Fisher Equation

In one sentence: The Fisher equation says the real growth of your money is what is left of your interest after prices have risen.

$(1 + i) = (1 + r)(1 + \pi)$

Read it out loud: “One plus i equals one plus r, times one plus pi. i is the nominal interest rate, the rate the bank quotes. r is the real interest rate, the growth in what your money can actually buy. pi is the inflation rate, how fast prices rise. Write each percent as a decimal, so 5 percent is 0.05.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Percent and decimal form, like 5% = 0.05: A percent is a part out of 100, so 5% means 5 parts out of every 100. To use a percent in a formula, write it as a decimal by dividing by 100: $5\% = 5 \div 100 = 0.05$ . Adding 1 to a decimal rate gives a growth factor. The number $1 + 0.05 = 1.05$ means an amount grows to 1.05 times its old size, which is 5% more.
Growth factor, like 1.05: A growth factor is the number you multiply an amount by to find its value after one period of growth. If 1000 dollars earns 5% in a year, the factor is 1.05, and the amount becomes $1000 \times 1.05 = 1050$ dollars. Growth factors are multiplied together when two growths happen one after the other or at the same time, which is why the Fisher equation uses multiplication and not addition.
Inflation: Inflation is the general rise in prices over time, so the same amount of money buys fewer goods than before. If a loaf of bread costs 2 dollars this year and 2.04 dollars next year, bread has risen by 2%. The inflation rate is how fast the general level of prices rises, measured as a percent per year, and it is the drag that eats into the value of savings.
Nominal rate and real rate: The nominal interest rate is the rate a bank or lender quotes, before any adjustment for prices. The real interest rate is the growth in what your money can buy after inflation has been taken out. A savings account can advertise a high nominal rate and still give a low real rate if prices are rising quickly. The Fisher equation is the rule that connects the two.
Approximately equal, $\approx$: The symbol $\approx$ means approximately equal. It says two numbers are close, but not exactly the same, which is useful when a small rounding difference does not matter. The Fisher equation has an exact form and a quick estimate written with this symbol, so the page explains when each one is good enough to use.

What each symbol means

Symbol	Meaning
$i$	The nominal interest rate, the rate a bank or loan quotes, written as a decimal per year. In the example it is 0.05, which is 5% per year. It has no money unit, because it is a rate, but it always means per year. Think of it as the headline growth of your money before prices are considered.
$r$	The real interest rate, the growth in what your money can buy each year after inflation is taken out, written as a decimal. In the example it works out to about 0.0294, which is 2.94% per year. A negative real rate means your money buys less at the end of the year than it did at the start.
$\pi$	The inflation rate, the percent by which the general level of prices rises over one year, written as a decimal. Here the Greek letter pi is used as the name of a rate, not as the number 3.14159. In the example $\pi = 0.02$ , which is 2% per year. It has no money unit, because it is a share of the old price level.

The formula piece by piece

$i$: Start with the nominal rate. $i$ is the rate the bank or loan quotes, written as a decimal per year, before prices are considered. It is the headline growth of your money.
$\pi$: $\pi$ is the inflation rate, a decimal per year, that says how much prices rise. It is the drag that prices put on what your money can buy.
$1 + i$: The growth factor for the money. Adding 1 to a rate turns 5% growth, which is the decimal 0.05, into the factor 1.05. One unit of money becomes 1.05 units after the year.
$(1 + r)(1 + \pi)$: Two growth factors multiplied together. The first is the real growth in buying power and the second is the growth in prices. Multiplying them gives the total growth of the nominal money.
$(1 + i) = (1 + r)(1 + \pi)$: The whole formula in one sentence: one plus the nominal rate equals one plus the real rate, times one plus the inflation rate, so the total growth of the money equals its real growth times the growth in prices.
$r \approx i - \pi$: The quick estimate: the real rate is about the nominal rate minus the inflation rate. It is exact only when the small product $r\pi$ is zero, so for larger rates it runs a little high.

What the formula is doing, and why

The Fisher equation connects three rates: the nominal rate $i$ that a bank quotes, the inflation rate $\pi$ that says how fast prices rise, and the real rate $r$ that says how much more your money can really buy. The formula is written with brackets as $(1 + i) = (1 + r)(1 + \pi)$ . Each bracket is a growth factor. The left side, $1 + i$ , is what one unit of money becomes after one year of interest, and $1 + \pi$ is how much prices grow over the same year.

Why multiply? Interest and inflation act on your money at the same time, and each one multiplies the amount. When a bank adds 5% to your savings, your money grows by a factor of 1.05. When prices rise by 2%, the cost of goods grows by a factor of 1.02. To find how much more your money can buy, you divide one factor by the other, which is why the real rate is $1 + r = \frac{1 + i}{1 + \pi}$ . Multiplying growth factors is the right way to combine two changes that happen together. Adding the percents only gives a rough answer.

Solving for the real rate gives $r = \frac{1 + i}{1 + \pi} - 1$ . This is the exact rule, and the example on this page uses it. The subtraction $r \approx i - \pi$ is a quick estimate that most people use in their heads. It works because multiplying out the brackets gives $(1 + r)(1 + \pi) = 1 + r + \pi + r\pi$ . Setting that equal to $1 + i$ and cancelling the 1 on both sides leaves $i = r + \pi + r\pi$ . The last piece, $r\pi$ , is a product of two small decimals, so it is small, about 0.0006 for the rates in the example. Leaving it out gives the estimate.

