The Ultimate Guide To Compound Interest

What is compounding and why should you care?

When you lend or invest a sum of money, you can expect to receive a certain sum in return. This is a charge paid to you for the use of your money, and it is known as interest. Interest is calculated over a specified time period, most commonly yearly, half-yearly, or quarterly.

 

Compounding is when the interest received on a sum of money – the principal – is accrued to that principal instead of being immediately withdrawn. As a result, the principal grows with each new interest period, as does the accrued interest, resulting in the highly lucrative snowball effect known as compounding.

 

Here’s what this looks like in visual terms:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Credit: Reddit

 

As you can see, thanks to the power of compounding $25 a week snowballs into a staggering $188,000 at the 40 year mark – the overwhelming majority of that attributable to compounded interest and not even the actual money deposits themselves! Not bad eh, and certainly enough to take a look at this concept in a little more detail.

 

The basics of compounding

Looking back at the graph above, we can already intuit what is arguably the most important thing to keep in mind when looking at the compounding phenomenon, which is the exponential nature of compounding and the primary importance of time in that calculation. This is reflected in the most basic compound interest formula, FV = P (1 + r)^t, in which the future value (FV) of an investment is expressed as an exponential with regard to time in years, t, with P as the principal and r as the interest rate.

 

E.g. Value of 500 dollars invested at 4% annual interest rate after 10 years

FV = 500 * (1+0.04)¹⁰ = 740.12 dollars

 

In practical terms, this means that the accrual of interest is like a huge train gathering momentum: It starts relatively slowly, getting faster and faster over time until it is an unstoppable juggernaut. And indeed, this is exactly what we see on the graph: Just look at the difference in the ratio of money deposited to interest earned at the 10 and 50-year marks to see what I’m talking about.

 

The importance of time in the investment equation, then, cannot be underestimated. To impress on you just how important, let’s take another example. Imagine we have two investors, Sally and Pete. Sally’s a sensible Sally and she begins investing at 21. She invests 1200 dollars a year over a 20 year period, a total investment of 24,000 dollars that nets her $59,295 by the age of 41. At this point, she stops, doesn’t invest another dime, and simply continues riding the compound train. Enter Pete. Pete’s not really thought much about his finances and doesn’t get started investing until 47. He puts in the same 1200 dollars a year, leaving him with $59,295 at age 67. But how much does our sensible Sally have by this point thanks to the extra 26 years of compounding? Well, $471,358 to be exact – almost 8 times as much as Pete even though they invested exactly the same amount of money over the same length of time! And here it is in a handy graphic:

 

Credit: My Investing Notes

 

Key takeaway: Understand that compounding is exponential with time as the key constraint, and get started as early as possible!

 

Optimal interest period

Now that we’ve covered compounding in its fundamentals, we can afford to incorporate a few refinements into our consideration.

 

Our most basic formula, FV = P (1 + r)^t, was missing a few subtleties, the first of which is how the period of interest repayments affects an investment’s value.

 

If we let n be the number of interest periods in the year, we can easily adapt our formula so that FV = P [1+(r/n)]^nt.

 

So what’s better – yearly interest, half-yearly, monthly?

 

Well, let’s put in some numbers. We have $10,000 dollars to invest and are being offered an interest rate of 10%.

 

Here’s what we’d have at the end of the year according to various interest periods:

 

Period	n=	Total at year end
Year	1	$11,000
Half-year	2	$11,025
Quarterly	4	$11,038.13
Monthly	12	$11,047.13
Weekly	52	$11,050.65

 

Key takeaway: Long story short, the more regular the interest repayment the better. However, note that if you were to calculate further than weekly (please feel free to do so if you are so inclined!), then the gains will quickly approach their limit and become so tiny as to be of negligible benefit.

 

Regular deposits

If we want to refine the formula still further, we can make it take into account regular deposits (as per the first graph, for instance, which showed a regular deposit of $25).

 

In this case, the formula is adapted as follows:

 

FV = P(1 + r)^t   +   c[ ((1 + r)^{t + 1} – (1 + r)) / r ] , where c is the regular deposit.

 

The reasoning behind this is described in detail here, and the site also offers a simple calculator that helps you factor in all the variables (repayment periods, deposits, principal, interest rate) without having to do the calculation yourself. You can access the calculator (screenshot below) via this link.

 

Key takeaway: By continuing to pay in and add to your principal on a regular basis, you accelerate the compounding effect.

 

Varying rates of return and the geometric mean

In the preceding sections, we have looked at the time, interest repayment period, and principal variables of the compounding equation; now, it’s time to take a closer look at the interest rate variable itself.

 

After all, however seductive the maths of compound interest looks on paper, any long term investment such as that required for successful compounding is going to be subject to quite a lot of variance and unpredictability, in which the rate of return you actually obtain on your investment plays a key role.

 

An investment portfolio might grow 50% one year and then drop 35% another, so how are you meant to evaluate a portfolio’s performance and make sure you’re heading in the right direction?

 

What we need is some method to standardise the rate of return in our calculations so we can get a more accurate picture of how we are doing.

 

This is where the geometric mean comes in. What the geometric mean allows us to do is to take varying rates of return over a time period and distribute them evenly to come up with a fixed number for our simple compounding calculation.

 

Here’s how it works in practice:

 

Let’s imagine this is how my investment has performed over the last 6 years:

 

Year	Rate of return
1	42%
2	-22%
3	14%
4	19%
5	7%
6	-28%

 

The geometric mean is calculated by multiplying all the rates of return to give a product, and then applying the 6^th root (because we are looking at a 6 year period).

 

Geometric mean = √6(1.42*0.78*1.14*1.19*1.07*0.72)

= 1.025 = 2.5% average return

 

Key takeaway: Find the average rate of return on your investment over a given time period by applying the geometric mean.

 

Real interest vs. nominal interest

The final factor that can come in and really mess up your on paper calculations is a little thing called inflation. It’s no good earning a return of 20% if inflation is at 30%. Your money is only as good as its buying power.

 

Real interest rates take this into account by adjusting for the impact of inflation. So if your nominal (non-adjusted) interest rate is 8%, for instance, and inflation stands at 2%, your real interest rate will be 6%. This is an important consideration to take into account when assessing your investments, especially over longer time periods.

 

Key takeaway: Your nominal interest rate and accumulation of wealth does not necessarily equate to your real world buying power. This depends on the real interest rate, adjusted for the effect of inflation.

 

The rule of 72

Finally, if you’ve had enough of variables, equations, and calculations, I’ll finish with a handy rule of thumb known as the rule of 72. The rule of 72 states that the time needed to double an investment by compounding can be roughly calculated by dividing 72 by your annual percentage interest rate. So, at 9% annual, 72/9 = 8 years, for instance.

 

I will finish here, because this introductory guide has already gone on long enough. I hope that it’s given you something of an appreciation for the power of compounding, and that you will be able to harness it to your advantage in the near future! This is what separates the wealthy from the rest – and the really good thing is, that if you can get started early, you can use this to your advantage even if you don’t have a huge amount to invest. Be smart, and start making your money work for you!