### Overview

- Covers the mathematics underpinning financial independence in a graphical fashion.
- Illustrates the impact of decisions on your long term financial health.
- Covers key FI concepts such as the Savings Rate, Safe Withdrawal Rate and the impact of expenditures on your ability to FI.

This section will cover the technical components of Financial Independence and the broad mathematics behind it. I will attempt to make it graphical as opposed to smashing you with the equations as I believe that it is more intuitive for most people. Nothing in this section will be novel and there is substantial prior art floating around on the internet for you to take advantage of.

The point of this section is to introduce the concept that there is nothing *impossible* about striving towards FI. Instead, by organising your life properly and setting up some basic structure you can make FI *inevitable*. A disciplined investing and expenditure control strategy over decades is almost guaranteed to make you a millionaire at the least through compounding of investment assets, but to take advantage we need to be active within the market itself.

- Part One – The Concept
- Part Two – Basic Financials, Expenditure Control and Debt
- Part Three – (This Article) The Financial Technicals behind FI
- Part Four – Investing and Tax
- Part Five – Designing a lifestyle to achieve FI
- Part Six – Essential Resources on the path towards FI

## Terms

To begin with, we need to define some basic terms which can be used throughout the document. This definition should promote a higher degree of precision as we continue the section. Feel free to come back to it at any time.

*i*– Annual Inflation Rate, that is, how the value of money is changing over time.*n*– Nominal Rate of Return received on an investment.*r*– Real Rate of Return of an investment. Defined as*n – i**t*– A period in time, used as a summarising variable.*P*– Principal*I*– Income*E*– Expenses*SR*– Savings Rate*SWR*– Safe Withdrawal Rate

# Compounding

Firstly, we need to cover the effect of *compounding* returns on your finances. Compounding occurs when you invest the proceeds of an investment back into the investment itself. This has the effect of increasing the principal invested. This principal can then earn you additional returns in what is termed a Virtuous Cycle.

The figure below illustrates this phenomenon. It represents a single dollar that has been left in an investment to grow at different interest rates over time. For simplicity I’ve assumed a constant rate as well as no little nasties like taxation. For inflation I’ve assumed that this is the *real* interest rate, *r*, as opposed to the nominal rate of investment *n*.

This figure highlights two things. 1) the value of *time* in investing. Even a subpar investment will often double or triple in value over a long period of time. However, it also shows us that investing one dollar now and waiting for 30 years will take a very long time to pay off. Instead we need some way of accelerating the process.

There are two ways this can work for us:

- Earn a higher rate of return on our investment
- Invest additional principal

For the purposes of this article I’m going to assume that you can’t expect to realistically beat the market year after year. I’m also going to assume that you’re not going into a highly speculative investment like digital tulips. Speculation has it’s place but this will be covered in Part 4. As such, your only *real* way to increase the amount you have invested is to start investing additional principal into your investments.

The figure below highlights the effect of investing an additional 10% of the investment starting principal every month for the entire time horizon.

Things look a little bit different now. Even the lower interest rates have principal at the end many multiples of the starting. Dumping additional money into solid investments over time is a good longer term strategy.

If we undertake a solid investment strategy in our formative years we should be able to compound these investments into an excellent final result. Now that we have some basics out of the way we can start to relate this back to financial independence.

# Savings Rate and Expenses

Probably the most important number in the FI universe is your savings rate. The proportion of your income that you’re able to save. There are many, many subtle variants upon this. What do you classify as income? What do you classify as expenditure? Is a mortgage expenditure? Is income that you cannot access for 40 years income? Are stock options and bonuses income? An important assumption in FI is that your expenditures will not blow out into the future. You cannot retire and then start living a very different lifestyle with different financial requirements. In all figures and calculations below expenditure is assumed to be constant.

