Most people own a savings account and understand how their money grows. Kind of. In fact, very few actually know the maths behind savings accounts; they check their accounts periodically but rarely invest any real thought into it.
However, understanding how these types of accounts work is crucial if you are serious about financial planning; they are particularly important if you are making big decisions like taking on a mortgage or striving for early retirement.
Knowledge about the saving and lending market helps you understand concepts like whether it is better to rent or buy a house. If you know how the market works, you can optimise how you engage with it. It is therefore extremely important that you master the maths behind compound interest, annuities and loan repayments.
That means understanding some pretty complex formulas. I know that they may look daunting but – trust me – it’s important enough to learn.
Let’s start with the basics: The annuity. An annuity is a regular payment you make or receive over a specified period of time (e.g. week, month, etc…).
Constant Annuity Formula
In the case of constant annuity, regular payments of a fixed amount are made over periods.
The future value is the total amount accumulated at the end of the period , in light of compound interest applicable at each period.
This is typically the formula banks use to grow savings accounts. This formula will thus help you understand how much money you save after 5 years, 10 years, etc, if you start from scratch and make monthly payments.
See below for an illustrative table that varies annuity amounts and periods. The deposits are assumed to be made monthly and we are using a savings rate of 5% per annum.
Now that you have the formula, you can adjust this to your circumstances. If you have never looked into this topic, you will be surprised by the exponential growth of your savings. Looks good, doesn’t it? And, if you do it right, this is how you can afford to retire early. Thank you compound interest!
Obviously the time it will take you to amass sufficient money to reach early retirement depends on the amount of money you can save and the savings rate you are able to lock in. So it really is up to you!
This formula can also be flipped around. For instance, you can find out the amount you would need to save every month if you want to reach $100,000 after 3 years with an interest rate of 5% per year.
The present value is the total amount you would need to have at in your bank account paying an interest rate of per period, such that you will be able to make constant payments over periods.
Such a formula allows you to plan for annuities you will receive in the future. For instance, if you have calculated that you need $10,000 per year to live off and you want to receive that amount every year for 50 years, it will tell you how much you need in capital to stop worrying about money and retire early.
Below are some example numbers showing how much money you will need in order to pay yourself a monthly allowance over a period of time. We are still assuming monthly annuities and a 5% interest rate per annum.
Looking at these figures, you’ll see that you need relatively little capital amount to sustain an ongoing stream of allowance. Again, this is thanks to compound interest working magic on your remaining capital.
Mathematical Growth Annuity Formula
If you are already impressed with the amount of money you could save from a constant annuity over time, you will be even more pleased with mathematical growth annuities. In a mathematical growth annuity, the amount invested is not fixed but rather grows at each period.
Arithmetic growth means that the annuity grows by adding an extra amount on top of the initial payment at every period. For instance, if you start with an initial amount = $300 per month into your savings account and can put an extra = $10 on top every month, it assumes you will make a payment of $330 after 3 months, $420 after a year, $1500 after ten years, etc…
While this might look ambitious at first glance, it more closely resembles the pattern of our savings. Indeed, our earnings tend to increase over time as we gain experience and move up the salary scale.
On the other hand, arithmetic growth is also a useful tool for understanding and accounting for future expenses – Like having kids and sending them to childcare, university, etc… (yikes!!) – That would reduce the amount we are able to save over time (i.e. would become negative in the formula and slowly erase our savings).
Just like with constant annuities, there is a future value and a present value:
Geometric growth assumes a multiplicative factor that will grow our initial annuity at every period. It is mainly used to take inflation into account. For instance, for a yearly annuity, we would have where is the yearly inflation rate.
The future value and present value of this type of growth is computed with the following formulas. There are two possible scenarios to avoid the big fat zilch:
How Do We Use Those Formulas?
You now have all the important formulas you need. So, let’s start putting them into practice by running some numbers.
You plan to retire early and live off your savings. You own your house and have calculated that you need $1,000 a month to live comfortably. You are currently 35 and want to play it safe, so need to make sure you look after yourself until you are 100 years old. You expect the savings rate to stay at 5% per year while the inflation rate is 2%. How much money do you need right now in order to quit your job?
To answer this question, we need to calculate the present value of a geometric growth annuity using monthly periods.
Now assume you would like to know when you can retire. You have figured out that if you were to retire right now, you would need $1,500 per month in allowance to cover all your expenses. You would also need to buy a house in a small town with an estimated cost of $200,000.
You currently have $100,000 in savings, are turning 30 and plan to live until 100. You are putting aside $1,000 a month but will continuously increase this amount by $100 every month thanks to your new-found Monkeyism mindset. We still assume a savings rate of 5% and an inflation rate of 2% per year. When will you be able to retire?
To help us with this question, let’s call the number of years you still need to work and save money. By the time you retire, your total net worth would be the sum of your $100,000 savings plus the additional monthly payments you make, compounded month after month until you can retire:
On the other hand, you have to figure out how much money you need in order to pay yourself the monthly retirement allowance, adjusted by the inflation rate. The needed capital will be the present value of an inflation-adjusted geometric annuity at year , and the inflation-adjusted future value of the house you want to buy:
The right year to retire will be the value that solves the following equation: Net Worth = Required Capital. Solving such an equation could be done using Excel’s solver functionality (Goal seek on the year such that Net Worth – Required Capital = 0). In this case, we would find = 8.26, which means you would be able to retire shortly after turning 38.
If you have more savings, started planning younger, need less money to live off, etc…, you could reach early retirement even sooner. I would encourage you to take the time to review all the formulas, understand the context in which they should be used and apply them to your personal situation.