Monte Carlo Information

"Monte Carlo gets people thinking about investment and long-term planning in terms of probability rather than certainty. [...] And once you think through the odds, you can make a smarter calculated bet." — Christopher Farrell, Newsweek

Deterministic Tools

We are all familiar with retirement planning calculators widely available on the Internet. You enter your salary, future annual savings, retirement age, existing assets, and other inputs. The projections are based on assumptions you select, most importantly the future expected return on your investments. After churning for a fraction of a second, the calculator tells you to the dollar how much you'll have at retirement, and how long your money will last.

These planning tools are called "deterministic": all inputs are fixed and they only reveal one single outcome.

Markets Don't Move in a Straight Line

Forecasting future investment returns is already quite a challenge. But assuming that they will go in a straight line is downright unrealistic. The reality is that investment returns occur in a random fashion over time, not in the straight line "average" that is used by most financial calculators. When you factor in cash flows — contributions and withdrawals over time - in an investment portfolio, results become even more problematic.

How accurate are the results? Are your assumptions realistic? If you retired in the last five years, chances are that the equity component of your retirement account got seriously depleted.

The assumptions of the retirement calculator now seem unrealistically high and very imprecise. Even if you selected a very low expected rate of return, you cannot reproduce these worst case scenarios using the traditional deterministic approach. A low conservative rate of return assumption only makes it easier to meet your average with spectacular returns in future years.

What Is Monte Carlo Simulation?

Imitating Real Life

Monte Carlo simulation is a mathematical method used to estimate the most likely outcome and the odds that certain events will occur, given a mathematical model of the problem.

The Monte Carlo method has actually been around for almost a century. Like the roulette wheels associated with the casinos of Monte Carlo, Monte Carlo simulation reproduces random outcomes by generating random numbers to obtain results. Unlike a roulette wheel, the Monte Carlo method uses random numbers to measure and quantify uncertainty and chance events.

Monte Carlo applications are used to solve complex science, engineering and finance problems that cannot be solved with traditional mathematical approaches. Financial and retirement planning falls in this category of complex nonlinear problems for which an analytical solution cannot be achieved or produces a very inaccurate approximation.

Risk and Return

A Monte Carlo simulation of capital market returns takes into account expected returns from each asset class (i.e., cash, fixed income, equities, etc.), their volatility, correlations between them, and other factors, all based on historical statistics. Random rates of return are generated by sampling values from a probability distribution such as a bell curve (a common one is a "lognormal" distribution). Returns from equity asset classes (such as Canadian, US or international equities) are typically higher than returns from low risk or risk-free investments (e.g., cash, GICs and fixed income). But higher equity returns also have greater risk, that is a greater range of outcomes, from complete loss of capital to appreciation many times over the initial purchase price. And they also experience greater volatility.

The last condition to reproduce historical conditions is that asset classes are not independent of each other, they display a certain level of correlation between each other. These mathematical relationships can be quantified and fitted in the model.

Brute Force Calculations

A Monte Carlo simulation uses these random numbers to calculate thousands of potential scenarios of the future. Results are sensitive to underlying assumptions, but with a degree of conservatism, it makes it possible to determine whether future cash flow requirements and retirement goals are realistic. If the overall likelihood is that money will run out, then factors such as retirement age, retirement income and planned savings are adjusted until the overall probability of achieving life goals is satisfactory.

For example, if the portfolio does not run out of money 800 times out of 1,000 scenarios, we can say that the probability of success is 80%.

Monte Carlo simulation is a "brute force" approach that is able to solve problems for which no other solutions exist.

The Shortcomings of Deterministic Projections

The graph below shows how investment volatility can result in outcomes that are dramatically different from the traditional deterministic approach. If you have $100,000 invested in equities and expect an 8% annual return, the funds should last 20 years if you withdraw $9,500 per year. This assumes the funds earn exactly 8% each and every year in the future.

10 simulations

But equities come with a fair dose of volatility. Assume returns will average 8% and follow a bell curve distribution, with a volatility of 18% per year. The correct term is "standard deviation". A standard deviation of 18% per year means that you can expect two thirds of the time an annual return between —10% (8% - 18%) and 26% (8% + 18%).

Using Monte Carlo simulation we get startling results with just 10 trials, in 6 of our ten simulations we run out of money before the 20 year period is over.

Two Hundred Trials

The next two graphs compare the deterministic single rate calculation with a graph (called a "spaghetti" graph) showing 200 simulations. Already, you can see a trend in these trials and uncommon extreme outcomes, such as very early bankruptcy and hyper wealth. (Note the scale of each graph is different.)

Single Rate Deterministic Projection
200 simulations

Each outcome has one chance of taking place in the total number of trials. Increasing the number of trials to thousands mean that the results will reproduce very closely the probability distribution (the distribution of all possible results).

What Are the Benefits of Using Monte Carlo Simulations?

There Is Risk in Investing

The central premise of using Monte Carlo simulations is that there is risk in investing. Long-term averages don't capture this concept. Monte Carlo simulations are a far more realistic measure of financial risk and reward than any of the traditional financial planning alternatives.

We may never again see a bear market like we experienced a few years ago. Then again, we might. But straight-line projections don't adequately account for the possibility.

When deciding on a portfolio asset allocation, it helps understand the potential impact of risk that comes with higher equity returns, the uncertainty of future returns and the possibility of loss. The probability of getting the desired outcome helps visualize the variability of potential results when greater risk is assumed.

Know the Odds of Achieving Your Goals

The goal is to raise the comfort level knowing the odds of achieving pragmatic financial lifestyle goals, and feel comfortable with a financial plan, even in periods of negative market performance.

If a probability of 80% percent seems too risky and you would like to increase the odds to 85% or 90% , you can adjust your annual savings, retire later, reduce your income requirements or change your portfolio allocation.

Conservative Approach

The risk of hitting a string of bad years early on can easily upset a retirement plan. For instance, if you retired during a period as bad as the stock-market returns of the mid-1970s, you would run out of money very fast. What traditional planning ignores is the timing of the returns. A Monte Carlo simulation highlights some of the problems that might arise in a down market shortly after retirement.

If there is not a high enough probability for success in achieving retirement goals, changes such as retiring older, saving more, adjusting income expectation or a combination of these become clear.

Success in Early Retirement Can Set the Tone

It is important never to lose sight that the type of investments and allocation within a portfolio have a direct impact on the amount of volatility that can occur. Small foreign companies will have large amounts of variation in returns, while high-grade short term Government will have very little.

Monte Carlo should not be viewed as a certainty test. It is a probability test. Ultimately, there will be only one outcome, but knowing not to take more risk than necessary and finding a safe spending level is invaluable information.

Discriminating consumers realize the importance of sophisticated analysis and are increasingly looking for this capability when considering a service provider. Financial planners using Monte Carlo simulations in their analysis are turning uncertainty into a competitive advantage!

How Does Monte Carlo Simulation Work?

Range of Solutions

In Monte Carlo simulation, the random selection process is repeated many times to create multiple scenarios. Together, these scenarios give a range of possible solutions, some of which are more likely and some less likely.

There are two kinds of results. The first is the probability of success: the percentage of trials that resulted in a successful outcome.

The second result is a graph showing "percentile lines". The lines show the outcomes at various points of success and failure in terms of percentiles. For example, the 95th percentile shows the outcome that ranked in the top 5%. The 5th percentile shows the outcome that ranked in the bottom 5%. The 50th percentile shows the outcome that ranked in the middle 50%.

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