Nominal and Real: the Fisher Relation

The exact relation, and why the approximation is not one

Fisher’s relation says the nominal rate is approximately the real rate plus expected inflation, and the word approximately is doing real work. The exact version is (1 + nominal) ÷ (1 + inflation) − 1, because the two rates compound rather than add. At 5% and 4% the two forms give 0.96% and 1.00%, a difference of four basis points; at 15% and 12% they give 2.68% and 3.00%, and at 100% and 90% the subtraction is arithmetically meaningless because it ignores the compounding of the price level entirely. The practical point is not pedantry but direction: the approximation always overstates the real return when the rates are positive, so a portfolio plan built on subtraction is systematically a little too optimistic, and a high-inflation regime makes the error large. The same logic applies to any real quantity: real wages, real revenue growth, real GDP, real policy rates. A real policy rate is the policy rate less expected inflation, which is why a committee holding a nominal rate constant while inflation falls is tightening without moving — the exact version of the same observation. There is a second-order effect worth knowing because it changes the shape of the yield curve. If investors care about their real return rather than their nominal one, then a rise in expected inflation has to be matched by a rise in nominal yields to keep the real rate constant — which is why the whole curve shifts when inflation expectations move, and why the long end is more sensitive to them than the short end, whose level is set by policy rather than by expectations. The same three instruments, in purchasing power — 10-year Treasury, 4.75% nominal: 1.0475 ÷ 1.031 − 1 = 1.60% real ← · Subtracting the rates instead: 1.65% — overstates, and the error grows with the rates · Money-market fund, 4.10% nominal: 0.97% real before tax · Same fund after 24% tax: 3.12% nominal, 0.02% real The inflation figure in all of this is an expectation, not an outcome, and it is an average across a basket nobody buys. Your personal inflation rate depends on rent or mortgage, health care, education and energy — for a household in a coastal city with children, the realised rate can be two or three points above the index, which is a real return of minus two percent in the same arithmetic.

Taxes, money illusion, and what “safe” means

Tax is applied to the nominal gain, which is the mechanism that makes inflation expensive for a saver and a benefit to the government that taxes the gain. A 24% tax on the 4.10% money-market yield costs 98 basis points of nominal return, and against 3.10% inflation the whole real return has gone. In a higher-inflation regime the effect is worse, because the taxable gain includes the compensation for inflation: at 8% nominal with 6% inflation and a 30% tax, the after-tax nominal return is 5.6% against 6% inflation, a real loss of 0.4% from an instrument that appeared to yield 8%. That is the arithmetic behind tax-advantaged accounts and behind the preference for deferral, and it is also why governments with large debts have a structural incentive to tolerate inflation: it reduces the real value of the debt and increases nominal tax receipts at the same time. Money illusion is the behavioural counterpart. Investors compare nominal returns to nominal yields and feel safe holding an instrument whose price does not move, while the purchasing power of that instrument declines at the inflation rate. The aggregate consequence is a wealth transfer from savers to borrowers that happens without a single transaction being mispriced in nominal terms — the mechanism that makes inflation politically potent and analytically subtle. So the useful definition of a safe asset is one whose real return is predictable rather than one whose nominal price is stable. A short Treasury bill has a known nominal payoff and an unknown real one; a long Treasury inflation-protected bond has a known real payoff and an unknown nominal one; equities have neither, and their claim to protect against inflation rests on pricing power rather than on the discount rate. Which is worth owning depends on the liabilities being funded rather than on the asset alone — a pension paying a nominal annuity and a household funding real consumption are not the same investor. • Tax is levied on the nominal gain, so inflation raises the tax on a real return of nothing. • Money illusion makes a stable nominal price feel safe while purchasing power declines. • Inflation is a wealth transfer from savers to borrowers and to the government. • Safety means a predictable real return, which is a different asset for a different liability. Break-even inflation is the market’s own estimate of expected inflation: the nominal yield of a bond less the real yield of the equivalent inflation-protected bond. It is available in real time and it is the cheapest way to see what the market is pricing, rather than what a survey says.

