When Monte Carlo analysis meets a black swan

In early November 2007, two investors met in the waiting room of their financial adviser’s office. By chance, both Robert, 62, and Sandra, 78, had appointments to discuss their retirement financial plans.

As part of the process, each went through a Monte Carlo simulation, which has become ubiquitous in financial planning circles, and were given a “success rate” figure. Robert and Sandra were told, independently, that based on their current investment asset allocations, desired spending rate, age and health, their retirement sustainability was approximately 80%.

In other words, they had an eight-in-10 chance of having enough money to sustain their retirement, which meant that their plans were in good, albeit not great, shape. Stated differently, their probability of running out of money in retirement was 20%. Thus, according to the MCS analysis, while Robert and Sandra differed in many ways, they shared the same retirement risk.

Now, 17 months later, Robert and Sandra share very little. Robert has seen his nest egg decline by almost half and has very little hope of sustaining his original retirement plans. By contrast, Sandra — who also lost money — is in much better shape. Yes, she may have to tighten her spending belt, but her life will be affected to a degree nowhere near as dramatic as Robert’s.

If both Robert and Sandra had the same retirement profile just 17 months ago, what happened?

As you might suspect, Robert had most of his nest egg allocated to equities, while Sandra was invested primarily in U.S. government bonds. The full impact of these differences typically is not revealed in Monte Carlo simulations, in which algorithms generate many different financial scenarios for the future, but report only the percentage of cases in which a certain income or spending target is met. This is the success rate.

Only rarely do MCS software packages provide any sort of sensitivity analysis. And when it is available, it is seldom used. As we’ve seen with Robert and Sandra, two investors can have relatively similar sustainability forecasts for their retirement income, while having wildly different real-world outcomes based on their portfolios and changes in the market value of their holdings.

So how should advisers measure, report and make sure these differences are clear to individual clients?

I propose the following: Whenever conducting a Monte Carlo simulation or using a similar analytic technique, an adviser should compute the sustainability number using two different sets of assumptions.

First, use a baseline set of assumptions representing the client’s current situation. Then assume that three years have gone by and the underlying investment portfolio has experienced a 1-in-100 event. This is a sustainability analysis assuming the occurrence of rare, often devastating events explained by Nassim Nicholas Taleb in his best-selling book, “The Black Swan: The Impact of the Highly Improbable” (Random House, 2007).

If we perform a sustainability analysis using both assumptions, we can come up with a very useful measure I call the sequence-of-returns downside exposure — or Sordex — ratio. The ratio is the current baseline sustainability number divided by the black-swan sustainability number minus one. The greater the Sordex ratio, the more vulnerable the plan is to abnormal market conditions.

To see how this ratio can be useful, let’s go back to Robert and Sandra, and add some detail. Robert had a $950,000 investible portfolio in November 2007 and wanted to consume $37,110 per year in inflation-adjusted terms. That’s equivalent to a spending rate of 3.91%. At 62, he was relatively young, considered himself to be risk-tolerant and therefore allocated only 10% of his portfolio to safe bonds, which earned a meager inflation-adjusted real return of 2%. The remaining 90% of his portfolio was allocated to equities, which, according to the embedded economic assumptions, were expected to earn a 7% real rate of return with a standard deviation of 25%.

Sandra, who is 16 years older than Robert, had only $330,000 of investment assets at the starting point of our analysis. From that, she wanted to consume $25,780 per year, also in real terms. This was a spending rate of 7.81%, which was much higher than Robert’s but understandable given her age.

More important, Sandra was very conservative and risk-averse. Her asset allocation consisted of 75% in safe bonds and only 25% in diversified equities. The economic return and volatility assumptions for Sandra’s MCS analysis were exactly the same as Robert’s. Under the baseline analysis, both were in the same risk category, yet we intuitively sense that Robert took on more risk.

Now let’s do a second sustainability analysis and assume that three years have elapsed. Robert is 65, and Sandra is 81. We’ll also assume that during those three years, the markets produced returns with a one-in-100 chance of occurring based on the same Monte Carlo assumptions — that is, an event that we might see once per century. Under those assumptions, Robert’s wealth declines to $342,400, while Sandra’s wealth declines to $197,700.

The assumptions also produce a hypothetical 29.6% sustainability for the 65-year-old Robert and 53.7% for the 81-year-old Sandra. Note that our black-swan assumptions produce much lower numbers than our baseline numbers. The important question is, how much lower?

Let’s go back to our sequence-of-returns downside exposure ratio. In our examples, the ratio of the baseline sustainability value (80% for Robert and Sandra) to the black-swan sustainability value (29.6% for Robert and 53.7% for Sandra) is 2.7 and 1.49, respectively. Subtract one from both numbers to convert them into marginal percentages, and we see that Robert’s Sordex ratio is 1.7 units, while Sandra’s is a much lower 0.49. Ergo, Robert’s sustainability is about 3.5 times more vulnerable to black-swan events than Sandra’s, even though her spending rate is double his.

I believe that the insight into retirement vulnerability provided by the Sordex measure can be helpful to advisers and their clients. The ratio can be easily computed in any Monte Carlo simulation or analytic software tool and thus requires no new mathematical tools or skills. The technique also can be extended to inflation and longevity black swans. Some MCS packages allow for such scenario analysis, making implementation easy.

Producing a value between zero (very good) and infinity (very bad), the Sordex ratio summarizes the vulnerability of a sustainability number to statistical outliers that are defined within the same simulation.

A Sordex ratio greater than 1 should raise alarm bells, while anything above 2 should set off ear-piercing sirens. Instead of condemning Monte Carlo analyses for missing the meltdown, let’s properly harness the full power of stochastic methods to give us tools that provide clear utility.

Moshe A. Milevsky is a professor of finance at the Schulich School of Business at York University in Toronto, the executive director of its IFID Centre and president and chief executive of The Quantitative Wealth Management Analytics Group Inc., a Toronto-based company that creates financial models, algorithms and calculators. He acknowledges the helpful suggestions of Dr. Peng Chen of Ibbotson Associates Inc. in Chicago and Brett Cavalieri of Morgan Stanley in New York.