What Is a Monte Carlo Simulation and Why Do We Need It?

Analysts can assess possible portfolio returns in many ways. The historical approach, which is the most popular, considers all the possibilities that have already happened. However, investors shouldn't stop at this. The Monte Carlo method is a stochastic (random sampling of inputs) method to solve a statistical problem, and a simulation is a virtual representation of a problem. The Monte Carlo simulation combines the two to give us a powerful tool that allows us to obtain a distribution (array) of results for any statistical problem with numerous inputs sampled over and over again. (For more, see: Stochastics: An Accurate Buy And Sell Indicator.)

Monte Carlo Simulation Demystified

Monte Carlo simulations can be best understood by thinking about a person throwing dice. A novice gambler who plays craps for the first time will have no clue what the odds are to roll a six in any combination (for example, four and two, three and three, one and five). What are the odds of rolling two threes, also known as a "hard six?" Throwing the dice many times, ideally a several million times, will give one the representative distribution of results which will tell us how likely a roll of six will be a hard six. Ideally, we should run these tests efficiently and quickly, which is exactly what a Monte Carlo simulation offers.

Asset prices or portfolios' future values don't depend on rolls of the dice, but sometimes asset prices do resemble a random walk. The problem with looking to history alone is that it represents, in effect, just one roll, or probable outcome, which may or may not be applicable in the future. A Monte Carlo simulation considers a wide range of possibilities and helps us reduce uncertainty. A Monte Carlo simulation is very flexible; it allows us to vary risk assumptions under all parameters and thus model a range of possible outcomes. One can compare multiple future outcomes and customize the model to various assets and portfolios under review. (For more, see: Find The Right Fit With Probability Distributions.)

Applications of Monte Carlo Simulation in Finance:

The Monte Carlo simulation has numerous applications in finance and other fields. Monte Carlo is used in corporate finance to model components of project cash flow, which are impacted by uncertainty. The result is a range of net present values (NPVs) along with observations on the average NPV of the investment under analysis and its volatility. The investor can thus estimate the probability that NPV will be greater than zero. Monte Carlo is used for option pricing where numerous random paths for the price of an underlying asset are generated, each having an associated payoff. These payoffs are then discounted back to the present and averaged to get the option price . It is similarly used for pricing fixed income securities and interest rate derivatives. But the Monte Carlo simulation is used most extensively in portfolio management and personal financial planning. (For more, see: Capital Investment Decisions - Incremental Cash Flows.)

Monte Carlo Simulation and Portfolio Management:

A Monte Carlo simulation allows an analyst to determine the size of the portfolio required at retirement to support the desired retirement lifestyle and other desired gifts and bequests. She factors in a distribution of reinvestment rates, inflation rates, asset class returns, tax rates and even possible life spans. The result is a distribution of portfolio sizes with the probabilities of supporting the client's desired spending needs.

The analyst next uses the Monte Carlo simulation to determine the expected value and distribution of a portfolio at the owner's retirement date. The simulation allows the analyst to take a multi-period view, and factor in path dependency; the portfolio value and asset allocation at every period depends on the returns and volatility in the preceding period. The analyst uses various asset allocations with varying degrees of risk, different correlations between assets and a distribution of a large number of factors including the savings in each period and the retirement date, to arrive at a distribution of portfolios along with the probability of arriving at the desired portfolio value at retirement. The clients' different spending rates and life span can be factored in to determine the probability the clients will run out of funds (the probability of ruin or longevity risk) before their deaths.

A client's risk and return profile is the most important factor influencing portfolio management decisions. The client's required returns are a function of her retirement and spending goals; her risk profile is determined by her ability and willingness to take risks. More often than not the return and risk profile of clients are not in sync with each other; for example, the level of risk acceptable to them it may make it impossible or very difficult to attain the desired return. Moreover, a minimum amount may be needed before retirement to achieve her goals, and the clients' lifestyle would not allow for the savings, or she may be reluctant to change it.

Let us consider an example of a young working couple who work very hard and have a lavish lifestyle including expensive holidays every year. They have a retirement objective of spending $170,000 a year (approx. $14,000/month), and leaving a $1 million estate to their children. An analyst runs a simulation and finds that their savings-per-period is insufficient to build the desired portfolio value at retirement; however, it is achievable if allocation to small cap stocks is doubled (up to 50% - 70% from 25% - 35%), which will increase their risk considerably. None of the above alternatives (higher savings or increased risk) are acceptable to the client. Thus, the analyst factors in other adjustments before running the simulation again. He delays retirement by 2 years, and decreases their monthly spend post retirement to $12,500. The resulting distribution shows that the desired portfolio value is achievable by increasing allocation to small cap stock by only 8%. With the available insight, he proposes the clients to delay retirement and decrease spending marginally, to which the couple agrees. (For more, see: Planning Your Retirement Using The Monte Carlo Simulation.)

Bottom line

A Monte Carlo simulation allows analysts and advisors to convert investment chances into choices. The advantage of Monte Carlo is its ability to factor in a range of values for various inputs; this is also its greatest disadvantage in the sense that assumptions need to be fair because the output is only as good as the inputs. Another great disadvantage is that the Monte Carlo simulation tends to underestimate the probability of extreme bear events like a financial crisis, which are becoming too frequent for comfort. In fact, experts argue that a simulation like the Monte Carlo is unable to factor in the behavioral aspects of finance and the irrationality exhibited by market participants. It is, however, an able servant at the disposal of advisors who need to ask smart questions from it.

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