Archive / Investment Adviser (Level 1) / Evaluating the Financial Position of Clients

nism-series-xa-real-rate-of-return-fisher-equation[ INVESTMENT ADVISER (LEVEL 1) ]

An investment portfolio generates a nominal annual return of 9.0% during a year when inflation averages 5.0%. What is the EXACT Real Rate of Return using the Fisher Equation?

[ A ]4.00% p.a. (Approximate linear difference: 9.0% - 5.0%)

[ B ]3.81% p.a. [ (1 + Nominal) / (1 + Inflation) - 1 = (1.09 / 1.05) - 1 ][ CORRECT ]

[ C ]14.00% p.a. (Adding nominal return and inflation)

[ D ]2.00% p.a.

Takeaway

Option B is correct: 3.81% p.a. [ (1 + Nominal) / (1 + Inflation) - 1 = (1.09 / 1.05) - 1 ]

Derivation

Nominal growth factor = 1 + 0.09 = 1.09. Inflation growth factor = 1 + 0.05 = 1.05. The exact Fisher relationship divides one factor by the other rather than subtracting the rates: Real Return = (1.09 / 1.05) − 1 = 1.038095 − 1 = 3.81% p.a. The commonly used shortcut, simply subtracting inflation from the nominal return (9.0% − 5.0% = 4.0%), is a linear approximation that ignores the compounding interaction between the two rates, and it overstates true purchasing-power growth by roughly 19 basis points here. The gap between the approximation and the exact figure widens further as either the nominal return or inflation rate climbs higher.

Why the others fail

A — 4.00% p.a. (Approximate linear difference: 9.0% - 5.0%)

This is the approximate linear subtraction (9.0% - 5.0%), which overstates real purchasing power growth by ignoring the compounding drag between nominal return and inflation.

C — 14.00% p.a. (Adding nominal return and inflation)

Adds the inflation rate to the nominal return instead of discounting the nominal growth factor by the inflation growth factor, producing a figure far too high.

D — 2.00% p.a.

Understates the real return produced by the Fisher equation; dividing the nominal growth factor by the inflation growth factor correctly yields 3.81%, not this lower figure.