"Even the most brilliant of mathematical geniuses will never be able to tell us what the future holds. In the end, what matters is the quality of our decisions in the face of uncertainty."

So said Peter Bernstein, the first editor of The Journal of Portfolio Management (currently a consulting editor). And since with investing, we are dealing with uncertainty (not risk, where we can measure the odds of success), portfolio construction is more an art than a science. However, we are not without tools to help us design portfolios that improve the odds of achieving our financial goals, whether they be leaving an estate of a minimum size or minimizing the risk of outliving our assets. The tools we have are the historical evidence of the risk and returns of various asset classes, their historic correlations and Monte Carlo simulators.

Monte Carlo simulations require a set of assumptions regarding time horizon, initial investment, asset allocation, withdrawals, savings, retirement income, rate of inflation, an asset's volatility and average expected return, and correlations among the different asset classes. Based on these inputs, a Monte Carlo program generates a pool of random returns from which one return is applied in each year of the simulation. This process is repeated thousands of times to calculate the likelihood of possible outcomes.

One of the main benefits of Monte Carlo simulations is that they allow an investor to view the potential outcomes of various strategies and see how marginal changes in asset allocations--as well as withdrawal rates--affect the odds of various outcomes.

Let's examine how this tool can be used to help you make better decisions as you assist clients in designing their portfolios. To do this, we will look at three case studies, each of which involves a 65-year old investor with a beginning portfolio value of $5 million and a 30-year investment horizon.

Case Study 1

In this example the 65-year old investor would like to be able to withdraw 6% a year in real terms--adjusted each year for inflation--for 30 years. We looked at a portfolio that is 100% stocks (S&P 500 Index) and found that the odds of his attaining at least $1, $500,000 and $1 million at the end of the period are 80%, 79% and 77%, respectively. However, if we add a 5% allocation to commodities in the form of a fully collateralized commodities futures investment that replicates the S&P GSCI (Goldman Sachs Commodity Index), the odds of success rise to 86% (versus 80%), 84% (versus 79%) and 82% (versus 77%).

Now let's look at the odds of success if the investor chooses a 4.5% withdrawal rate. With a 100% stock portfolio the odds of having at least $1, $500,000 and $1 million are 95%, 95% and 93%, respectively. The odds increase to 98% (versus 95%), 97% (versus 95%) and 96% (versus 93%) when we add a 5% allocation to the S&P GSCI.

In both the 4.5 % and 6% withdrawal rates the addition of a small allocation to commodities enabled us to lower the volatility of the portfolio without lowering its return. Consequently, the odds of successfully achieving the three target balances in the Monte Carlo runs increased. Considering the high returns that commodities earned in the past, there seems to be no tradeoff when incorporating them into an equity portfolio, thus creating an easy decision to add this asset class.

Case Study 1-A

Here we will run Monte Carlo simulations very much like the above but this time using expected returns and volatility assumptions for equity and commodities instead of historical returns. The return estimates used in the Monte Carlo simulations of this case study were lower than the ones used in the previous case study. The difference is especially significant on the commodities side.

As in the first example, the 65-year-old investor would like to be able to withdraw 4.5% annually in real terms for 30 years--adjusted each year for inflation. We found that the odds of a 100% stock portfolio having at least $1, $500,000 and $1 million at the end of the period are 61%, 58% and 54% respectively. However, when we added a 5% allocation to commodities, the odds of success changed to 62% (versus 61%), 58% (unchanged) and 55% (versus 54%). The addition of commodities slightly improved our odds of success even though the expected return of the new portfolio was lower than the expected return of the all-stock portfolio. This seems like an odd result, but consider how the expected risk and reward mechanisms of a portfolio work together. The fact that commodities have a negative correlation to equities resulted in a significant reduction in the volatility of the portfolio--reducing its negative impact on compound returns, which in turn more than compensated for the reduction in expected return. It is the combination of expected return and volatility that determines the performance of a portfolio over the long term, not just the expected return. This is why it is important not to make the mistake of considering the risk and return of an asset in isolation. Instead, one should consider how the addition impacts the risk and return of the total portfolio. It is also the very reason why diversification is one of the most important concepts in finance.

