Inflation and the Real Rate of Interest

Your account balance can grow while what it actually buys shrinks. The real rate of interest \(i'\) strips inflation out of a nominal (current-dollar) return so you can see how much more you can actually buy: \(1+i' = (1+i)/(1+r)\).

By the end you'll be able to compute the exact real rate \(i'\) from a nominal rate i and an inflation rate r, and explain why the quick shortcut \(i'\approx i-r\) is only an approximation.

Predict: does the quick "i − r" rule over- or understate the true real rate? Raise r and check the gap.

Drag i and r. The exact real rate \(i'=(1+i)/(1+r)-1\) and the quick approximation \(i-r\) agree at \(r=0\), but as inflation rises, the shaded gap between them widens — the approximation drifts away from the truth.

exact i′ = (1.080/1.030) − 1 = 4.854% · approx i−r = 5.000% · gap = 0.146 pts

1 invested today buys 1.04854 in real terms after one year.

Real rate \(i'\) vs. inflation r — exact curve vs. the i−r approximation
exact i′ = (1+i)/(1+r) − 1 approx i′ ≈ i − r gap
Formal

One dollar invested grows to \((1+i)\) dollars in a year, but each of those dollars then buys only \(1/(1+r)\) as much, since prices rose by factor \((1+r)\). Multiplying the two factors gives the real accumulation factor: \(1+i' = (1+i)/(1+r)\). Solving for the rate itself, \[ i' = \frac{1+i}{1+r} - 1 = \frac{i-r}{1+r}. \] This is the Fisher relationship. A useful rearrangement, \(1+i=(1+i')(1+r)\), shows the nominal rate as real growth compounded together with inflation — the same "compose two growth factors" idea you've seen for accumulation and discount factors elsewhere in this section.

Applied

Expanding the exact formula gives \(i = i' + r + i'r\); when \(i'\) and r are both small, the cross term \(i'r\) is tiny and drops out, leaving the familiar shortcut \(i' \approx i - r\). That shortcut is fine for a back-of-envelope check, but the exam standard — and the honest number when rates aren't tiny — is the exact ratio, not the subtraction.

Worked example

An investment earns \(i=8\%\) over a year in which prices rise \(r=3\%\). Exact real rate: \(i' = 1.08/1.03 - 1 = 1.048544 - 1 = \) 4.854%. The rough estimate \(i-r = 5\%\) is close but overstates the real return — purchasing power grew by about 4.85%, not the full 5%.

Your turn

Same relationship, different rates: \(i=5\%\), \(r=1\%\). \(1+i = 1.05\), \(1+r = 1.01\). Dividing, \((1+i)/(1+r) = 1.05/1.01 = 1.039604\). Now finish it: \(i' = 1.039604 - 1 = \) ____

Reveal the answer

\(i' = 1.039604 - 1 = \) 3.96%. Compare it to the rough shortcut \(i-r = 4\%\) — the exact rate is a little lower, exactly the pattern the diagram above shows for any \(r>0\). Set i = 5% and r = 1% on the sliders to see this same gap.

More info — pitfalls to watch for

Two mix-ups come up often. First, treating \(i-r\) as exact instead of an approximation — always divide by \((1+r)\) on the exam. Second, confusing this "nominal" (current-dollar) rate with a nominal rate convertible m-thly from elsewhere in this section — they're unrelated uses of the same word, one about inflation, the other about compounding frequency. Finally, don't assume the formula breaks down under deflation (\(r<0\)): it still holds, and the real rate then exceeds the nominal rate, since dividing by \((1+r)<1\) increases the ratio instead of shrinking it. See the Wikipedia real interest rate link in Dive deeper for more on this.

Check your understanding

Question 1 of 4

An investment earns a nominal (current-dollar) rate \(i = 10\%\) during a year when prices inflate at \(r = 4\%\). What is the exact real rate \(i'\)?

Question 2 of 4

For positive inflation (\(r > 0\)), how does the quick shortcut \(i' \approx i - r\) compare to the true real rate?

Question 3 of 4

Prices fall over the year — deflation, so \(r < 0\). How does the real rate \(i'\) compare to the nominal rate i in that case?

Question 4 of 4

You deposit 1000 today at nominal \(i = 7\%\) for one year while inflation runs \(r = 2\%\). Writing the equation of value in real (purchasing-power) terms, what is the real value of the ending balance?

Recap

  • The real rate satisfies \(1+i' = (1+i)/(1+r)\), so \(i' = (1+i)/(1+r) - 1 = (i-r)/(1+r)\).
  • The shortcut \(i' \approx i-r\) drops the tiny cross term \(i'r\) — it's an approximation, not the exact formula, and it overstates the real rate whenever \(r>0\).
  • A rearrangement \(1+i = (1+i')(1+r)\) shows the nominal rate as real growth compounded with inflation.
  • The formula holds for deflation (\(r<0\)) too — then the real rate exceeds the nominal rate.

Dive deeper

Sources

  • Inflation and the Real Rate of Interest