The Boggy Marsh Part 2 – O(2^m+t)

In part 1 of The Boggy Marsh we discovered that there was a secret financial blueprint that would allow even lower-income individuals to get ahead; if only life was approached the correct way.

Today we’re going to delve into the specifics. How exactly will careful planning and execution help aid your journey through The Boggy Marsh, and how big will the impact be?


What does this icky Math formula mean?

It’s the Mathematic equation for the growth of your money. The number one most important factor when aiming for financial independence.

It’s written in a language called Big O Notation; an engineering invention that is normally used to determine the time cost of an algorithm depending on the input variables.

I’m currently studying the theory in preparation for a Google interview and when formulating the equation for FIRE I realised that time is just as important as money. In fact, it’s completely equal.

The time complexity of financial growth is 2 to the power of time + money.

The more time you have, and the more money you have, your pot growth will double after each iteration. It will grow exponentially.

If you prepare early enough, the amount of time that you will have on your side will allow your pot to venture up the exponential curve to the point of indifference, the money that you save at a later stage will hardly matter.

A trial run through The Marsh

Before your adventure begins, let’s simulate a few scenarios for tackling The Marsh. Remember, if you dive in headfirst you’ll get stuck.

For these illustrations the target income in retirement is 20k per year, the return on investment is 6%, and we can assume that at the beginning of year 1 the person is 22 years old.

The traditional method

The first chart maps the approach that most people take, I call this the ‘don’t worry about your pension’ model. These are the people that save barely anything into their pensions (circa £100 per month) until the age of 45, they then commence ‘panic mode’ and begin to save £20k per year.

If you followed this model, by the time you’re 45 years old you would have only saved a meager 58k from depositing £100 per month since you started your career at 22.

You’ll then engage ‘panic mode’ at 45 after realising that at this rate, you’ll never retire. At great discomfort to yourself and your family (as you’ve been big spenders for 23 years), you manage to deflate your life enough to save £20k per year.

You have to live ‘struggling’ like this for a whopping 13 years before eventually retiring at 58. Over these 13 years, you’ve had to part with 260k of your after-tax earnings.

If instead, you tried your hardest to save as much as you could upon graduating and managed to save 20k per year for 5 years, your life would have turned out a whole lot different.

You ‘sacrificed’ the first 5 years of your career and saved almost all of your earnings. You knew that the time complexity of compound interest was O(2^m+t) and that it was paramount to do everything you could to save as much as you could at this early stage of your life. Whether this meant living with your parents or in shared accommodation, you were all in!

Doing so allowed you to amass 116k by the age of 27. You could then kick back, relax, and save nothing but a scant £100 per month (have to get that employee pension match) for the rest of your life, you’re saving journey is now over.

That 116k, like yeast in bread dough, took on a life of its own after you had finished with it. That little cyan coloured bar indicates the amount of interest your money is earning every year, and after being cultivated, you allowed it to continue to grow on its own.

By the time you’re 51 years old your little nest egg that you saved all of those years ago has grown into 504k. You’re now ready to retire.

‘Sacrificing’ the first 5 years of your career allowed you to retire 7 years earlier and you only had to save 100k over 5 years instead of 260k over 13.

You used time to your advantage and treated your future self to a life of leisure.

Better yet, saving for those first 5 years wasn’t even hard! You had zero commitments back then; no children or house to care for; you simply deferred your life and focused on saving until your nest egg was ready to grow on its own.

The extreme method

Followers of TSN will know that my own savings are actually way higher than 20k per year. A lot of family and friends (and even people from the FIRE community) scrutinise my 80% savings rate; they say that I’m under 30, I should be traveling more; living a little, “Whilst I’m still young!”

If I plug my figures into the simulator, you can soon see why these are the most important years that I should be saving.

I’ve currently just passed the year 2 line and I’m 27 years old.

If I continued with my current savings rate for 3 more years, I could save absolutely nothing ever again and retire on 20k (tax-free) per year at the age of 41. By choosing to live frugally instead of ‘live life whilst I’m young,’ for the first 5 years of my career, I will have earned myself an extremely early retirement for not much struggle.

I could even choose to work part-time and merely cover my expenses at the age of 30 and I’d still get to retire at 41.

Of course, I don’t plan to stop saving in 3 years time as I believe that spending less makes you happier. My pathway will most likely look more like this:

After my front-loaded first 5 years, I’ll probably look to severely reduce my workload but I’ll still try to save at least 20k per year.

