How Wrong Is Nominal Minus Inflation? Exactly One Year's Inflation Wrong

One line of algebra replaces the whole argument

Start from what the real return is supposed to mean: how much more you can buy, not how many more units of currency you hold. If a basket costs 100 today and 103 next year, and your money grows from 100 to 108, then next year you can buy 108 ÷ 103 = 1.048544 baskets — a gain of 4.8544 percent in things. That division is the definition, not a refinement of one. Rearranging it gives (n − i) ÷ (1 + i), which shows exactly where the subtraction comes from and exactly what it is missing: the subtraction is the numerator on its own, with the denominator forgotten. Since the denominator is (1 + i), forgetting it multiplies the answer by (1 + i), and multiplying by (1 + i) is the same operation as failing to convert last year's money into this year's.

The markets that have to get this right do not subtract. Inflation-indexed government bonds work by scaling the principal by an index ratio built from the consumer price index — a multiplication, applied daily, precisely so that the security's real coupon is a real coupon and not a real coupon minus a rounding convention. Statistical agencies deflate nominal series by dividing them by a price index for the same reason. Anywhere the answer has to survive contact with a settlement system, the division is the operation, and the subtraction survives only where nobody has to pay out on the difference.

Why the product of the two rates is the wrong intuition

The folk rule says the shortcut is fine when rates are low and breaks down when they are high, and it usually explains this with the cross term n × i. The cross term is genuinely in the expansion — the absolute error is (n − i) × i ÷ (1 + i), which for small i is close to n × i − i squared — but treating the product as the driver produces a false prediction that is easy to construct. Hold n × i fixed at 0.0060 and walk along it. At n = 60 percent with i = 1 percent the error is 0.5842 points. At n = 20 percent with i = 3 percent it is 0.4951. At n = 6 percent with i = 10 percent it is −0.3636, and at n = 2 percent with i = 30 percent it is −6.4615. Same product, errors spanning seven points and both signs. The product is not the driver; the gap between n and i is, scaled by i.

The rule that does hold is the relative one, and it holds without exception because it is an identity rather than an approximation: the shortcut is always high by exactly i of the true answer. That gives you a mental correction you can apply without a calculator. Compute n − i, then take that much inflation off it. At 8 percent nominal and 3 percent inflation, 5 percent minus 3 percent of 5 percent is 4.85 — the exact answer to three figures. Deflation flips it: with a nominal 2 percent and prices falling 1 percent, the exact real return is 3.0303 percent and the shortcut says 3.0000, so the shortcut is now low by 1 percent of the answer, because i is negative. The identity holds in both directions and does not care about the sign.

Fifteen hundredths of a point becomes $1,762.61 over thirty years

A gap of 0.1456 percentage points sounds like a rounding argument until it is compounded. Put $10,000 to work at 8 percent nominal for thirty years with inflation at 3 percent. The nominal pot is $100,626.57. Deflate it properly — divide by 1.03 to the thirtieth, or equivalently compound at the exact real rate — and it is worth $41,456.81 in today's money. Compound the shortcut's 5 percent instead and you get $43,219.42. The shortcut overstates the outcome by $1,762.61, or 4.25 percent of the real pot. The reason the relative error grows from 3.00 percent in a single year to 4.25 percent over thirty is that the two growth factors differ by a constant ratio, and a constant ratio compounds: 1.05 × 1.03 ÷ 1.08 is 1.0013889, and 1.0013889 raised to the thirtieth is 1.0425.

The same compounding logic wrecks a second shortcut that looks harmless: averaging inflation over several years and applying it once. Take three years of 8 percent nominal returns against inflation of 2, 7 and 3 percent. Chain the exact real returns year by year and cumulative real growth is 12.0599 percent, a geometric mean of 3.8684 percent a year. Apply the arithmetic mean inflation of 4 percent instead and you get 3.8462 percent. The gap is only 0.0222 points here, and it is always in the same direction — averaging the inflation understates the real return — because the deflator is convex in the rate. It is small when inflation is stable and grows with the variance, which is precisely when people reach for the average.

