Today, I’m here to talk about common misconceptions people have about compound interest.

Let’s start with a quick question to see how much you understand compound interest:

There is a single microscopic cell in a bowl. Every day, that cell splits/doubles. After 100 days the bowl is full of cells. How many days did it take to fill half the cup?

If you said 50- congratulations, you’re human (and wrong).

If you said 99- You are correct and either very good at maths, or understand compound interest, or have heard this one before.

Humans find it very hard to understand the concept of compound/exponential growth, we are used to thinking in linear terms.

Before we dive into the misconceptions, let’s just level the playing field and ensure we know the difference between linear growth and Compound/exponential growth.

## Definitions

### Linear growth

Linear growth is when you have the same increase (value) in every step. For example: 10, 12, 14, 16, 18….

In this example, we increase the previous number by 2 in each step.

In the finance world, this happens when only the principal earns interest. Think about a £100 deposit that yields £4 each year, you’ll have £100, £104, £108, £112… and so on.

Linear relationships (and linear growth) are pretty easy for most people to comprehend.

### Compound/exponential growth

Compound (or “exponential” as we learned during COVID) growth is when you have the same increase (per cent) in every step. For example: 2, 4, 8, 16, 32…

In this example, we double (or increase by 100%) the previous number in each step.

In the finance world, this happens when not only the principal but also the interest earns interest. I know that was a confusing sentence so let’s use an example.

We’ll use (again) a £100 deposit but this time it earns 4% on the closing balance.

At the end of year 1, we’ll have £100 + £100*4% = £104 (same as in the linear example).

However, at the end of year 2, we’ll have £104 + £104 *4% = £108.16. Here, the increase is £4.16: £4 from the original £100 and £0.16 from the £4 interest earned in year 1.

The increase will become bigger and bigger as we go on, I’ll save you the examples (because we have more than enough examples coming up).

This is known as “compound interest” because the interest earns interest, which is interesting.

Cool, now we’re aligned.

## Misconceptions about compound interest

Three variables affect the final value you’ll end up with:

1. Interest rate

2. Cashflows (initial amount and/or periodic amount)

3. Time

We’ll go through each one of these to determine if increasing them will result in linear growth or compound (exponential) growth.

The best way to check if a relationship is linear is to increase a variable (we’ll double it) and see if the result increases by the same rate (will it double?).

I know several people in the FI community who think they understand compound interest but still get these wrong. Have no fear, Lazy FI Dad is here (my future bumper sticker). I want to ensure we all understand the components of compound interest and how they work.

For each of these variables, we’ll use a few examples to ensure we’re happy with the conclusion. I’ll also give an example at the end of each variable to show where the misconception is.

**Disclaimer for the finance/Excel nerds (my people!): I will assume the payments/interest are made at the end of each period. Also, all the numbers below can be calculated using the FV formula in Excel (or Google Sheets, yuck).**

### 1. Interest rate in compound interest

The first variable that affects the final value you’ll end up with is the interest (or growth) rate.

#### First example- Initial amount with no periodic deposits

We will assume an amount of £100 invested over 20 years. Once with 4%, once with 8%.

£100 invested over 20 years with 4% annual growth will result in £219.11 at the end, or a growth of £119.11.

£100 invested over 20 years with 8% annual growth will result in £466.10 at the end, or a growth of £366.10.

366.10 / 119.11 = 3.07

As you can see, we doubled the growth rate but more than tripled the growth!

#### Second example- Initial amount and periodic deposits

The same £100 invested over 20 years, once with 4%, once with 8%. However, now we will also invest/deposit an additional £5 each period. Total investments: £100 + £5 * 20 = £200 so the total growth will be the final amount minus £200.

£100 invested over 20 years with an annual investment of £5 and 4% annual growth will result in £368.00 at the end, or a growth of £168.

£100 invested over 20 years with an annual investment of £5 and 8% annual growth will result in £694.91 at the end, or a growth of £494.91.

Here the rate is (494.91/168) = 2.95 (almost triple)

**Conclusion: Interest rate has an exponential relationship with the final value.**

*“But wait, how come the first example was more than triple and the second example was less than triple”*

Great question! (I love talking to myself). Let’s assume 1 year with £100 and a 4%/8% growth rate.

After 1 year you’ll have £104/£108, which means a growth of £4/£8, exactly double, this seems linear. That’s the whole point. For compound interest rate to do its magic, it needs time. When we deposit £5 each year, those first £5 have “only” 19 years to grow, the next £5 have “only” 18 years to grow and so on.

