Core Path · L1.3.07

Real vs nominal rates

10 minInterest ratesInflation

Know these first

Math Atoms

By the end you will be able to

  1. 1Use the Fisher relationship to separate inflation from real return.
  2. 2Explain the difference between nominal and real rates in terms of purchasing power.
  3. 3Use the shortcut real ≈ nominal − inflation, and know when it fails.
  4. 4Read breakeven inflation from a nominal bond and an inflation-linked bond.

Why it matters

In October 2022, UK consumer prices were rising at 11.1% a year. A saver earning 3% on a deposit saw the balance go up, yet could buy 7.3% less with it than a year before. The number on the statement and the value of the money had parted company.

Core explanation

Money versus what money buys

Every rate you have seen so far in this module is a nominal rate: it tells you how many more pounds, dollars or yen you will have. But people do not want pounds for their own sake. They want what pounds buy. If prices rise, each pound buys less, and the gain in purchasing power is smaller than the nominal rate suggests.

The rise in prices is inflation, written π. The growth in what your money buys is the real rate. Roughly, real rate = nominal rate − inflation: at 5% interest and 2% inflation you are about 3% better off each year.

That shortcut works when the numbers are small. It goes wrong when inflation is high. The exact real rate comes from comparing growth factors, not subtracting rates:

Nominal rateInflationShortcutExact real rate
5%2%3.00%2.94%
40%35%5.00%3.70%

At 5% and 2% the shortcut is off by 6 basis points (a basis point is 0.01%). At 40% and 35%, rates seen in high-inflation economies, it overstates the real return by more than a full percentage point.

Over many years the gap compounds. £10,000 saved at 4% for 10 years grows to £14,802. But with prices rising 3% a year, that buys only what £11,014 buys today.

Did the saver get richer?

Decide, then see why

  1. Interest 2.0%, inflation 0.5%. Can the saver buy more at the end of the year than at the start?

  2. Interest 4.0%, inflation 6.0%. Can the saver buy more at the end of the year than at the start?

  3. Interest -0.5%, inflation -1.0%. Can the saver buy more at the end of the year than at the start?

The formal version

Three symbols: rnomr_{\text{nom}} for the nominal rate, rrealr_{\text{real}} for the real rate and π\pi for inflation. The idea is to count baskets of goods, not pounds.

The formulas, for reference

1+rnom=(1+rreal)(1+π)1 + r_{\text{nom}} = (1 + r_{\text{real}})(1 + \pi)
rreal=1+rnom1+π−1  ≈  rnom−πr_{\text{real}} = \frac{1 + r_{\text{nom}}}{1 + \pi} - 1 \;\approx\; r_{\text{nom}} - \pi

Over TT years, the real value of a nominal amount is FVT/(1+π)T\mathrm{FV}_T / (1+\pi)^T: discount it by inflation to express it in today's money.

Breakeven inflation. Some governments issue inflation-linked bonds whose payments rise with a price index, so their yield is a real rate. Put a nominal bond and a linked bond of the same maturity into the Fisher relationship and solve for π\pi:

πbreakeven=1+ynominal bond1+ylinked bond−1\pi_{\text{breakeven}} = \frac{1 + y_{\text{nominal bond}}}{1 + y_{\text{linked bond}}} - 1

That is the breakeven inflation rate: the market's price for future inflation (plus some extra for risk and liquidity).

Try it

Start at 4% interest and 3% inflation. Then push inflation above the interest rate and watch the line for what your money buys fall below the dashed line where you started. Try 15% and 12% to see the shortcut error grow.

Try itWhat your savings really buy

Open the inflation calculator →

Real rate (exact)

0.97%

Shortcut r − π1.00%
Shortcut error3 bp
£10,000 saved today balance (nominal) what it buys, in today's money
10.0k12.4k14.8ktoday10y

After 10 years

Balance£14,802
Buys what today costs£11,014

The dashed line is the £10,000 you started with. Below it, you can buy less than you could today.

Worked example

Treasuries against TIPS

A pension fund in Canada is choosing between two 10-year US government bonds: a nominal Treasury yielding 4.20%, and a TIPS (Treasury Inflation-Protected Security) yielding 1.80% real. These yields are illustrative; both are annual rates. Try each step before revealing it.