My advice, is to treat expenditure as *anything that isn’t invested into an asset with future value*. As such. I define expenditure as the *remainder* not as the sum in an off itself. For me:

Expenditure = Income – Investments

I have decided to represent expenditure this way as I believe it is a fairer representation of the savings rate that takes into account *delayed purchases*. If you save for six months in a separate account with the intention of making a large purchase this shouldn’t be factored into your savings rate. Those expenses have already been allocated it just hasn’t come through yet. This helps to avoid the *lumpy* issues that consumption based saving introduces. How much you’re investing is essentially the *surplus* cash you have available from your income, this is what governs how quickly you’ll reach your goals.

Your savings rate and expenses are what gets you closer to FI. The basic math in this section has been inspired by the work of Jacob Lund Fisker in Early Retirement Extreme First, some basic definitions:

SR = (Income – Expenditure) / Income

Or. Using my definition as above:

SR = Investments / Income

There are all sorts of wonderful definitions out there in the world about the savings rate (SR). These vary, predominately because people wish to pad the number for either internet points or to prove a slightly different point. You can do this if you want, but it’s simpler to just use a basic definition. One thing to keep in mind here is whether you should use pre-tax or post-tax income as the denominator. This one is up to you here.

A Savings Rate of 50% thus means that you’re dumping 50% of your income into investments. Or, thinking about it in an alternative fashion you’re saving one years worth of expenses every year. A savings rate of 66% means we’re saving two years worth of expenses. We can view our savings rate vs our expenditures as follows:

This is another *exponential* style function. You should be getting used to these by now at this stage. What this illustrates is that as our savings rate increases our ability to live off invested earnings increases accordingly. At a savings rate of 75%, at the extreme end but still possible for the most frugal amongst us, it is possible to save *three* years of expenses each year. After ten years of saving at this rate it will be possible to live for 30 years without considering any investment returns at all.

Now this is extreme, most *retirement* strategies have people saving somewhere between 5-20% of their income each year. This isn’t a large amount at all and is based off life expectancy and capital draw down. If you retire at 65 and you can only expect to live to be 80 then you’ll only need enough assets to live off for 15 years. As such, you can get away with saving such low amounts by working for a lot longer and having a much shorter retirement. If you want a nice long retirement of 40-50 years then such a savings rate will be insufficient.

What this should also tell you is that there is nothing magical about the 60-65 year old range that we’ve arbitrarily introduced for retirement. We can retire *much* earlier if we’re able to save *much* more.

So, let’s take another slightly more complex view of the savings rate. In this example, instead of looking at a principal amount we’re instead going to look at the *cumulative number of years of expenses* we have saved against different savings rates over time. I have once again ignored any investment returns in this example for simplicity. I’ll introduce them in in the following sections.

I find the figure here very powerful. If you’re able to reach a high savings rate then you’ll be able to reach many multiples of expenditure years saved in a very short period of time, *before we even consider compounding*. However, a somewhat distressing component is that the higher our savings rate the less the interest rate matters. If we refer to the figures above we see that the major benefits of compounding occur in the final years of the compounding horizon. A high savings rate means we’ll reach our principal requirements before this compounding even comes into effect.

We know from the section on compounding above that we can broadly expect our investments to grow over time. But to understand what that is we need to understand the Safe Withdrawal Rate (SWR).

# The Safe Withdrawal Rate

The Safe Withdrawal Rate is the percentage of your principal you can reasonably expect to withdraw every year, taking into account varying annual returns, that will not cause the principal to be depleted. That is, a safe withdrawal rate converts a lump sum into a risk adjusted annual income up to a certain level.

The most commonly quoted safe withdrawal rate is somewhere between 3 and 4% which is based off a study called “The Trinity Study”. The study is a *survival analysis*, it looks at under a given set of assumptions what withdrawal rates could be expected to *survive* the ups and downs of the stock market by preserving their principal over time. For example, if you retired in the year 1955 the maximum *safe withdrawal rate* that would have guaranteed safety might have been 7%. In 1985 it may have been 5.1%, in 1993 it may have been 4%.

What the authors of the study determined was *assuming the future returns may look similar to past returns* what withdrawal rate would allow the principal to *survive* to the end of the time horizon. I don’t want to flog the dead horse of SWR for too long here but it’s an important consideration in your investment planning. The chart below, from *PortfolioCharts* illustrates the impact of a 4% withdrawal rate over the time period 1972 to 2015.

