Presenter

1
An ALM Linear Stochastic Programming Model for a
Brazilian pension fund

• Presenter
• Davi Michel Valladão
• Pontifícia Universidade Católica do Rio de
  Janeiro - Brazil
• davi_michel_at_yahoo.com.br
• Co-authors
• Álvaro de Lima Veiga Filho, PUC-Rio
• Ana Tereza Vasconcellos Estellita Pessoa, PUC-Rio
• Camila Spinassé, PUC-Rio

2
Summary

1. Motivation
2. Asset
3. Liability
4. Optimization
5. Illustration
6. Conclusion and future works

3
Motivation

• ALM
• Linear Stochastic Programming
• Literature
• Model General Description

4
ALM

• Asset and Liability Management
• Definition It is process of strategic
  formulation, implementation and revision of
  investments and liabilities to reach the
  institutional financial goals given some
  restrictions.

• ALM

• Markowitz

• Multi-period solution
• Scenario tree structure
• Linear stochastic programming

• One-period solution
• Mean-variance model
• Quadratic programming

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Linear Programming

• Deterministic

• Stochastic

• Known asset returns
• Known liability cash flows
• Optimization problem
• max c.x
• s.t. A.xb

• Stochastic returns
• Unknown liability cash flows
• Scenario tree structure
• Optimization problem
• max Ec.x
• s.t. Ai.xb , i1,..N

6
Literature

• Drijver, Klein and Vlerk OR, 2000
• Scenario tree generation
• Cost of funding minimization
• Chance constraint and transaction costs
• Kouwenberg OR, 2001
• Scenario tree generation
• Cost of funding minimization
• Maximum allocation constraints and transaction
  costs
• Hilli, Koivu and Pennanen OR, 2004
• Scenario tree generation
• Maximum expected wealth utility
• Maximum allocation constraints, transaction costs
  and regulatory constraints
• Gulpinar, Rustem and Settergren JEDC, 2004
• Scenario tree generation methods comparison

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Model General Description

• Strategic allocation ? indexes instead of assets
• Long run (20 years)
• asset classes stocks, interest rate, inflation
• Scenario tree generation
• Maximum final wealth with underfunding penalty
• Transaction costs
• Regulatory constraints (Brazilian law)
• Liquidity constraints

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Model General Description
Stochastic forecast of economic variables
Stochastic asset returns
Asset Scenario Generation
VAR Model
Optimal Allocation
Inflation Scenarios
Stochastic nominal cash flow
Liability Scenario Generation
Deterministic real cash flow
Liability Model
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Asset

• VAR Model
• Scenario Tree Generation

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VAR Model

• Based on Brazils Central Bank working paper
  number 33 (Minella RBE, 2003)

All series computed as log first difference

• A, B, C and S estimated with past data

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VAR Model

• Estimated coefficients

12
VAR Model

• Estimated coefficients

13
VAR Model

• Estimated coefficients

14
VAR Model

• Estimated coefficients

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VAR Model

• Residual Normality test

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VAR Model

• Residual serial correlation test

17
VAR Model

• Impulse Response

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Scenario Tree Generation

• Tree Structure 1-10-6-6-4-4

4X
Yt A B.Yt-1 C.Yt-2 et(j)

4X

6X
Adjusted Random Sampling

6X
..

.


10X

6X

6X
Initial Allocation

4X

4X
1 year
1 year
3 year
5 year
10 year
t
(10 branches)
(60 branches)
(360 branches)
(1440 branches)
(5760 branches)
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Scenario Tree Generation

• Adjusted Random Sampling (Kouwenberg OR,2001)
• For each root or branch node
• Generate k/2 values of et(j) , j1, k/2
• Compute antithetic values et(j k/2) - et(j)
• Variance adjustment for each tree stage
• et(j)Std. dev.(VAR)/Std. dev.(et(j )) , j1,k
• Compute Yt A B.Yt-1 C.Yt-2 et(j)

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Liability

• Liability Model
• Liability Scenario Generation

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Liability Model

• Artificial data
• Defined benefit plan
• No new participants
• Risk factors
• Mortality
• Retirement time
• Monte Carlo Simulation
• Deterministic output (simulation average real
  value)

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Liability Scenario Generation

• Input
• Deterministic real cash flows
• Inflation scenarios (new risk factor included)
• Output
• Stochastic Nominal cash flows

23
Optimization Model

• Objective Function
• Constraints

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Objective function

• Maximize the expected final wealth with
  underfunding penalty

• probN scenario N probability
• yN wealth at the end of the studied period
  (scenario N)
• wN deficit at the end of the studied period
  (scenario N)
• b bonus
• p penalization

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Constraints

• For each pair of linked nodes (A,B),

B

A

• there are the following constraints
• Balance
• Transaction

• Liquidity
• Maximum allocation on stocks (regulatory)

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Constraints

• Balance constraints

• xi (N) value () in asset i at node N
• ri asset i return
• L liability nominal cash flow
• TC Transaction cost

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Constraints

• Balance constraints (for B as a final node)

• xi (N) value () in asset i at node N
• ri asset i return
• L liability nominal cash flow
• TC Transaction cost
• yN wealth at the end of the studied period
  (scenario N)
• wN deficit at the end of the studied period
  (scenario N)

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Constraints

• Transaction constraints

• Liquidity constraint

• buyi (N) how much was bought from asset i at
  node N
• selli (N) how much was sold from asset i at node
  N

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Illustration

• Assumptions
• Deficit Probability X Initial Wealth
• Liquidity Problem X Initial Wealth
• Transaction costs X Initial Wealth
• Optimal Initial Allocation X Initial Wealth
• Optimal Expected Allocation

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Assumptions

• Based on one of the most important sponsored
  pension fund in Brazil
• 161 thousands lives simulated
• 83 thousands active participants
• 59 thousands retired participants
• 19 thousands pensioners
• Standard family concept
• Real wage growth 2 per year
• Contribution 16 of wage (8 from participant
  and 8 from sponsor)
• Benefit 90 of the wage average of the last 12
  months

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Deficit Probability X Initial Wealth
Related wealth
Deficit level
100 44 billions of dollars
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Liquidity Problem X Initial Wealth
100 44 billions of dollars
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Transaction Costs X Initial Wealth
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Optimal Initial Allocation X Initial Wealth
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Optimal Expected Allocation
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Optimal Expected Allocation
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Optimal Expected Allocation
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Conclusion and Future Works
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Conclusion and Future Works

• ALM is an important decision instrument (not
  exploited in Brazil)
• Model characteristic
• Linear Stochastic Programming
• Transaction cost and Brazilian law incorporated
• Liquidity problem included
• Future works
• Check VEC Model for economic variables
• Include end-effects
• Make a Stochastic liability model
• Use Wage as risk factor

40
Thank You!