Now let's examine the same Monte Carlo simulation but change the third portfolio target balance from $1 million to $4 million:

Ending Portfolio ValueS&P 500S&P 500/Commodities (100/0) (95/5)

> $161%62%

> $500,00058%58%

> $4 million37%36%

While the addition of commodities to the equities portfolio in this simulation increased our chances of not running out of money, it reduced our chances of achieving the higher target balance of $4 million. Again, these results can be explained by the fact that the tradeoff from the addition of commodities (lowering volatility while lowering expected return) leads to a tradeoff between increasing the chances of preserving a portfolio and increasing the chances of accumulating high net worth. Unlike Case Study 1, this tradeoff makes our decision to include commodities in the portfolio neither clear nor obvious. Having to make a decision between reaching a high goal and securing a lower goal is sometimes a difficult one to make. It is a decision that needs to take into account the client's personal situation and the relative importance of the various possible financial objectives in that client's life. There is no right or wrong answer. Having said that, the addition of commodities would likely be the preferred choice for the risk-averse investor.

The choice between the 6% and 4.5% withdrawal rates is also a personal one. Lowering the withdrawal rate increases the odds of success, but at the price of lower current consumption. Faced with this type of decision, the investor should consider what options he has should the "risk of failure" appear. Those with options (e.g., selling a second home, moving to a lower cost of living area, cutting spending) and the willingness to exercise them can accept more risk--use a higher withdrawal rate. Those without options should consider minimizing risks since the price of being alive without adequate financial assets is high.

Case Study 2

In this example, our 65-year old investor is willing to withdraw 4.5% from his portfolio annually. This time we will explore his exposure to two different model portfolios: Portfolio A has a moderate tilt to small-cap and value stocks, and Portfolio B has a strong tilt to small-cap and value stocks. Initially, our investor decided that he prefers a moderate tilt to small-cap and value stocks and an allocation of 80% equities and 20% bonds. The expected return of this portfolio is 7.3%, and its expected standard deviation is 13.4%. By giving the portfolio the stronger tilt to small-cap and value stocks, we can achieve the same expected return with just a 50% allocation to stocks. That portfolio has an expected standard deviation of only 9.0%.

Now let's look at the odds for success produced by the Monte Carlo simulation:

End Portfolio Value Portfolio A (80/20) Portfolio B (50/50)

> $172%88%

> $500,00067%84%

> $1 million63%77%

50% of having over$2.89 million$4.16 million

The investor can lower the volatility of his portfolio and consequently, significantly increase the odds of success. That seems like a tradeoff that most--but perhaps not all--investors would make. The lesson to be learned here is that for any given portfolio, regardless of its size, the asset allocation that enables us to reduce the standard deviation of the overall portfolio while keeping its expected return at a given level should be expected to perform better in the long run.

Case Study 3

Our same 65-year-old investor now determines that he needs to take enough risk to attempt to earn a 10% rate of return. As it turns out, the expected rate of return for Portfolio B is 10.1%. However, that is only the mean expected return; other outcomes are not only possible, they are also likely. Furthermore, we see that its standard deviation is expected to be 17.4%. As an alternative, our investor could consider a portfolio that is tilted even further to the riskiest stocks--small-cap value stocks. A portfolio that holds all its equity in small-cap value stocks (domestic and international) could hold less equity than the original portfolio. It turns out that to achieve the 10.1% return, an investor holding only small-cap value stocks would need to have an equity allocation of just 70%. That portfolio's expected standard deviation is 16%.

Let's now look at the results of the Monte Carlo simulation.

End Portfolio ValuePortfolio B Small-Cap Value Stocks

(100% Equity)(70% Equity/30% Fixed )

>$1`85%90%

>$500,00083%88%

>$1 million81%85%

Investors who fully understand the nature of the risks (especially the risk of tracking-error regret), face situations that require them to own risky portfolios, or they are more concerned about the risk of failure than the potential for accumulating a large estate. They should at least consider portfolios that have stronger tilts to the risk factors of size and value while lowering their equity exposure overall. Doing so reduces the odds of the portfolio "failing," which for most people--to use a Star Trek term--is the "prime directive."

Larry E. Swedroe (bamservices@bamstl.com) is a principal and director of research for the Buckingham Family of Financial Services and author of seven books, including The Only Guide To Alternative Investments You'll Ever Need.