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**Using stationary
distributions to model returns**

Fortunately for the
financial services software industry, the use of *stationary *
distributions in their conventional models provides great cover for their
indefensibly optimistic return expectations.

One of the bizarre
consequences of modeling returns with a stationary distribution is that the*
*distribution of possible outcomes* increases *with time.
Dramatically.

Take, for instance,
a $1000 investment modeled to achieve an optimistic 12% arithmetic average
return with a highly volatile 30% standard deviation. * If
*you model returns with a

*, then after the first year, there are equal, 1 in 6 chances of falling outside a boundary between -1 and +1 standard deviations from the mean, with resulting portfolio values of either less than $835 or above $1522. To put it another way, the model predicts a 2 in 3 chance of the portfolio ending up*

**stationary normal distribution***inside*, and 1 in 3 chance of the portfolio ending up

*outside*, the $835-$1522 range. The ratio between the -1 and +1 standard deviation thresholds, after 1 year, is 1.8. Over the short time frame of 1 year, that's not too unreasonable for a highly volatile asset.

But over the long
term, the stationary return distribution model * blows up*.
After 30 years, the stationary normal distribution model predicts a 1 in 6
chance that the $1000 portfolio will grow to a value less than $6,657 and a
1 in 6 chance that it will grow

*above*$125,490. To put it another way, the model predicts a 2 in 3 chance of the $1000 investment (after 30 years) growing to a value inside a yawning $6,657 to $125,490 range. There's still a 1 in 6 chance it will be less, and a 1 in 6 chance it will be more. The ratio between the -1 and +1 standard deviation thresholds, after 30 years, is

*! After 70 years, the ratio grows to an astronomical 88.6. With just typical financial planning time frames, conventional simulations spit out essentially meaningless outcomes – anything can happen.*

**18.9**After 30 years, the -1 S.D. threshold represents a 6.5% annualized return, which is considerably less than the originally forecast 12% annualized return. The +1 S.D. threshold represents a 17.5% annualized return, with phenomenal compounding.

The inevitable long-term blow up of Monte Carlo simulations using stationary distributions provides excellent cover for the industry's indefensibly optimistic return expectations. Assume the stock market just barely keeps up with an average 3% inflation rate over a 30-year period. The financial services industry can claim that even with their 12% expected return assumption, their model predicted a 5% chance of such a dismal long-term result.

The stationary distribution modeling assumption leads to an absurdly wide distribution of multi-period outcomes. The model itself is based on a fundamentally unsound and statistically unlikely proposition: that returns are completely independent of past returns; that there is no serial correlation between returns; and that returns following an astronomical stock market bubble have the same expected distribution as returns following a protracted bear market.

"Monte Carlo simulation is generally an oversimplification of the real world.... Monte Carlo variables assume that the processes being studied are independent of each other and that each value is a random draw from a distribution, or serially independent.... Monte Carlo simulation homogenizes away the factors that drive stock returns." –David Nawrocki, Ph.D., "Finance and Monte Carlo Simulation," Journal of Financial Planning, Nov. 2001. |

In their July, 1987 NBER Working Paper entitled “Mean Reversion in Stock Prices: Evidence and Implications,” authors James Poterba and Lawrence Summers developed evidence that stock returns are positively serially correlated over short periods and negatively autocorrelated over long periods. Over long time frames, such as a typical investor's remaining life expectancy, the completely random walk model of stock returns is a bad one.

Twenty-two years later, that
knowledge continues to be ignored. The financial services software
industry *continues* to model returns with stationary distributions.
Interestingly, Financeware's CEO recently criticized mean-reverting models
in an April 2009
White Paper. And why not? Adopting a mean-reverting model
would force Financeware to temper its return expectations. *That *
would be bad for business.

**Blaming it on the
tail**

The chief problem with the traditional Monte Carlo models employed by the financial services industry is not that their models underestimate the risk, but rather that their models overestimate – and grossly so – the expected reward.

Yet since the Great Panic of 2008, the financial media has asked nary a question, and published nary a critique, of the optimistic return assumptions that most financial software providers employ in their Monte Carlo models. Rather, practically all of the mainstream press and industry attention has been directed to "black swans" and "fat tails." A repeated criticism lobbed against Monte Carlo simulation models used by the financial planning industry is that the conventional bell-curve distribution used to model returns underestimates the risk. The tails, these financial journalists assure us, are not fat enough.

For example, a May
2, 2009
*Wall Street Journal* article entitled "Odds-On Imperfection: Monte
Carlo Simulation – Financial Planning Tool Fails to Gauge Extreme Events" attacked conventional Monte Carlo simulators
for "assum[ing] that market returns fall along a bell-curve-shaped
distribution," and having ** tails **that assign "negligible odds
to a 54% decline."

Likewise, a February 2009 Morningstar article quotes Roger Ibbotson as arguing that the main lesson of 2008 is that "the standard deviation doesn't capture all the risk." Another same-dated article notes that none of the "most widely used risk measures" – "standard deviation, kurtosis, and Morningstar Risk" – "predicted the dire outcomes that the funds faced." Yet another same-dated article blames the lognormal distribution used in many models for not having a fat enough tail.

How has the financial software services industry responded? Predictably. Oh yes, the problem is in the tails. We'll fatten those tails right up. The WSJ article reports that in 2008, Morningstar "tweaked its asset-allocation software" to allow "users to choose a bell-curve-shaped distribution or a 'fat-tailed' distribution." Likewise, ESPlanner was also "considering offering clients Monte Carlo scenarios that incorporate fatter-tailed distributions."

Which misses the
point. Unless those distributions are centered around *realistic *
return expectations, fattening the tails will do as much good as putting
lipstick on a pig. Models that use unrealistic return assumptions will
continue to set up millions of advisor clients for disappointment.

Sure, minor
improvements can be made to fatten the tails of conventional Monte Carlo
distributions used to model equity returns. **But tweaking the tails
does not matter nearly as much, in the long run, as centering the
distribution of returns about a reasonable expected mean.**

To the relief of the
financial services industry, the financial press simply does not get it.
So you won't find much discussion from the financial software services industry about *
re-centering *their bell curves around more realistic forward-looking equity return
assumptions. *That*, after all, would spoil the fun.
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TIP$TER avoids both the "stationary" distribution and "fat tail" problems by employing "exploratory simulation" of past S&P 500 return data. It also avoids unrealistically optimistic forward-looking return assumptions by scaling that return data to match your forward-looking expected returns.

But even if you prefer Monte Carlo simulation to exploratory simulation, TIP$TER offers a fat-tailed, non-stationary "double lognormal" distribution model with which to model returns. For more information, consult the user manual.

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