# Accounting for the cost of volatility: alternatives to Monte Carlo

Last time I talked about the “volatility tax” and how just using the simple arithmetic mean can give a misleading picture of future returns. I also showed how using Monte Carlo can give a clearer picture, since it will include volatility.

What if you like the simplicity of a deterministic (and spreadsheet friendly!) model but want some way to account for volatility?

That’s where Gregory Kasten’s model comes to the rescue.

In the July 2013 issue of the *Journal of Financial Planning* he published “Using a Simplified Deterministic Model to Estimate Retirement Income Sustainability”.

The simplified method presented in this paper uses standard deviation for conversion of arithmetic average return to the derived geometric average return for each level of confidence.

I’ll spare you the details of how it is derived and jump directly to the practical uses.

## Select your confidence level

First you need to select the confidence level. This is entirely up to you. Returns are volatile. Are you trying to account for a “slightly conservative” case? Or a “world is nearly ending” case? If you’re not sure, the 80th percentile confidence level is a good place to start.

Pick your confidence level and use the corresponding polynomial formula. Let’s assume we’ve chosen 80, so we’ll use the formula

*y = -1.1820*𝑥*² *− *0.0929*𝑥* *− *0.0010*

(Note that the table can be used to calculate the 10, 20, 25, and 33 percentiles if you’re interested in happy scenarios; I’ll let you read the paper to see how that works.)

## Plug in the standard deviation

What do you do with this polynomial formula? You set 𝑥 to the volatility of the portfolio you want to account for. In our case, that is the standard deviation of US equities. According to the 2012 Credit Suisse Global Investment Returns Yearbook, US equities have historically had a standard deviation of 20.4%.

*y = -1.1820(.204)² *− *0.0929(.204) *− *0.0010*

*y = -0.0691 = -6.91%*

This *y* is your “reduction factor”.

## Subtract the reduction factor from the arithmetic mean

The same Credit Suisse publication says the historical US equity mean return has been 6.2%. So we subtract 6.91% from 6.2%

*6.2% − 6.91% = -0.71%*

This is your “derived geometric return”. Plug that number into your spreadsheet.

## Holy crap that number is low!

Yep. Both the confidence interval and the volatility — standard deviation — plays a big role in the final result.

If we are willing to be a little less conservative our final “derived geometric return” will be higher:

- 66th percentile confidence level: 3.65% derived geometric return
- 75th percentile confidence level: 1.33% derived geometric return
- 80th percentile confidence level: -0.71% derived geometric return
- 90th percentile confidence level: -15.27% derived geometric return

But volatility is the real culprit here. Kasten’s model is assumes standard deviations between 0% and 20%. We’re right at the upper-end of that range with 100% equities so we’re pushing the limits of the model. But, really, the model is just saying that volatility has a huge impact.

Duh.

To see the impact of volatility, let’s keep everything the same (80th percentile confidence level, 6.2% arithmetic mean) and try some different standard deviations:

- 20.4% standard deviation: -0.71% derived geometric return
- 18% standard deviation: 0.60% derived geometric return
- 15% standard deviation: 2.05% derived geometric return
- 12% standard deviation: 3.28% derived geometric return

We didn’t change anything except lower the volatility: this helps show why worrying about volatility isn’t *totally* a behavioural issue.

Instead of using historical US equity returns, what if we use historical world equity returns? A lot of people think future US equity returns will be lower and more like the world historical average.

- 66 confidence level: 3.31% derived geometric return
- 75 confidence level: 1.52% derived geometric return
- 80 confidence level: -0.05% derived geometric return
- 90 confidence level: -10.44% derived geometric return

Going back to the Early Retirement Now spreadsheet, we can use some derived geometric returns and see how it changes the results. We’ll try different numbers for the projected future real returns of stocks (cell B9):

- 5% (the default): gives a safe withdrawal rate (1 percentile) of 3.59%
- 3.31% (66 confidence level): gives a withdrawal rate of 3.39%
- 1.52% (75 confidence level): gives a withdrawal rate of 3.09%
- -0.05% (80 confidence level): gives a withdrawal rate of 2.58%

A spread of 2.58% to 3.59% just from trying to take volatility into account. And now you know how to include volatility into your own spreadsheets.