# The Monte Carlo Simulation: Understanding the Basics

## What Is a Monte Carlo Simulation?

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.

### Key Takeaways

• The Monte Carlo method uses a random sampling of information to solve a statistical problem; while a simulation is a way to virtually demonstrate a strategy.
• Combined, the Monte Carlo simulation enables a user to come up with a bevy of results for a statistical problem with numerous data points sampled repeatedly.
• The Monte Carlo simulation can be used in corporate finance, options pricing, and especially portfolio management and personal finance planning.
• On the downside, the simulation is limited in that it can't account for bear markets, recessions, or any other kind of financial crisis that might impact potential results.

## 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 several million times, would provide a 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.

A Monte Carlo simulation can accommodate a variety of risk assumptions in many scenarios and is therefore applicable to all kinds of investments and portfolios.

## Applying the Monte Carlo Simulation

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.

## Uses in Portfolio Management

A Monte Carlo simulation allows an analyst to determine the size of the portfolio a client would need at retirement to support their desired retirement lifestyle and other desired gifts and bequests. She factors into a distribution of reinvestment rates, inflation rates, asset class returns, tax rates, and even possible lifespans. 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 depend 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 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 client's different spending rates and lifespan can be factored in to determine the probability that the client will run out of funds (the probability of ruin or longevity risk) before their death.

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 desired return and the risk profile of a client are not in sync with each other. For example, the level of risk acceptable to a client may make it impossible or very difficult to attain the desired return. Moreover, a minimum amount may be needed before retirement to achieve the client's goals, but the client's lifestyle would not allow for the savings or the client may be reluctant to change it.

## Monte Carlo Simulation Example

Let's consider an example of a young working couple who works very hard and has a lavish lifestyle including expensive holidays every year. They have a retirement objective of spending \$170,000 per 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 the allocation to small-cap stocks is doubled (up to 50 to 70% from 25 to 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. the analyst delays their retirement by two 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 percent. With the available insight, the analyst advises the clients to delay retirement and decrease their spending marginally, to which the couple agrees.

## The 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. 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, a useful tool for advisors.

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1. Commodity Futures Trading Commission. "CFTC Glossary."