The power of time

No matter what I choose, I’d have a plethora of options; those first 5 years would have been worth it. Nurturing that little cyan line and allowing it to grow on its own has given me this superpower.

Instead of buying a big house, a nice car, and having children straight away; I’ve waited patiently.

This is what everyone should be doing, but no one is. Everyone is forgetting that time is half of the equation! O(2^m+t)

If you have time on your side you should be trying your hardest to save as much as you can. The younger you are, the more urgent; as the money you save will be worth exponentially more.

What if you don’t earn enough?

Even though I believe that anyone can make the wage of a CEO, I know that some people aren’t capable of saving a lot due to financial commitments. This is the reason why it’s even more important to save earlier before you pick up these commitments.

There are no bounds to the length you should go to save when you are young. You could live with your parents or pick up extra jobs. These years will define the rest of your working life.

All of the simulations in this post work exactly the same way if you cannot save the amount shown straight away, you just have to push back your numbers by N amount of years.

Why people don’t talk about this

The subject of time isn’t discussed much in the FIRE community. This is likely because a lot of people on this pathway found our little group later in life. As the power of time is exponential, they can’t use it to its full advantage.

Not starting earlier is the number one regret from most FIRE devotees, it’s mine too. Regret usually manifests with self-affirmation, and as blogs are portals to a persons’ inner-mind, this attitude bleeds through into their writing.

But make no mistake, if you are young and reading this, you should save right now! It’s an emergency. Yes, you could defer your savings in the hope of earning more and saving more later, but time is equal to money.

Time is the one thing that you have right now, don’t sacrifice it in the hope of earning more later.

Here’s an extra little simulation to mull over:

I was working at 16 years old on minimum wage. If instead of going to University I continued living with my parents and invested all of my minimum-wage earnings until the age of 22…

I would have enough to retire at the age of 40.

Yep. For most people with early retirement as a goal, they would have reached their goal earlier if they didn’t go to University and if they never earned more than minimum wage. This stuff absolutely blows my mind.

TIME is just as important as earning more. In fact, it’s better than earning more, because it’s a commodity that is guaranteed.

If you’ve got time on your hands, don’t waste it.


19 thoughts on “The Boggy Marsh Part 2 – O(2^m+t)

  1. Nice graphs and different way of putting an old argument.
    What you point out is that a lot of people leave it until 40 to start saving – but the reality is not as simple as that.
    By the time I am 40, I’ll have two kids coming to school age. Family holidays then are in peak season and out household spending may be stretched!
    Of course we could have had kids earlier – but that’s a whole different compound problem!
    If you are childfree and have kicked your expensive habits by 40 – then you can start saving.

    What I would like to point out is that whilst you are young you spend it (if you have it and borrow it if you don’t) and when you are older you realise you are fogged! and try to save again.
    There’s a lot going on (career paths, inheritance, house price inflation, births, deaths, marriages, divorces, illness/death, good luck, misfortune.
    The message is simple – save earlier but it’s a little like telling dieters to “just eat less” – it’s not the whole story.

    1. What you talking about old stuff GFF, this is REVOLUTIONARY information! 😉

      Your last point is what I want to drive home. If kids were given the correct information, they truly knew the impact, I think at least some would choose the saving route. The thing is, most adults don’t learn this until way later in life, something is wrong in society, but that’s what we’re trying to preach in the FIRE movement right?

  2. Those graphs are amazing but hindsight is a wonderful thing!

    The problem with the brilliant plan you lay out is that had it been presented to me when I was the tender age of 22, I would have just laughed in your face and shouted ‘Impossible’!

    Despite my folks being savers who retired early, I ‘rebelled’ and developed into someone who succumbed to peer pressure, who was lazy with their finances, who spent all their money (most of which they didn’t have). I have to say that I had a lot of fun doing so but what wasn’t fun was the time it took to pay off my debts so the best years of compounding were lost to me, pretty much 2 decades!

    Had my mind been open to such ideas, I think it would have still been very hard to implement such plans as none of my peers were into saving or investing. Perhaps it would have been different had the guy I was seeing then also embraced the idea, but he was a bigger spender than I was! You’re lucky Mrs Ninja is onboard with your lifestyle.

