FOUNDATION 03 / Money & mathematics
Percentages, rates & purchasing power
Read returns correctly and separate money growth from purchasing-power growth.
By the end of this lesson
- Distinguish percent changes and percentage points.
- Compound returns across time.
- Calculate a real return under explicit assumptions.
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Percentages need a denominator
Percentage change = (new−old)/old. From 80 to 100 is a 25% gain, but from 100 back to 80 is a 20% loss. The starting base matters; percent changes are not symmetric.
From a 5% rate to a 7% rate is an increase of 2 percentage points, or a 40% relative increase. One basis point is 0.01 percentage point; 25 basis points equals 0.25 percentage point.
Holding-period and multi-period returns
For one holding period without interim reinvestment issues, return = (ending price−starting price+cash income)/starting price. Buying at 100, receiving 3 and ending at 105 gives 8%. State whether fees and taxes are included.
Across periods, multiply growth factors: total return = ∏(1+Rₜ)−1. A +20% year followed by −20% leaves 1.2×0.8=0.96, a 4% loss. Adding the two rates would give the wrong answer.
Annualize comparable periods
A geometric average reflects compounded growth: [(ending value/starting value)^(1/n)]−1 for n years without external cash flows. An arithmetic mean describes the average of individual period observations; it does not generally reproduce the wealth path.
Do not annualize a short exceptional return as though it were a forecast. If deposits or withdrawals occur, a simple start-to-end ratio is insufficient; time-weighted and money-weighted methods are later topics.
Real return measures purchasing power
Exact real return = (1+nominal return)/(1+inflation)−1. Subtracting inflation is only an approximation. Use rates for the same period and a relevant inflation measure; a household’s actual spending basket may differ.
A positive nominal return can still mean a loss of purchasing power. Currency translation can also change an investor’s result. Do not compare returns in different currencies without explaining the conversion.
Money grows 8%, prices grow 3%
- Start with 10,000; ending money is 10,800.
- Deflate by 1.03: purchasing-power equivalent = 10,485.44.
- Real return = 1.08/1.03−1 = 4.8544%, not exactly 5%.
Try before looking at the explanation
Change one input in the practice tool below. Predict the direction first, calculate second, then explain why it changed.
Work the numbers
Common mistakes to catch
- Adding successive returns rather than multiplying factors.
- Confusing 1 percentage point with a 1% relative change.
- Treating inflation adjustment as a guarantee about personal expenses.
Knowledge check · 6 original questions
Answer all six. At least five correct marks the lesson complete for self-study. You can retry; this is not a credential.
Keep these ideas
- Always state the base, period, currency and treatment of income.
- Use the exact ratio when the distinction between real and nominal matters.