Compound Interest

Compound interest is interest calculated on an amount that already includes interest previously earned.

The arithmetic is genuinely powerful and it is also routinely overstated. Both statements are true and this page covers both.

What is compound interest?

Interest earned on principal and on interest already added to it.

Simple interest is calculated only on the original amount. Compound interest is calculated on a base that grows each period, which is why the two diverge slowly at first and then substantially.

The same mechanism operates on debt. An unpaid balance accrues interest, the interest is added to the balance, and the following period's interest is calculated on the larger figure. Compounding is indifferent to which direction it runs.

How does compounding work?

Through repeated application to a growing base.

According to Jason Fernando writing on Investopedia, compound interest is calculated by multiplying the principal by one plus the annual rate, raised to the number of compounding periods, with the original principal then subtracted to isolate the interest.

The formula is A = P (1 + r/n)^(nt), and each term matters.

P is the principal, the amount you started with.

r is the annual rate expressed as a decimal, so six percent is 0.06.

n is the number of times interest compounds each year. Annually is 1, monthly is 12, daily is 365.

t is the number of years.

A is the total at the end, including the principal. Subtract P to get the interest alone.

Why frequency matters less than people expect

Moving from annual to monthly compounding at the same nominal rate improves the outcome modestly. Moving from monthly to daily improves it very little.

The rate and the time dominate. Frequency is a refinement, and marketing that emphasises daily compounding is drawing attention to the least consequential of the three variables.

The distinction to ask about is nominal versus effective. A nominal rate compounded more often produces a higher effective rate, and the effective figure is the one that lets two arrangements be compared. Where only a nominal rate is quoted, the comparison is incomplete.

The rule of 72

Divide 72 by the annual rate to approximate how many years an amount takes to double.

At six percent, roughly twelve years. At nine percent, eight. At three percent, twenty-four.

It is an approximation rather than a calculation, and it is accurate enough for mental arithmetic across the range of rates most people encounter. Its value is that it makes the effect of a rate difference immediately visible without a spreadsheet.

It works against you identically. A debt at eighteen percent doubles in about four years if nothing is paid.

What are the benefits of compounding?

Growth accelerates without additional effort. The base grows, so each period produces more than the last from the same rate.

Time contributes more than rate for most people. A modest rate over a long period frequently outperforms a higher rate over a short one, which is why starting matters more than optimising.

Regular contributions compound too. Each addition begins its own compounding from the date it arrives, which is why consistency outperforms timing for most households.

It is arithmetic rather than a product feature. Any arrangement where earnings are added to the base rather than withdrawn compounds. Nothing needs to be purchased for the mechanism to operate.

What are the risks and the overstatements?

This section is longer than the benefits section, deliberately, because the overstatement is where the harm occurs.

A nominal rate is not a real rate. Inflation erodes purchasing power every year. A projection across thirty years in nominal dollars describes a number, not what it buys. The real rate is what the nominal rate leaves after inflation, and it is substantially lower.

Tax and cost are usually omitted. A return quoted before tax, fees and trading costs is not a return anybody received. Those deductions compound too, in the wrong direction.

Uninterrupted is an assumption, not a fact. Every compound projection assumes contributions continue and nothing is withdrawn. Real financial lives include job changes, illness, and years when contributions stop. A projection that never accounts for interruption describes a life nobody has.

Sequence matters once withdrawals begin. Two arrangements with identical average returns produce very different outcomes if one has poor years early in a withdrawal phase, because each withdrawal during a decline locks the loss in. Averages describe the past; sequences describe what actually happened to a person.

Compounding runs against you on debt with exactly the same arithmetic, and household debt is where most people encounter it first and most expensively.

None of this makes compounding untrue. It makes any presentation of compounding worth reading carefully, and it makes the guaranteed portion of any arrangement more interesting than the projected portion.

What arrangements involve compounding?

