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From Index Level to Inflation Rate

Price indices are published as levels, not rates. The level itself is meaningless — it is anchored to an arbitrary base period set to 100. What matters is the change. For an index $I$ :

π_{t}^{yoy} = \frac{I_{t}}{I_{t - 12}} - 1 π_{t}^{ann} = {(\frac{I_{t}}{I_{t - 1}})}^{12} - 1

The first is the year-over-year rate quoted in headlines. The second annualizes a single month, and the gap between them is where most bad inflation commentary lives. Year-over-year is smooth but backward-looking: it still contains eleven months of history, so it keeps reporting last year’s shock long after prices have stopped moving. The annualized monthly rate is current but noisy enough that a single month tells you almost nothing. The compromise professionals use is the three- or six-month annualized rate:

π_{t}^{3 m} = {(\frac{I_{t}}{I_{t - 3}})}^{4} - 1

When the 3-month annualized rate sits well below the year-over-year rate, disinflation is already underway and the headline has not caught up. That divergence, not the headline, is the signal.

To convert a nominal return $R$ into a real one, do not subtract. Deflate:

1 + r_{real} = \frac{1 + R_{nominal}}{1 + π} ⟹ r_{real} = \frac{R_{nominal} - π}{1 + π}

At low inflation the subtraction shortcut is close enough. At 8% inflation and a 10% nominal return, subtraction says 2% and the truth is 1.85% — and compounded over thirty years that rounding error is a 5% difference in terminal wealth. Use the ratio.

The mirror image is purchasing power. A dollar today is worth

PV = \frac{1}{{(1 + π)}^{n}}

in $n$ years. At 3%, a dollar buys 74 cents in ten years and 55 cents in twenty. The rule of 70 gets you there mentally: prices double in roughly $70 ∕ π$ years, so 3% inflation halves your money in about 23 years. That is well inside a retirement.

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