This is the question that changed everything for me. It’s such a simple question. How many years do you want to work? Most people never think this way. When they do, the popular answers are usually “You work until you are 65” or “You work until you win the lottery”.

Why do we spend so much time thinking about where we want to work, where we want to live, the car we want to drive, and the house we want to live in, but not this more fundamental question?

If we take MMM’s shockingly simple math behind early retirement and flip it around, we can see the following options emerge:

Working Years | Savings Rate Required |
---|---|

40 years | 10% savings rate |

30 years | 20% savings rate |

20 years | 40% savings rate |

15 years | 50% savings rate |

10 years | 65% savings rate |

5 years | 80% savings rate |

*How long do you want to work for??*

The trend is pretty dramatic. Save the recommended 10% of your income and you’ll be doing better than the majority of the United States… but you’ll still be on track for a 40 year mandatory sentence. Up your savings rate to 65% and you could be free in less than a decade! The concept behind it is simple: when your annual investment returns cover your expenses, you’re done- you’ve reached financial independence!

But how does this math work? What are the underlying assumptions? And how can it be independent of my salary? As an engineer, I like to fully understand the math behind charts like these before drastically changing my lifestyle.

I decided to search for the underlying equation derivation online, and as it turns out, it isn’t easy to find. MMM discusses the result, but not the actual math. He defers to NetWorthify, which has a great calculator and ‘behind the math‘ page, but their final equation is totally wrong! I turned to Early Retirement Extreme, but even Jacob doesn’t do the full derivation in his book.

For me, this was a “Challenge Accepted” moment! I’ll give you the final equation here, but head to the bottom of the post for the full derivation.

# The Early Retirement Equation

Where n = number of years you have to work before retiring r = market rate of return, after taxes and inflation s = annual savings rate w = annual withdrawal rate

Here it is, the early retirement equation. Plug in an expected market rate of return, a savings rate, and a withdrawal rate, and out pops the total number of years in your working career. Sounds great- but what values should you plug in?

# The 4% Rule

The FI community makes the case for a 4% safe withdrawal rate, or SWR, and I agree this is a great starting point for your FI planning. The 4% rule comes from the Trinity Study, which performed a Monte Carlo analysis withdrawing different amounts from portfolios across well over a century of stock market history. The results showed that 96% of the time, a portfolio with a 4% annual withdrawal rate lasted at least 30 years. Those are pretty good odds!

The Mad Fientist wrote an excellent post about how the success of the portfolio is primarily predicted by the volatility in first 10 years of withdrawals, so you’ll want to build in some flexibility in your FI plans if you choose 4% withdrawals. Dropping to a 3% will hedge that volatility, but you’ll have to save quite a bit longer.

Choose a 5% withdrawal rate instead to reach FI quicker, but remember that this requires even more flexibility, particularly if your first 10 years of FI are rough market years. The 4% withdrawal rate, then, is a nice middle ground and is what I’m planning for my retirement withdrawals, so I can be your guinea pig. I’ll document my transition to FI and the flexibility required right here on this blog over the next year to two!

Note that across the internet you’ll stumble upon many articles discussing why the 4% rule is flawed, and why many early retirees will likely fail. While some of the points seem valid, the writers are usually making two invalid assumptions: (1) early retirees will never earn another dime in their 50+ year retirement, and (2) their spending is completely inflexible. These assumptions may be true for traditional retirees, but are ridiculous for most of us in the FIRE community!

The truth is that most FIRE types could pull 5% and still be fine. I’m not planning on sitting in front of my TV and ordering pizzas for the next 50 years. I’m using FI to transition to a lifestyle where I can work on creative and meaningful endeavors without focusing on income. After a few years, many early retirees end up making additional income in FI as a natural side effect of having the freedom to pursue meaningful work on their own terms. I think you’ll find this true of nearly everyone who has reached the milestone and quit their traditional job.

