*Last Updated: September 25, 2022*

There are a couple questions that pop up frequently in the comments sections of posts and forum threads about computing time to Financial Independence (FI):

- What about inflation?
- What about salary raises?

Good news: these factors are essentially already accounted for in most “Time To FI” analyses. Why/how though?

A couple reasons:

First, the expenses, income, and savings rate values used in “Time to FI” calculations are assumed to increase by exactly inflation each year.

Second, the assumed investment return rate is the long term average AFTER accounting for inflation.

With a long term average stock market investment return rate of 10% and a long term average inflation rate of about 3%, we use 7% for the investment return rate in our “time to FI” calculations.

In other words, all values we use in our “time to FI” calculations are in “present day” dollars. And we can easily map those values forward or backward in time using published inflation rates (typically the consumer price index (CPI) inflation rates published by the U.S. Bureau of Labor Statistics).

BUT: what if your expenses or income DON’T match inflation? In other words, what if your expenses or income grow slower or faster than the average inflation rate each year?

Of course it’s extremely unlikely your expenses or income will grow by EXACTLY inflation, but what if they are substantially different?

For example, what if you put significant effort into lowering your expenses each year, or increasing your income faster than inflation through smart and proactive hard work and investing in your own skills and knowledge? I.e. you focus on your circle of control (affiliate link) instead of worrying about national inflation rates?

Let’s find out!

## The Math

Now for the fun part – the math! That really is the best part of my posts, right? Excellent, I’m glad you agree.

As usual, I decided to put all the detailed mathematical derivations in a PDF using LaTeX, rather than putting 8 pages of equations on this post.

I know, that makes you very sad. I mean, who wouldn’t want to review all my derivations with a fine tooth comb, replicating and verifying all the math on their own?

But there are some strange people that don’t like doing that for some reason, so I’ve spared them the derivations.

The final expression to compute the number of years to FI when you have an expense growth rate (relative to inflation) and an income growth rate (relative to inflation) has the form:

where

where is your initial annual expenses, is the withdrawal rate, is the investment return rate, is the initial value of the portfolio, and is your initial annual income.

Unfortunately you’ll notice something about this equation: it’s not a direct expression for . Unfortunately there is no closed form direct analytical solution for . Boo! (Though if you are aware of one, please let me know!)

Fortunately we can use a very simple classic numerical method called the Newton Rhapson method to quickly solve for – see the math spec for more details.

One other note: you can see that if or equal , then we have 0 in the denominator – bleh! This is a wonderful little singularity in the math, but fortunately we can derive a different expression for these corner cases, as shown in Appendix A and B in the math spec. It still requires employing the Newton Rhapson method to solve, but the solutions are consistent with solutions from the above equation when considering values immediately before and after the singularity.

## Pretty Plots

As usual, I gotta make some neat plots. Fortunately Python makes that pretty easy. It always feels like I’m exploring new territory when I create these plots – I never know exactly what I’ll find.

Let’s kick things off with a plot of the number of years to FI versus your expense growth rate relative to inflation, for five different initial savings rates (SR), starting with $0:

The values where each of the curves intersect the “0% growth rate” line in the middle of the plot represent the standard times to FI for each of those savings rates.

But you can see how quickly a) your time to FI extends to forever if your expenses go up each year (relative to inflation) and b) your time to FI drops if your expenses go down each year (relative to inflation). And the lower your savings rate, the more pronounced these changes are.

In fact, with a 10% savings rate, if your expenses increase by just 0.66% per year relative to inflation, you’ll NEVER achieve FI! Bleh!

Essentially the asymptotes seen on the right side of the plot represent the scenarios where your expenses overrun your income (though pushed out a bit because of early savings investment returns).

Thus you can see 1) why it’s so vital to not let your expenses grow substantially relative to inflation, and 2) the power of reducing your expenses to drop your time to FI.

Next up, we have the number of years to FI versus your income growth rate relative to inflation, for five different initial savings rates (SR), starting with $0:

This plot is almost the inverse of the previous plot: as your income increases relative to inflation, your time to FI drops (assuming your expenses grow at the inflation rate). And if your income drops relative to inflation your time to FI grows (or becomes infinite for smaller initial savings rates).

And if your initial savings rate is low (e.g. 10%), then even a minor negative income growth rate (e.g. getting a 2% raise with 3% inflation every year) makes achieving FI impossible!