This practice cannot tell you which to choose, and the boundary is a licence rather than a preference: it is not registered with the Canadian Investment Regulatory Organization and does not provide securities advice.

What can be said is descriptive. Compounding operates wherever earnings are added to a base rather than taken out. That includes interest-bearing deposits, registered accounts where growth is not withdrawn, reinvested distributions, and the accumulating value inside a permanent insurance contract, subject to that contract remaining exempt under the Canadian rules.

Which of those suits you depends on your circumstances, your horizon and your tax position, and it is a question for a person qualified to answer it who knows those facts.

What increases the effect?

Four factors, in order of how much they matter.

Time. The largest single contributor, and the only one that cannot be recovered later.

Rate, with the caution that a higher rate generally accompanies higher uncertainty, and a projection comparing rates without comparing risk is comparing one variable.

Contributions, both their size and their consistency.

Not interrupting it. Withdrawals reset the base, and a base reset in year eight loses the compounding that would have occurred from year eight onward, not merely the amount withdrawn.

What is a sound approach to compounding?

Stated as principles rather than as a recommendation, because a recommendation requires knowing you.

Start, rather than optimise. The years lost to deciding are years the arithmetic cannot recover.

Prefer consistency to timing. Regular contributions compound from the date each arrives, and consistency is a behaviour you control.

Use the sheltered room available first. For most Canadian households, unused registered contribution room is where compounding operates most efficiently, because growth is not reduced by annual taxation.

Address compounding debt before pursuing compounding growth. A debt compounding at a high rate defeats growth compounding at a lower one, and the comparison is rarely close.

Read every projection for its assumptions, particularly whether it is nominal or real, gross or net, and whether it assumes no interruption.

Working the formula, step by step

The formula is easier to trust once it has been applied once by hand, so here it is with round illustrative numbers rather than any actual arrangement.

Take $10,000 at six percent, compounded annually, for ten years.

P is 10000. r is 0.06. n is 1. t is 10.

The bracket is one plus 0.06 divided by 1, which is 1.06. The exponent is n times t, which is 10. So the calculation is 10000 multiplied by 1.06 to the power of ten, giving roughly $17,900. Subtracting the original $10,000 leaves about $7,900 of interest.

Simple interest on the same terms would be $600 a year for ten years, so $6,000. The difference of roughly $1,900 is compounding.

Over thirty years the same terms produce roughly $57,400, of which about $47,400 is interest. Simple interest over thirty years would produce $18,000. That widening gap is the whole point of the concept, and it is why time contributes more than rate for most people.

Now adjust it. At three percent inflation the thirty-year figure is worth roughly $23,600 in today's purchasing power rather than $57,400. Deduct tax on the growth and cost, and it falls again. The arithmetic did not change; the presentation did.

These are round illustrative numbers chosen for clarity. They are not a projection of any product, and no arrangement is being described.

Compounding against you

The section most often left out, and the one with the largest effect on most households.

Consumer debt compounds. A balance carried at a high annual rate, with interest added to the balance each period, grows on exactly the arithmetic described above. The rule of 72 applies: at eighteen percent, an unpaid balance doubles in roughly four years.

Minimum payments are structured around this. A payment covering interest and a small fraction of principal extends the period over which compounding operates, which is why the total repaid can substantially exceed the amount borrowed.

The comparison that matters. Growth compounding at a modest rate does not outrun debt compounding at a high one. Where both exist, addressing the second generally beats pursuing the first, and the arithmetic is rarely close enough to require a calculation.

Interest paid is capital not recovered. It leaves and does not return, which is the connection between this concept and capital recovery.

Why time beats rate, demonstrated

The claim gets repeated without being shown, so here it is.

Someone contributing for ten years and then stopping, leaving the amount to compound, frequently ends with more than someone starting ten years later and contributing for twice as long. The first has more years of compounding on early contributions, and early contributions are the ones with the most time to work.

