Actuarial Science and Financial Mathematics: Doing Integrals for Fun and Profit

Actuarial Science andFinancial Mathematics:

Doing Integrals for Fun and Profit

Rick Gorvett, FCAS, MAAA, ARM, Ph.D.

Presentation to Math 400 Class

Department of Mathematics

University of Illinois at Urbana-Champaign

March 5, 2001

Presentation Agenda

• Actuaries -- who (or what) are they?

• Actuarial exams and our actuarial science courses

• Recent developments in

– Actuarial practice

– Academic research

What is an Actuary?The Technical Definition

• Someone with an actuarial designation• Property / Casualty:

– FCAS: Fellow of the Casualty Actuarial Society– ACAS: Associate of the Casualty Actuarial Society

• Life:– FSA: Fellow of the Society of Actuaries– ASA: Associate of the Society of Actuaries

• Other:– EA: Enrolled Actuary– MAAA: Member, American Academy of Actuaries

What is an Actuary?Better Definitions

• “One who analyzes the current financial implications of future contingent events”

- p.1, Foundations of Casualty Actuarial Science

• “Actuaries put a price tag on future risks. They have been called financial architects and social mathematicians, because their unique combination of analytical and business skills is helping to solve a growing variety of financial and social problems.”

- p.1, Actuaries Make a Difference

Membership Statistics (Nov., 2000)

• Casualty Actuarial Society:– Fellows: 2,061– Associates: 1,377– Total: 3,438

• Society of Actuaries:– Fellows: 8,990– Associates: 7,411– Total: 16,401

Casualty Actuaries

• Insurance companies: 2,096• Consultants: 668• Organizations serving insurance: 102• Government: 76• Brokers and agents: 84• Academic: 16• Other: 177• Retired: 219

“Basic” Actuarial Exams

• Course 1: Mathematical foundations of actuarial science– Calculus, probability, and risk

• Course 2: Economics, finance, and interest theory

• Course 3: Actuarial models– Life contingencies, loss distributions, stochastic

processes, risk theory, simulation

• Course 4: Actuarial modeling– Econometrics, credibility theory, model estimation,

survival analysis

U of I Actuarial Science Program:Math Courses Beyond Calculus

Exam #• Math 210: Interest theory 2• Math 309: Actuarial statistics Various• Math 361: Probability theory 1• Math 369: Applied statistics 4• Math 371: Actuarial theory I 3• Math 372: Actuarial theory II 3• Math 376: Risk theory 3• Math 377: Survival analysis 4• Math 378: Actuarial modeling 3 and 4

U of I Actuarial Science Program:Other Useful Courses

• Math 270: Review for exams # 1 and 2

• Math 351: Financial Mathematics

• Math 351: Actuarial Capstone course

• Fin 260: Principles of insurance

• Fin 321: Advanced corporate finance

• Fin 343: Financial risk management

• Econ 102 / 300: Microeconomics

• Econ 103 / 301: Macroeconomics

CAS Exams -- Advanced Topics

• Insurance policies and coverages• Ratemaking• Loss reserving• Actuarial standards• Insurance accounting• Reinsurance• Insurance law and regulation• Finance and solvency• Investments and financial analysis

The Actuarial Profession• Types of actuaries

– Property/casualty– Life– Pension

• Primary functions involve the financial implications of contingent events– Price insurance policies (“ratemaking”)– Set reserves (liabilities) for the future costs of

current obligations (“loss reserving”)– Determine appropriate classification structures

for insurance policyholders– Asset-liability management– Financial analyses

Table of Contents From a Recent Actuarial Journal

North American Actuarial JournalJuly 1998

• Economic Valuation Models for Insurers• New Salary Functions for Pension Valuations• Representative Interest Rate Scenarios• On a Class of Renewal Risk Processes• Utility Functions: From Risk Theory to Finance• Pricing Perpetual Options for Jump Processes• A Logical, Simple Method for Solving the Problem of

Properly Indexing Social Security Benefits

Actuarial Science and Finance

• “Coaching is not rocket science.” - Theresa Grentz, University of Illinois

Women’s Basketball Coach

• Are actuarial science and finance rocket science?

• Certainly, lots of quantitative Ph.D.s are on Wall Street and doing actuarial- or finance-related work

• But….

Actuarial Science and Finance (cont.)

• Actuarial science and finance are not rocket science -- they’re harder

• Rocket science:– Test a theory or design– Learn and re-test until successful

• Actuarial science and finance– Things continually change -- behaviors, attitudes,….– Can’t hold other variables constant– Limited data with which to test theories

Recent Developments inActuarial Practice

• Risk and return– Pricing insurance policies to formally reflect risk

• Insurance securitization– Transfer of insurance risks to the capital markets

by transforming insurance cash flows into tradable financial securities

• Dynamic financial analysis– Holistic approach to modeling the interaction

between insurance and financial operations

Dynamic Financial Analysis

• Dynamic– Stochastic or variable– Reflect uncertainty in future outcomes

• Financial– Integration of insurance and financial

operations and markets

• Analysis– Examination of system’s interrelationships

U/WInputs

Investment& Economic

Inputs

U/W GeneratorPayment Patterns

U/W Cycle

CatastropheGenerator

InvestmentGenerator

U/WCashflows

InvestmentCashflows

TaxOutputs

& SimulationResults

DynaMo (at www.mhlconsult.com)

Interest RateGenerator

Key VariablesFinancial

• Short-Term Interest Rate• Term Structure• Default Premiums• Equity Premium• Inflation• Mortgage Pre-Payment

Patterns

Underwriting

• Loss Freq. / Sev.• Rates and Exposures• Expenses• Underwriting Cycle• Loss Reserve Dev.• Jurisdictional Risk• Aging Phenomenon• Payment Patterns• Catastrophes• Reinsurance• Taxes

Sample DFA Model Output

Distribution for SURPLUS /Ending/I115

PR

OB

AB

ILIT

Y

Values in Hundreds

0.00

0.03

0.06

0.10

0.13

0.16

6.8 13.9 21.1 28.2 35.4 42.5 49.7

Year 2004 Surplus DistributionOriginal Assumptions

0

0.05

0.1

0.15

0.2

0.25-3

2.9

1.3

35.5

69.7

103.

