Bonds, returns and business math Updated November 2026

Real vs nominal return

A nominal return is the stated percentage gain; the real return is what it buys after inflation. The Fisher equation is (1 + nominal) = (1 + real) x (1 + inflation), so real = (1 + nominal) / (1 + inflation) - 1. The shortcut nominal minus inflation is only approximate.

Syntax

real = (1 + nominal) / (1 + inflation) - 1   (approx: real = nominal - inflation)

Part	What it is
`nominal`	Stated return as a decimal
`inflation`	Inflation rate over the same period
`real`	Return in purchasing-power terms

Examples

Nominal 8%, inflation 3%
1.08 / 1.03 - 1

Result: Real = 4.85% (approximation: 5%)

Savings at 5%, inflation 4%
1.05 / 1.04 - 1

Result: Real = 0.96%

Nominal 2%, inflation 5%
1.02 / 1.05 - 1

Result: Real = -2.86% (losing purchasing power)

Common mistakes

Subtracting inflation from the nominal rate for large numbers: at 20% nominal and 10% inflation the real return is 9.09%, not 10%.
Discounting real cash flows at a nominal rate (or nominal flows at a real rate) in an NPV; keep both on the same basis.
Forgetting taxes: tax is charged on the nominal gain, so the after-tax real return is lower still.

Practice Real vs nominal return

These drills check your answer on live data, free, no account needed.

Real return after inflationYour investments earned 4.25% last year while inflation was 2%. What was your exact real rate of return?
Holding period returnYou buy 490 shares at $12.00 each, collect dividends of $2.90 per share, and sell all of them a year later at $9.50. Ignoring commissions, what is your holding period return? (Enter a negative number for a loss.)

Questions

What is the Fisher equation?

(1 + i) = (1 + r)(1 + pi), linking the nominal rate i, the real rate r and expected inflation pi. The approximation i = r + pi is called the Fisher approximation.

Why does the real return matter more than the nominal?

Because you spend money on goods whose prices rise. A 5% return with 4% inflation leaves you only about 1% better off in what you can buy.

Holding period returnHolding period return is (ending value - beginning value + income) / beginning value. Stock and bond examples, how to annualize an HPR and common mistakes.
CAGRCAGR = (ending value / beginning value)^(1/years) - 1. Examples for investments and revenue growth, Excel RRI, and why CAGR differs from the average return.
Rule of 72The rule of 72 estimates doubling time as 72 divided by the rate. Examples at 6%, 9% and 18%, how accurate it is, the exact ln(2) formula and Excel NPER.
NPV (net present value)Net present value is the sum of discounted cash flows minus the initial investment. Examples at 10% and 8%, the accept rule, Excel NPV and the sign mistake.