With the introduction of Monte Carlo simulation methodologies, the technology of retirement planning for individuals now rivals that of corporate pension planning. Monte Carlo analysis is computer and data intensive, so its availability for personal retirement planning at affordable cost is a direct result of the availability of inexpensive computing power. Such methodologies are now readily available to individual investors and their investment managers, from a variety of vendors.

Monte Carlo simulation is the process by which probability ''distributions'' are arrayed to create path-dependent scenarios to predict end-stage results. The methodology is useful when trying to forecast future results that depend on multiple variables with various degrees of volatility. Its use in projecting retirement wealth is valuable because the prediction of future wealth depends on multiple factors (investment returns, inflation, etc.), each with a unique distribution of probable outcomes. Monte Carlo simulation is generally superior to steady-state, or deterministic, forecasting because it incorporates the consequences of variability across long-term assumptions and the resulting path dependency effect on wealth accumulation. Merely using long-term averages for capital market returns or inflation assumptions oversimplifies their variability and leads to the clearly unrealistic implication of linear wealth accumulation. There is also an inherent assumption when using deterministic forecasting that performance in future periods will more or less replicate historical performance. Monte Carlo estimation, in contrast, allows for the input of probability estimates over multiperiod time frames and generates a probability distribution of final values rather than a single point estimate. This approach allows the investment adviser to view projections of possible best- and worst-case scenarios and leads to better financial planning over long time frames (Traditional Finance).

The ultimate objective of probabilistic approaches, such as Monte Carlo simulation, for investment planning is to improve the quality of managers' recommendations and investors' decisions. A brief look at the distinction between traditional deterministic analysis and probabilistic analysis reveals how the latter approach seeks to achieve that objective. In both approaches, the individual supplies a similar set ofpersonal information, including age, desired retirement age, current income, savings, and assets in taxable, tax-deferred, and tax-exempt vehicles. In a deterministic analysis, single numbers are specified for interest rates, asset returns, inflation, and similar economic variables. In a Monte Carlo or probabilistic analysis, a probability distribution of possible values is specified for economic variables, reflecting the real-life uncertainty about those variables' future values.

Suppose an individual investor is 25 years away from her desired retirement age. A deterministic retirement analysis produces single-number estimates of outcomes for stated objectives, such as retirement assets and retirement income at the end of 25 years. Using the same inputs, a Monte Carlo analysis produces probability distributions for those objective variables by tabulating the outcomes of a large number (often 10,000) of simulation trials, each trial representing a possible 25-year experience. Each simulation trial incorporates a potential blend of economic factors (interest rates, inflation, etc.), in which the blending reflects the economic variables' probability distributions.

Consequently, whereas deterministic analysis provides ayes/no answer concerning whether the individual will reach a particular goal for retirement income, or perhaps retirement wealth, mirroring a single set of economic assumptions, a Monte Carlo analysis provides a probability estimate, as well as other detailed information, that allows the investor to better assess risk (e.g., percentiles for the distribution of retirement income). Thus, Monte Carlo analysis is far more informative about the risk associated with meeting objectives than deterministic analysis. The investor can then respond to such risk information by changing variables under her control. An advisory module may present a range of alternative asset allocations and the associated probabilities for reaching goals and objectives.

A probabilistic approach conveys several advantages to both investors and their investment advisers. First, a probabilistic forecast more accurately portrays the risk—return trade-off than a deterministic approach. Until recently, advisers nearly exclusively used deterministic projections to inform their recommendations and communicate with their clients. Unfortunately, such projections cannot realistically model how markets actually behave. The probability of observing a scenario in which the market return is constant each year is effectively zero. Fundamentally, deterministic models answer the wrong question. The relevant question is not ''How much money will I have if I earn 10 percent a year?'' but rather ''Given a particular investment strategy, what is the likelihood of achieving 10 percent a year?'' By focusing on the wrong question, deterministic models can fail to illustrate the consequences of investment risk, producing, in effect, a misleading ''return—return'' trade-off in investors' minds whereby riskier strategies are always expected to produce superior long-term rewards.

In contrast, a probabilistic forecast vividly portrays the actual risk—return trade-off. For example, an investor considering placing a higher percentage of his portfolio in equities might be told that the average forecast return of the S&P 500 Index is 13 percent. Given an average forecast money market return of 5 percent, it may seem obvious that more equity exposure is desirable. This choice, however, should take into account the risk that the S&P 500 will not achieve its average return every year. Moreover, the median simulation outcome of the S&P 500, using the average return of 13 percent, is likely to be substantially lower because of return volatility. For example, a 20-year forecast of $1,000 invested in the S&P 500, using a riskless average return of 13 percent, yields ending wealth of $11,500. If a simulation is performed assuming normally distributed returns with an annual standard deviation of 20 percent, the median wealth after 20 years is only $8,400. In addition, a simulation-based forecast shows that there is substantial downside risk: The fifth percentile of wealth after 20 years is only approximately $2,000, even before adjusting for inflation.

