Fisher Real vs Nominal Rate

Fisher Equation: Real vs Nominal Interest Rates

Irving Fisher laid this out in The Theory of Interest (1930). His equation ties the nominal rate, the real rate and expected inflation together: (1 + i_nominal) = (1 + i_real) · (1 + π). Rearrange it for the real rate and you get i_real = (1 + i_nominal) / (1 + π) − 1. The nominal rate is the number your bank advertises. The real rate is how much your purchasing power genuinely grows.

Plenty of people fall back on the linear shortcut i_real ≈ i_nominal − π, and it holds up fine while both rates stay in the single digits. Push inflation higher, though (Brazil in the 1980s, Argentina today, Turkey in 2022), and that shortcut understates the gap, so you have to go back to the exact compound form. Take a 12% nominal rate against 8% inflation: the real rate is 3.70%, not 4%.

Applications

Retirement planning is where it shows up most. A portfolio paying 10% nominally with inflation at 6% is really only compounding wealth at about 3.77%. In Brazil it's the tool you need to weigh Tesouro IPCA+ (a real coupon on top of IPCA) against Tesouro Prefixado (a flat nominal rate). Central banks reach for it when they set policy rates aimed at a target real rate, and corporate treasurers use it to discount long-dated cash flows.

FAQ

When does the approximation break down? Once either rate passes roughly 10%, the linearisation error climbs above 0.5 percentage point. Under hyperinflation, stick with the exact formula every time.

Should I use expected or realised inflation? When you're deciding ahead of time which investment to pick, go with expected inflation (the Focus survey, breakevens). When you're looking back to judge what happened, use realised IPCA/CPI.

Can the real rate be negative? Yes, any time inflation runs above the nominal rate. Brazilian savings (poupança) handed out negative real returns through 2021–2022, and US Treasuries did the same in the years after the 2008 crisis.