Hint: it's all about your savings rate

Mathematical Side

The classical approach to financial independence (FI)

Mr Money Mustache (MMM) popularised the “shockingly simple maths behind retirement” back in 2012, and it’s generally seen as the ‘classical’ way to become financial independent (FI).

Classical FI doesn’t care about ratios between pensions and ISAs, about whether you’re LeanFI or FatFI (yes, those are apparently things), or what age you’re thinking about retiring. It’s a really simple way to let you know when you can retire.

To quickly recap, financial independence, is a state where you have enough wealth to live on, or assets generating enough income, such that you don’t need to work a “normal job” anymore.

As long as you’re investing your savings, and getting a decent interest rate, they’ll grow and eventually generate enough returns that you can live off the returns alone. At that point you’re financially independent and can stop working if you want.

The percentage you withdraw every year should be slightly less than the interest you’re earning, known as a safe withdrawal rate (SWR), that way the money shouldn’t run out, and you’ve got a little margin for poor performing years.

Savings rate is key, not the absolute numbers

MMM explains that the absolute numbers (how much you earn, and how much you save) aren’t as as important as your savings rate which = amount you save / amount you earn after tax.

To illustrate why, here are the two extremes:

  • Save 0%, and spend 100% of your income, you’ll never save a penny, and you’ll have to bank on the state pension at 60something (or more likely 70something) when you eventually retire
  • Save 100%, and spend 0% of your income (i.e. live for free) and you could stop working and retire now (assuming you can maintain this lack of spend if you stopped working)

If your savings rate is somewhere in the middle, you’ll become financially independent somewhere in the middle too. Here’s a chart that shows a little more - years to FI vs savings rate.

A higher savings rate accelerates the time to FI

This chart below shows that the time it takes to get financially independent does not scale linearly with your savings rate.

The beauty of having a higher savings rate is that you’re not only saving more money, faster but you’re also spending less, and need less money overall to retire. These two things have a compounding effect, and can rapidly accelerate financial independence. That’s why the graph isn’t linear, it’s exponential.

What does this mean for you, in practice?

This grid, for UK audiences, is appropriated and adapted from this one by US-based Four Pillar Freedom.

It tells you, for each post-tax income (the x-axis) and spend (y-axis) combination, how long it’ll take you to retire (in years).

Net pay per hour

The assumptions I’ve used are:

  • 3.5% interest rate (above inflation) on savings
  • 3.0% safe withdrawal rate (see end of article for references on why)
  • £0 initial savings

If you already have some savings, ping me an email :email: and I can send you a spreadsheet that will let you factor this in.

It’s all about widening the gap

The key thing this chart should tell you is that the wider the gap between what you earn and what you spend, the quicker you can retire. Exponentially quicker in fact.

For example, if you’re in the £35k to £45k post-tax region, spending £15k a year vs £30k could mean you retire in 13 years rather than 35 years. That’s an enormous difference.

If you’re starting at 25, that’s retirement at 40, not 60 :open_mouth: . Those are some peak years.

Read on for some maths…

How does the maths work?

I’ve dug into this to understand more.

We want to work out how long will it take to become financially independent (FI) for different savings rates. Let’s call this time period “n”.

Financially independent means your expenses are covered by the safe amount you can withdraw from your “futureSavings”. So we’re going to write an equation that helps us to find ‘n’ when the futureSavings have grown big enough to support a withdrawal rate of 3% that covers your expenses.

There are six key variables

startingPot: £X,000
interestRate: 3.5%
withdrawalRate: 3%
annualIncome: variable
savingsRate: variable
futureSavings: variable

Equation 1: your future expenses

FIExpenses = how much you’ll spend per year when you’re financially independent.

Withdrawal rate = how much of your savings pot you’ll be safe to withdraw per year

FIExpenses  = withdrawalRate * futureSavings

We’ll assume you’ll spend the same in retirement as you will now. So FIExpenses = currentExpenses

currentExpenses = withdrawalRate * futureSavings

Our current expenses are also a function of our savings rate:

currentExpenses = (1 - savingsRate) * annualIncome

Equation 2: your future savings

After n years, your savings will be a combination of the pot of money you started with, and the money you saved along the way (a function of your savings rate). Both of these will be growing with a particular interest rate.

futureSavings after n years = [Starting pot after n years] + [Ongoing monthly payments after n years]

We can calculate how the starting pot will grow, using a compound interest calculation. We’ll call this futureSavings A.

The ongoing payments are a little tricker (a future value series, see the reference below. We’ll call this one futureSavingsB.

Here are those equations:

futureSavings = futureSavingsA + futureSavingsB

futureSavingsA = startingPot * (1+interestRate)^n
(this is just compound interest)

futureSavingsB = (annualIncome * savingsRate) * ((1+interestRate)^n - 1) / interestRate
(this is a future value series, see reference below)

Combine the two equations and rearrange

We take this, combine it with the expenses formula earlier, and rearrange to remove FutureValue and CurrentExpenses.

Equation 1) currentExpenses = futureSavings * SWR%
Equation 2) currentExpenses / withdrawalRate = [startingPot * (1+interestRate)^n] + [(annualIncome * savingsRate) * ((1+interestRate)^n - 1) / interestRate]
Combined Equations) ((1 - savingsRate) * annualIncome)) / withdrawalRate = [startingPot * (1+interestRate)^n] + [(annualIncome * savingsRate) * ((1+interestRate)^n - 1) / interestRate]

We do some fairly heavy rearranging, and then take the natural logarithm of either side (see reference below).

An equation to find ‘n’ (years to FI) based on savings rate

n = (ln((annualIncome * (interestRate * (-savingsRate) + interestRate + savingsRate * withdrawalRate)) / (withdrawalRate * (annualIncome * savingsRate + initialValue * interestRate)))) / (ln(interestRate + 1))

^^NOTE: this uses logarithms

And that’s what you pop into Excel to generate the chart above :thumbsup: .