# Market Rate vs. Savings Rate

Compound interest is a powerful thing. Numerous articles and even books exist on the topic. But there’s a problem with compounding: It takes a long time to get going! Here in the FI community, we don’t want to wait 20+ years for compounding to work its magic. To retire in ten years or less, the most important number is actually your *savings rate*.

Let's test this equation out on Wolfram Alpha and chart some results. Assuming a 4% SWR:

Market Rate | Savings Rate | Years Required |
---|---|---|

8% | 80% | 5.2 |

2% | 80% | 5.9 |

8% | 25% | 25.3 |

8% | 10% | 38.3 |

We see that someone withdrawing 4% per year, earning 8% per year in the market, and saving 80% of their money each year can retire in 5.2 years. But what if we change our market rate of return down to only 2%?

In the chart above, our retirement date only pushes forward about 8 months. The market rate of return matters a little for long term wealth preservation, but is practically insignificant during the wealth accumulation phase. What if we put our market rate back to 8% but lower our savings rate to 25% instead?

Ouch- This pushes our retirement date out over 20 years! As you can see, the savings rate has much more impact than the market rate in determining your retirement date. This is good news, too, because unlike the market, your savings rate is *completely* within your control! Traditional advice is to save 10% of your income. How does that look in the chart above?

40 years?! No thanks. For me, the thought of sitting in a cubicle for 40 years terrifies me. I want to get out and explore the world. I want to go on hour long walks in nature, and still have time for fitness, cooking, and creativity every day. That’s why I advocate for very high savings rates! (See this MMM post for details on exactly how to calculate your current savings rate and net worth, and sign up for Personal Capital to start tracking your net worth over time.)

# Summary

The early retirement equation does a great job in showing how your savings rate is the single most important factor in determining how many years are in your working career. Recent college graduates just entering the workforce can use this powerful tool to literally plan how long they’d like to work. For those like myself who have made bad financial decisions for years and need a financial 180, changing your views on lifestyle and spending is difficult. It is almost ‘common knowledge’ that times are always tough, and saving money is always an impossible challenge.

But the truth is that in 1st world countries like the U.S., at almost any income level, you can save 50% of your income if you really want to. Yes, even on a teacher's salary! What it requires is for you to change your thinking about money: Most people think a 2500 square foot home and two to three new cars in the driveway is normal, or worse, something they deserve. Same with eating out at restaurants multiple times per week. When you realize the opportunity costs, you realize these purchases are actually fancy luxuries! Start cooking at home! Sell those new cars! Downsize that new house!

Once you change your thinking, you can challenge other norms as well. Commuting an hour to work? Nope, that's a luxury you can’t afford. Move closer to work. Too expensive? Move to another state. Can't? Change careers. Get creative! There are so many options to increase your savings rate, and the opportunity costs for those dollars are enormous! It's only when you tighten constraints and shut down all these options that the pessimistic complaint of “I could never save 50%” comes true.

OK. You get it. You need to save more. But once you do, where do you put your money so it can grow? CDs? The stock market? The real estate market? All this and more… on the next Financial 180!

# Extra Credit: Deriving The Equation

Bonus reading material for those interested! I couldn't find this derivation *anywhere* else online, so I figure I'm adding some value deriving it here. Let’s start with two equations you may remember from high school: The compound interest equation, and the annuity equation.

Here, a = annual savings in dollars, r = market rate of return, n = number of years, and p = your current savings. Let’s assume our compounding period is annual, to keep the math as concise as possible. We can add the two future values together, because while saving for FI you will be contributing new money to your investments every year (equation 2) while your current savings simultaneously grows (equation 1). This simple equation is equation 3:

Let’s make a key assumption that our expenses in retirement will be the same as our expenses today. This actually bakes in a safety margin because for most folks, expenses go down when they stop working (via reduced transportation, clothing, & outsourcing costs). This gives us equation 4, which states that current expenses (annual income – annual savings) are equal to future expenses (our withdrawal rate in retirement multiplied by the our total future portfolio value). Expressed mathematically:

Where j = annual income, a = annual savings (same as in equation 2), w = withdrawal rate, and F is our Future Portfolio Value from equation 3. We need one final equation so that we can express things in terms of savings rate. This is equation 5, which simply states that savings rate is equal to annual savings divided by annual income.

