Real rate of return

Why subtracting inflation is not the answer

A return in currency and a return in buying power are different quantities. If a balance grows while prices grow by the same proportion, the currency figure rises and nothing has been gained. The real rate is the part of a nominal return that survives that comparison.

The correct relation is a ratio, not a difference. Growth and inflation both apply to the same base over the same period, so removing one from the other means dividing the growth factor by the inflation factor — not subtracting one percentage from another.

The subtraction shortcut always overstates the real rate, and the error grows as both rates grow. At low single-digit rates it is small enough to ignore in conversation and not in a long projection, because the overstatement compounds along with everything else. At high rates it is large enough to reverse the conclusion.

The result can legitimately be negative, and that is the most useful thing it says. A nominal return below inflation is a loss of buying power even though the balance rose, which is the exact situation a currency-denominated statement is incapable of showing.

When this calculation is used

• Deciding whether a savings or deposit rate is actually gaining ground or merely keeping pace.
• Converting a nominal projection into today’s money by adjusting the rate once instead of deflating every figure afterwards.
• Comparing returns across periods or countries with different inflation, where nominal rates are not comparable at all.
• Checking how far the familiar subtract-inflation shortcut has moved an answer you already have.

Worked example

Take an ordinary deposit or bond rate together with a typical inflation assumption for the same period.

The output is the real rate: the annual growth in buying power. Compare it with the difference between the two inputs and note that the real rate is the smaller of the two — always, and by a margin that widens as the rates rise.

Now raise both inputs substantially while keeping the gap between them the same. The subtraction shortcut would return an unchanged answer; the real rate falls. That divergence is the whole reason this is a calculation rather than a subtraction.

What each input represents

Assumptions and limits

• Both rates cover the same period and are quoted on the same basis.
• The inflation rate is a general measure and may not match the prices you personally face.
• The result is a rate, not an amount, and says nothing about tax or fees — both of which apply to the nominal return first.
• A single constant rate for each is assumed; no path or variability is modelled.

What the guards protect against

• The inflation rate cannot reach the point where the relation has no meaningful value; such an input is refused rather than answered with something that would look like a rate.
• Both inputs are bounded to realistic ranges, so a figure entered in the wrong units is refused instead of quietly producing a plausible-looking answer.

Provenance

Fisher relation between nominal rate, inflation and real rate

The ratio of the nominal growth factor to the inflation factor, expressed as a rate — the exact form rather than the additive approximation.

Educational reference. The signed pack carries its own citation; the page reports the verification state of the release it mounted rather than asserting one.