In some of the scenarios determined even 4% was too aggressive. In others, it was more than safe. As such, there is still *risk* in using a SWR, in particular. Right at the *beginning* of the retirement period the risk is the greatest. If you were to retire and then suddenly go into a bear market, liquidating investments when they’re trading at a substantial discount is not ideal. Likewise, if you retire at the beginning of a raging bull market then you’ll probably be okay as your investments will be growing year over year.

Another thing to consider is the investments that have been made. Different countries will have different taxation structure and different investment opportunities with different risk. You should do your own research here and be aware that these will all impact the SWR that can be used.

Now. Why is the Safe Withdrawal Rate important to our calculations? Simple. It influences the amount of principal we’re required to save in order to reach Financial Independence. The SWR is the amount we can safely withdraw from our principal each year. That is, in a primitive sense the SWR multiplied by our principal gives us the amount available for expenditure.

As such, your principal requirements are governed by:

Principal = Expenses / SWR

A SWR of 4% indicates that we need a principal of 25x our annual expenditures. One of 3% means we need principal of 33x our expenditure.

To put this simply, to reach Financial Independence you will need to have investment earning principal of at least 25x your annual expenditure.

# Combining Savings Rate and Safe Withdrawal Rates

So far we’ve covered the Savings Rate and the SWR separately, but now we need to contextualise them.

- The SWR governs how much principal we need in terms of our annual expenditures to support our lifestyle.
- The SR governs how quickly we will reach that principal

You can now see why the SR is considered the most important number, though the SWR is a close second. From our figure above, which I’ll reproduce here we can see how long it will take us to reach different principal amounts, *relative to our expenditure*.

That is, if our savings rate was 75% we would be able to work for 10 years and then *never* need to work again as our investments would support us at the constant expenditure level. As such, our Savings Rate is the primary factor which governs when we’re able to retire early. The difference between a 3% and 4% SWR at a SR of 75% is just two years. Small enough in the scheme of things to not actually matter. However, at a SR of 50% this will be five years.

But, the figure above doesn’t take into account any compounding effects of our investments. As such, we need to update our graphs. In the figure below I’ve reproduced the above figure but have used an investment return of 5 and 7% at this stage, separated across the two figures.

And this in a nutshell is Financial Independence. Increase your savings rate high enough, find some solid long term investments that will pay you the market average over time and stick with it for a number of years. If you follow this disciplined proposition you should be able to reach Financial Independence sometime in your 30s or 40s for most people. Granting you an additional 20-30 years of retirement as compared to the retire at 65 age bracket.

How fast you actually get there will of course depend upon a number of factors. High income individuals have a leg up, their savings rate will naturally be higher. Likewise, those amongst us who have embraced frugality to an extreme will also be able to achieve high savings rate. The individuals who practice the systems approach of Early Retirement Extreme come to mind. Likewise, the markets will also have an impact. If the return over the period was *only* 3% as in the figure below then the time horizon will get pushed out further. But it is still possible.

Interestingly, in a low return environment the savings rate becomes even more important as a higher principal accumulation rate will buffer you from any variance in the investment returns over time.

This concludes the theoretical sections of this article. In the next section I’ll go into some of the practicalities that these imply. It’s all very well and good understanding the theory but without the ability and discipline to follow a strategy over the longer term it will all be for naught.

# Practical Considerations and Useful rules of thumb

Obviously it isn’t reasonable to undertake a full first principals analysis from scratch for every decision in our lives. If we believe the psychological research then our brain has two modes, a thinking mode and an instinctual mode. To achieve FI we want to internalise a few useful heuristics that shift us from the thinking mode into the instinctual mode. Luckily these fall out from the mathematics above.

The journey towards FI is a slow one, even in the best case scenario you’ll still be looking at 5-7 years worth of ultra disciplined investing and expenditure reduction before you can reach FI.

