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**Research Sponsored by the**

**Casualty Actuarial Society and the**

**Society of Actuaries**

*Investigators:*

Kevin Ahlgrim, ASA, PhD, Illinois State University

Steve D’Arcy, FCAS, PhD, University of Illinois

Rick Gorvett, FCAS, ARM, FRM, PhD, University of Illinois

Western Risk and Insurance Association

January 2004

We wish to thank the Casualty Actuarial Society and the Society of Actuaries for providing financial support for this research, as well as guidance and feedback on the subject matter.

Note: All of the following slides associated with this research project reflect tentative findings and results; these results are currently being reviewed by committees of the CAS and SoA.

• Motivation for Financial Scenario

Generator Project

• Short description of included economic variables

• Using the model and motivating this research

• Methodology

• Conclusions

• CAS/SOA Request for Proposals

– Stems from Browne, Carson, and Hoyt (2001) and

Browne and Hoyt (1995)

• Goal: to provide actuaries with a model for projecting economic and financial indices , with realistic interdependencies among the variables.

• Wilkie, 1986 and 1995

– Widely used internationally

• Hibbert, Mowbray, and Turnbull, 2001

– Modern financial tool

• CAS/SOA project (a.k.a. the Financial Scenario

Generator) applies Wilkie/HMT to U.S.

• Inflation

• Real interest rates

• Nominal interest rates

• Equity returns

– Large stocks

– Small stocks

• Equity dividend yields

• Real estate returns

• Unemployment

*q*

*• Modeled as an Ornstein-Uhlenbeck process dq t*

*= k q*

*( m q*

*– q t*

*) dt + s q dB q*

*r*

• Two-factor Vasicek term structure model

*• Short-term rate ( r ) and long-term mean ( l ) are both stochastic variables dr t dl t*

*= k r*

*= k l*

*(l t*

*( m l*

*– r t*

*) dt + s r*

*– r t*

*) dt + s l dB r dB l*

*s*

•

Model equity returns as an excess return

*( x t*

*) over the nominal interest rate s t*

*= q t*

*+ r t*

*+ x t*

• Empirical “fat tails” issue regarding equity returns distribution

• Thus, modeled using a “regime switching model”

1. High return, low volatility regime

2. Low return, high volatility regime

•

*Equity dividend yields ( y) and real estate*

– O-U processes

•

Unemployment ( u )

*– Phillip’s curve: inverse relationship between u and q du t*

*= k u*

*( m u*

*– u t*

*) dt + a u dq t*

*+ s u e ut*

Inflation Real Interest Rates

Unemployment Nominal Interest Real Estate

Lg. Stock Returns Sm. Stock Returns

Stock Dividends

• Model is meant to represent range of outcomes possible for the insurer

• Parameters are chosen from history (as long as possible)

• Of course, different parameters lead to different

– This research: How much does this affect a life insurer?

• Browne, Carson, and Hoyt (2001) only indicate important variables to consider

• What is the potential model risk, if using the

Financial Scenario Generator?

• Specific question: what is the impact of parameter

“errors” on projected life insurer results?

• Contribution: Which processes require more attention? Which processes should sensitivity analysis be performed?

• Dynamic financial analysis

• Insurers can project operations under a variety of economic conditions

• Useful for demonstrating solvency to regulators

• May propose financial risk management solutions

• Use Financial Scenario Generator to project life insurance product

• Calculate PV of projected surplus/shortfall

• Vary underlying parameters of various processes to determine valuation sensitivity

• Annual payment whole life product

– May be interest sensitive whole life

• No expenses

• EOY DB and lapse

• Cash value = required reserve (NLP)

– Again, may be interest rate sensitive

• 10,000 new policies, issue age 35

• 20 year projection

• NLP

• Lapses: Base and interest sensitive

• Discount any remaining surplus back to time 0

5

4

3

2

7

6

9

8

1

0

-30 -20

Distribution for PV Surplus/T26

X <=-18028052

5%

M ean = 1000127

X <=10914911

95%

-10

Values in Millions

0 10 20

Base case

*Short Rate*

Mean Rev Speed

Volatility

*Long Rate*

Mean Rev Speed

Volatility

Mean Rev Level

*Mean*

1,000,127

*Real Interest Rates*

1,007,893

975,504

1,586,248

1,004,527

2,374,030

*Stdev*

9,505,468

9,413,302

9,537,642

7,880,753

15,076,870

8,803,360

Base case

*Mean*

1,000,127

*Stdev*

9,505,468

*Regime Switching Equity Model*

Avg Monthly Return 1,517,555 9,464,937

Volatility 1,002,727 9,538,019

• Even with “minor” investments in equities, assumed average return has major impact on profitability

• Reversion of long-term interest rates is crucial

– Level and speed of reversion

0.25

0.20

0.15

0.10

0.05

0.00

**Figure 12 **

**Actual 1 Year Interest Rates (4/53-4/03) versus Model 1 Year Interest Rates**

**Interest Rate**

Model

Actual

**Figure 16 **

**Actual S&P 500 (1871-2002) versus Model Large Stock Returns**

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

-0.8 -0.5 -0.3

0 0.25 0.5 0.75

1 1.25 1.5 1.75

2

**1 Year Return**

Model

Actual

**Figure 17**

**Actual Small Stock Returns (1926-1999) versus **

**Model Small Stock Returns**

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

-0.8 -0.5 -0.2 0.1 0.4 0.7

1 1.3 1.6 1.9 2.2 2.5

Model

Actual