We now have all the equations we need to solve the big question: how long do you want to work for? To do this, we need to perform some substitution and solve for n. Let’s start by taking equation 4 and re-writing the left hand side in terms of savings rate using equation 5:

Next, plug equation 3 into the right hand side to get:

Next, substitute the FV values from equations 1 and 2:

But we can rewrite annual savings (a) in terms of savings rate (s) using equation 5 again:

Now, we just need to isolate n and perform the natural logarithm to solve. This requires a bit more algebra and is completely procedural… so to save time, let’s copy and paste the following text in to Wolfram Alpha and let it solve for n for us:

j - j s = w p (1 + r)^n + w j s×((1 + r)^n - 1)/r, solve for n

Ignoring the imaginary portion of the result (see assumptions section below), we get equation 6:

Wow! At this point I’m beginning to think Pete’s title ‘Shockingly Simple Math’ is a bit misleading! To test things out, plug the formula into Excel, or let Wolfram Alpha plug in values for us. Try pasting the following:

(ln((j (r (-s) + r + s w))/(w (j s + p r))))/(ln(r + 1)), j=100000, s=.8, p=100000, r=.08, w=.04

And you should get this result: 4.03003. This means you need to work for 4 more years if you have an 80% savings rate, a current portfolio value of $100K, and earn $100K a year, with a market rate of 8% and a withdrawal rate of 4%.

# Simplifying the Equation

But didn’t we say we could calculate all of this using savings rate instead of salary? We can, if we assume we are starting from $0 and eliminate p from the equation. Let’s go back to simplification step 4 and set p=0:

Now let’s move j*s over to the right hand side…

…then divide both sides by j:

Sweet! We were able to remove both p and j completely from the equation. Solving for n back on Wolfram Alpha and doing one final simplification gives us a formula that should look familiar:

There it is! The Early Retirement Equation. It is a bit easier to digest in comparison to equation 6 above. Taking it further, an interesting thing happens if we bake in a safe withdrawal rate of 4% and a market return rate after inflation of 8%:

Use this handy one variable FIRE equation to impress your friends!*

# Assumptions

- Solution for n is over the reals
- n, j, s, p, r, w are > 0
- Withdrawal rate should be less than market return
- Current annual expenses are equal to annual expenses in retirement
- You want the money to last forever, so no drawing down of principal
- Annual income is after taxes
- Rate of market return is after taxes and inflation
- Interest is compounded annually

# Behind the Equation

Now that you understand the equation, you can plug in numbers to recreate your favorite graphs. JL Collins uses this chart from Darrow Kirkpatrick’s book ‘Retiring Sooner‘:

By plugging into equation 6 above, we can now confirm they assumed a 4% SWR and an 8% market rate of return.

MMM references this handy graph from NetWorthify, which assumes a more conservative 5% market return with the same 4% SWR:

Jacob from ERE takes things a step further and graphs the retirement date curves for multiple rates of return:

If you've read this far, you must be as passionate about this stuff as I am… Cheers!

**This will probably only impress your really cool friends. *

Interested in starting your own Financial 180? You've come to the right place. The math is easy: create a gap between what you earn, and what you spend. If you can save half your income, your working career will only be around a decade long! Want to shorten it even more? Read on to see exactly what expenses the wife and I cut from month to month. Track your progress against the milestones of FI, and gradually build up your own savings snowball. Check out the books and links in our resources section and jump-start your journey to FI. The you ten years from now will be glad you did! Ready? Start here.

I’m impressed! I must be really cool.

I can confirm this.

Thoroughly impressed as well! I <3 math! Keep the posts rolling, just like your savings snowball!

