Illustration by Brian Ajhar

A technique that is revolutionizing the way investment advisors and financial planners analyze investment portfolios carries a somewhat paradoxical name: Monte Carlo simulation. To many, Monte Carlo conjures up images of Princess Grace, the idle rich, and gambling at the famous casino in that tiny nation in the south of France

Monte Carlo simulation, however, represents almost the exact opposite of this high-stakes, high-risk world. It is a powerful statistical tool that allows investors, managers, and planners to examine the performance of portfolios under thousands of different scenarios, increasing the accuracy and reliability of financial projections. The advent of the personal computer makes this computationally intensive analysis more accessible to financial professionals. Consequently, creative investment advisors have been able to apply this technique, originally used for specialized options strategies, to different aspects of financial analysis.

Investment managers and financial planners have started to use this analysis as a standard part of their services. The more investors know about Monte Carlo analysis the better able they will be to interpret the results and make decisions accordingly. And the more that planners use Monte Carlo as part of their standard service offerings to clients, the better served those clients will be.

But there also exists the possibility of misunderstanding the technique, resulting in inappropriate interpretation of the results and/or inappropriate advice. We'll try to lessen that possibility.

**Pitfalls of Traditional Analytical Approaches**

To put it bluntly, the traditional model of financial planning is fatally flawed. When meeting with investment managers or financial planners, investors typically express their goals in one of two ways (or a combination of the two):

- How can I maximize the amount of money that can be withdrawn from the portfolio (without the risk of outliving my money!), thereby achieving the highest possible standard of living?

- How do I maximize my net worth, while maintaining an acceptable lifestyle, so that my heirs and/or charitable endeavors receive the highest possible amount when I die?

Given this charge, the typical financial planner begins to assemble the information necessary to develop a portfolio that he or she believes will achieve the client's goal. Included in the initial analysis will be information about the investor's current net worth, anticipated cash flow or after-tax income requirements, planned expenditures, life expectancy, and assumed inflation and tax rates.

The planner then puts together an investment portfolio that, based on the portfolio's expected return, will grow and/or deliver enough income to meet the client's stated objectives.

This is where the problem begins in a traditional analysis. When using projections to show how the prescribed portfolio meets the investment demands, it is common to use the expected rate of return as a constant rate of return throughout the assumed life of the portfolio. For projection purposes, a portfolio that is expected to return, say, 10% per annum over the life of the portfolio is assumed to generate a 10% return every year, and cash flows and investments are planned accordingly.

What this method ignores, of course, is the risk of the portfolio. In a typical discussion with an investor the issue of risk is expressed by the statistic standard deviation, which is a measure of expected volatility. However, this notion of risk is usually addressed in a superficial manner (i.e., only to determine a subjective pain threshold and tolerance for portfolio volatility). Rarely is the risk of the portfolio factored meaningfully into the analysis. Specifically, where the traditional analysis falls down is in its failure to understand how the volatility of the portfolio in the early years can have a dramatic affect on the ability of the portfolio to achieve its goals in later years. For an investor with ongoing income or cash flow needs, this is a critical oversight. A simple example helps illustrate this point.

Assume an investor is looking to invest a $1 million portfolio, and expects the portfolio to return 10% per annum over its life. Further, the investor wishes to pull $50,000 per year out of the portfolio. There is an almost infinite number of "paths" the portfolio might take and still end up with a 10% average return over its life. The table below illustrates how a small variation in the actual rate of portfolio return from year to year can have a dramatic effect on the ability of the portfolio to meets its cash flow goals.

As the chart clearly shows, after only three years, the two sample portfolios have a $21,000 (2.1%) difference in market value despite having identical total returns and standard deviations over the three-year period. For the investor with no cash flow needs, the yearly portfolio returns and market values are irrelevant. But for investors with regular income requirements the annual variation in returns may become even more important than the average return earned during the overall time period.

The Effect of Volatility on Portfolio Performance | |||
---|---|---|---|

Account Value at the Beginning of the Year |
Actual Rate of Return in Each Year |
Withdrawal Taken at the End of Each Year |
Portfolio Value at the End of Each Year |

Scenario One | |||

$1,000,00 | 0.00% | $50,000 | $950,000 |

$950,000 | 10.00% | $50,000 | $995,000 |

$995,000 | 20.00% | $50,000 | $1,144,000 |

Scenario Two | |||

$1,000,000 | 20.00% | $50,000 | $1,150,000 |

$1,150,000 | 10.00% | $50,000 | $1,215,000 |

$1,215,000 | 0.00% | $50,000 | $1,165,000 |

So how do we factor this portfolio volatility into the analysis? Enter Monte Carlo.

**The Basics of Monte Carlo**

In its essence, Monte Carlo analysis is simply a technique for simulating the possible paths (series) of returns that a portfolio might take over its lifetime in order to achieve the overall expected return. The risk of the portfolio (expressed as its standard deviation) determines the nature and variability of the different portfolio paths.