Overall these results are not surprising: if your income is going up, you’re earning more money and getting to FI faster. If your income is dropping, of course it will be harder and longer to get to FI.

Here’s something that might be surprising to some though: looking at the 50% initial savings rate curve, you can see that increasing your income by 1% relative to inflation each year gets you to FI in 13.7 years; but if instead you reduce your expenses by 1% relative to inflation each year, you’re looking at 13.1 years to FI. A full six months earlier!

Thus reducing your expenses is MORE POWERFUL than increasing your income. You can see that by comparing the values for every curve in the above plots – reducing your expenses by a particular percentage ALWAYS gets you to FI faster than increasing your income by the same percentage.

Now what if we turn the knobs for income growth rate and expense growth rate at the same time? We need to select a particular initial savings rate though, so let’s pick the 50% initial savings rate, and then plot number of years to FI versus expense growth rate (relative to inflation) for five different income growth rates (relative to inflation):

For a standard 50% savings rate scenario, where both your expenses and income match inflation each year, you’re looking at 15.0 years to FI, which matches previous plots.

But if you let your expenses grow by 3% a year relative to inflation (as happens to many folks as they experience lifestyle inflation and try to keep up with the Joneses), and your income just keeps up with inflation, your time to FI leaps up to 33.1 years – more than doubling!

If you also increase your income by 3% a year relative to inflation (e.g. getting 6% raises each year with 3% inflation), fortunately that brings it back down to 17.8 years, but it’s still not as good as the 15.0 years you were originally facing with expenses and income matching inflation.

Obviously the most painful line is on the far left in red: a -6% income growth rate (vs inflation) (i.e. if inflation is 3%, your earnings are dropping by 3% each year in that year’s dollars).

If you’re in this situation, and your expenses are exactly tracking inflation (0% growth rate), even if you’re saving 50% of your income each year, you’re looking at 61.2 years to FI – not very quick!

BUT, if you can drop your expenses by 3% each year (i.e. just keep them constant instead of tracking a 3% inflation rate), you’ll be FI in just 16.5 years! And if you can drop your expenses by 6% each year (relative to inflation), you’ll be FI in just 11.5 years.

## Example Scenarios

### You And Your Friend Jen

Let’s say you and your friend Jen are both kicking butt with a 50% savings rate by getting good jobs after graduating from school and by sticking to your student level lifestyle. You’re both also really pumped about buying your freedom from mandatory work.

One night over some beers you get into an argument with Jen though: she thinks the best way to get to FI is to focus on income, as long as your expenses don’t go crazy (i.e. just track inflation). “You just need to focus on your work and getting those raises! And switch jobs if you have to!”

On the other side of the table you argue that focusing on expenses is more important: “But Jen, your final FI number is based on your expenses! If you bring that down, then you don’t need to obsess about income, as long as your raises keep up with inflation.”

So you and Jen make a beer bet right then and there: she’ll focus on income, and you’ll focus on expenses, and whoever gets to FI first wins the bet (and a beer!).

Jen gets to work hustling at her job, with minimal thought to reducing her expenses (i.e. they increase by the inflation rate each year). As a result, she manages to consistently receive 6% raises each year, which is 3% each year relative to the average 3% inflation rate. With all this hard work she reduces her time to FI from 15.0 years to 12.1 years – a little less than 3 years reduced off her mandatory work period, or 19%! Not bad at all!

You get to work on your expenses each year instead. As inflation pushes up costs of goods by 3%, you find ways to keep your total nominal expenses roughly the same year to year. E.g. if you spend $50K in 2022, you also spend $50K in 2023 (instead of 1.03*50000 = $51500). Thus your expense growth rate is -3%, relative to inflation. That might mean reducing your expenses in one category (e.g. your cell phone plan), while other parts of your budget go up with inflation. You also make sure you’re working hard at the office, but you’re putting in less hours than Jen and so you’re seeing 3% raises each year – matching inflation. The result: you reduce your time to FI from 15.0 years to 10.9 years – a greater than 4 year reduction off your mandatory work period, or 27%! Excellent!

So after 11 years you meet up with Jen to collect your winning beer from her and gloat a little. She still has about a year of work ahead of her, but you’ll both be FI only a dozen years after graduating, so really you’re both winning in the end. You just get to give her a hard time about your extra year of freedom the rest of your lives – what else are friends for?