What will trip you up on this journey is not the big things. Those are often easy to manage. Instead, it is the little one off costs that arrive, the small pleasures that so many of us take for granted and the poor longer term decisions that we may make. For a better understanding part two of this series which covers the effects of expenditures on FI should be read. In the remaining sections I’ll simply cover some technical heuristics to consider.

## Calculating the True Cost of a one off cost

A one off cost is often something which is a *want* rather than a need in this case. It is something which is additional on top of your day to day expenditures. I classify these expenditures as “If I didn’t buy it then I would invest the money”. That is, an item which has a 1:1 trade off with an investment.

Unfortunately for most people the *true cost* of an item depends upon where you are on the compounding curve. If you’re right at the beginning and you’re expecting to have that money compounding in your favour for at least the next ten years while you accumulate capital. On the other side of the coin, if you’ve got one year left to live then the *true cost* of an item can be less than it’s sticker price, you may have surplus capital and no other use for it.

## Expense to Principal Requirements

Often when we’re going about our day to day lives we have a series of daily or monthly expenses. These expenses, such as a morning coffee, bus fare, buying our lunch at a store all seem small in aggregate. However, they have the nefarious effect of both.

- Delaying the time to reach FI by decreasing your savings rate
- Increasing the capital requirement required to sustain a lifestyle

A daily expenses of $5/day or $150/month at the 3% withdrawal rate will require a principal of $60,000 to support continuously. This is $60,000 which you will have to save on top of your other lifestyle expenses. If, for example you are able to save $30,000 of principal per year, then it would take you two years of additional saving just to satisfy the $5/day habit. On the other hand, if we have multiple habits like this which compound to $20-25/day (very easy to do in my city, Coffee at $4, Public Transport at $8, Lunch at $10) then you will not only be spending an additional $8,000/year on just these three expenses. Your savings rate will also decrease by $8,000/year and the principal requirements balloon up another $250,000.

This is in essence a double whammy. Not only do you need an extra $250,000/year in order to satisfy the coffee, lunch and transport costs into perpetuity. You’ll also be delayed on the savings journey quite considerably. The figure below illustrates the *value* of $8,000 which has been invested annually for the first ten years and how it grows over time. Over the first ten years the interest rate doesn’t matter as much. But it’s still $100,000 of additional investment income that you’re forgoing over this time horizon in order to work.

What this illustrates towards us is that our expenditure has the single greatest impact upon our ability to enjoy financial independence. The fitness analogy is that you can’t out work a bad diet, it will always catch up to you. So to will our expenses.

A monthly expense can be converted into a principal figure by the following formula:

P = 12/SWR * Monthly Expenditure

For a SWR of 3% this is a multiplier of 400, for a SWR of 4% it is *only* 300. It requires $2,000 in capital to offset a $5/month habit. At $100/month this rises to $40,000 and so on.

Simply put, to realistically achieve FI you must be able to control your expenses, starting with the biggest ones first. A $300/month car payment which could have been $100/month. A $3000/month mortgage that could have been $2000/month. These expenses will move the lever much much more than cutting out a little treat here and there. Tackle the biggest items first and then slowly move down the track.

A useful tool here is the concept of *Lent* from the religious sphere. During *lent* worshippers abstain from particular luxuries as well as fasting for a period of 40 days in order to cleanse and purify the soul. You can use this concept, go without a particular luxury for 40 days and assess the impact it has upon your life. Trivial? Good. Don’t reintroduce it. Essential? Okay, bring it back into your life and don’t feel guilty about it any more.

Learning to appreciate the little luxuries is a key component of the FI journey. We must master the hedonic treadmill

# Conclusions

In this article I’ve gone through the basics technical underpinnings of Financial Independence. I’ve shown how compounding can work in your favour and how the journey towards FI is predicated upon control of your expenditure, not just having a high income. In the remaining parts of this series I am going to cover basics of investing, understanding how financial independence can fit in your lifestyle and cover some resources that I have found particularly helpful in my journey.

## References

- Cooley, Philip L., Carl M. Hubbard, and Daniel T. Walz. 1998. “Retirement Savings: Choosing a Withdrawal Rate That Is Sustainable.” AAII Journal 10, 3: 16–21

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