Financial modeling, actuarial valuation and solvency in insurance

Springer Finance

Editorial Board

Marco Avellaneda

Giovanni Barone-Adesi

Mark Broadie

Mark H.A. Davis

Emanuel Derman

Claudia Klüppelberg

Walter Schachermayer

Springer Finance

Springer Finance is a programme of books addressing students, academics and prac￾titioners working on increasingly technical approaches to the analysis of financial

markets. It aims to cover a variety of topics, not only mathematical finance but

foreign exchanges, term structure, risk management, portfolio theory, equity deriva￾tives, and financial economics.

For further volumes:

http://www.springer.com/series/3674

Mario V. Wüthrich Michael Merz

Financial Modeling,

Actuarial Valuation

and Solvency

in Insurance

Mario V. Wüthrich

RiskLab

Department of Mathematics

ETH Zurich

Zurich, Switzerland

Michael Merz

Faculty for Economic and Social Studies

Department of Business Administration

University of Hamburg

Hamburg, Germany

ISSN 1616-0533 ISSN 2195-0687 (electronic)

ISBN 978-3-642-31391-2 ISBN 978-3-642-31392-9 (eBook)

DOI 10.1007/978-3-642-31392-9

Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013936252

Mathematics Subject Classification: 62P05, 91G30

JEL Classification: G22, D52, D53, D82, E43, G12, G17, G32, G38

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Acknowledgements

This book is the product of an ongoing project we have been working on for sev￾eral years. As such it was not really defined as a project but it is rather the re￾sult of many activities we have been involved in. These include our own prac￾tical experience; discussions with regulators, scientists, practitioners, politicians,

other decision-makers, colleagues and students; continuing education at confer￾ences, workshops, working groups and our own lectures. We are deeply grateful

to ETH Zürich and to University of Hamburg. During all these times we were very

generously supported by our departments at these universities, and we have and

continue to experience these environments as stimulating and motivating. Special

thank-you’s are reserved for Prof. Hans Bühlmann and Prof. Paul Embrechts for

their continued support.

We greatly appreciate that the present manuscript profits from various inspir￾ing discussions, continuative thoughts, helpful contributions and critical comments

with and by several people: Hansjörg Albrecher, Peter Antal, Philipp Arbenz,

Manuela Baumann, Hans Bühlmann, Bikramjit Das, Catherine Donnelly, Karl￾Theodor Eisele, Paul Embrechts, Peter England, Vicky Fasen, Damir Filipovic,´

Alois Gisler, Sebastian Happ, Enkelejd Hashorva, Frank Häusler, John Hibbert,

Laurent Huber, Philipp Keller, Roger Laeven, Alexander McNeil, Christoph Möhr,

Antoon Pelsser, Enrico Perotti, Eckhard Platen, Simon Rentzmann, Robert Salz￾mann, Marc Sarbach, Urs Schubiger, Pavel Shevchenko, Werner Stahel, David Ste￾fanovits, Josef Teichmann, Andreas Tsanakas, Richard Verrall, Frank Weber, Armin

Wolf and Hans Peter Würmli. We especially thank Manuela Baumann for coding

Example 3.21. Moreover, we appreciate that several anonymous reviewers have read

previous versions of this manuscript very carefully. They have approached the sub￾ject from several different angles which has led us to provide a more comprehensive

and complete description of the topic and helped us bridge a few gaps in previous

versions of the manuscript.

Special thanks go to Alessia, Luisa, Anja, Rosmarie, Valo, Coral, Jürg, Gior￾gio, Matthias, Stephan, Ted, Juvy, Ursin, Francesco, Peter and Peter, Fritz, Reini.

v

vi Acknowledgements

Last but not least we thank Dave, Martin and Andy for endlessly enjoying the

silence.

