# It doesn’t matter how much you earn for your financial independence

This is not a financial blog, but let’s use an example to apply some maths. Remember that this is just a thought experiment, so take it with a grain of salt.

## 0. FIRE

If you’re not familiar with FIRE movement, I’ll explain it quickly. The idea is easy – save and invest substantial amount of money to become financially independent in an early age, meaning you can retire early and live off the capital.

The 4% rule refers to an influential paper called Trinity study. It states, that yearly consumption of 4% of your capital will not exhaust it in a long time horizon. In other words, you need to save and invest reasonably 25 times your yearly needs to live off your capital. Here I assume the rule is currency-agnostic: if you spend 40k$a year, you need to save up 1m$, to cover yearly costs of living of 40k zł, you need 1m zł, and so on.

To keep it simple, I will not go into detail. There’s a lot of resources on that topic to be found online. Further, for brevity, I will refer to the number of years to achieve FIRE as the “FIRE number”. It’s a common practice to use this term to express the value of the assets required to achieve FIRE, but since I will not touch on this topic, I’ve decided to redefine this term.

## 1. How long should I keep saving?

If you’re scared of maths, feel free to jump to the next section :>

Let’s denote by $s$ the savings rate, $y$ – yield, $c$ – consumption rate. If you save 10% of your yearly income, $s=0,1$. If you invest it and earn 4% yearly, then $y=0,04$. Notice, that we’re not taking inflation into account, so we can consider $y=0$ as keeping your funds on a savings account. That would require your savings account to beat the inflation (tax included), which is unlikely, but let’s keep it simple. If we follow the 4% rule, then we’re going to consume $c=0,04$ of our capital. Let’s also introduce multiplier a $m = y + 1$.

Next, let’s denote our yearly income by $x$ and our savings after $i$ years of saving as $x_i$. The following hold: $x_1 = sx$ and $x_{n + 1} = (1 + y)x_n + sx$. Unfolding the recursive form yields the formula for $x_n$: $x_n = sx + msx + ... +m^{n - 1}sx = sx \sum_{i = 0}^{n - 1}m^i = sx \frac{1 - m^{n - 1}}{1 - m} = \frac{sx}{y} \left( (1 + y)^{n - 1} - 1\right)$.

Since every year we save $sx$, then the rest we consume: $(1 - s)x$. We want our consumption to stay the same, so $x_n c = (1 - s) x$ (*). Now we only need to solve this equation for $n$.

We have $\frac{csx}{y}((1 + y)^{n - 1} - 1) = (1 - s)x$. Notice how $x$ can be crossed out on both sides of the equation. After some operations we get the result: $n = 1 + \log_{y + 1} (1 + y \frac{1 - s}{cs})$

Notice that this only holds for $y > 0$. The formula for $x_n$ for the corner case $y = 0$ is even simpler: $x_n = nsx$. After substituting this in the equation (*) and performing some transformations the formula for $n$ is obtained: $n = \frac{1 - s}{cs}$

## 2. Simulations

Now that we have found the analytic solution, we can proceed to writing the code. In python we can write a simple function:

import math
def years_to_fi(s, y, c):
if y == 0:
return (1 - s) / (s * c)
else:
return math.log(1 + y * (1 - s) / (c * s), y + 1)

Suppose we’re only beating inflation, so the yield equals 0. Let’s depict how many years we need to save in terms of the savings yield. Number of years to FIRE in terms of the savings yield (yield = 0%).

By saving 30% of our income or less we probably have no chance for FI (this doesn’t mean we shouldn’t do that, our retirement will be much greater anyway). To retire after 30 years of work, we would have to save up at least 45% of our income.

Let’s see how would it look like if we invested our money earning 4% per annum (note that in reality this means beating inflation by 4% net, so even with 0% inflation with the capital gains taxes nearing e.g. 20%, this should be around 5%). Number of years to FIRE in terms of the savings yield (yield = 4%).

This looks substantially different. Earning additional yield on our savings greatly eases achieving FI. First, reaching FI after 30 years of work requires from us only 30% of savings rate instead of 45%, this already looks more reasonable. Second, saving and investing 20% of your income results in FIRE after 41 years, which means that by sacrificing 1/5 of your income you can already retire a couple of years earlier and enjoy double retirement (funded by government) in short time.

The following graph depicts the effect of earnings yield of your investments on the number of years to FI (assuming 4% rule and saving quarter / half of your income). Number of years to FIRE in terms of the yield on investments (s = 25%). Number of years to FIRE in terms of the yield on investments (s = 50%).

The last parameter is consumption rate. Depending on the plans for the future, the environment one lives in and other factors, conservative investor would probably opt for a safer variation of the 4% rule, for example by assuming consumption rate of 3,5% or 3%. This is depicted on the graph below. Number of years to FIRE in terms of the consumption (s = 50%, y = 4%).

By saving half of our earnings we only have to work for 18 years to achieve FIRE according to 4% rule, but lowering the consumption rate to 2% almost doubles that time. This is no wonder, the capital you need doubles (50 times instead of 25 times of your yearly costs of living).

## 3. Discussion

First, let me comment on that clickbait title – it stems from the calculations in section 1. Intuitively the more you earn, the easier it should be for you to save up for your (earlier or not) retirement, but the salary cancels out very nicely in our equations. Indeed, the more you earn, the easier it is to increase the savings rate and this is what really matters in our model.

To make this statement more explicit, consider two exemplary people: “rich” man earning 10k$a month and spending 8k$ and “poor” man earning 5k$a month and spending 3k$. Assuming they have the same saving / investment strategy, they will be able to save up the same amount of money in given period of time, but the costs of living of the “rich” man increase the FIRE number substantially. Unless they lower the living standards, they will have to save up almost 3 times the amount the “poor” man has to.

Of course, following this “fixed savings” logic, a person earning 3k$would have to live off 1k$ a month, which is, in most cases, impossible or very hard. This only shows the more you earn, the easier it is for you to save up more.

I’d like to share a couple of conclusions from this exercise:

1. As it is stated in the title and commented on above – in this simplified model it all boils down to how much of your income you can save up. This number carries the relationship between the growth of your investments and your costs of living, which is the key in determining your FIRE capabilities (not the absolute value of your assets).
2. In general, higher earnings should not imply higher consumption. Of course, if you earn more, you can increase your standards of living, but there’s no rule that you need to spend fixed percentage of your income. This phenomenon has its own name – lifestyle inflation.
3. Investing your money makes a huge difference. No matter how aggressively you save up, reasonable investment of your capital will heavily reduce the number of years you need to work if you want to achieve FIRE.
4. By only keeping your money on the deposits / buying bonds (beating the inflation) you would have to save up half of your income to have a chance of retiring earlier. And this will only give you a couple of years, not to mention if you aim for retiring in your 40’s or 50’s.
5. The 4% rule may be questionable for you depending on your conditions. Fixing consumption to a more conservative value (e.g. 3% or 3,5%) can noticeably increase the FIRE number.

I have created a jupyter notebook if you’d like to play around this model or put your own numbers in.