A second benefit of a probabilistic approach is that a simulation can give information on the possible trade-off between short-term risk and the risk of not meeting a long-term goal. This trade-off arises when an investor must choose between lowering short-term volatility on one hand and lowering the portfolio's long-term growth because of lower expected returns on the other hand.

Third, as already discussed, taxes complicate investment planning considerably by creating a sequential problem in which buy and sell decisions during this period affect next-period decisions through the tax implications of portfolio changes. Through its ability to model a nearly limitless range of scenarios, Monte Carlo analysis can capture the variety of portfolio changes that can potentially result from tax effects.

Finally, an expected value of future returns is more complicated than an expected value of concurrent returns, even in the simplest case of independent and normally distributed returns. For concurrent returns, the expected portfolio return is simply the weighted sum of the individual expected returns, and the variance depends on the individual variances and covariances, leading to the benefits of diversification with lower covariances. In this case, the $1 invested is simply divided among several investment alternatives. The future return case, however, involves a multiplicative situation; for example, the expected two-period return is the product of one plus the expected values of the one-period returns, leading to the importance of considering expected geometric return. As Michaud (1981) demonstrates, the expected geometric return depends on the horizon of the investment. The stochastic nature of the problem can be summarized by recognizing that the $1 invested now will then be reinvested in the next period and possibly joined by an additional $1 investment. This scenario clearly differs from the simple one-period case of spreading the dollar among several asset classes. Again, Monte Carlo analysis is well suited to model this stochastic process and its resulting alternative outcomes.

Monte Carlo simulation can be a useful tool for investment analysis, but like any investment tool, it can be used either appropriately or inappropriately. What should investors and managers know about a particular Monte Carlo product in order to be confident that it provides reliable information? Unfortunately, not all commercially available Monte Carlo products generate equally reliable results, so users should be aware of product differences that affect the quality of results.

First, any user of Monte Carlo should be wary of a simulation tool that relies only on historical data. History provides a view of only one possible path among the many that might occur in the future. As previously mentioned, it is difficult to estimate the expected return on an equity series using historical data, because the volatility of equity returns is large in relation to the mean. For example, suppose we are willing to assume that the expected return of the S&P 500 is equal to the average historical return. Annual data from 1926 through 1994 would yield an average return of 12.16 percent. Adding just five more years of data, however, would produce an average return of 13.28 percent. For a 20-year horizon, this relatively small adjustment in the input data would lead to a difference of more than 20 percent in ending wealth, given returns every year that were equal to the assumed average.

Second, a manager who wants to evaluate the likely performance of a client's portfolio should choose a Monte Carlo simulation that simulates the performance of specific investments, not just asset classes. Although asset class movements can explain a large proportion of, for example, mutual fund returns, individual funds can differ greatly in terms of their performance, fees, fund-specific risk, and tax efficiency. Failing to recognize these factors can yield a forecast that is far too optimistic. As an example of how much fees can affect performance, consider the case of a hypothetical S&P 500 index fund that charges an annual fee of 60 basis points; expected return is 13 percent with annual standard deviation of 20 percent and normally distributed returns, and capital gains are taxed at 20 percent. A Monte Carlo simulation shows that a $1,000 investment will grow to a median after-tax wealth of $6,200 after 20 years, if that fund pays no short-term distributions. In contrast, an investor with access to an institutional fund that charges only 6 basis points will see her after-tax wealth grow to a median of $6,800 after 20 years.

Third, any Monte Carlo simulation used for advising real-world investors must take into account the tax consequences of their investments. Monte Carlo simulation must and can be flexible enough to account for specific factors such as individual-specific tax rates, the different treatment of tax-deferred versus taxable accounts, and taxes on short-term mutual fund distributions. To understand the importance of short-term income distributions, take the previous example of the institutionally priced index fund. If the same fund were to pay half of its annual return as a short-term distribution taxed at a rate of 35 percent, the $6,800 median wealth after 20 years would shrink to just $5,600.

Certainly, no forecasting tool is perfect, and Monte Carlo simulation has drawbacks that create challenges in relying on it solely as a window to the future. Inputting distributions in determining probability outcomes for the simulations can be biased by historical perspective and the perceptions of the analyst. The process can be quite rigorous and still produce estimates that vary widely from actual result

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