You can see in the first example that the growth more than tripled. You also saw above that in 1 year, it exactly doubled (which seems linear). Also, the last payment (at the end of the 20th year) didn’t grow at all as it was done on the last day. Anything between 1-20 years will be somewhere in the middle, so less than triple, weighing it down.

#### Example from personal finances

Some people think that doubling (or tripling) the growth rate will double (or triple) their total returns. We saw that’s not true, it will more than double (or triple)!

I want to give one more example with the risk of confusing you a little bit, let’s call it “advanced material”.

While this will confuse some of you, it will give some of you a much better understanding of the relationship between interest (or growth) rate and the final amount you’ll end up with.

Let’s talk about management fees. People (outside the FI community) tend to think that the difference between a 0.5% management fee and a 1% fee is small because it’s “only” 0.5%.

Let’s take an example from above but now we’ll add the fees (again- at the end of the period).

The annual fee is calculated from the closing balance at the end of the period.

##### Example of fees

We’ll use £100 initial amount, no periodic deposits, 4% annual growth and 0.5%/1% annual fee.

The final amount with no fees (above) was £219.11.

The final amount with a 0.5% annual fee is £198.21, a decrease of £20.90 compared to no fees. Total fees were £14.68.

“*Hold on, how come the decrease is larger than the fees?!*“

Did Lazy FI Dad get his maths wrong? That could happen but not today.

While the fees decrease the final amount due to money leaving your investment account, they also decrease the total growth. Think about the first year- because fees were taken, the closing amount of year 1 is smaller and the interest in year 2 was earned on a smaller amount. Fees also decrease the growth!

Now, let’s use 1%.

The final amount with a 1% annual fee is £179.21, a decrease of £39.90 compared to no fees.

Total fees were £27.83. Here, it’s less than double compared to the £14.68 in the 0.5% fee scenario.

*“Huh?! how is it now LESS than double?”*

Again, this makes sense when you think about it. Because you take higher fees, the amount used to calculate the fees gets smaller each year.

But… let’s look at the final amount with a 0.5% fee and the final amount with a 1% fee.

With a 0.5% fee, we got £198.21. With a 1% fee, we got £179.21.

The amount with a 1% fee is almost 10% (!!!) smaller than the amount with 0.5%. I’m sure that doesn’t sound so small now, does it?

### 2. Cashflows

The second variable that affects the final value you’ll end up with is the amount of these cashflows.

This is probably the biggest misconception about compound interest (and the reason I am writing this post).

#### First example- Initial amount with no periodic deposits

We will assume a 20-year investment period with a 4% annual growth. Once with an initial amount of £100 and once with £200.

£100 invested over 20 years with 4% annual growth will result in £219.11 at the end.

£200 invested over 20 years with 4% annual growth will result in £438.22 at the end.

438.22 / 219.11 = 2

As you can see, the final amount exactly doubled.

#### Second example- Initial amount and periodic deposits

We will assume a 20-year investment period with a 4% annual growth. Once with an initial amount of £100 and periodic deposits of £5 and once with an initial amount of £200 and periodic deposits of £10 (doubling the previous scenario).

An initial amount of £100 invested over 20 years with £5 periodic deposits and 4% annual growth will result in £368.00 at the end.

An initial amount of £200 invested over 20 years with £10 periodic deposits and 4% annual growth will result in £736.01 at the end (the one penny is due to rounding, I’ll use £736.00).

736 / 368 = 2

Once again, the final amount exactly doubled.

**Conclusion: Cashflows have a linear relationship with the final value.**

#### Example from personal finances

The biggest misconception with cashflows is when people say that all the money should go into one account because of “compound interest”. We just saw that’s wrong!

2 separate accounts with £100, 4% growth invested over 20 years (no annual deposits) each will result in 2 accounts with a £219.11 value in each of them, or a total of £438.22.

1 account with £200, 4% growth invested over 20 years (no annual deposits) will result in an account with £438.22, the exact same amount.

Stop lying to yourself! Pooling all your money into one account will not leave you with more (or less) money!

Except for a few weird scenarios (that aren’t related to compound interest)…

##### Advantages of putting all of your money in one account

Some investment platforms cap their fees. For example, Vanguard charges a 0.15% (of your portfolio value) annual platform fee up to a value of £250,000 or £375 platform fee a year.

That means that they’ll charge the lower of 0.15% or £375 per year.

Assume you and your spouse have £400,000 to invest.

If you and your spouse each put £200,000, you’ll pay an annual fee of £300 each (£200,000 * 0.15%) or a total of £600.

