annuity writers merely act as the steward of holders’ investments and the financial risks of subaccounts are entirely transferred to

policy-1.2 Variable annuities 11policyholders In the early 2000s, there have been some changes to the US tax codewith a result that many tax advantages of variable annuity start to diminish In order tocompete with mutual funds, nearly all major variable annuity writers introduced var-ious types of investment guarantees, such as guaranteed minimum maturity benefit(GMMB), guaranteed minimum death benefit (GMDB), guaranteed minimum accu-mulation benefit (GMAB), guaranteed minimum withdrawal benefit (GMWB), etc.These investment guarantees are considered riders to base contracts and in essencetransfer the downside risk of financial investment from policyholders to back insur-ers We shall explore the details of these riders in the next few subsections.

Policyholders

Separate

Account

General Account Advisors

Fund Managers

withdrawals guarantee

pay-ments

advising

commission

inv mgmt charge ins charge

rider charges surrender charge

investment financialreturns

inv.

mgmt fees

revenue sharing

Insurer

premium

· account-value based

· guarantee basedFIGURE 1.1: Diagram of cash flows for variable annuity contracts

Sources of revenue

In order to compensate for all necessary business costs and to maintain healthyand profitable operations, insurers levy on policyholders a range of fees and charges,which are typically unbundled and disclosed in a transparent manner on productprospectus While there are one-time or annual fixed fees by policy, most fees andcharges are assessed and deducted from policyholders’ subaccounts on a daily ba-sis as percentages of account values or guarantee values While terminology variesgreatly in the market, the common fees and charges are usually in the following cat-egories

• Rider charge: These charges compensate insurers for the costs of various ment guarantees and usually split by benefits These are often stated as percent-ages of guaranteed values

invest-• Insurance charge: This charge is used to compensate insurers for providing a baseGMDB and also quoted as a percentage of the account value

• Administrative expense charge: this charge covers the costs associated with ing and distributing the products It can be an annual per policy fixed rate or a

serv-percentage of the account value Expense charge is often combined with ance charge and known as mortality and expenses fee (M&E fee).

insur-• Investment management fee: This fee is paid to professional fund managers fortheir services It is typically stated as a percentage of the account value

• Surrender charge: This is also known as contingent deferred sales charge(CDSC) These charges are imposed in the event of policy lapse and intended

to offset upfront costs associated with policy issue The charges are typically sethigh on the first year and gradually decline to zero The ramp structure is used

to discourage early lapsation They are often a percentage of the account value.This may also include withdrawal charge, which apply to a policyholder whomakes a partial withdrawal in excess of a maximal penalty-free amount

While there is a great variety of product development in the industry, the focus

of this book is to provide an understanding of the modeling and risk management ofcommon product features Therefore, we shall always take a minimalist approach toconsider simplified models in order to bring out the essential elements and generalideas

To facilitate the discussion of stochastic models for variable annuities, we duce the alphabet soup of notation to be used throughout the notes Details will beprovided as they appear in the following sections

intro-• St, the market value of the underlying equity index or fund att If more than onefund is involved, this is considered to be the portfolio value of all funds

• Ft, the market value of the policyholder’s subaccounts att ≥ 0 F0is considered

to be the initial premium invested at the start of the contract

• Gt, the guaranteed base used to determine the amount of payment to the icyholder from various riders at timet ≥ 0 Examples will follow in the nextsection

pol-• n, the number of valuations per year

• m, the nominal annualized rate at which asset-value-based fees are deductedfrom subaccounts Although we do not explicitly indicate the frequency of fee

1.2 Variable annuities 13payments, we typically assume that the payment period matches the valuationperiod.

The portion available for funding the guarantee cost is called margin offset orrider charge and is usually split by benefit In the notes, we denote the annualizedrate of charges allocated to the GMMB bymeand that of the charges allocated

to the GMDB bymd Note that in generalm > me+ mdto allow for overheads,commissions and other expenses

• h, the annualized rate at which guarantee-based fees are deducted from counts

subac-• r, the continuously compounding annual risk-free rate This typically reflects theoverall yield rate of assets in the insurer’s general account backing up guaranteedbenefits

• T , the target value date (or called maturity date), typically a rider anniversary onwhich the insurer is liable for guarantee payments

