Compound interest and percentages: the maths of saving

Compound interest is described as the most powerful force in finance for a reason: it is the only mechanism that makes money grow without further effort on your part. Understanding it matters twice over, because it works just as relentlessly in your favour when you invest as against you when you carry debt.

Simple vs. compound interest

With simple interest, interest is always calculated on the initial capital: €10,000 at 5% generates €500 every year. With compound interest, the previous year's interest is added to the capital and generates new interest: the second year it is calculated on €10,500, the third on €11,025, and so on.

Over one year the difference is trivial; over thirty it is enormous. Those €10,000 at 5% simple interest become €25,000 after thirty years. Compounded, they become about €43,200. Same capital, same rate, same time: the only difference is whether the interest itself earns interest.

The rule of 72

A practical rule: divide 72 by the annual interest rate to get approximately the number of years it takes for your money to double. At 6%, your investment doubles every 72/6 = 12 years. At 9%, every 8 years. This illustrates why even small differences in return have an enormous long-term impact.

The time effect: two investors

Investor A starts at 25 and contributes €200/month for 10 years (until 35), then stops. Investor B starts at 35 and contributes €200/month for 30 years. At 65, assuming 7% annual return, Investor A has more money despite having contributed three times less. Time in the market beats the amount contributed.

Compounding frequency

Monthly compounding produces more than annual compounding at the same nominal rate. A 6% rate compounded monthly is equivalent to an APR of 6.168%. For long-term investments in index funds, compounding is continuous and automatic as dividends are reinvested in the fund.

Inflation: the silent enemy

To calculate real return, you must subtract inflation. If your investment returns 7% but inflation is 3%, your real return is 4%. Over the long term, keeping money in a non-interest-bearing current account means losing purchasing power every year.

Compound interest working against you

The same mechanism operates in reverse on debt, and there it is merciless. Revolving credit cards, which charge around 20% APR, compound the unpaid balance month after month. Paying only the minimum instalment can mean a €3,000 balance takes over a decade to clear and costs more in interest than the original purchase.

This is why any sensible financial plan clears expensive debt before starting to invest. Earning 7% on an investment while paying 20% on a card is a guaranteed net loss, however well the investment performs.

Percentages: the basis of every financial calculation

Percentages are the piece of mathematics most used outside a maths classroom: sales, mortgages, pay rises, election results, VAT. They are also the one that generates the most errors, because nearly all of them are variations on the same four operations. Master these four and you have covered practically everything you will meet.

Case 1: What is X% of a number?

Formula: result = number × (X/100). Example: what is 15% of 340? → 340 × 0.15 = 51. Common uses: calculating a tip, a sale discount, an agent's commission or loan interest.

Case 2: What percentage is A of B?

Formula: % = (A/B) × 100. Example: 45 out of 180 → (45/180) × 100 = 25%. Common uses: calculating market share, pass rates, investment return or what a cost represents as a share of total budget.

Case 3: Percentage change between two values

Formula: change = ((new − old) / old) × 100. If a flat's price rose from €200,000 to €230,000: change = ((230,000 − 200,000) / 200,000) × 100 = +15%. If it fell to €190,000: −5%. Essential for interpreting economic data.

Case 4: Applying an increase or discount

Final price after X% discount: P_final = P_base × (1 − X/100). Final price after X% increase: P_final = P_base × (1 + X/100). Beware of chained percentages: a 20% discount followed by another 20% is not the same as a 40% discount (it is 36% in total).

The costliest error: rising and falling by the same percentage does not return you to the start

If a share falls 50% and then rises 50%, you have not recovered your money: you are down 25%. Starting from €100, the fall leaves you at €50, and a 50% rise on €50 is only €25, so you end at €75. The reason is that each percentage applies to a different base: the first to 100, the second to 50.

This asymmetry is why recovering from a loss is harder than suffering it. Getting back to €100 from €50 requires a 100% rise, not 50%. It is the same reason a 20% discount on an already 20%-reduced price adds up to 36% and not 40%.

Percentage points versus percentages

If an interest rate goes from 2% to 3%, has it risen by 1% or by 50%? Both, depending on how you measure — and from that comes a confusion that appears constantly in financial journalism. It has risen one percentage point in absolute terms and 50% in relative terms. When someone says a tax «goes up by 2%», it is worth asking whether they mean points or proportion, because the effect on your pocket is radically different.

• •From 2% to 3%: +1 percentage point, but +50% in relative terms
• •From 20% to 22%: +2 percentage points, +10% relative
• •Unemployment falling from 12% to 9%: −3 points, −25% relative