Tax lands on the nominal return, and that dwarfs the Fisher correction

Almost every tax system taxes the number of currency units you gained, not the purchasing power you gained. That has a consequence with a clean closed form. If tax takes a fraction t of the nominal return, the real return you keep drops by t × n ÷ (1 + i), and expressed as a share of the real return you would have had, the effective tax rate on your real return is t × n ÷ (n − i). At 8 percent nominal, 3 percent inflation and a 30 percent tax rate, that is 0.30 × 8 ÷ 5 = 48 percent: a headline rate of 30 percent takes 48 percent of what you actually gained. Run the ladder and the arithmetic is unforgiving — 15 percent nominal tax is 24.0 percent of the real return, 26.375 percent is 42.2 percent, 30 percent is 48.0 percent, 40 percent is 64.0 percent.

The same formula gives the inflation rate at which your after-tax real return reaches zero: solve for i and you get i = n × (1 − t). At 8 percent nominal with a 30 percent tax rate, inflation of 5.6 percent leaves you exactly where you started in purchasing power despite a positive, taxable, reportable gain every year. That is the honest ranking of the three effects on this page. The Fisher correction is worth 0.15 points on our example. The tax is worth 2.33 points. And the tax and inflation interact rather than adding, because the tax is charged on a return that includes the compensation for inflation. Get the tax wrapper right first, then get the formula right, and never let the second argument crowd out the first.

Frequently asked questions

Is the subtraction ever the right thing to use?

Yes, in two honest cases. The first is mental arithmetic where you know the direction of the bias and it does not change the decision: at 2 percent inflation the shortcut is high by 2 percent of the answer, which on a real return of 3 percent is six hundredths of a point, and no decision you make about a savings account turns on that. The second is when you are working in continuously compounded returns, where the relation really is a subtraction: if you define the log return as the natural logarithm of one plus the simple return, then the real log return is exactly the nominal log return minus the log inflation rate, with no correction term at all, because logarithms turn the division into a difference. That is why continuously compounded figures are the convention in a lot of fixed-income and academic work. The trap is mixing conventions — subtracting a log inflation rate from a simple nominal return produces a number that is neither.

Which inflation rate should I put in — the published index or my own?

It depends on what the answer is for, and the two uses genuinely need different inputs. If you are comparing your result against a benchmark, a fund, a bond yield or another investor, use the published index for your currency area, because that is what everyone else is deflating by and a comparison is only meaningful in a common unit. If you are asking whether you can afford something in the future, the published index is a weighted average of a national basket, and your basket is not that basket: someone whose spending is dominated by rent in one city, or by childcare, or by energy, faces a personal inflation rate that can sit a long way from the headline for years at a time. Run both. The published figure tells you how you did; your own tells you what it will buy. And note the asymmetry the identity above implies — since the error scales with i, a personal rate a few points above the national one does not merely shift the answer, it also makes the shortcut worse by the same proportion.

What if inflation is negative?

The formula does not change and the bias reverses. With a nominal 2 percent and prices falling 1 percent, the exact real return is 1.02 ÷ 0.99 − 1 = 3.0303 percent, while the shortcut says 3.0000 — the shortcut is now too low, by exactly 1 percent of the answer, which is i again with its own sign. Deflation flatters savers in exactly the way inflation punishes them, and the effect on a real return is larger than people expect: prices falling 5 percent turns a nominal 2 percent into a real 7.3684 percent, where the shortcut would say 7. The one place the arithmetic genuinely breaks is inflation of exactly minus 100 percent, where the denominator is zero and the real return is undefined — a mathematical curiosity rather than an economic scenario, but the reason a well-built calculator refuses that input rather than returning an infinity.

How does this differ from just deflating the final amount at the end?

It does not differ at all, and that is the best check on the whole exercise. Compounding at the exact real rate and deflating the nominal result by the price index are the same operation written two ways, because dividing once at the end by (1 + i) raised to the number of years is algebraically identical to dividing by (1 + i) inside every year. Our figures show it: $10,000 at 8 percent for thirty years is $100,626.57, divided by 1.03 to the thirtieth gives $41,456.81, and compounding $10,000 at the exact real 4.8544 percent for thirty years gives $41,456.81 as well. If your two routes disagree, one of them has a subtraction in it. The practical reason to prefer the real rate is that it makes the comparison to a target legible: a real 4.8544 percent tells you immediately that purchasing power roughly quadruples over thirty years, whereas a nominal 8 percent and a separate inflation series tell you nothing until you have done the division anyway.