## Understanding MaxiFi’s Monte Carlo Risk Analysis

MaxiFi’s risk analysis recognizes that you are investing at risk, i.e., that you can’t tell precisely what real rates of return (returns adjusted for inflation) you will earn each year on your regular and retirement account assets. We call a particular sequence of annual real returns that you may receive a trajectory of returns.

Your investment strategy – the assets you tell MaxiFi you'll invest in through time – determines what trajectories of real returns you may experience. Suppose your investment strategy is to hold stocks for 15 years and then switch to bonds. On average, stocks yield higher annual real returns than do bonds. But their annual real returns are far more variable. Hence, your trajectories of real returns will be generally high, but very variable for the first 15 years and generally low and smooth thereafter.

Or your investment strategy may entail holding half of your assets in stocks and half in bonds on an ongoing basis. In this case, your real return trajectories will generally lie midway between the typical all-stocks or all-bonds return trajectories. This mixed investment strategy would exhibit less real return variability than the all-stocks strategy, but more variability than the all-bond strategy.

For any given investment strategy, the particular trajectory of real returns you end up receiving will differ from the average real returns trajectory because of the random nature of the real returns you earn on your investments. In other words, you can’t expect to earn the average real return every year. Instead, there will be high and low real returns along any trajectory. You can also experience lots of low real returns or lots of high real returns in a row. This is called *sequence of return risk*.

MaxiFi can’t tell you what trajectory of real returns you’ll receive through time. It can and does show you the range (distribution) of real return trajectories you may face given your investment strategy. Keep in mind that the historical real return data we use to help predict future real returns are no guarantee of future returns. It’s also important to bear in mind that you only go through life once and will only experience one trajectory of real returns.

## Translating a Real Returns Trajectory Into a Living Standard Trajectory

The trajectory of real returns you experience will help determine, along with your other economic and demographic inputs, the living standard your household will experience each year. We call a trajectory of annual living standards a *living standard trajectory*. For each real return trajectory you may experience, there is a corresponding living standard trajectory.

There is another key factor that enters into the determination of your living standard trajectory given your real return trajectory and other inputs. This other factor is your spending behavior, which you can control by setting an annual safe rate of return. This is the return you can count on receiving, on average. We ask you to be cautious in setting your safe rate of return as well as all other inputs.

## MaxiFi’s Monte Carlo Living Standard Risk Analysis

Once you’ve specified your investment strategy and safe rate of return, MaxiFi produces hundreds of possible living standard trajectories to show you the range of living standard trajectories you may experience.

MaxiFi repeats the following steps to generate the hundreds of living standard trajectories. In this description, we’ll assume you are 50 years old to make things concrete. But the same process is followed regardless of your current age. We’ll also assume you specified a 3.5 percent safe rate of return together with a 2.5 percent inflation rate. This implies a roughly 1 percent safe annual real return.

- MaxiFi draws, at random, a real return trajectory – a trajectory of annual real returns for your regular assets, your retirement account assets, and your spouse/partners’ retirement account assets (if applicable). These draws are based on your specified
*investment strategy*, which can entail holding different portfolios (combinations) of assets over time. [see note #2 at the bottom of this page for information about how these returns are drawn.] - MaxiFi calculates what you should spend this year (at age 50) if you knew for sure you were going to earn, in each future year, the safe real return you specified. MaxiFi records your household’s living standard at age 50. Note that if you set a lower safe real rate of return, MaxiFi will calculate a smaller spending amount. Hence, your choice of the safe real rate of return controls your spending behavior. The lower the safe rate you set, the less you will spend in the present and, therefore, the lower the chances of having very low spending in the future.
- MaxiFi advances you by one year to age 51, but in calculating what assets you’ll have at age 51, it uses the actual real returns that it drew for you for age 50 in Step 1. Each annual step forward will create new saving & withdraw amounts, and new retirement withdraw amounts and of course new living standard & discretionary spending level simulating the way things would work in real life use of the program over time.
- MaxiFi calculates what you should spend this year (at age 51) if you knew for sure you were going to earn, in each future year, the safe real return you specified. MaxiFi records your household’s living standard at age 51 as well as new asset balances.
- MaxiFi advances you by one year to age 52 and so on. In other words, it runs your household forward in this manner, calculating a living standard for your household for each year through your household’s last possible year of life. If you were 60 at the time and the plan runs to age 100, the program will rebuild the plan 40 times, walking through these calculations year by year to complete just one trajectory out of 500.
- MaxiFi collects all your annual living standards and stores them as the first of your living standard trajectories.
- MaxiFi draws a new return trajectory and repeats steps 1 through 6.
- After generating 500 living standard trajectories, MaxiFi ranks them based on their average annual living standard. It then displays five living standard trajectories (as well as their associated discretionary spending levels) – the trajectory with the 95th highest average annual living standard, the trajectory with the 75th highest average annual living standard, the median trajectory with the 50th highest average annual living standard, the trajectory with the 25th highest average annual living standard, and the trajectory with the 5th highest average annual living standard.

