A couple of months ago, I discovered Mr. Money Mustache (MMM). He’s somebody who “retired” at the age of 30 by saving a ~3/4 of his income and living a frugal lifestyle. A unique and enthralling way of living the American Dream. I have been reading through his different posts and like what I read. As you could likely guess, many of MMM’s blogs deal with investing. I like his posts because they encourage me to think about topics and ideas that I may otherwise not think about.

One of these topics is the “safety factor” when dealing with investing. Safety factors are commonly used in engineering and MMM was trained as an engineer. In a nutshell, safety factors often use the worst case scenario and then include a multiplier (e.g., 10 times that value) as a “safety factor”. As an example, perhaps a type of chair breaks with 1,000 lbs during a test. A safety factor of 2 would limit the chair to 500 lbs max in case the wood deteriorates or other stressors are added or not accounted for.

In contract, I have been trained and an ecotoxicologist with a healthy mix of statistics tossed in to my education. We usually deal with messy data and safety factors are often discouraged because they often do not perform well. One alternative approach is probabilistic risk assessment. This approach assigns a probability distribution to inputs (e.g., market returns) and, conditional on the quality of the data, allows one to estimate the probability of an even occurring. In this post, I apply a a probabilistic approach to explore MMM’s investment assumptions.

MMM usually assumes a 5% return rate on the stock market (or more specifically, on broad index funds) after inflation and uses a 4% withdrawal rate as an added layer of safety. He uses this to define the “safe withdrawal rate” as

“the maximum rate at which you can spend your retirement savings, such that you don’t run out in your lifetime.”

Besides applying this to retirement, he also applies this to idea to understanding how investments grow. Overall, I like his approach. It is a nice approximation that is easy to understand. That being said, I use probabilistic tools at work and I want to explore this investment concept more.

Motivating my own case study is my pending switch from a contractor to Federal Term employee. I will have a ~$3,000 in my 401k when I leave and am wondering how much money will be there in 30 years when I can access it. I’ll do my simulations in R because I am familiar with the program and it is free so anybody can “follow along at home” if they would like (Note that you’ll need to not copy the “>” into R).

To start off, I define two variables:

> nYear = 30 # number of years
> growth = 1.05 # annual growth after inflation

and then use this to calculate how much money I will have in 30 years. This is a difference equation that MMM likes to use and the math is simple:

> growth^nYear
[1] 4.321942

Based upon this worst case scenario, I will should expect to have 4.3 times my starting investment. Therefore, my $3,000 will be come about $13,000 when I can access it in 30 years. However, what about the variability in this simulation and uncertainty? This year, I might get 1.05 growth, but next year could be 1.03 and the year after 1.05. We could simply take the product of 30 years:

1.05 * 1.04 * 1.03 * 1.6 …

but this is tedious, error prone, and boring. Programing and simulations come to the rescue for situations such as this. Instead of generating the numbers by hand, we can assume that they come from a normal distribution. This requires me to guess a standard deviation. I’ll guess 0.05 because it seems reasonable and I do not know a great deal about quantitative finance. In R, this looks like:

> rnorm(n = nYear, mean = 1.05, sd = 0.05)
[1] 1.0460267 1.0851230 0.9941926 1.0551473 1.0233291 1.0695875 1.0287381 1.1444693
[9] 1.0428234 1.0557862 1.0947897 1.0367700 1.0036091 1.0797985 0.9861632 1.0154970
[17] 0.9705547 1.1681693 1.0083213 1.1077836 1.0220802 1.0551286 0.9268657 0.9814797
[25] 0.9921165 1.1120307 1.0640827 1.0439697 1.0824330 1.0698607
and this gives us 30 different growth rates. We then take the product of this and save it as x:
> x = prod(rnorm(n = nYear, mean = 1.05, sd = 0.05))
>
> x
[1] 3.318441

In this example, my investment become 3.31 times larger. But, this is just one possible trajectory. To check out more, I created a function that also plots a histogram and tells me where 95% of the simulations fall:

marketReturn <- function(nYear, growth, sd, reps = 10000, title = "Histogram"){
x <- numeric(reps)
for(i in 1:reps){
x[i] <- prod(rnorm(nYear, growth, sd))
}
hist(x, main = title)
return(list(
min = min(x),
max = max(x),
ci95 = quantile(x, c(0.025, 0.975))
)
)
}

I can now use this function to run 10,000 simulations:

> marketReturn(nYear = nYear, growth = growth, sd = sd, reps = 10^4, title = paste("sd =", sd))
$min
[1] 2.070814
$max
[1] 9.194219
$ci95
2.5% 97.5%
2.996903 6.019339

In case you are unfamiliar, a historgram shows the number of counts on the y-axis and the observations on the x-axis. In this case, the x-axis is how many times my money has grown at the end of 30 years (e.g., 4 times).

Thus, assuming a worse case scenario, I can expect my money to grow between 3 and 6 times it current value. However, what if we dig around on the web, perhaps we can do better with the guess for the standard deviation:

This page list 0.0346 as a standard deviation:

> sd = 0.034
> marketReturn(nYear = nYear, growth = growth, sd = sd, reps = 10^4)
$min
[1] 2.360319
$max
[1] 7.885858
$ci95
2.5% 97.5%
2.989520 6.035744

Which yields a similar result. Another site uses 0.15, but this seems high

> sd = 0.15
> marketReturn(nYear = nYear, growth = growth, sd = sd, reps = 10^4, title = paste("sd =", sd))
$min
[1] 2.191817
$max
[1] 7.982334
$ci95
2.5% 97.5%
2.977771 6.008814

Regardless, the 5% annual growth after inflation is a worst case scenario from MMM. A more optimistic return might be 7%:

> growth = 1.07
> sd = 0.05
>
> marketReturn(nYear = nYear, growth = growth, sd = sd, reps = 10^4, title = paste("Mean growth of 7% with sd =", sd))
$min
[1] 2.830627
$max
[1] 18.28947
$ci95
2.5% 97.5%
4.491611 12.106446

Thus, if I am lucky, or at least average, I will most likely have 6-7x my initial amount when I retire. Thus, my $3,000 will likely be $18,000 to $21,000 in today’s dollars! If I am unlucky, I will only have ~3x my current amount (actually my worst case was ~2.8, which is more pessimistic that MMM’s worst case). And if the market does very well, I could have 12-18x my current amount!

Einstein is often quoted as saying:

“The power of compound interest the most powerful force in the universe.”

Although this quote may or may not be from him, these simulations demonstrate the power of compound interest.

Although, past performance is no indicator of future gains, the future does look promising! If you’d like to explore different scenarios, please feel free to adapt and share this code. If you find anything cool or any bugs with my code, please share it in the comments.