The shortcut gets worse as the rates grow. With 25% interest and 20% inflation, the subtraction gives 5%, but the exact real rate is $\frac{1.25}{1.20} - 1 \approx 0.0417$ , which is about 4.17%. So the exact formula is the one to use for large rates, and the shortcut is fine for the small rates most savers see.

An everyday picture

Think of a school lunch card that adds 5% extra credit each year, while the cafeteria raises its prices by 2%. The card’s balance grows, but the lunches it buys cost more too. Your real gain is how many more lunches you can buy at the end of the year than at the start. That gain is the card’s growth divided by the cafeteria’s price growth, not the card’s growth minus the price growth. Dividing the two growth factors is exactly what makes the Fisher equation multiply rather than subtract.

Worked example: How much more can savings really buy?

A household puts 1000 dollars into a savings account that pays 5% a year. Over the same year, a basket of groceries that costs 100 dollars today rises in price by 2%, to 102 dollars. The 5% is the nominal rate $i$ and the 2% is the inflation rate $\pi$ . These figures are made up for illustration. We want the real rate $r$ : how many more baskets the savings can buy at the end of the year than at the start.

Write the rates as decimals. Percents are easiest to use in decimal form, so we divide each one by 100: $i = 5\% = 0.05$ and $\pi = 2\% = 0.02$ . A growth factor is one plus the rate, so the bank’s growth factor is $1 + i = 1.05$ and the price growth factor is $1 + \pi = 1.02$ .
Rearrange the Fisher equation to find the real rate. We start from $(1 + i) = (1 + r)(1 + \pi)$ and divide both sides by $1 + \pi$ , which gives $1 + r = \frac{1 + i}{1 + \pi}$ . Dividing by the price growth removes the effect of prices, so what is left is the growth in buying power.
Calculate the ratio. Dividing 1.05 by 1.02 gives about 1.0294, so $1 + r \approx 1.0294$ . Subtracting 1 from both sides gives $r \approx 0.0294$ . Written as a percent, that is 2.94% per year, which is the real rate for this household.
Check the answer with real money. The savings grow to 1050 dollars after one year, while the basket grows to 102 dollars. Today the savings buy 1000 divided by 100, which is 10 baskets. At the end of the year they buy 1050 divided by 102, which is about 10.294 baskets, so the buying power has grown a little.
Find the gain in baskets. The number of baskets went from 10 to about 10.294, so the gain is the ratio $\frac{10.294}{10} - 1 \approx 0.0294$ . That is the same 2.94% we found from the formula, so the two methods agree.
Compare with the quick subtraction. The shortcut $i - \pi$ gives $0.05 - 0.02 = 0.03$ , which is 3%. That is about 0.06 percentage points above the exact 2.94%. For rates this small the shortcut is close enough for a quick estimate, but the exact formula is the one to use when the rates are large.

Answer: The real rate is about 2.94% per year. After one year the savings buy about 10.29 baskets instead of 10. The bank advertised 5%, but because prices rose 2%, the real gain in buying power is about 2.94%.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Treating $r = i - \pi$ as the exact answer. The subtraction is a quick estimate, and it runs a little high. The exact real rate comes from dividing the growth factors.
Typing percents as whole numbers inside the formula, so 5 instead of 0.05. Then $1 + 5 = 6$ would mean 600% growth, which is nonsense for a 5% rate.
Mixing a nominal rate with a real rate in the same calculation, such as comparing a bank’s advertised rate with a price index without adjusting for inflation. The two numbers answer different questions.
Forgetting that the rates must cover the same period. A nominal rate per year and an inflation rate per month cannot be put into the formula together.

Where you will meet it

Judging whether a savings account, bond or loan really makes money after prices rise, rather than only looking at its advertised rate.
Economic discussions of monetary policy, where the real interest rate is watched as a sign of how much borrowing is being discouraged.
Inflation-protected government bonds, where the return that matters to the buyer is the real return after the price adjustment.
Retirement and pension planning, where future savings are measured in what they will buy, not in the number of dollars shown on the statement.

Questions people ask

Why is the real rate not exactly the nominal rate minus inflation?

For small rates the subtraction is close, which is why people use it. The exact formula multiplies the two growth factors, and multiplying produces a small extra piece, $r\pi$ , that the subtraction leaves out. For rates of a few percent that piece is tiny, so the shortcut works well.

What happens if inflation is higher than the interest rate?

The real rate becomes negative, so your money buys less at the end of the year than at the start. For example, a 5% rate with 6% inflation gives a real rate of about $-0.0094$ , or minus 0.94%. Savings still grow in dollars, but they lose ground against prices.

Why does the real rate matter more than the nominal one?

Money is only useful for what it buys. A 5% rate in a year with 2% inflation lets your money buy about 3% more, not 5% more. The nominal rate tells you how many dollars you will have, and the real rate tells you how much more you can actually do with them.

Inflation Rate: The inflation rate says how much more prices rose this year than last, as a share of last year’s prices, measured with a price index.
Compound Interest: Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.
Present Value: Present value says how much money you must put aside today, at a steady interest rate, so that it grows to a known amount by a later date.