Optimizing Multi-Period DFA Systems

Professor John M. Mulvey

Department of OR and Financial Engineering

Bendheim Center for Finance

Princeton University

July 2000

Strategic Asset and Liability Systems (DFA)

Towers Perrin-Tillinghast CAP:Link/OPT:Link, TAS significant impact (e.g. US West -- $450 to 1001 Million)

American/Munich Re-Insurance – ARMS Financial planning for individuals

– Home Account, Financial Engines KontraG bill in Germany

W. Ziemba and J. Mulvey, eds., World Wide Asset and Liability Modeling, Cambridge University Press, 1998

• Single models

Limitations of Traditional Mean-Variance

Single period– Transaction and market impact costs

– Cannot compare short-term and long-term

Ignores liabilities– Misses contribution patterns

– Risks are asset-only

Assumes symmetric returns

Asset Only Downside Risk Efficient Frontier 5 Year Time Horizon

1

2

3

45

67

89

1011

1

2

3

4

5

6

7

89

1011

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Shortfall Under 6.0000%No

min

al C

ompo

und

Retu

rn

w/out

w/ MITTS & MBS

Model Uncertainties

Simulate Organizationscenarios

Risk aversion

Calibrate and sample

What ifs

Basic Technology

Optimize

Purpose of a Scenario Generator

Construct a representative set of scenarios: plausible paths over planning period – S– Economic factors

– Asset returns

– Liabilities

– Business activities

Use in financial simulator and optimizer

1 2 3 4 ... Ttime

Horizon

Structural models are well placed to support DFA

Company Strategy

Asset Mix Product Mix Capital

Structure Reinsurance

Economic Scenario Generator

Projected FinancialsRisk Profile = Distribution of FutureFinancial Results

Prob

abilit

y

Asset Behavior ModelAsset Behavior Model

Product Behavior ModelProduct Behavior Model

Noise

Noise

Optimization

Inflation Interest Rates Credit Costs Currency

Exchange GDP

Generating Scenarios

Employ stochastic processes for key economic factors:

– interest rates

– inflation

– currencies

Sample with discrete time and discrete scenarios

Examples:

Towers Perrin’s global CAP:Link (Tillinghast TAS)

Calibrated in 21 countries

Siemens Financial Services

Tree generator

Model Uncertainties

Simulate Organizationscenarios

Calibrate and sample Optimize

Corporate Simulations

Project state of company across multi-year horizon– Decisions at beginning each stage

– Uncertainties during periods

– Policy rules guide system

– Iterate over all scenarios

1 2 3 4 ... Ttime

Horizon

Decisions Examples: American Re, Renaissance Re,

Tillinghast TAS-PC

Basic Constructs1 2 3 4 ... T

time

Horizon

Also decisions regarding corporate structure

Asset allocation

Investment Network with Borrowing (each scenario)

STOCK

BOND

LOAN 1

CASH

InterestPayment

InterestPayment

InterestPayment

TerminalNode

Time 1 Time 2 Time 3

Contribution and pay pension benefits

Model Uncertainties

Simulate Organizationscenarios

Calibrate and sample Optimize

Optimization Framework

Surplust = market value (assetst - liabilitiest) Grow economic surplus over planning

period, pay liabilities, reduce insurance costs– t = {1, 2, …, T}– maximize risk-adjusted profit– analyze over representative set of scenarios {S}

Policy constraints, plus risk measures, e.g. sufficient capital to meet 100-200 year losses

Dynamic Optimization Approaches

Dynamic stochastic control (Brennan-Schwartz-Lagnado) relatively simple stochastic model small state-space, few general constraints

Multi-stage stochastic programming (Frank Russell) realistic decision framework, sample scenarios large-size due to # conditional variables

Optimize decision rules (Towers Perrin/Tillinghast) understandable, generate confidence estimates non-convex

Stochastic Programs

1 2 3

time

HORIZON

Xj,ts

Structure of Multi-stage Models

A1

A2

As

Non-anticipativity constraints

scenarios

Optimize over Policy

Decision rules satisfy non-anticipativity conditions Example -- surplus management strategy -- Goals-at-

RiskTM

Intuitive, easy to implement Generates small, highly non-convex optimization problem Employ stochastic program to inspire good decision rules

Non-Convexity

Asset/Liability Efficient Frontier 50 Year Time Horizon

12

3

4

6

7

8

9

10

5

6.5

7.0

7.5

8.0

8.5

9.0

2.22 2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.42

Ave

rag

e C

om

po

un

d P

ort

folio

Ret

urn

Payout On

Current

Conclusions Multi-period DFA systems are operating today

– Better linkages needed with tactical systems

Customized products will grow from integrated risk management systems

Implementation in various applications– Pension planning– Insurance companies– Coordinated risk management for divisions– Individuals