### Reducing Expenses Example

OK, all that sounds great, but how do you actually reduce your expenses by a meaningful percentage each year? Especially without your life getting uncomfortably spartan?

Well, to start, you can go after the low hanging fruit, like your cell phone bill.

The largest wireless communications service provider in the U.S. is Verizon, with 142.8 million subscribers at the end of 2021. This subscriber count represents a staggering 43% of the total U.S. population at the end of 2021.

If you look at Verizon’s cell phone plans, the average for one line is $80, and for four lines (family plan) it’s $180.

That’s $960 a year for one line, and $2160 a year for four lines. Ouch!

But if you recognize how crazy this is and switch to a low cost carrier like Mint Mobile (referral link), then you’re paying just $15/month for each line.

Thus you’d save $65/month for a single line, and $180-$15*4 = $120/month for a family of 4. So that’s $65*12 = $780/year for single person, and $120*12 = $1440/year for a family. Incredible.

So if you’re spending $37500 on a $75K salary (50% savings rate), and you save $780/year by switching phone carriers, that’s a full 2% reduction in your expenses! And if you’re a family of four, saving $1440/year is a 4% reduction in your expenses. Nice!

If all your other expenses remain the same, you’re looking at a 5% to 7% reduction in your expenses relative to 3% inflation. All from just a few minutes of work switching the sim card in your phone(s)! And even if your other expenses rise with inflation, this kind of savings can partially or fully offset those increases, and you’re still looking at a 3% reduction in your expenses relative to inflation.

The effort to switch your phone carriers might take more than a few minutes: maybe a couple hours in the end. And yet that small effort gets you to FI faster than someone working their tail off to get a 5% to 7% raise (on top of inflation, so really 8% to 10% overall). Even if you love to work, I hope you can see how much more effectively reducing your expenses gets you to FI!

## Our Income and Expense Growth Rates

So how did our income and expenses change each year on our path to FI from 2016 to 2021? And how did those rates compare to inflation?

First let’s look at expenses:

Year | Percentage Change In Expenses From Previous Year | Inflation (CPI) |

2017 | -0.6 | 2.1 |

2018 | -8.8 | 2.4 |

2019 | -12.7 | 1.8 |

2020 | 11.8 | 1.2 |

2021 | 21.6 | 4.7 |

Average: | 2.3 | 2.4 |

Average without 2021: | -2.5 | 1.9 |

We reduced our expenses quite a bit each year from 2017 to 2019 (and well below inflation), and then in 2020 and 2021 we saw increases in our expenses well above inflation. What’s going on?

In 2020 and 2021 we had a variety of novel large expenses, including the birth of our daughter in 2021, my first surgery involving anesthesia (fun!), big vehicle expenses, some home appliances requiring replacement, giant property tax increases as home values in Austin skyrocketed, and of course good ol’ inflation acting on many things we bought.

Thus I think of 2020 and 2021, especially 2021 with the birth of our daughter, as somewhat anomalous for us. We plan to bring our expenses back down in 2022 (or at least as much as inflation and increased property taxes will let us). And by substantially reducing our expenses in the earlier years, the increases in our expenses in 2020 and 2021 were based on smaller expense totals.

Overall though, if you average the rates for all 5 years, we actually pretty closely match the average inflation rate across the entire time period. If you remove 2021 with its (rare and final for us) child birthing expenses, we averaged -2.5% a year expense growth rate, vs 1.9% average inflation, for a growth rate relative to inflation of -4.4%.

Now let’s look at income:

Year | Percentage Change In Income From Previous Year | Inflation (CPI) |

2017 | 78.2 | 2.1 |

2018 | 16.3 | 2.4 |

2019 | 5.9 | 1.8 |

2020 | 9.2 | 1.2 |

2021 | -1.9 | 4.7 |

Average: | 21.5 | 2.4 |

Average without 2017 increase or 2021 decrease: | 10.5 | 1.8 |

Like for the expenses table, we don’t have that many years of income to consider, since I graduated with my PhD in 2016 and we hit FI in 2021.

I didn’t start full time work until Sept 2016 after graduating, so the giant 78.2% increase from 2016 to 2017 is kinda comparing apples to oranges.

2021 was also not a traditional year for us: my wife went down to half time in May of that year after our daughter’s birth, and I left my job in October of that year.

So really just 2018, 2019, and 2020 are years where we both worked full time the entire year following a year when we both worked full time.