The relation holds in the long run, which is not the run you invest in

The Fisher relation is usually taught as an accounting identity, and it is — for the market’s *expectations* of inflation. The empirical version, in which the nominal interest rate moves one-for-one with realised inflation, holds over long periods and fails badly over the horizons investors actually experience. Nominal rates adjust to inflation slowly, because the adjustment runs through expectations, through policy, and through contracts that were signed before the change. That slowness is the reason real rates can go deeply negative for a while: when inflation ran near 9% with a policy rate below it, the realised real return on cash was substantially negative, and savers holding nominal deposits were funding the adjustment. That has a direct and expensive consequence for the supposedly safe asset. A nominal government bond is a promise of fixed dollars, and its real value depends on a variable the bond does not control. In a period where inflation and policy rates rise together, the price of the bond falls at the same time as its real coupon is being eroded — the losses in long-dated government bonds during the 2022 repricing were among the largest on record for an asset class routinely described as risk-free, and the risk was never default. The relevant frame is not “safe versus risky” but “which variable does this asset have no protection against”, and for nominal bonds that variable is inflation. The useful way to hold all of this is as a **hurdle that every asset has to clear**. A real return is what changes purchasing power, and the same market return is a different outcome depending on the inflation path and on the tax treatment. That is why the comparison this lesson builds — nominal, after-tax, and real — matters more than the headline yield on any instrument: an asset that appears to return 5% while inflation runs at 4% and the gain is taxed annually can be close to a promise to stand still, and the same figure in a tax-deferred account can be a different number entirely. The real hurdle is the only one an investor can spend. Index-linked government bonds exist precisely because the Fisher relation is a long-run average rather than a short-run guarantee: their quoted yield is a real yield, and it is the cleanest observable price of the real hurdle at that maturity. Where a market offers them, compare your expected real return against them rather than against a nominal yield.

The index is a construction, and the construction has a lag

Every statement in this lesson depends on a number that is itself a manufactured statistic, and the manufacturing shows up in ways a learner can use. A price index is a fixed basket of goods with weights, sampled monthly, adjusted for quality, and then aggregated. Four of those steps matter for how the number behaves. **The weights**: shelter is the largest single component of the consumer price index at roughly a third of the basket, and it enters through owners’ equivalent rent — an estimate of what a homeowner would pay to rent their own house, derived from surveyed rents rather than from any transaction. Because leases are signed for a year, the survey captures a rolling average of rents agreed months earlier, so shelter inflation responds to a change in market rents with a lag measured in quarters and keeps contributing to the index long after rents have stopped rising. That single fact explains most of the episodes in which the headline and the market’s sense of the economy disagreed. **Substitution and quality.** A fixed-weight index assumes households keep buying the same basket while prices change, which overstates inflation when people switch away from what got expensive; chain-weighted and Fisher-ideal constructions address that, which is one reason the index the Federal Reserve targets is the personal consumption expenditures measure rather than the consumer price index. Quality adjustment is the other half: a phone that is twice as fast at the same price is deflation in the index’s terms, so the statistician’s hedonic adjustments decide how much of a price change is real. These are not tricks — a price index that ignored them would be measuring the wrong thing entirely — but they mean the number is a model of the cost of living rather than a direct reading of it, and two agencies building it slightly differently will publish different inflation rates from the same economy. **The wedge.** That is exactly what happens in the United States, where the CPI and the PCE index — the measure the central bank targets — diverge by a few tenths of a percentage point in a typical year and by substantially more when the divergence comes from shelter, medical care and the treatment of housing. The PCE index covers a broader set of expenditures, gives shelter about half the weight, and updates its weights as consumption patterns shift, so it is usually the lower and less volatile of the two. When they disagree, the disagreement is informative: a gap that widens because of shelter is telling you the CPI is carrying a lagged rental signal the PCE is not, and a trader watching only one of them is watching a construction rather than the economy. The practical version for this lesson is a rule about vigilance: when you convert a nominal rate to a real one, name which index you converted with, remember that the target is stated against one of them, and treat a single month’s print as a sample from a series that will be revised — because the first estimate of a monthly figure carries a revision history, and the revisions are not random. • Shelter is about a third of CPI and enters through an estimate with a lease-length lag that runs for quarters. • Substitution and quality adjustment are corrections to a fixed basket, not embellishments. • Chain-weighted and Fisher-ideal constructions respond to what households actually buy. • The CPI–PCE wedge is usually a few tenths, and it widens when shelter and medical care diverge. • Name the index when you compute a real rate, and treat a first print as a revision in progress. The deeper point connects this lesson to the whole subject: the Fisher relation is exact, and every input you feed it is an estimate. The real rate you compute is therefore a real rate under a stated index, which is why a dispute about whether the policy rate is restrictive can survive long after both sides agree on the nominal number.

Sources

• The relation between nominal and real interest ratesFisher (1930), “The Theory of Interest”
• Inflation expectations and real returnsFederal Reserve education materials; standard real-return methodology
• Taxes, inflation and the “money illusion” in saving behaviourStandard personal-finance literature; Modigliani & Cohn (1979) on inflation and valuation