913

8.2

172.

420

6.6

240.

827

5.0

309.

2

Millions

Pro

babi

lity

Year 2004 Surplus Distribution Constrained Growth Assumptions

0

0.05

0.1

0.15

0.2

0.2567

.7

94.4

121.

114

7.8

174.

620

1.3

228.

025

4.7

281.

430

8.1

334.

8

Millions

Pro

babi

lity

Model Uses

Internal

• Strategic Planning• Ratemaking• Reinsurance• Valuation / M&A• Market Simulation

and Competitive Analysis

• Asset / Liability Management

External

• External Ratings• Communication with

Financial Markets• Regulatory / Risk-

Based Capital• Capital Planning /

Securitization

Recent Areas of Actuarial Research

• Financial mathematics

• Stochastic calculus

• Fuzzy set theory

• Markov chain Monte Carlo

• Neural networks

• Chaos theory / fractals

The Actuarial ScienceResearch Triangle

Mathematics

ActuarialScience

Finance

Stochastic Calculus /Ito’s Lemma

Financial Mathematics

PortfolioTheory

ContingentClaimsAnalysis

Fuzzy SetTheory

Markov ChainMonte Carlo

Chaos Theory /Fractals

Theoryof Risk

DynamicFinancialAnalysis

InterestRateModeling

InterestTheory

Financial Mathematics

Interest Rate Generator

Cox-Ingersoll-Ross One-Factor Model

dr = (-r) dt + r0.5 dZ

r = short-term interest rate = speed of reversion of process to long-run mean = long-run mean interest rate = volatility of processZ = standard Wiener process

Financial Mathematics (cont.)

Asset-Liability Management

Duration

D = -(P / r) / P

Convexity

C = P / r2

r

P

Price-YieldCurve

Stochastic Calculus

Brownian motion (Wiener process)

z = (t)0.5

z(t) - z(s) ~ N(0, t-s)

Stochastic Calculus (cont.)

Ito’s Lemma

Let dx = a(x,t) + b(x,t)dz

Then, F(x,t) follows the process

dF = [a(F/x) + (F/t) + 0.5b2(2F/x2)]dt + b(F/x)dz

Stochastic Calculus (cont.)

Black-Scholes(-Merton) Formula

VC = S N(d1) - X e-rt N(d2)

d1 = [ln(S/X)+(r+0.52)t] /t0.5

d2 = d1 - t0.5

Stochastic Calculus (cont.)

Mathematical DFA Model

• Single state variable: A / L ratio• Assume that both assets and liabilities follow

geometric Brownian motion processes:

dA/A = Adt + AdzA

dL/L = Ldt + LdzL

Correlation = AL

Stochastic Calculus (cont.)

Mathematical DFA Model (cont.)

• In a risk-neutral valuation framework, the interest rate cancels, and x=A/L follows:

dx/x = xdt + xdzx

where

x = L2 - AL AL

x2 = A

2 + L2 - 2AL AL

dzx = (AdzA - LdzL ) / x

Stochastic Calculus (cont.)

Mathematical DFA Model (cont.)

Can now determine the distribution of the state variable x at the end of the continuous-time segment:

ln(x(t)) ~ N(ln(x(t-1))+x-(x2 /2), x

2 )

or

ln(x(t)) ~ N(ln(x(t-1))+(L2 /2)-(A

2 /2), A2+L

2-2AL

AL )

Fuzzy Set Theory

Insurance Problems

• Risk classification– Acceptance decision, pricing decision– Few versus many class dimensions– Many factors are “clear and crisp”

• Pricing– Class-dependent– Incorporating company philosophy / subjective

information

Fuzzy Set Theory (cont.)

A Possible Solution

• Provide a systematic, mathematical framework to reflect vague, linguistic criteria

• Instead of a Boolean-type bifurcation, assigns a membership function:

For fuzzy set A, mA(x): X ==> [0,1]• Young (1996, 1997): pricing (WC, health)• Cummins & Derrig (1997): pricing• Horgby (1998): risk classification (life)

Markov Chain Monte Carlo

• Computer-based simulation technique• Generates dependent sample paths from a distribution• Transition matrix: probabilities of moving from one

state to another• Actuarial uses:

– Aggregate claims distribution– Stochastic claims reserving– Shifting risk parameters over time

Neural Networks

• Artificial intelligence model

• Characteristics:– Pattern recognition / reconstruction ability– Ability to “learn”– Adapts to changing environment– Resistance to input noise

• Brockett, et al (1994)– Feed forward / back propagation– Predictability of insurer insolvencies

Chaos Theory / Fractals

• Non-linear dynamic systems

• Many economic and financial processes exhibit “irregularities”

• Volatility in markets– Appears as jumps / outliers– Or, market accelerates / decelerates

• Fractals and chaos theory may help us get a better handle on “risk”

Conclusion

• A new actuarial science “paradigm” is evolving– Advanced mathematics– Financial sophistication

• There are significant opportunities for important research in these areas of convergence between actuarial science and mathematics

Some Useful Web Pages

• Mine– http://www.math.uiuc.edu/~gorvett/

• Casualty Actuarial Society– http://www.casact.org/

• Society of Actuaries– http://www.soa.org/

• “Be An Actuary”– http://www.beanactuary.org/