## Using MaxiFi’s Monte Carlo Risk Analysis

Once you’ve run MaxiFi’s risk analysis based on your specified investment strategy and spending behavior (determined by your specified safe real rate of return), you can try other investment strategies and spending behaviors. You’ll see that less aggressive investing will lead to living standard trajectories that are lower, but less variable. You’ll also see that less aggressive spending, generated by a lower assumed safe real rate of return, will produce living standard trajectories that start out lower, but have less chance of being low in the future.

Bear in mind that you will experience just one trajectory of returns through time and one living standard trajectory based on that trajectory of returns and your spending strategy. Consequently, you face three risks when it comes to deciding how aggressively to invest and save:

- You’ll end up on a living standard trajectory that’s lower, on average, than you’d like.
- You’ll end up with more annual variability in your living standard than you’d like.
- You’ll experience more downside living standard risk toward the end of your life than you’d like.

MaxiFi doesn’t suggest what investment strategies and safe rate of return you should adopt. This depends on your attitudes toward risk. What MaxiFi does is show you the implications of investing and spending more or less aggressively to assist you in deciding how to invest and spend through time.

A Final Comment on Range Reports vs Trajectory Reports

When viewing a range report like the following (this one is a range of living standard report, but there are other range reports as well), you can slide your arrow pointer over the report from left to right and observe the range of possible living standards (in this case) that are possible given your asset allocation and spending behavior assumptions.

Note that the range of possible living standards gets wider as time goes on. The lower you set your spending behavior setting percentage in the MC set up, the more likely you will see this chart tip upward from left to right. Note also that the dark blue area (25th-75th) can be interpreted to mean that 50% (75-25 = 50) of the time, the living standard will fall within this range. The dark grey area (10th-90th) can be read to mean that 80% of the time the living standard would fall within this range for any particular year that you select. In the example above, the year 2037 has been selected. Note that you can click on the labels in the chart legend to toggle them on or off. Users are typically concerned with what is possible along the bottom edge (10th percentile) of the 10th-90th percentile range.

Whereas the range report focuses on an independent vertical slice of probabilities across any one year that you select, the trajectory reports focus on a connected sequence or living standard threads across the full life of the plan. The trajectories shown in the trajectory report are just 5 of the 500 that were generated.

The "95th Percentile" trajectory means 95% of all the trajectories MaxiFi produced have a lower average living standard and 5% have a higher average living standard. The "50th Percentile" means 50% of all trajectories have a lower and 50% have a higher average living standard.

The trajectories may cross. Indeed, they may cross many times. For example, the 50th Percentile trajectory may be higher than the 95th Percentile trajectory in some years. The reason is that investment returns won't always be high, moderate, or low along any trajectory. Trajectories that are high on average won't be high in every year, and those that are low on average won't be low in every year. This is called sequence of return risk.

Your actual living standard trajectory could look like any of these trajectories, with up years and down years. When returns are high, you'll spend more and enjoy a higher living standard. When they are low, you'll spend less and have a lower living standard. Thus, the living standard trajectories not only show the ups and downs you may experience, but also how well or poorly you may fare, on average, over your lifetime. While most will locate around the middle, ending up on a higher or lower trajectory with lots of ups and downs over the years is a possibility you should not ignore.

**NOTES**