Mario V. Wüthrich

Michael Merz

Zurich, Switzerland

Hamburg, Germany

February 2013

Contents

1 Introduction ................................ 1

1.1 Full Balance Sheet Approach . ................... 3

1.2 Solvency Considerations . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Further Modeling Issues . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Outline of This Book ........................ 6

Part I Financial Valuation Principles

2 State Price Deflators and Stochastic Discounting ........... 11

2.1 Zero Coupon Bonds and Term Structure of Interest Rates ..... 11

2.1.1 Motivation for Discounting ................. 11

2.1.2 Spot Rates and Term Structure of Interest Rates ...... 12

2.1.3 Estimating the Yield Curve . . . . . . . . . . . . . . . . . 15

2.2 Basic Discrete Time Stochastic Model ............... 18

2.2.1 Valuation at Time 0 . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Interpretation of State Price Deflators . . . ......... 22

2.2.3 Valuation at Time t > 0 ................... 23

2.3 Equivalent Martingale Measure . . . . . . . . . . . . . . . . . . . 26

2.3.1 Bank Account Numeraire .................. 26

2.3.2 Martingale Measure and the FTAP . . . . . . . . . . . . . 27

2.4 Market Price of Risk . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Spot Rate Models ............................. 35

3.1 General Gaussian Spot Rate Models ................. 35

3.2 One-Factor Gaussian Affine Term Structure Models ........ 38

3.3 Discrete Time One-Factor Vasicek Model .............. 41

3.3.1 Spot Rate Dynamics on a Yearly Grid . . . ......... 42

3.3.2 Spot Rate Dynamics on a Monthly Grid . . ......... 45

3.3.3 Parameter Calibration in the One-Factor Vasicek Model . . 47

3.4 Conditionally Heteroscedastic Spot Rate Models . ......... 56

3.5 Auto-Regressive Moving Average (ARMA) Spot Rate Models . . . 60

3.5.1 AR(1) Spot Rate Model ................... 61

vii

viii Contents

3.5.2 AR(p) Spot Rate Model ................... 62

3.5.3 General ARMA Spot Rate Models ............. 63

3.5.4 Parameter Calibration in ARMA Models . ......... 64

3.6 Discrete Time Multifactor Vasicek Model .............. 65

3.6.1 Motivation for Multifactor Spot Rate Models ........ 65

3.6.2 Multifactor Vasicek Model (with Independent Factors) . . . 67

3.6.3 Parameter Estimation and the Kalman Filter ........ 72

3.7 One-Factor Gamma Spot Rate Model ................ 87

3.7.1 Gamma Affine Term Structure Model . . . ......... 87

3.7.2 Parameter Calibration in the Gamma Spot Rate Model . . . 90

3.8 Discrete Time Black–Karasinski Model ............... 92

3.8.1 Log-Normal Spot Rate Dynamics .............. 92

3.8.2 Parameter Calibration in the Black–Karasinski Model . . . 93

3.8.3 ARMA Extended Black–Karasinski Model ......... 95

4 Stochastic Forward Rate and Yield Curve Modeling ......... 97

4.1 General Discrete Time HJM Framework .............. 98

4.2 Gaussian Discrete Time HJM Framework . . . . . . . . . . . . . . 100

4.2.1 General Gaussian Discrete Time HJM Framework ..... 100

4.2.2 Two-Factor Gaussian HJM Model .............. 102

4.2.3 Nelson–Siegel and Svensson HJM Framework ....... 105

4.3 Yield Curve Modeling . . . . . . . . . . . . . . . . . . . . . . . . 106

4.3.1 Derivations from the Forward Rate Framework . . . . . . . 106

4.3.2 Stochastic Yield Curve Modeling .............. 109

Appendix Proofs of Chap. 4 ....................... 125

5 Pricing of Financial Assets ........................ 131

5.1 Pricing of Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . 132

5.1.1 General Cash Flow Valuation in the Vasicek Model .... 132

5.1.2 Defaultable Coupon Bonds ................. 135

5.2 Financial Market ........................... 137

5.2.1 A Log-Normal Example in the Vasicek Model ....... 139

5.2.2 A First Asset-and-Liability Management Problem ..... 143

5.3 Pricing of Derivative Instruments . . . . . . . . . . . . . . . . . . 146

Appendix Proofs of Chap. 5 ....................... 149

Part II Actuarial Valuation and Solvency

6 Actuarial and Financial Modeling .................... 155

6.1 Financial Market and Financial Filtration .............. 155

6.2 Basic Actuarial Model ........................ 157

6.3 Improved Actuarial Model . . . ................... 164

7 Valuation Portfolio ............................ 169

7.1 Construction of the Valuation Portfolio . . . . . . . . . . . . . . . 170

7.1.1 Financial Portfolios and Cash Flows . . . ......... 171

7.1.2 Construction of the VaPo . . . . . . . . . . . . . . . . . . 171

Contents ix

7.1.3 Best-Estimate Reserves . . . . . . . . . . . . . . . . . . . 174

7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.2.1 Examples in Life Insurance ................. 177

