Inflation and Real Returns





Kerry Back

CPI and inflation

  • Standard U.S. price index is the Consumer Price Index for all Urban Consumers.
  • Basket of goods with prices sampled monthly in 87 urban areas
  • % change in the index is the inflation rate
  • Inflation in 2021 was 7%, so a typical item that cost $100 on Jan 1 cost $107 on Dec 31.
  • We say that the $107 on Dec 31 is $100 in Jan 1 dollars.
  • Any price \(x\) on Dec 31 is \(x/1.07\) in Jan 1 dollars.

Multiple years

  • If there is 7% inflation in 2021 followed by 5% inflation in 2022, then
    • the cost of a $100 item grows to $107 and then another 5%, which is \(100 \times 1.07 \times 1.05\).
  • Any price \(x\) at the end of 2022 would be

\[\frac{x}{1.07 \times 1.05}\]

in beginning of 2021 dollars.

Constant dollars

  • For any dates 0, 1, …, T and inflation rates \(i_1, \ldots, i_T\), we say that a price \(x\) at date \(T\) is

\[\frac{x}{(1+i_1) \cdots (1+i_T)}\] in date 0 dollars.

  • We might refer to “date-0 dollars” as “constant dollars.”
  • The denominator in the above expression is \(\text{CPI}_T/\text{CPI}_0\).

Real rate of return

  • Suppose you make 10% on your portfolio and there is 7% inflation. Consider an item that cost $100 at the beginning of the year.
  • Each $100 of your portfolio at the beginning of the year would buy 1 of them.
    • The $100 grows to $110.
    • The cost of the item goes to $107.
    • So you could buy 1 and 3/107 items.
  • Your real rate of return is 3/107.

General formula

The general formula for the real rate of return \(r^*\) is

\[r^* = \frac{r-i}{1+i}\]

Compounding real rates of return

  • The formula \(r^*=(r-i)/(1+i)\) implies

\[(1+r_1^*) \cdots (1+r_n^*) = \frac{(1+r_1)\cdots (1+r_T)}{(1+i_1) \cdots (1+i_T)}\]

  • The denominator is \(\text{CPI}_T / \text{CPI}_0\), so compounding real rates of return “takes inflation out” of compounded nominal rates.
  • It is the accumulation from $1 in date–0 dollars.