If you average across ALL the years, including the massive 2016 to 2017 income increase, we averaged 21.5% a year increase in income – blasting well past the 2.4% average inflation in that time span.

But if you average over the less anomalous time frame from 2018 to 2020, we averaged 10.5% a year, vs 1.8% inflation. Still pretty good! Thus a 8.7% average growth rate relative to inflation.

So we did attack the FI challenge from both ends, decreasing our expenses a good amount (ignoring 2021’s child birthing event) and increasing our income vs inflation a good amount.

With a national median salary of $75K and a 50% savings rate, a -4.4% growth rate in expenses relative to inflation drops your time to FI from 15.0 years to 9.8 years. A 8.7% income growth rate relative to inflation drops your time to FI from 15.0 years to 9.2 years – not that much less time! Even though the income growth rate is nearly double in magnitude the expense “shrink” rate!

Now if you have BOTH a -4.4% growth rate in expenses and a 8.7% income growth rate relative to inflation, you’re looking at 7.3 years to FI. Nice!

Our average savings rate was actually closer to 70% during our path to FI, so starting from that savings rate and having BOTH a -4.4% growth rate in expenses and a 8.7% income growth rate relative to inflation, time to FI is just 5.1 years – right in line with how long it took us.

## Code

If you’re wondering how different expense or income growth rates may affect your FI projected date, or if you just want to dig into the code I used to generate the plots and values above, feel free to play around with the python script I used in this analysis. See the embedded python interpreter below. You can fill in your own values in the user inputs section at the top to see what your situation looks like. Hit the Run/Play (sideways triangle) button to generate the plots. To go back to the script to make any changes, hit the Pencil icon. If you want the text larger, hit the hamburger menu button, then scroll to the bottom to see larger font options. In that same menu you can also Full Screen the window, and other actions.

## Summary

If you can consistently increase your income each year more than inflation, you’ll get to FI faster – something I suspect is obvious to most folks. Similarly obvious: shrinking your expenses each year relative to inflation will also reduce your time to FI.

But what may not be obvious: reducing your expenses each year by a certain percentage relative to inflation is MORE POWERFUL than increasing your income by that same percentage relative to inflation each year.

What does this mean for you?

Well, in general you always want to keep an eye out for ways to increase your income and reduce your expenses (especially in categories hit hardest by inflation).

But if you have to choose where to focus your energies, and you want freedom from mandatory work as soon as possible, I strongly recommend you focus on reducing your expenses. Especially in the early days of your journey to FI, when there is likely lots of low hanging fruit that you can pick to easily and (nearly) painlessly reduce your expenses.

In other words, instead of fighting the inflation wave by working tons of extra hours year round in an effort to substantially “beat” inflation with larger income increases, ride it to freedom by nixing all those pesky unnecessary expenses that are slowing you down!

Derek MorrisonThanks for the well-researched post!

I’d have to think about it, but I feel conflicted about the “you and your friend Jen” scenario. I wonder if it would be realistic for me (or me and my family rather) to cut expenses by 6% (before inflation) each year over about a 10-year time frame without feeling too spartan.

Another angle I could see is that Jen didn’t have to work that much longer to achieve FI, and she ended up with a more prestigious (and hopefully satisfactory) job role.

On the other hand, she would have had to definitely work hard to hit a 9% (before inflation) raise each year.

EngineeringYourFIThanks Derek!

Yeah, reducing expenses by 6 percent each year in today’s dollars (so reducing nominally by 3% each year) could prove tougher in the later years as you squeeze down more and more into a smaller expense footprint. Though fortunately 3% of a smaller number is a smaller nominal reduction each year.

Perhaps a more realistic comparison would have been -3% and 3%: You keep your expenses constant each year (instead of increasing by 3% with inflation), and Jen gets a nominal 6% raise each year (so 3% after inflation). In that case, you’d be FI in 10.9 years, and Jen would be FI in 12.1 years – so you’re still finishing over a year before her. Maybe I should update the example with those numbers in fact…. that scenario seems more realistic.

Also a very good point about Jen’s job satisfaction. Though probably quite a few folks would argue that the higher paying roles end up MORE stressful, but they just can’t lose the higher paycheck, or don’t want to feel like they’re going “backward” by taking a lower status role (even if they’d enjoy it much more). Wonder if there’s some research on this…

Thanks again for the comment!