Compound Interest

What compound interest is

Compound interest describes the phenomenon that interest already earned itself earns interest once it is added to the principal. Over long horizons this turns seemingly modest annual returns into exponential growth that far outstrips the intuitive linear projection. Albert Einstein is said to have called compound interest the "eighth wonder of the world" — though the attribution is unverified.

Mathematically, the ending capital K_n after n periods at a constant periodic rate i is:

K_n = K_0 × (1 + i)^n

By contrast, simple interest (no reinvestment) grows the capital only linearly: K_n = K_0 × (1 + n × i).

The rule of 72

A useful rule of thumb for mental estimates: the number of years it takes for capital to double at a given rate is approximately 72 divided by the rate in percent.

• At 3% per year: doubling in roughly 24 years
• At 5% per year: doubling in roughly 14 years
• At 7% per year: doubling in roughly 10 years
• At 10% per year: doubling in roughly 7 years

The rule follows from the Taylor-series approximation of the natural logarithm and is accurate to within a few months in the 4-12% range.

A worked example: ETF savings plan

An investor pays €200 per month into an ETF savings plan with an assumed average return of 6% per year net of costs.

| Term | Contributed | Final value | Of which compound | |------|-------------|-------------|-------------------| | 10 years | €24,000 | €32,776 | €8,776 | | 20 years | €48,000 | €92,408 | €44,408 | | 30 years | €72,000 | €200,903 | €128,903 | | 40 years | €96,000 | €398,197 | €302,197 |

After 40 years the compound element exceeds the contributions by more than three to one. The final ten years of the example generate more wealth than the first thirty combined — a direct consequence of the exponential growth function.

The Austrian tax brake

The KESt of 27.5% on capital income acts as a brake on compounding because it reduces the reinvested gross return. The effect is strongest with annual taxation (e.g. deemed distributions from non-Austrian accumulating ETFs); it is weaker for instruments where tax is deferred until sale (classic accumulating Austrian equities).

Cutting the gross return from 6% to 4.35% (= 6% × 72.5%) reduces the 40-year savings-plan ending value in the example by around €100,000 — a concrete reminder that tax-efficient structuring matters enormously over long horizons.