CAS 1999 Dynamic Financial Analysis SeminarChicago, Illinois

July 19, 1999

Calibrating Stochastic Models for DFA

John M. Mulvey - Princeton University

François Morin - Tillinghast - Towers Perrin

Bill Pauling - Towers Perrin

Basic financial modeling architecture

EconomicScenarioGeneration

Optimization Objectives

Vs. Risks

Results Regulatory GAAP Tax Economic Cash

Financial Model Asset Performance Liability Performance

Global CAP:Link

TAS: P/C

Opt:Link

Towers Perrin’s Global CAP:Link

Used by institutional investors around the world

Calibrated for 10 currencies

Widely recognized and published model

Named honoree for Edelman Award

Global CAP:Link : General cascade structure

Currencies

Real Yields

Stock DividendGrowth Rate

DividendYields

FixedIncomeReturns

Stock Returns

Other Asset Classes

General PriceInflation

TreasuryYield Curve

ExpectedInflation

Wage Inflation

Global CAP:Link : Multi-currency links

Japan Europe

U.S.A

Other Countries

Key links are: Currencies Dividend yields Bond yields

Overview:Stochastic differential equations

Stochastic differential equations generate time series for each variable:

drt = f1(ru - rt)dt + f2(rt, pt,…)dt + f3(rt)dZ1

Initial conditions

set initial values for variables to current levels

set ‘normative’ values to long-term expected values

Normative conditions

set initial and normative values equal to each other

MeanReversion

VariableLinks

RandomElement

Overview:Assumptions and Calibration

Assumptions refer to the mean value of key economic variables: Bond yields Inflation Dividend yields Equity risk premium

Well researched in a multi-period time frame

We offer ‘Basic Expectations’ approach, but will implement other approaches

Calibration refers to more subtle aspects of the models behavior Degree of mean

reversion Probability of

‘extreme’ values Key linkages between

variables (average correlations, etc.)

The calibration comes as a part of the implementation of the model

Calibration Problem

Stochastic differential equations (55 equations, 220 parameters)

Non-convexity

Many targets must be satisfied simultaneously

Non-Convexity

Calibration

Independent parameter estimation (regression) produces inconsistent and unrealistic results

Past is not a central estimate of the future

Calibration follows a different approach Define behavioral characteristics of scenarios

emergent properties Set parameters in such a way as to produce

required characteristics

Calibration Solution - Theory

Generalized Method of Moments

Simulated Moments Estimator

Integrated Parameter Estimation

Calibration Solution - Integrated Parameter Estimation

Extends simulated moment estimation

Target vector can include a variety of descriptive statistics standard deviation correlation serial correlation distribution percentages other range estimates (inter-quartile ranges) frequency of inversion probability of extreme values

Parameters are bounded

Calibration Solution - Calibration Tool

Interface to Global CAP:Link

Objective function target ranges target weights

Non-convex optimizer

Example: Calibrating Scenario Generator with Both Assets and Liabilities

Calibrate Global CAP:Link to produce liability growth as well as asset returns

Determine target statistics & importance weights

Use non-convex optimization model - based on Integrated Parameter Estimation approach

Solve for best set of parameters

Step 1: Analyze Historical DataInflation

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

Year

CPI

Med CPI

Legal CPI

Bill Pauling:Bill Pauling:

Step 2: Set Targets

Medical CPI Legal ServicesCPI

Standarddeviation

1.9 - 2.2% 0.8 - 1.0%

Correlation toCPI

0.60 - 0.70 0.45 - 0.55

Average spreadover CPI

0.90 0.70

Step 3: Use Calibration Tool

Enter target types, ranges and importance weights

100 scenarios per iteration

Calibrate to normative conditions

Step 4: Review Model Output

Use optimal parameters to generate 500 scenario run

Critically examine full set of Global CAP:Link results

Adjust targets and parameter ranges, if necessary

Step 4: Review Model Output

EXAMPLE1999-2008 Price inflation

1.791.82

1.831.881.91

1.951.98

1.98

1.98

1.94

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Year

2/23/99 18:01:59

Mean

90th%

75th%

50th%

25th%

10th%

Step 4: Review Model Output

EXAMPLE1999-2008 Legal Services CPI

2.652.702.762.84

2.933.043.18

3.363.623.96

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Year

2/23/99 18:00:08

Mean

90th%

75th%

50th%

25th%

10th%

Step 4: Review Model Output

EXAMPLE1999-2008 Medical CPI

2.67

2.70

2.722.76

2.802.84

2.86

2.85

2.862.92

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Year

2/23/99 18:00:32

Mean

90th%

75th%

50th%

25th%

10th%

Example: Linking Assets and Liabilities for DFA

Insurance line of automobile policies liabilities driven by both Medical CPI and Legal

Services CPI

Reward measure: ending surplus

Risk measure: volatility of ending surplus

Use OPT:Link to produce asset/liability efficient frontier

Surplus Optimization Framework

Surplust = market value (assetst - liabilitiest)

Grow economic surplus over planning period t = {1, 2, …, T} maximize risk-adjusted profit for entire

company analyze over representative set of scenarios

Internal and external constraints on asset mix GAAP income other financial statement measures

Asset Liability Efficient Frontier (ALEFsm)

Asset Mix %: 1 2 3 4 5 6 7 8 9 10 Current

Cash-U.S.A 74.0 38.4 12.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0Eqty-U.S.A 1.9 13.3 22.1 32.6 45.8 62.9 73.2 82.7 91.6 100.0 20.0Bond Index 24.1 48.3 65.4 67.4 54.2 37.1 26.8 17.3 8.4 0.0 75.0

Reward 43.5 60.2 73.2 84.2 93.9 106.8 114.7 122.0 128.9 135.4 73.7Risk 11.4 16.1 24.0 32.3 41.7 56.5 66.6 76.6 86.6 96.6 24.5

Example Surplus Efficient Frontier 10 Year Time Horizon

10

9

8

7

6

5

4

3

2

140

50

60

70

80

90

100

110

120

130

140

150

10 20 30 40 50 60 70 80 90 100 110

Standard Deviation

ALEF

Current

Conclusions

Assets and liabilities should be calibrated together

Integrated parameter estimation can be used with non-convex optimization techniques to calibrate the model