The Fisher equation
The nominal rate links the real rate and expected inflation, set when the loan is priced: i ≈ r + πᵉ, exactly i = (1+r)(1+πᵉ) − 1 = r + πᵉ + r·πᵉ. The cross-term r·πᵉ is the gap the approximation drops, and it grows with inflation. Later, realized inflation decides the actual real return r = (1+i)/(1+π) − 1.
Nominal rate iapprox 5.00%exact 5.06%
Exact − approx gap (cross-term) +0.06%Inflation surprise π − πᵉ +0.00%
Real rate r2.00%
Expected inflation πᵉ (sets the nominal contract)3.00%
Realized inflation π (decides the real return)3.00%
The nominal rate is locked at 5.06% (exact), versus 5.00% from the approximation. The +0.06% gap is the dropped cross-term r·πᵉ, which widens as inflation rises. With no inflation surprise, the ex-post real return equals the ex-ante real rate 2.00%.
Try this.Hold r = 2% and πᵉ = 3% (nominal locked at 5.06%), then push realized inflation above 3%. The lender's real return drops below the 2% they expected: an inflation surprise transfers purchasing power from lender to borrower. Now raise expected inflation toward 12% and watch the exact-vs-approx gap widen well past the 0.06-point gap seen at 3%.