7.2.2 Example in Non-life Insurance ............... 181

7.3 Claims Development Result and ALM . . . . . . . . . . . . . . . 187

7.3.1 Claims Development Result . . . . . . . . . . . . . . . . . 187

7.3.2 Hedgeable Filtration and ALM ............... 188

7.3.3 Examples Revisited . . . . . . . . . . . . . . . . . . . . . 192

7.4 Approximate Valuation Portfolio . . . . . . . . . . . . . . . . . . 197

8 Protected Valuation Portfolio ...................... 205

8.1 Construction of the Protected Valuation Portfolio . . . . . . . . . . 205

8.2 Market-Value Margin . . . . . . . . . . . . . . . . . . . . . . . . 207

8.2.1 Risk-Adjusted Reserves . . . . . . . . . . . . . . . . . . . 207

8.2.2 Claims Development Result of Risk-Adjusted Reserves . . 209

8.2.3 Fortuin–Kasteleyn–Ginibre (FKG) Inequality ........ 211

8.2.4 Examples in Life Insurance ................. 213

8.2.5 Example in Non-life Insurance ............... 223

8.2.6 Further Probability Distortion Examples . ......... 230

8.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 234

8.3.1 Non-life Insurance Run-Off ................. 234

8.3.2 Life Insurance Examples .................. 244

9 Solvency .................................. 261

9.1 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

9.1.1 Definition of (Conditional) Risk Measures ......... 261

9.1.2 Examples of Risk Measures . . . . . . . . . . . . . . . . . 265

9.2 Solvency and Acceptability . . ................... 268

9.2.1 Definition of Solvency and Acceptability . ......... 268

9.2.2 Free Capital and Solvency Terminology . . . . . . . . . . 274

9.2.3 Insolvency . . . . . . . . . . . . . . . . . . . . . . . . . . 277

9.3 No Insurance Technical Risk . ................... 278

9.3.1 Theoretical ALM Solution and Free Capital ........ 278

9.3.2 General Asset Allocations .................. 283

9.3.3 Limited Liability Option ................... 286

9.3.4 Margrabe Option . . . . . . . . . . . . . . . . . . . . . . . 291

9.3.5 Hedging Margrabe Options ................. 296

9.4 Inclusion of Insurance Technical Risk ................ 299

9.4.1 Insurance Technical and Financial Result . ......... 300

9.4.2 Theoretical ALM Solution and Solvency . ......... 302

9.4.3 General ALM Problem and Insurance Technical Risk . . . 309

9.4.4 Cost-of-Capital Loading and Dividend Payments ...... 313

9.4.5 Risk Spreading and Law of Large Numbers ......... 321

9.4.6 Limitations of the Vasicek Financial Model ......... 325

9.5 Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . 326

9.5.1 Standard Deviation Based Risk Measure . ......... 327

9.5.2 Estimation of the Covariance Matrix . . . ......... 333

x Contents

10 Selected Topics and Examples ...................... 337

10.1 Extreme Value Distributions and Copulas .............. 337

10.2 Parameter Uncertainty ........................ 339

10.2.1 Parameter Uncertainty for a Non-life Run-Off ....... 339

10.2.2 Modeling of Longevity Risk ................. 352

10.3 Cost-of-Capital Loading in Practice ................. 356

10.3.1 General Considerations ................... 356

10.3.2 Cost-of-Capital Loading Example .............. 358

10.4 Accounting Year Factors in Run-Off Triangles . . ......... 366

10.4.1 Model Assumptions . . ................... 366

10.4.2 Predictive Distribution . . . . . . . . . . . . . . . . . . . . 368

10.5 Premium Liability Modeling . ................... 369

10.5.1 Modeling Attritional Claims ................. 371

10.5.2 Modeling Large Claims . . . . . . . . . . . . . . . . . . . 375

10.5.3 Reinsurance ......................... 376

10.6 Risk Measurement and Solvency Modeling . . . . . . . . . . . . . 381

10.6.1 Insurance Liabilities . . ................... 381

10.6.2 Asset Portfolio and Premium Income . . . ......... 385

10.6.3 Cost Process and Other Risk Factors . . . ......... 387

