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Real versus nominal rates

Ratio, rates and money · level 8, lesson 4 of 6

A question to start

Money invested at 6% a year, compounded yearly, doubles in about 12 years: \(1.06^{12} \approx 2.01\). Suppose that over the same 12 years, prices in the shops rise by 3% a year.

Has the investor doubled what their money can buy? No. Prices have risen by a factor of \(1.03^{12} \approx 1.43\), so the doubled money buys only about \(2.01 \div 1.43 \approx 1.41\) times as much as before. The money doubled; its buying power grew by about 41%. This lesson separates the two ideas, the nominal rate (what the number on the statement does) and the real rate (what the money can do), and shows exactly how they are linked.

The idea

Nominal, inflation and real

  • The nominal rate \(n\) is the stated rate at which an amount of money grows: the interest rate, the return.
  • The inflation rate \(f\) is the rate at which prices in general rise, usually measured by a price index such as Canada's Consumer Price Index (CPI), from level 6.
  • The real rate \(q\) is the rate at which buying power grows: how much more stuff the money can buy.

(The letter \(n\) here is a rate, not a number of periods. Rates in this lesson are yearly, written as decimals.)

In one year, an amount of money is multiplied by \(1 + n\). The price of a typical basket of goods is multiplied by \(1 + f\). The number of baskets the money can buy is money divided by price, so it is multiplied by

\[1 + q = \frac{1 + n}{1 + f}.\]

This is the Fisher equation, named after the economist Irving Fisher. It is exact, and it is just a ratio of two multipliers.

The approximation and its error

Rearrange the exact equation:

\[q = \frac{1 + n}{1 + f} - 1 = \frac{(1 + n) - (1 + f)}{1 + f} = \frac{n - f}{1 + f}.\]

When inflation is small, \(1 + f\) is close to 1, so

\[q \approx n - f.\]

"Real rate ≈ nominal rate minus inflation" is the everyday rule. The exact value is the approximate one divided by \(1 + f\), so the approximation slightly overstates a positive real rate. At low inflation the difference is tiny; at high inflation it is not. With 50% nominal and 40% inflation, the rule says 10%, but the exact real rate is \(\tfrac{1.5}{1.4} - 1 \approx 7.1\%\).

Over many years

Over \(t\) years with steady rates, buying power is multiplied by

\[\left(\frac{1 + n}{1 + f}\right)^{t} = (1 + q)^{t}.\]
start = 1 × 2.01 × 1.43 × 1.41 money prices buying power

To describe a future amount in today's dollars (also called real dollars or constant dollars), divide it by the growth of prices, \((1 + f)^t\). This is the same as the "what 100 dollars from 1990 buys today" calculation in level 6, run forwards.

Negative real rates

If the nominal rate is below inflation, the real rate is negative: the account balance goes up, but what it buys goes down. A savings account at 2% with prices rising 4% a year loses about 1.9% of its buying power each year. The statement shows more dollars each year; the shopping basket shows fewer things.

Tax comes first

Income tax on interest is charged on the nominal interest, the whole of it, even though part of that interest only makes up for inflation. If interest is taxed at rate \(\tau\) (the Greek letter tau), the after-tax nominal rate is \(n(1 - \tau)\), and the real after-tax rate is

\[\frac{1 + n(1 - \tau)}{1 + f} - 1.\]

Tax rules on investment income differ by type of account and type of income, and change over time; check the current ones. The examples below use a made-up flat rate.

Worked examples

1. The basic case. A GIC pays 5% when inflation is 3%. Find the real rate, approximately and exactly.

  • Approximate: \(5\% - 3\% = 2\%\).
  • Exact: \(\dfrac{1.05}{1.03} - 1 \approx 0.01942\), about 1.94%.