• Txis the future lifetime of the policyholder of agex at inception

• L, the net present value of future liabilities at the start of the contract The niques to be introduced in the notes can be used to analyze liabilities at any othervaluation date We shall focus on the start of the contract for simplicity

tech-Let us first consider the cash flows of a stand-alone variable annuity contract Thelife cycle of a variable annuity contract can be broken down into two phases.Accumulation phase

Policyholders’ investment accounts grow in proportion to certain equity-indices

in which policyholders choose to invest at the inception In practice, a policyholder’account value is often calculated by two measurements

• (Accumulation unit) The policyholder’s account is credited with a number ofaccumulation units based on the policyholder’s net asset value The number ofaccumulation units is equal to the initial purchase payment divided by the ac-cumulation unit value Additional purchase payments and transfers into the sub-account increase the number of accumulation units, while withdrawals, transfersout of the sub-account and deduction of certain contract charges often decreasethe number of accumulation units

• (Accumulation unit value) The initial accumulation unit value for each account is often set arbitrarily Each sub-account’s accumulation unit value isthen adjusted each business day to reflect income and capital gains of underlyingequity index/fund, and the deduction of M&E fees, etc

sub-For example, if you make an initial purchase payment of$1, 000 on Monday andthe accumulation unit value at the end of day is$10, then your account is creditedwith100 units If you take a withdrawal of $100 before the end of Tuesday, then the

number of accumulation units reduces to90 units After the stock exchange closes onTuesday, it is determined that each unit value increases from$10 to $10.25 for yourselection of investment fund Then your account is worth$10.25 × 90 = $922.5 onTuesday night In other words, the policyholder’s total value for each sub-account isthe product of the number of accumulation units and accumulation unit value, each ofwhich fluctuates for various reasons These measurements are intuitive and necessaryfor practical purposes, such as easy-to-explain for policyholders and transparency

of fees and charges, etc However, they unnecessarily produce lengthy notation forthe mathematical model to be introduced Therefore, in order to bring out a concisemodel, we ignore the nuances of changes to accumulation unit versus those to unitvalue and only consider the evolution of total values of a sub-account throughout thebook, rather than the breakdown of accumulation unit and unit value

First consider the discrete time model with a valuation period of1/n of a timeunit, i.e.t = 1/n, 2/n, · · · , k/n, · · · , T The fees and charges by annuity writersare typically taken as a fixed percentage of the-then-current account values for eachperiod The equity-linked mechanism for variable annuity dictates that at the end ofeach trading day, the account value fluctuates in proportion to the value of equityfund in which it invests and deducted by account-value-based fees

Observe that the income from the insurer’s perspective is generated by a stream ofaccount-value-based payments The present value of fee incomes, also called marginoffset, up to thek-th valuation period is given by

Before diving into details of various riders, we take a moment to discuss the mon types of guarantee base, from which all benefits are determined Guarantee base

com-is the notional amount used solely for the purpose of determining guarantee benefits.Variable annuities generally provide guaranteed appreciation of the guarantee baseregardless of the performance of the policyholder’s account value

1.2 Variable annuities 151.2.2 Resets, roll-ups and ratchets

Resetoption is typically associated with the automatic renewal of variable annuitycontracts with fixed terms It is intended to allow a policyholder to lock in investmentreturns When the contract is renewed, the new guarantee base is reset at the levelwhich is the greater of the guarantee base from the previous term and the accountvalue at the renewal date Let{T1, T2, · · · , } be a sequence of renewal dates withthe understanding thatT0 = 0 With a reset option, the guarantee base at time Tk isgiven by

GT k= max{GTk−1, FT k}, k = 1, 2, · · ·

Roll-upoption allows the guarantee base to accrue interest throughout the term

of the policy For example, if the roll-up rateρ is a nominal rate payable n times peryear, then the guarantee base is determined by

G(k+1)/n = Gk/n



1 + ρn

, fork = 0, 1, · · · Note that this recursive relation implies that

Gk/n= G0



1 + ρn

k

Step-upoption, which is also known as ratchet option, is in essence a form of resetoption The guarantee base can increase with the policyholder’s investment account

at the end of each period However, the guarantee base would never decrease, even

if the investment account loses value There are two common types of methods tostep-up the guarantee base

• Lifetime high step-up: This first type of step-up is based on lifetime high If the

current account value exceeds the guarantee base from the previous period, thenthe guarantee is reset to the current account value Otherwise, the guarantee baseremains the same