10.6.4 Accounting Condition and Acceptability . ......... 388

10.6.5 Solvency Toy Model in Action ............... 390

10.7 Concluding Remarks ......................... 402

Part III Appendix

11 Auxiliary Considerations ......................... 407

11.1 Helpful Results with Gaussian Distributions . . . . . . . . . . . . 407

11.2 Change of Numeraire Technique .................. 408

11.2.1 General Changes of Numeraire ............... 408

11.2.2 Forward Measures and European Options on ZCBs .... 410

11.2.3 European Options with Log-Normal Asset Prices ...... 415

References ................................... 419

Index ...................................... 427

Notation1

m ≥ 0 Maturity of zero coupon bonds (ZCBs)

R(t,m) Continuously-compounded spot rate at time t for

maturity m>t

L(t,m) Simply-compounded spot rate at time t for maturity

m>t

Y(t,m) Annually-compounded spot rate at time t for maturity

m>t

r(t) Instantaneous spot rate (short rate) at time t ≥ 0

rt = R(t,t + 1) Continuously-compounded spot rate at time t for

maturity t + 1 (one-year risk-free rollover)

F(t,s + 1) Forward interest rate at time t for s ≥ t

f (t,m) Instantaneous forward interest rate at time t<m

β Parameter of Svensson and Nelson–Siegel modeling

n ∈ N Final time horizon

J = {0,...,n} Set of all points in time

J− = {0,...,n − 1} Set of points in time

F = (Ft)t∈J Filtration on measurable space (Ω,F) with

F0 = {∅,Ω} and Fn = F

P Real world probability measure on measurable space

(Ω,F)

(Ω,F,P,F) Filtered probability space

P∗ ∼ P Equivalent martingale measure on measurable space

(Ω,F)

1We give some notational conventions we are using. We do however stress that it is not always easy

to find good and consistent notation throughout the text. It may therefore happen that the same letter

is used for different objects. This we cannot avoid completely because we join concepts and models

from three different subject areas, namely actuarial science, financial mathematics and economic

theory.

We mainly work in a discrete time and finite time horizon model. The interval between two

points in time typically is one year and t ∈ R+ measures time in yearly units.

xi

xii Notation

(ξt)t∈J , (ζt)t∈J Density processes

X = (X0,...,Xn) Discrete time cash flow

L2

n+1(Ω,F,P) Hilbert space of (n + 1)-dimensional square integrable

cash flows X

L2

n+1(Ω,F,P,F) Hilbert space of (n + 1)-dimensional square integrable,

F-adapted cash flows X

L1

n+1(Ω,F,P,F) Space of (n + 1)-dimensional integrable, F-adapted

cash flows X

Lϕ Set of priceable cash flows X for state price deflator ϕ

k ∈ J Index for single cash flow Xk

t ∈ J Today’s time point used for price processes

A = (At)t∈J Financial filtration on measurable space (Ω,F)

T = (Tt)t∈J Insurance technical filtration on measurable space

(Ω,F)

H = (Ht)t∈J Hedgeable filtration on measurable space (Ω,F)

ϕ = (ϕt)t∈J State price deflator

ϕ˘ = (ϕ˘t)t∈J Span-deflator

ϕA = (ϕA

t )t∈J Financial deflator

ϕT = (ϕT

t )t∈J Probability distortion

I Financial market of basis financial instruments

A(i) Basis financial instrument i ∈ I

(A(i)

t )t∈J Price process of basis financial instrument A(i), i ∈ I

Z(m) ZCB with maturity m

Z(m) Cash flow of ZCB with maturity m

P(t,m) Price of ZCB Z(m) at time t ≤ m

U Financial portfolio

(Ut)t∈J Price process of financial portfolio U

U(k) Financial portfolio sold at time k ∈ J

(U(k)

t )t∈J Price process of financial portfolio U(k)

B Bank account

(Bt)t∈J Price process of bank account B

M(t) Margrabe option with maturity t

(M(t)

s )s∈J Price process of Margrabe option

Callt(A,K,T) Price at time t of European call option on instrument A,

with strike K and maturity T

Putt(A,K,T) Price at time t of European put option on instrument A,

with strike K and maturity T

Λ = (Λ(0)

