Monte Carlo Simulation: a beginning… not the end result

Frank Sortino & Hal Forsey

 

Monte Carlo Simulation (MCS) is one of the oldest and most widely used statistical procedures for making inferences based on a small sample.  The procedure was developed by Stanislaw Ulam in 1946 as a way of estimating the probability of winning at a game of solitaire while working on the Manhatten Project.  It is cited prominently in the marketing literature of Morningstar, Financial Engines and T. Roe Price, among others.  MCS is part and parcel of the Bootstrap Procedure used by Sortino Investment Advisors (SIA).[1]  We believe that the way most practitioners use MCS creates some problems that are overcome by employing the SIA methodology.

 How MCS is used:

1.  To estimate the probability returns will be above or below a certain number

2.  To estimate the probability of having enough money to retire at a certain date.

3.  To estimate the probability the withdrawal rate at retirement is not excessive.

 This is equivalent to using a confidence interval to asses the probability of returns falling below a certain level and equating that with risk. 

 Problems:

This application of MCS confuses a procedure for generating a probability distribution with a methodology for estimating risk.  It equates risk with the probability of a bad outcome.  Probability is only one component of risk.  There is a magnitude component as well.  An investment strategy may only have a 10% chance of failure, but like Long Term Capital Management, failure could result in financial disaster. Standard deviation is a probability weighted function of returns about the mean.  Downside risk is a probability weighted function of returns below the required return (RR) an investor needs to accomplish his or her goal.

 MCS assumes the individual’s RR is irrelevant.  Therefore, all investors agree on the degree of risk for all assets.  In terms of a Fishburn utility function, investor’s who make their decisions based on probabilities of a bad outcome are risk takers and the exponent in equation 1 is equal to 0. Behavioral finance shows investors tend to ignore a very small chance of a very bad outcome.  Thus, they become risk takers for events like the dot com bubble.  Thus, it would seem that the way Monte Carlo simulation is used supports a flaw in the decision making ability of investors that leads them to buy into bubbles at the top.

At least value at risk assumes the exponent in equation 1, is equal to 1.  In which case, investors are risk neutral and their utility function is linear below the RR.  Risk neutral investors believe losing all their money is only twice as painful as losing half of it.  Firms that use Monte Carlo Simulation focus on the average return and standard deviation when evaluating managers or determining the asset allocation.  SIA focuses on upside potential and downside risk.

 SIA Methodology:

1.  The bootstrap procedure is used to estimate a distribution of annual returns that could

     have happened based on a sampling procedure developed by Bradley Efron in 1993.

2.  A three parameter log normal is fit to the data to provide a continuous asymmetric

     distribution and partially correct for end point sensitivity. 

3.  Upside potential & downside risk for each manager are then estimated from the

     distribution for ranking managers.     

4.  Portfolios of active managers and passive indexes are then formed that maximize the

     added value of active managers for a given RR while managing downside risk and

     reducing costs. 

 

None of this is available in standard Monte Carlo Simulations.

 

Results:

1. The farther returns fall below an investor’s RR, the greater the pain.  Returns above the RR are not painful and not risky.

2. Risk of a given asset will appear different to investor’s with different RR’s.

3. Investors are risk averse.  Losing all their money is many times more painful than  loosing half of it.  The exponent in equation #1 equals 2 not 0 or 1.