However, to an investor with ongoing income requirements, the critical issue is not where the portfolio ends up but the path it takes to get there. If the portfolio was to generate negative returns early in its life it could have a significant impact on the market value later on, and the investor may need to dip into principal sooner than expected to meet cash flow requirements. This necessitates a higher-than-expected return in later years to make up the shortfall, and also increases the risk that the investor will outlive his or her money.

What Monte Carlo simulation does is help to analyze not just the overall outcome of the portfolio's performance, but also the probabilities of certain paths occurring. What is of most concern are those paths that result in the client running out of money. This allows the planner to quantify the risk that the portfolio will not meet expectations because of poor performance in early years.

To run the analysis properly, the planner will need to know or make reasonable assumptions about several variables that will affect performance. Among these are:

- The expected return and standard deviations of the portfolio being analyzed

- The timing and amount of cash flow withdrawals

- The expected rate of inflation

- The marginal and capital gains tax rates

- The nature of the portfolio account (taxable, IRA, 401(K), etc.)

It is also necessary to assume what sort of distribution the series of returns will follow (normal, lognormal, etc.).

We typically assume a normal, or "bell-shaped," distribution, which is the distribution assumed for many statistical analyses. This simply means that most (two-thirds) of the portfolio paths will fall within one standard deviation of the expected outcome, with a decreasing number of paths falling further and further away from the "most likely" path.

These variables are then plugged into the Monte Carlo program , which is available off-the-shelf from a variety of providers, though we created our own using Excel.

The Monte Carlo program takes these variables and uses a "random number generator" to create thousands of possible paths the portfolio might take over its lifetime. Because the paths have been defined to have a specific expected return and standard deviation, they will all end up at the same terminal value (the same one you would get if the portfolio really did generate its expected return each and every year). But there is an infinite number of ways to reach that terminal value.

What this analysis allows the investor to examine is the probability that he or she will not be able to achieve his or her income or cash flow goals because of an early dip in the portfolio value. As an example, we had a new client who had the following assumptions with respect to her portfolio:

- She had $1.5 million in cash to invest.

- She wanted to withdraw $100,000 per annum from her portfolio.

- A 3.5% inflation rate.

- A 40% tax rate on withdrawals from her portfolio.

The client expressed general concern regarding the equity markets. When we performed an initial, subjective, risk analysis, we recommended that she invest in a portfolio with an expected return of 10 percent per year. Taking $100,000 per annum amounted to a 6.7% withdrawal from the portfolio, so we had a 3.3% "cushion" in terms of withdrawal rate versus return rate. This analysis, however, quickly seemed unrealistic as well as over-simplistic.

Portfolio #1 | Portfolio #2 | Portfolio #3 | |

Yearly Withdrawal |
Return 10.0% Standard Dev. 9.5% |
Return 11.9% Standard Dev. 12.3% |
Return 13.1% Standard Dev. 14.7% |

$80,000 | 91.30% | 97.90% | 100% |

$100,000 | 80.50% | 91.70% | 99.60% |

$120,000 | 66.00% | 85.10% | 95.90% |

Using Monte Carlo analysis, we created the following matrix for the client. On the left hand side we see possible annual payouts, with $100,000 being the client's target. We also considered three portfolios for the client, with Portfolio #1 (the most conservative) as the portfolio we initially considered.

Two other, "more aggressive," portfolios were also examined. The percentage figures in the table represent the probability that the client will have enough money at the end of her life (Put another way, 100% minus the percentages in the table show the probability that our investor would run out of money before she died.)

Had our client implemented Portfolio 1 (the "more conservative" portfolio), she had a 20% chance of going broke in her lifetime. Needless to say, both our client and we thought this was unacceptable. Moving to a more aggressive portfolio (Portfolio 2) reduced the probability she would outlive her money to a mere 8%, and the most aggressive portfolio of all (Portfolio 3) reduced her risk of going broke to less than 1%. What was counter-intuitive to the client (but true nonetheless) was the notion that by accepting more volatility in her portfolio, she actually decreased her real investment risk: the risk of out-living her money.

Conversely, we frequently find that this analysis results in the investor realizing they do not need to take as much risk in their portfolio as they think they do in order to meet their goals. If a portfolio with a lower standard deviation still results in the investor meeting his goals 100% of the time, regardless of the early performance of that portfolio, then why take more risk?

In conclusion, we believe the use of Monte Carlo simulation should be a standard part of every asset allocation and portfolio creation exercise. It is not necessary for an investor to understand Monte Carlo simulation or the mathematics behind it, but its use does allow for easy visualization of the effect portfolio volatility can have on the investor's ability to reach investment and income goals. It also transforms and quantifies the discussion of risk into a tangible, meaningful part of portfolio design.

Investors whose financial planners are not using Monte Carlo simulation as a part of their analysis and service may be facing more risk than they realize. Moreover, planners who themselves do not use Monte Carlo simulation in analyzing client portfolios may be shortchanging their clients as well.