The useful math is small: compound growth, present value, and adjustments for fees or inflation. Complexity usually arrives from products, not from the core identities.
Worked examples below use round numbers for teaching. They are not forecasts, quotes or personalised projections.
Formulas
- Future value (lump sum) — FV = PV × (1 + r)^n — r is the periodic rate, n the number of periods.
- Future value (annuity-ish savings) — With regular end-of-period contribution C: FV ≈ C × [((1+r)^n − 1) / r] when r ≠ 0.
- Real rate (approx.) — r_real ≈ r_nominal − i, with i the inflation rate; prefer the Fisher form for precision.
- Rule of 72 — Years to double ≈ 72 / (100 × r) when r is a decimal rate expressed as percent — e.g. 8 → ~9 years.
Worked examples
- Lump sum path — €10,000 at 6% for 20 years → about €32,100 before fees/tax. Halve the rate to 3% and the same horizon yields roughly €18,100.
- Contribution path — €300 monthly at 5% for 25 years is on the order of €175k contributed-plus-growth (illustrative; timing conventions vary).
- Fee wedge — A 1% annual fee on a 7% gross path leaves ~6% net — over 30 years the terminal gap versus a 0.1% fee is large, not “one percent”.
- Inflation haircut — 8% nominal with 3% inflation is roughly 5% real — the statement can look rich while purchasing power compounds slower.
Rules of thumb
- 72 / rate — Quick doubling-time check for steady compound rates in a mid single-digit to low double-digit band.
- Savings rate first — Before optimising 0.2% of expected return, ask whether the contribution rate can move by 2–5 points.
- Cost budget — Keep all-in investing costs boringly low when the edge is time, not stock-picking.
- Buffer years — Near spending phases, hold a cash/bond buffer sized in years of withdrawals, not vibes.
Variables
- PV / principal — Starting balance that growth acts on.
- r (rate) — Periodic growth rate — always ask gross vs net, nominal vs real.
- n (periods) — How many compounding intervals remain.
- C (contribution) — Recurring additions; often the most controllable input.
- Costs — Fees, spreads, loads and taxes that reduce r.
If a product cannot be translated into PV, r, n, C and costs, you do not yet understand what will compound.