Lecture Notes for Simulation

19 January 2005 - Introduction

401(k) Programs

• A company offers a 401(k) program with the following rules:

  • Each 401(k) member has X% of their monthly salary available for investing each month.

  • A member may buy shares in any of three mutual funds.

  • Members must be fully invested at all times.

  • Account transactions are performed once a month.

The Investment Plan Problem

• An investment plan is a plan for spending money and managing assets in a 401(k) account.

  • One simple investment plan always buys the mutual fund with the highest price per share (growth investing).

  • Another simple plan always buys the mutual fund with the lowest price per share (value investing).

• The investment plan should maximize long-term return.

• The problem is:

  What's the best investment plan to maximize long-term return?

Understanding The Investment Problem

• Is it really a problem? It depends.

  • A general rule: always take free money if offered.

    • But make sure it's really free money.

• A better investment plan is free money over a poorer one.

  • But the better plan has to be no more expensive than the poorer one.

Solving The Investment Problem

• How can you solve the investment plan problem?

  • Think of a plan.

  • Use it.

  • See how it works.

• There are some problems with this, though.

  • See how it works compared to what? (What's your benchmark?)

  • What if it doesn't work? (The opportunity-cost problem.)

• Might there be some better approach to a solution?

Resolving The Investment Problem

• Solving the investment problem requires knowing the future.

  • Which most of us don't.

• However, we do (or can) know the past.

  • Assuming the future will be like the past may be reasonable.

• Try out investment plans on historic data.

  • Now we can try lots of plans and go with the best one.

  • The benchmark is the set of plans used.

  • Bad choices don't cost us anything (except a little time).

The Investment Plan Solved?

• Have we really solved the investment-plan problem?

  • Not really.

• The plans considered may not be the best plans.

  • And there are infinitely many plans to consider.

• The future may not be like the past.

• There's lots of other problems.

Recapitulation

• A problem that can't be easily (or practically) solved directly.

• Solve a problem that's like the original problem, but is easier or more practical) to solve.

• Hope the same solution can cover both problems.

  • Or perhaps do better than hope.

Simulation

Simulation builds a model of a system and conducts experiments on the model. The experiment results are used to estimate answers to questions about the system.

• The three keys words are model, experiment, and estimates.

Model

• An abstraction of the system of interest.

  • Abstraction is important.

    • Include the necessary system features.

    • Ignore the unnecessary system features.

    • Wisdom lies in knowing the difference between necessary and unnecessary.

• A model comprises three parts: the input, the output, and the relation between input and output.

Experiment

• Given a system model, play with the inputs and see what happens with the outputs.

  • Sometimes you may hold the inputs constant and vary the model.

  • It's a bad idea to vary both the inputs and the model.

• The result of experimenting with the model is a set of input-output pairs.

Estimation

• Determine the system characteristics of interest from the set of input-output pairs produced by the experiments.

• These are not answers to the problem.

  • They are estimates drawn from a small number of experiments.

  • They are characteristics of a model of the problem, not the problem itself.

The Investment Plan Again

• Let's reconsider the last investment plan solution as a simulation.

  • What's the model?

  • What's the experiment?

  • What's the estimation?

This page last modified on 20 January 2005.