    My one redeeming factor was that I joined the company pension scheme as soon as I could and it’s only because of this that I have been able to aim for FIRE at all.

    I will certainly want to mention the powers of compounding to my nephew although will need his mum’s buy in, in case she feels I’m putting ‘fancy ideas’ in his head!

    1. Thanks for the insight Weenie 🙂

      If you had been sat down and explained to beautifully what the full impact of saving now vs later would be, do you think it would have changed your mind?

      Even though a lot of people say statements like this (“Yeah right, I wouldn’t have cared at that age!” etc.) I feel it’s at least the duty to educate younger people so they know what ‘sacrifice’ they’re making. Then at least they’ll have the choice on their own, instead of not finding out about the blueprint until later in life. At least then the education system wouldn’t have failed them.

      Of course, a lot of young people, like you say, probably wouldn’t give a second thought to it (even if taught!) But it may change some peoples lives.

      My partner, for example, started working at 14!! At 16 she had 3 jobs. She spent all of her money on clothes and accessories, in her words she was ‘rolling in it’ at that age as she had zero expenses. If she had invested all of what she had earned, she’d have more money invested than me right now; she’s said she regrets spending it all.

      1. I think if someone had told me during a particular time in my life, then yes, I could have lived my life differently and considered the saving/investing path earlier. The ‘window of opportunity’ was probably just after I had graduated, with only a little debt and I was unemployed for nearly a year. Things only started to go wrong when I was earning money and I enjoyed spending it too much! I have a friend in her 40s who like your wife also started work at 14 – her mortgage will be paid off in 5 years but she has no retirement savings or pension…different decisions, different priorities!

  3. If the Order is time plus money squared your formula on the graph and title should be O((t+m)^2). Yours currently says 2 to the power of money plus time!

    1. Oops! Thanks for that spot, I spelt out the formula incorrectly.

      It’s still O(2^(m+t)) as I wanted to represent exponential growth, O((m+t)^2) is ‘only’ quadratic.

  4. Like the graphs. Must have taken a lot of time.

    I started saving at 33 and only really kicked on from 41. With knowledge would I have done different, hard to say.

    Think fire will always be a small minority pursuit as most will dismiss as for high earners only, too extreme or unrealistic. So I think preaching to the masses pointless, but educating like minded helpful.

  5. I think the tricky bit with your calculations is to make sure you maintain that 6% interest and keep track of the investments. Making sure that your investment choice maintains that 6% over 20 years is very hard. You need to constantly watch and switch to smth else if the 6% is not kept. There’s a fine line between deciding to switch to smth else or sticking with your initial choice. And then there are the dips in the stock market.
    Haven’t even mentioned about the fees. One investment platform might increase their fees over 5- 10 years time. It just requires constant monitoring.
    And when you think you have it under control, somehow 1 year slips by, and things change again.
    It’s easier if you have one currency and one investment platform. But if you have 2 or 3 and they are all invested in different countries… OH GOD.
    It is an amazing realization of the power of compound interest and time. Great post !

  6. I got compound interest reasonably early but only ever saved into a pension. I’m probably the embodiment of this. I’d say my pension now is pretty much sorted (circa 220k at 38) even if I never put another penny in and that’s from never having put more than 12% in myself. It’s why I get so frustrated with colleagues who say well pensions aren’t worth it my pensions useless. Like it’s some outside force that’s caused it. I’m like Well yes because you’ve never put any money in it numbnuts

    1. Yea, there are way too many people that say pensions are worthless!

      That’s a hefty chunk at such a young age. I’m going to have to do some investigations into when to actually stop depositing myself.

  7. Food for thought, Ninja!

    When I was 9 or 10, my mum explained interest and compounding as I opened a bank account through school. It captured my imagination. Money for free, I thought, and free money on top of that! The rate was 10% back then! I always saved since then. At the least, I broke even when I was a student without work for a while but mostly worked part time. I opened tax free savings before I was a tax payer haha and had shares before 18 by asking my parents to buy them under their name! So, overall, I’m a savings advocate, self taught, and it was very much a secret life I led. However, deep down I was also driven by a fear of not having financial security as well as enjoying saving.

    While educating about money among other crucial life skills is something I feel passionate about, I can’t agree that saving a tonne early is the correct way. There are probably several sensible strategies, in my opinion. Not everyone wants to retire early. Certainly my hubby doesn’t want to retire before 57!