Thanks John, glad you are enjoying the content! Let me know if there are any topics you are interested in me covering more in depth.

Teachers saving 50% of their salaries? Get out of town lunatic! I’m going to have to report you to the Internet Retirement Police. Good day Sir. Ed

OK. You got me. Everyone knows that the only people who can save a million dollars are doctor-lawyer DINKs. And even then, a million dollars isn’t enough to retire anymore in this crazy world! 😉

Very impressed with the math and great to see it tested! Our goal is to save 50%+ of pre tax income this year with a goal of achieving FIRE. Agree we need to start reframing how we think about saving and redefining needs vs wants.

Excellent post, thanks!

Thank you! I’ve been looking for this equation everywhere!

I love equation 6 because it answers the question: How long from NOW until FI? This is much more useful to me than all of the tables that only answer the question: How long from ZERO until FI.

I just love your awesome brain, Joel!

Haha Thanks! 🙂 It’s not that awesome, it just loves challenges. And any excuse not to work on actual cubical work… :-p

I tried the early retirement equation and maybe I’ve bumbled it in excel…

When I lower the withdrawal rate, the length to retirement goes up and vice versa; seems backwards. I also kept the withdrawal rate below the market return as later assumed.

Anyone else having this problem?

While it may seem counter-intuitive at first, the equation is actually working correctly. If you want a lower SWR, you need to save more money, because the assumption is your annual spend doesn’t change. So if your annual spend is $40k, that’s 4% of $1M, but that’s 3% of $1.333M, hence you need to save more. And saving more takes longer..

A 4% withdrawal rate is equivalent to 25x your annual spend, as 1/25 is 0.04. For a 3% SWR we do 1/0.03 = 33.333… which implies just over 33 times your annual spend. (In my milestones of FI post, I refer to this as Fat FI. Likewise, a SWR of 5% is equivalent to 20x annual spend, which I call Flex FI.) Hope this helps!

Hi Joel! This is great. Thanks for doing the heavy lifting and posting this up. I’m afraid I’m not following the use of the assumption for income in what I’ll call the “long form” equation (shown just before you describe simplifying the equation). Admittedly, I’m not a math guy and am just now educating myself on FI, so I appreciate your patience and any help you can provide!

The assumption is that j is annual income *after* taxes. How does one account for pre-tax contributions and employer matches to retirement accounts when determining annual income after taxes? Such contributions reduce one’s tax liability. The employer match is additional income on top of one’s annual salary.

The contributions and employer matches are also part of one’s savings rate. I’m not understanding how the equation accounts for that. For instance, let’s say my salary is $100k, I contribute $5k pre-tax to a 401(k) and my employer matches my contribution for another $5k. How would that fit into the long form equation?

Hey Ryan! If you look at equation 4, you can see the left hand side defined as (j – a), which is annual income minus annual savings. We are then assuming that this difference is equal to our annual expenses; namely, annual income minus annual savings equals annual expenses. This model is actually agnostic about taxes. If you use annual income before taxes, you should expect the resulting annual expenses to include tax expenses as well, though, these are different for everyone and dependent on the annual income, so it is much easier to just model this after taxes instead, and handle tax expenses separately. If you like, you can add things like employer matching and other incentives to your annual income (j), since these are actual dollars you are earning.

As you pointed out, other tax benefits can make the final “how long will you work” estimate longer than it will actually be… I like to think of this as icing on the cake rather than banking on it, since tax laws can (and often do) change over time.

Even more icing on the cake: assuming future expenses in retirement equal expenses while working. Often, expenses in retirement are considerably lower than while employed full time, so this formula actually gives a very conservative estimate overall, which helps make up for the fact that we often aren’t very good at estimating our expenses consistently over time.

Even more icing: potential social security benefits, raises over time, etc. If the equation says you have 10 more years of work left, you might actually have a few years less than that. It’s just an estimate, and it is best used to show deltas: namely, how much shorter would your working career be if you lowered your expenses from A to B.

Hope this helps!