Stochastic Optimization ESI 6912

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Stochastic OptimizationESI 6912
NOTES 3 FINANCIAL PLANNING

• Instructor Prof. S. Uryasev

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Financial Planning
Financial Planning
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Notations

• to invest
• years before college
• tuition
• number of investment
  periods
• investment index
• ( , - stocks
  , - bonds)
• period number

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Notations (contd)
Utility
Wealth
Utility function of wealth at year for a
goal
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Notations (contd)

• return on
  investment
• decision
  vector
• space of random elements
• scenario number
• probability of
  scenario

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Scenario Tree
Tree of scenarios for three periods
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Scenario Scenario
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Scenario Representation
Returns (scenario representation)
for
otherwise
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Tree Representation of Variables
Decisions in nodes
decisions decisions decisions
Similar, we denote returns
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Objective and First Period Constraint
First-period constraint budget constraint
invest initial wealth
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Constraints for Further Periods
Constraints for periods
, for each
Constraints for period
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Model Data

• initial wealth
• years before college
• surplus reward
• storage penalty

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Stochastic Programming ProblemTree
Representation
s. t.
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Optimal Solution with Three-Period Stochastic
Problem
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Value of Stochastic Solution
Expected value (EV) solution
Average return on stocks 1.155 each period
Average return on bonds 1.13 each period
Optimal investment strategy buy only stocks
EV -3.84
Stochastic solution (recourse problem (RP))
RP -1.52
VSS
VSS RP - EV -1.52 - (-3.84) 2.32
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Stochastic Programming ProblemScenario
Representation
s. t.
set of scenarios with the same random outcomes
up to time .
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Algorithms
- nonanticipativity constraints links
scenarios - problem can be decomposed without
these constraints