Which bond, and when?

Step 1 of 4
  1. 1

    Breakeven inflation

    1.042÷1.018−1=2.3576%1.042 \div 1.018 - 1 = 2.3576\% a year. The shortcut says 2.40%.

Regulatory lens

Central banks are given inflation targets, which anchor the π in the Fisher relationship. The Bank of England targets 2% consumer price inflation, set by the government. The European Central Bank aims for 2% over the medium term, and the US Federal Reserve for 2% inflation as measured by the personal consumption expenditures (PCE) price index. When markets believe these targets, long-term nominal rates settle around the real rate plus about 2%.

Looking ahead: insurers and pension funds must allow for inflation in their liabilities, and stress tests include inflation shocks (Level 4).

Sparks

SparkIrving Fisher, 1896
The American economist Irving Fisher set out the link between interest and expected changes in the value of money in Appreciation and Interest (1896), and developed it in The Theory of Interest (1930). The relationship has carried his name ever since.
SparkInflation-linked bonds are older than you think
During the American War of Independence, Massachusetts paid its soldiers with notes, issued in 1780, whose value was tied to the price of a basket of goods, because paper money was losing value so fast. Modern linked bonds came much later: UK index-linked gilts in 1981 and US TIPS in 1997.

Key takeaways

  • Nominal rates count money; real rates count what money buys.
  • Fisher: (1 + nominal) = (1 + real)(1 + inflation), so real = (1 + nominal) ÷ (1 + inflation) − 1.
  • The shortcut real ≈ nominal − inflation is fine at low rates and misleading when inflation is high.
  • To express a future amount in today's money, discount it by inflation.
  • Breakeven inflation, from nominal and inflation-linked bonds, is the market's price for future inflation.

Reaction

Check what stuck

Question 1 · objective 1

A deposit pays 6% a year and inflation is 4%. What is the exact real rate, in %, to two decimals?

%

Question 2 · objective 1

An investor wants a 2% real return and expects 3% inflation. What nominal rate does she need, in %?

%

Question 3 · objective 3

When is the shortcut real rate ≈ nominal rate − inflation least accurate?

Question 4 · objective 4

A 10-year nominal government bond yields 3.5% and a 10-year inflation-linked bond yields 1.0% real. What is the breakeven inflation rate, in %, to two decimals?

%

Question 5 · objective 2

Your savings balance grows from £5,000 to £5,150 in a year in which prices rise 4%. What happened to your purchasing power?

Glossary

Nominal rate
The interest rate in money terms, as quoted by a bank or on a bond.
Real rate
The interest rate in terms of what money buys: the growth in purchasing power after inflation.
Inflation (π)
The rate at which the general level of prices rises, usually measured by a consumer price index. Written π.
Purchasing power
How much a sum of money can buy.
Fisher relationship
(1 + nominal rate) = (1 + real rate)(1 + inflation).
Inflation-linked bond
A bond whose coupons and repayment rise with a price index, so it pays a known real return.
Breakeven inflation
The inflation rate at which a nominal bond and an inflation-linked bond of the same maturity give the same return.
Basis point
One hundredth of a percentage point: 0.01%.

Deep Dives & Origins

Deep Dives

  • Frederic S. Mishkin, The Economics of Money, Banking, and Financial Markets (Pearson), chapter 4, on real and nominal interest rates. Beginner.
  • Irving Fisher, The Theory of Interest (Macmillan, 1930), chapter 2. Intermediate, historical.

Origins

  • Irving Fisher, Appreciation and Interest (American Economic Association, 1896).
  • Office for National Statistics, Consumer price inflation, UK: October 2022.
  • Bank of England, monetary policy remit; ECB, monetary policy strategy statement (2021); Federal Reserve, Statement on Longer-Run Goals and Monetary Policy Strategy.
  • Robert J. Shiller, “The invention of inflation-indexed bonds in early America”, NBER Working Paper 10183 (2003).
  • UK Debt Management Office (index-linked gilts); US Treasury, TreasuryDirect (TIPS).

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Day counts and rate conventions

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The yield curve: spot and forward rates