2. The doubled money. At 6% for 12 years with 3% inflation:

  • Money: \(1.06^{12} \approx 2.012\). Prices: \(1.03^{12} \approx 1.426\).
  • Buying power: \(2.012 \div 1.426 \approx 1.411\), a 41% increase. (Equivalently, the real rate is \(\tfrac{1.06}{1.03} - 1 \approx 2.91\%\), and \(1.0291^{12} \approx 1.41\).)

3. A negative real rate. Savings earn 2% a year while inflation is 4%.

  • Real rate: \(\dfrac{1.02}{1.04} - 1 \approx -1.92\%\) a year.
  • After 10 years the money buys \(\left(\tfrac{1.02}{1.04}\right)^{10} \approx 0.824\), about 82% of what it bought at the start.

4. After tax. Interest at 5% is taxed at a made-up rate of 30%; inflation is 3%.

  • After-tax nominal: \(5\% \times 0.7 = 3.5\%\).
  • Real after tax: \(\dfrac{1.035}{1.03} - 1 \approx 0.49\%\). Most of the 5% was used up by tax and inflation.

5. The rate you need. To earn 3% real when inflation is 2.5%, what nominal rate is needed?

  • \(1 + n = 1.03 \times 1.025 = 1.05575\), so \(n \approx 5.575\%\), a little more than \(3 + 2.5 = 5.5\).

Try it

  1. Nominal 7%, inflation 2%. Find the real rate approximately and exactly.
  2. Nominal 3%, inflation 3.5%. Find the exact real rate. What does its sign mean?
  3. What nominal rate gives a real rate of 4% when inflation is 3%?
  4. 10 000 dollars grows at 4% a year for 20 years while inflation is 2.5%. Find the final amount in dollars, and in today's dollars.
  5. A salary rises from 50 000 to 56 000 dollars over 4 years, while prices rise 12% in total. What happened to the salary's buying power?
  6. At 3% inflation, how many years until money kept in a drawer buys half as much? Use the rule of 72, then the exact formula.
  7. Show that the exact real rate is \(\dfrac{n - f}{1 + f}\), and explain why the rule \(n - f\) overstates it when \(f > 0\) and \(n > f\).
  8. Nominal 30%, inflation 25% (as can happen in a country with high inflation). Compare the approximate and exact real rates.
Answers
  1. Approximate \(7 - 2 = 5\%\). Exact \(\dfrac{1.07}{1.02} - 1 \approx 4.90\%\).
  2. \(\dfrac{1.03}{1.035} - 1 \approx -0.48\%\). Negative: the money buys a little less each year, even though the balance grows.
  3. \(1.04 \times 1.03 - 1 = 0.0712\), so 7.12%.
  4. Dollars: \(10\,000 \times 1.04^{20} \approx 21\,911.23\). Today's dollars: \(21\,911.23 \div 1.025^{20} \approx 13\,371.79\). The buying power grew by about 34%, not 119%.
  5. In starting-year dollars, \(56\,000 \div 1.12 = 50\,000\). Buying power is unchanged: the raise only kept up with prices.
  6. Rule: \(72 \div 3 = 24\) years. Exact: the drawer money's buying power is multiplied by \(\tfrac{1}{1.03}\) a year, so we need \(1.03^t = 2\), \(t = \dfrac{\ln 2}{\ln 1.03} \approx 23.4\) years.
  7. \(\dfrac{1 + n}{1 + f} - 1 = \dfrac{1 + n - 1 - f}{1 + f} = \dfrac{n - f}{1 + f}\). If \(n - f > 0\) and \(1 + f > 1\), dividing by \(1 + f\) makes it smaller, so \(n - f\) is too big.
  8. Approximate \(30 - 25 = 5\%\); exact \(\dfrac{1.30}{1.25} - 1 = 4\%\). At high inflation the error is a whole point.

Common mistake

Celebrating nominal growth. "My savings doubled" says nothing about buying power until you know what prices did. Always ask what the money buys.

Subtracting when inflation is high. "Nominal minus inflation" is an approximation for small rates. With high inflation, or over many years, use the exact ratio \(\dfrac{1 + n}{1 + f}\).

Where it leads