,...,Λ(n)) T-adapted insurance technical liability

(Λ(k)

t )t∈J Probability distorted process of insurance liability Λ(k),

k ∈ J

S Asset side of balance sheet

St Value of asset side S of balance sheet at time t ∈ J

S (t) Asset portfolio with allocation chosen at time t ∈ J

Notation xiii

S(t)

s Value of asset portfolio S (t) at time s ∈ J

S (t) = n

k=t+1 w(t)

k U(k) Cash flow representation of asset portfolio S (t)

S (t) =

i∈I w(t)

i A(i) Instrument representation of asset portfolio S (t)

VaPot(X) Valuation portfolio of cash flow X at time t ∈ J

VaPoprot

t (X) Protected valuation portfolio of X at time t ∈ J

VaPoapprox

t (X) Approximate valuation portfolio of X at time t ∈ J

Qt(X) Value of X at time t ∈ J

Q0

t (X) Undistorted value of X at time t ∈ J

X(t+1) Outstanding liabilities at time t ∈ J−

R0

t (X(t+1)) Best-estimate reserves at time t ∈ J−

Rt(X(t+1)) Risk-adjusted reserves at time t ∈ J−

Rnom

t (X(t+1)) Nominal reserves at time t ∈ J−

MVMϕ

t (X(t+1)) Market-value margin at time t with state price deflator ϕ

CDRt+1(X(t+1)) Claims development result for best-estimate reserves at

time t + 1

CDR+

t+1(X(t+1)) Claims development result for risk-adjusted reserves at

time t + 1

I Last observed accident year (non-life insurance)

i ∈ {1,...,I } Accident years

J Last development year (in non-life insurance)

j ∈ {0,...,J } Development years

Xi,j Claims payment in non-life insurance for accident year i

and development year j , i.e. accounting year k = i + j

Ci,j Nominal cumulative payments in non-life insurance for

accident year i and development year j

Ci,J Nominal ultimate claim in non-life insurance

fj Chain-ladder factor for development period j

f +

j Risk-adjusted chain-ladder factor for development

period j

f (t)

j Posterior chain-ladder factor at time t

f (+t)

j Posterior risk-adjusted chain-ladder factor at time t

Lx+k Number of people alive aged x + k at time k

Dx+k Number of people aged x + k that die within (k − 1,k]

px+k Second order survival probability within (k − 1,k] for

people aged x + k

qx+k Second order death probability within (k − 1,k] for

people aged x + k

p+

x+k First order survival probability within (k − 1,k] for

people aged x + k

q+

x+k First order death probability within (k − 1,k] for people

aged x + k

ρ Risk measure

xiv Notation

ρt Conditional risk measure

M Subset of a.s. finite random variables

VaR1−p(X) Value-at-Risk of X on security level 1 − p

ES1−p(X) Expected shortfall of X on security level 1 − p

CTE1−p(X) Conditional tail expectation of X on security level 1 − p

ADt+1 Asset deficit at time t + 1

Ft Free capital at time t

SCt Solvency capital at time t

TCt Target capital at time t

RBCt Risk bearing capital at time t

λ Market price of risk

δ Span of time grid (in yearly units)

spCoC Cost-of-capital spread

r

(t)

CoC Cost-of-capital rate at time t

rRoSC Return on solvency capital

SRt Sharpe ratio at time t

r0:T Observations {r0,...,rT } at time T

Vco(X) Coefficient of variation of random variable X

Ψβ1 (·), Ψβ1,β2 (·) Risk reward functions

Xrun-off

(I+1) Run-off liability cash flow at time I

Xnb

(I+1) Cash flow new business (premium liability) of year

I + 1

Xac

(I+1) Cash flow attritional claims

Xlc

(I+1) Cash flow large claims without reinsurance cover

Xlc,ri

(I+1) Cash flow large claims, including reinsurance cover

X†

(I+1) Run-off life-time annuity cash flow at time I

Xcosts

(I+1) Costs cash flow

Xincept

(I+1) Inception costs cash flow

Xclaims handling

(I+1) Claims handling costs cash flow

Xliability

(I+1) Total liability cash flow after time I

(Π0,Π1) Premium cash flow