A mathematical model for early retirement

Earlier I wrote about a mathematical model for beard growth and awesomeness. In that discussion I briefly described a similar model for net worth growth. It turns out, though, that your actual net worth number is not a particularly useful measure for determining when you can retire. If early retirement is your goal, the only thing you need to concern yourself about is your savings rate. In this post, I’ll describe why your savings rate is the most important measure and how you can use it to calculate how many years it will take you to retire. If you want to retire in the next 10 years, as Mrs. Beard and I have planned, then you’ll also learn how to determine what your savings rate needs to be.


How much do I need to retire?

When a discussion about retirement pops up, many people wonder about the amount of money they need to save in order to retire. If you travel around the internet, you’ll read supposedly educated people shooting off all kinds of answers to that question, usually ranging from $2 million to $5 million. Typically, it comes in the form of complainypants hysterics about how no one can retire without 40 gadzillion dollars, and no one can save that much because of Republicans, Obama, or the Illuminate.


But, this question is absolutely impossible to answer without knowing something very, very important. How much do you plan to spend? The “experts” never seem to ask this question. Usually they just assume you need to match your current salary multiplied by some factor in order to retire. However, your current expenditures have all kinds of crap built in that go away in retirement. Stuff like a mortgage, which you’ve hopefully paid off by then. Or savings. You don’t need to buy fancy suits all the time anymore, because you’re not working. Basically, your expenses go down when you retire. It’s a proven fact.

You’re salary actually has absolutely nothing to do with the amount of money you need to retire. Seriously. The only variable that matters is your savings rate. Let me explain.

Savings Rate

Your savings rate S is calculated like so:

S = \frac{I-E}{I},

where I is your take-home income (ignoring taxes) and and E is your expenditures. Basically, subtracting your spending from your take-home pay gives you an estimate of how much you are saving. Notice, your savings rate could be negative if you spend more than you earn.

If you want some more detail on how to calculate your savings rate, then Mr. Money Mustache goes into some very good detail.

Now that you have your savings rate, you can figure out how long it’s going to take you to retire. If you save 100% of your earnings and spend nothing, then congratulations! You can retire right now because you have found a way to live for free. If you spend every single penny that you earn, or worse, you spend MORE, then you will never ever ever never be able to retire. If you spend more than you earn, you create debt, and if you keep doing that then you get very, very poor. See the website WhyYourePoor.com for more information on this.

What does it mean to be retired? When your savings are generating the amount of money you need for expenses through interest, then you can retire. A insanely big and awesome study was conducted that showed that you can safely draw 4% from your investments each year pretty much forever without ever having to worry about running out of money.

If you need to spend $40,000 per year to maintain your awesomeness, then you will need $1 million to retire. Four percent of $1 million is $40,000.

If you only spend $25,000 per year, then you’ll only need about $625,000 to retire.

A small family can live pretty fancy on only $25,000 per year, especially if your house is paid off. My family of four has expenses of $36,486 per year, and that’s with a mortgage, a student loan payment, and other crazy fancy bullshit we don’t need. Assuming we still had that mortgage in retirement AND student loan in retirement (we won’t), we would need $912,150 to retire RIGHT NOW.


A Mathematical Model for Early Retirement

So where does savings rate come in? Well, the savings rate gives you a good handle on both how much you spend and how much you earn. Someone that earns $100,000 but spends $80,000 has a savings rate of 20%. Someone that earns half of that at $50,000 and spends $40,000 will have a savings rate of 20%, too. These two people will be able to retire IN EXACTLY THE SAME AMOUNT OF TIME!

Check out this handy tool at Networthify.com. Assuming a (low) ROI of 5% per year, and based on the trinity study, they are able to calculate the number of years a person will need to save until retirement based only on their savings rate. I have reproduced the results here and fit the data to a mathematical model:


This graph shows that your years until retirement if you start from ZERO is related to your savings rate S as follows:

R = -21.68 \ln{S} + 100.96

Grab your calculator, because all you need to do is plug in S, and out pops the number of years you have until retirement. If you’re not starting from zero, then just use the tool at Networthify.com.

Let’s say you have a target in mind for retirement. You want to know what your savings rate needs to be in order to hit that target. Simple.

S = e^{-\frac{R - 100.96}{21.68}}.

The shorter the time you have until your retirement age, the larger that savings rate needs to be.

The Beard family is looking to retire in 10 years minimum. Since we have a head start, we would need a savings rate of about 55% to meet that goal. And that is pretty much right were we are.


Early Retirement

Here is something cool. Let’s say you’re a bearded badass that recently graduated college and you landed an awesome job. You’re pulling down $80,000 per year. Hell yeah. If you can find a way to save 67% of your income, then you can retire at 30 friggin’ years old. I mean never having to work for money again. Ever. For reals.

Let’s say you’re a young married couple, 28 years old and earning $50,000 per year each. That’s not too terribly hard, right? That’s a combined income of $100,000. If you can find a way to spend only $33,000 of that per year, then you will both be retired by 38. I have a wife and two children and a house about 1,000 square feet more than we could ever need and I spend near that amount. You two can do that easily.

Let’s say you’re family makes the average household income of $52,000 per year. This one’s a tough one, because it could be hard to live on $17,200 per year. Possible, of course, but we like fancy living around here. Let’s say you find a way to live fancy on $25,000 per year. That’s a savings rate of 52%. You can retire in 15.7 years. If you’re 35 years old with some gray hairs in that glorious beard, then you’ll be able to retire at 50. I’m talking a straight up, average, regular Joe without a fancy degree and a fairly basic job. That dude can can easily … easily … retire by 15 years ahead of what is considered “normal.” This my friends is the power of math and Beard Magic.

The big takeaway here is that the math for early retirement is simple, and it’s surprisingly easy to figure out a way to retire super early. I’m not talking “early” retirement at 60, but as early as 30 for some of you young folks out there. And I’m not talking a life of deprivation, either.

Imagine retiring before you even manage to reach terminal beard. How badass.

A mathematical model for early retirement

3 thoughts on “A mathematical model for early retirement

  1. I think your savings rate formula is wrong. I believe it should be:

    S = (I – E) / I

    Just discovered your blog today through RockstarFinance and I love it!

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