The practical implication is uncomfortable. The most valuable years are the ones when income is lowest and competing demands are highest, which is precisely when contributions are hardest to make. That is a real tension rather than a motivational point, and any presentation treating it as simply a matter of discipline has ignored the circumstances of the people it is addressed to.

The honest version. Start what you can sustain rather than what a projection suggests. A contribution maintained for thirty years outperforms a larger one abandoned in year four, and the second is what happens when the amount is set by optimism.

A note on the language around this

The word investment appears throughout most writing on compound interest, including the title this page inherited. It is worth separating two things it runs together.

Compounding is arithmetic. It describes how a quantity behaves when earnings are added to it. It is not a product, a strategy or an asset class.

An investment is a decision about where to put capital, with risk, cost and tax attached. Compounding operates inside many of them and inside several things that are not investments at all.

Conflating the two allows any arrangement that compounds to borrow the persuasive force of the arithmetic without its risks being examined. A participating whole life contract accumulates value and is an insurance product, not an investment. The compounding inside it is real; so is its cost structure, weighed in the case against this product, and what it gets right.

Four questions to ask of any compounding projection

Applicable to a projection from anyone, including anything on this website.

Is this nominal or real? A thirty-year figure that ignores inflation overstates purchasing power substantially, and almost every projection is nominal unless it says otherwise.

Is this gross or net? Before or after tax, fees and cost. Those deductions compound too, and a projection quoting a gross rate against a net alternative is comparing two different things.

What does it assume about interruption? Almost all projections assume contributions continue and nothing is withdrawn. Ask what the figure becomes with three years of contributions missed, because that is a normal life rather than a pessimistic case.

What is guaranteed and what is assumed? Where an arrangement has a contractual floor, that floor is the part that does not depend on anyone's assumption. Any figure above it rests on a projection, and the two should never be presented as a single number.

A projection that answers all four is worth reading. One that answers none is a marketing document with a chart in it.

There is a fifth question worth adding where the projection concerns a long horizon, and almost nobody asks it. What happens to this figure if the assumption moves by one percentage point? Over thirty years a single point of difference in an assumed rate changes the outcome by a margin most people find startling, which is a useful reminder that a projection is a statement about an assumption rather than about the future. Where a presenter cannot show you the figure at a lower assumption, they have shown you one scenario and called it a plan.

Compounding you interrupt, and compounding you do not

The arithmetic above assumes the capital is left alone. Households rarely leave capital alone, because life requires financing.

Every withdrawal restarts the curve. A vehicle bought with savings removes the base that was compounding, and the years it would have taken to rebuild are the real cost of the purchase.

Every loan pays somebody else's curve instead. The interest leaves the household permanently.

The approach behind Infinite Financial Sovereignty®, a registered trademark of Jose Salloum, is aimed at that specific problem: capital that continues its own schedule while being used, with the flow repaid so the base rebuilds rather than restarting.

It is not a higher rate, and any description implying so has misdescribed it. It is a different answer to what happens when the money is needed, and it costs more than leaving capital untouched would. The arguments against it are at objections and risks.

What compounding cannot survive

Interruption. Every withdrawal restarts the curve from a lower base, and the years the base would have taken to rebuild are the real cost of the withdrawal.

Cost. A fee charged annually compounds against the base as reliably as growth compounds for it.

And an assumption held too confidently. A projection rests on a rate nobody can promise, and the longer the projection the more of it is assumption.

Read a projection as an assumption with arithmetic attached, rather than as a forecast. That habit survives every product and every market.

What this page will not do

It will not use the arithmetic as an argument for anything.

Compounding is a property of numbers. Whether any particular arrangement delivers it, at what cost, and whether it suits you, are separate questions requiring facts this page does not have.

The other ideas underneath financial decisions, explained the same way and without a product attached, are in money principles.

This page is general information and is not investment, tax or financial advice. All amounts referred to are Canadian dollars.