    I have spent thousands on certain things in my youth I don’t regret and they give me compounding in ways that money cannot buy if I tried to get them now instead of when I was younger. Education, reasonably cheap travel, the cost of spending time with friends in my youth. They have given me memories, developed me as i was still ‘malleable’ and made me the person i am in some way.

    In all honesty I have slightly regretted a couple of times being overly frugal, because in the end I lost out but I was stressed at not earning much (I didn’t get my first f/t job til 27). I wouldn’t have gone into debt but just spent more than what i felt i could afford. But hey ho, you live and learn what is worth what to you…

    Had I not spent those thousands, and had I been shown how to invest (better) and manage risk (I knew no one who invested so was operating very much in the dark and was surrounded by peers and boyfriends crap with money!) I’m sure my assets would be much, much higher now. But im glad I spent what I did, I just needed to know more than simply saving and compound interest. The internet has definitely helped there.

    1. Thank you for the insight Firelite, it’s a sobering read. I might regret certain underspending in the future when I look back. That’s why I’m thankful for this blog; it steers me in the direction of being mindful and looking within at what I should be doing.

      I think the things that I’ll regret the most (if I don’t stop being so frugal in these areas) will be seeing my parents more or buying them little things (or making) to show my appreciation. As people say: “They won’t be there forever.”

      Same goes for little gifts for my wife, I really hate spending any money, but sometimes a small bunch of cheap flowers goes a long way.

  8. Thank you for being so receptive in your response. Gifts to those you care about and spending to enable time together – yes!! I totally agree. I spend hundreds on my family for Christmas and took my mum away twice this year (she’s retired on the state pension). I have just offered to take her away in February too. It’s not easy, b/c I want to be seeing my savings rate going up!!! Plus having her around isn’t easy in itself. But if money is life energy, then spending on those close to you is very important to me. 🙂 It’s the difference between frugal and stingy/miserly. 😉

    I’d never expect a bouquet of (cheap) flowers from my hubby (and I’m not sat down long enough to appreciate them totally, which he knows), but he always gets me high quality cards on special occasions (not the cheap ones I often buy for people, lol) and thinks carefully of what to write inside (even if it’s all in capital letters) – and I love that! He’s otherwise very frugal. You can say, well, why spend an extra £2 on a card which is on show for all of a week or so, but love isn’t and shouldn’t be rational. 🙂 Also, if he makes me a cuppa without me asking, I really like that!! 😀

  9. This was an intriguing post SavingNinja, but there is more to the story being told.

    When you are projecting out into the future like this you need to adjust for inflation. Today those expenses may cost £20k, but in 20 years time it will cost more like £50k to purchase the equivalent quantity of goods and services. To strip out the effects of inflation, present the projections using a constant purchasing power such as the value in 2019 money.

    The future value of those £20k withdrawals also compounds. It is important to strip that out before calculating the compound growth, rather than just taking it off at the end. A real life illustration of this in action is the way overpayments on a mortgage make a huge difference in both the duration and total cost of the loan.

    Taxes are a complicated subject that will produce a different answer for everyone. However one thing most people will have in common is that if they plan to retire in their early 40s, then pension accounts are going to be a decade or more away from being accessible. That means taxable accounts and ISAs will need to fund living costs during that period. For many of us that means there will likely be a tax bill associated with this. It may be that in order to have that £20k available to pay the bills, the person needs to sell something like £24k of assets to cover the bills + income/capital gains taxes.

    You’ve done the easy part here, but it would be good to see a final chart added that presents the same story after taxes and fees, accounting for inflation and the compounding effects of the withdrawals. It will likely push out the timescales some, but also highlight why it is so important to manage fees and minimise tax expenses.

    1. In all of the other calculations that use 6% as a general rule of thumb for passive index funds (MMM, Smarter Investing, etc.) they state that this figure is inflation-adjusted – so all of the end figures shown in the post should represent today’s money.

      I’ve also assumed that all of the savings are in tax-sheltered accounts and that there is enough in a financial bridge to make it to a pension. (if applicable).

      Taxes would, of course, would still be incurred if saving for a large amount of income, but all of the figures are for gross. I’m hoping to not pay any tax as I’m aiming for 20k or less and I’d hope that at least 8k would be covered by my ISA.

      I’d be surprised if many people would need more than 20k per person in retirement 😮

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