If you put it all in one account, the fee will be £375 (the cap), saving you £225.

Notice this is all about the Vanguard fee cap, it has nothing to do with compound interest!

I just wanted to show that there are reasons to pool all the money into one account.

##### Advantages of spreading your money over several accounts

Let’s use the same example above of £400,000 between you and your spouse.

From the Vanguard website:

“Vanguard is covered by the Financial Services Compensation Scheme (FSCS). This means you may be entitled to compensation up to £85,000 in the unlikely event that we’re unable to meet our financial obligations to you.”

Let’s take this unlikely event for a second and play it out. If you put it all in one account (to save fees) you are only covered up to £85,000. If you split it into 2 accounts, each account is covered up to £85,000, or £170,000 in total. You can split it into 5 accounts (with other providers as well), £80,000 each and then it’s all covered 🙂

Again, this is a regulatory “insurance” and has nothing to do with compound interest!

I wanted to show that there are also reasons to split the money across several accounts.

OK, back to compound interest.

### 3. Time

The third and final variable that affects the final value you’ll end up with is time (or length of investment period). I’ll jump straight to the conclusion and then go through a few examples.

I opened this post with a question about a cell that doubles itself.

After 99 days half a cup was full but after 100 days the cup was completely full.

Going from 99 days to 100 days (1.01% increase) resulted in double the amount of cells (100% increase). Of course, it’s not linear. Still, let’s see some financial examples and then talk about the misconception.

#### First example- Initial amount with no periodic deposits

We will assume an amount of £100 invested with a 4% annual growth rate. Once with a 20-year investment period and once with 40.

£100 invested over 20 years with 4% annual growth will result in £219.11 at the end, or a growth of £119.11.

£100 invested over 40 years with 4% annual growth will result in £480.10 at the end, or a growth of £380.10.

380.10 / 119.11 = 3.19

As you can see, we doubled the investment period but more than tripled the growth!

#### Second example- Initial amount and periodic deposits

The same £100 invested with a 4% annual growth rate, once with a 20-year investment period and once with 40. However, now we will also invest/deposit an additional £5 each period. Total investments: £100 + £5 * 20 = £200 so the total growth will be the final amount minus £200.

£100 invested over 20 years with an annual investment of £5 and 4% annual growth will result in £368.00 at the end, or a growth of £168.

£100 invested over 40 years with an annual investment of £5 and 4% annual growth will result in £955.23 at the end, or a growth of £655.23.

Here the rate is (655.23/168) = 3.90 (almost quadruple!)

**Conclusion: Time (or investment period) has an exponential relationship with the final value.**

#### Example from personal finances

The biggest misconception with time is that people think the % of your FI number that you reached means the same % of your journey finished.

Let’s use a new example, with someone starting their journey to FI with no initial amount but with the ability to save/invest £10,074.53 per year over 20 years. We’ll also assume a 4% growth rate. The final amount after 20 years will be £300,000 (and 15p, fine), what a coincidence!

After 10 years that person will have £120,955.89 or 40.32% of their FI number. Does that mean that (from a time perspective) they are 40.32% of their journey done? of course not! (* Screaming this next part *) The relationship is not linear!!!!

I already told you that this person’s journey is 20 years long, so after 10 years they’ll be exactly halfway into their journey. 50%, not 40.32%.

The higher the growth rate you assume, the bigger the difference will become.

That’s because, in the second half, there is more money to apply the growth rate to, so more growth each year in the later part of the journey. That’s also why people say that the first £10k/£100k/£1m is the hardest, it’s maths.

In our July 2023 results post, I mentioned we are at 33.40% of our FI number, which means we are **MORE THAN** 33.40% into our FI journey (from a time perspective). Now that’s some optimism for you to take away from this very number-heavy post.

I hope you understand compound interest a little better now 🙂

And remember- splitting or pooling accounts will not increase or decrease your money’s growth!

There is a mistake in this calculation:

£100 invested over 40 years with an annual investment of £5 and 4% annual growth will result in £955.23 at the end, or a growth of £755.23.

Here the rate is (755.23/168) = 4.50 (more than quadruple!)

You are correct, because we invested £100 initially and £5 each period for 40 periods the total investment is £300 and not £200, corrected.

It now says “£100 invested over 40 years with an annual investment of £5 and 4% annual growth will result in £955.23 at the end,

or a growth of £655.23.Here the rate is (655.23/168) = 3.90 (almost quadruple!)”Thanks so much!!!