G(k+1)/n = max

Gk/n, F(k+1)/n

, fork = 0, 1, · · · (1.15)Observe that this recursive relation leads to the representation

Gk/n= max

j=0,1,··· ,k



Fj/n

• Annual high step-up: With the second type of step-up, the current account value

is compared to its value at the previous policy anniversary If the current accountvalue exceeds its previous value, then the guarantee base is increased by the samepercentage as the account value since the last anniversary

By rearrangement, the recursive relation can also be written as

Sometimes these options are combined to offer guaranteed compound growth

on the guarantee base and to allow the guarantee base to “lock in” gains from thepolicyholder’s designated investment:

G(k+1)/n = maxn

Gk/n



1 + ρn

, F(k+1)/n

o, fork = 0, · · · (1.17)Observe that this option also has a representation

Gk/n=

1 + ρn

−j

Fj/n

, F(k+1)/n

j=0,··· ,k



1 + ρn

−j

Fj/n

,

1 + ρn

−(k+1)

F(k+1)/n

,

which agrees with (1.18) withk replaced with k + 1

There are also other combinations in the market For example, a common practice

is to offer the greater of a step-up option and a roll-up option:

k

j=0,1,··· ,k{Fj/n}



While discrete time models are easy to explain, their continuous time parts are often more elegant in representations and more mathematically tractable

counter-in advanced modelcounter-ing If we divide each time unit counter-inton subintervals and let n go

to infinity in all models above, then we can obtain their continuous time analogues.Many of these representations use the elementary identity in (1.2)

• Roll-up option:

Gt= lim

n→∞G⌈nt⌉

n = G0eρt,where⌈x⌉ is the integer ceiling of x

1.2 Variable annuities 17

• Lifetime high step-up option:

Gt= sup

where the supremum is defined inAppendix C.2

• Combination of roll-up and step-up #1:

The continuous-time analogue of annual high step-up option requires the concept

of positive variation, which is to be introduced inSection 2.4 The continuous-timeanalogue of (1.16) is given by

1.2.3 Guaranteed minimum maturity benefit

The guaranteed minimum maturity benefit (GMMB) guarantees the policyholder aminimum monetary amountG at the maturity T As the insurer is only possibly liablefor the amount by which the guarantee exceeds the policyholder’s account balance atmaturity, the present value of the gross liability to the insurer is

e−rT(G − FT)+I(Tx> T ), (1.20)where(x)+ = max{x, 0} Consider the individual net liability of the guaranteed

benefits from the insurer’s perspective, which is the gross liability of guaranteedbenefits less the fee incomes The present value of the GMMB net liability is givenby

L(n)e (Tx) := e−rT(G − FT)+I(Tx> T ) − MT ∧T x,

wherex ∧ y = min{x, y} and the margin offset is given by (5.55)

We shrink the valuation period to zero by takingn to ∞, thereby reaching thelimiting continuous-time model Recall that

rate of total charges As a result, for each sample path, the continuous-time analogue

1.2.4 Guaranteed minimum accumulation benefit

It is increasingly common that GMMB riders are designed to be automatically newed at the end of their fixed terms Such a rider is often referred to as the guaranteeminimum accumulation benefit (GMAB)

re-Consider a two-period GMAB for an example Suppose that the original GMMBmatures at the end ofT1periods, when the policyholder is guaranteed to receive aminimum of a pre-determined guarantee amount, sayG0 In other words, the invest-ment account is worthmax(G0, FT 1), at which the new guarantee is reset Let T2bethe maturity of the renewed contract There are two cases to be considered, as shown

inFigures 1.2and1.3

1 The equity performs so poorly in the first period that the policyholder’saccount investmentFT 1 drops belowG0 at the renewal date Then theinsurer is responsible for injecting the additional cash(G0 − FT 1) intothe policyholder’s account and the new guaranteed levelG1remains thesame asG0

2 The equity performs so well in the first period thatFT 1exceedsG0 Thenthere is no payment from the insurer However, the guaranteed level forthe second period is reset toFT 1

In other words, the policyholder should never lose what has been accumulatedfrom the previous period

GT = max(G0, FT ),

FIGURE 1.3: GMAB gross liability - Case 2

for which the account value fluctuates in sync with equity index,

com-GT = max(GT , FT ),