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Percentage change

Measure a change relative to its old value, understand why a 25% gain reverses with a 20% loss, and distinguish price changes from investment returns.

Percentage change expresses a rise or fall relative to the old value. It answers “How large is the change compared with where we started?” A USD 50 gain means something different on a USD 200 investment than on a USD 250 investment.

For old value V0V_0 and new value V1V_1, define the fractional change rr by

r=V1−V0V0,V0≠0.r=\frac{V_1-V_0}{V_0},\qquad V_0\ne0.
The percentage change is 100r%100r\%: a fractional change of 0.250.25 means 25%25\%, not 0.25%0.25\%. With a positive baseline, a positive result indicates an increase and a negative result a decrease. Keep both observations on the same basis: the same currency, units, instrument and adjustment convention. This old-value denominator is also the convention used by the U.S. Bureau of Labor Statistics.

A gain and its reversal use different denominators

Suppose a share rises from USD 200 to USD 250. Its change is 50/200=25%50/200=25\%. If it then falls back to USD 200, the change is −50/250=−20%-50/250=-20\%. The cash movements have equal magnitude, but the second starts from a larger price.

A hypothetical share rises from USD 200 to USD 250: plus USD 50 divided by USD 200 is plus 25%. Returning to USD 200: minus USD 50 divided by USD 250 is minus 20%. Multiplying 200 by 1.25 and 0.80 gives USD 200 and a net price change of zero.
A hypothetical share rises from USD 200 to USD 250: plus USD 50 divided by USD 200 is plus 25%. Returning to USD 200: minus USD 50 divided by USD 250 is minus 20%. Multiplying 200 by 1.25 and 0.80 gives USD 200 and a net price change of zero.
Original hypothetical prices, USD per share. Each move uses its own opening price as the denominator; no distributions or costs.

Successive fractional changes combine through compounding:

rcombined=(1+r1)(1+r2)−1.r_{\text{combined}}=(1+r_1)(1+r_2)-1.

Here 1.25×0.80−1=01.25\times0.80-1=0. Adding 25%−20%=5%25\%-20\%=5\% would incorrectly report a gain. More generally, after a change r>−1r>-1 from a positive starting value, the fractional change needed to return is

rreverse=11+r−1=−r1+r.r_{\text{reverse}}=\frac{1}{1+r}-1=-\frac{r}{1+r}.

A complete loss takes the value to zero; there is then no finite percentage gain that restores the original value from that zero base.

Percentage change is not every kind of return

For an unchanged share holding, the percentage change in its price is its price return. Total return includes distributions too. If the USD 200 share finishes at USD 250 and pays a USD 4 cash dividend at the end of the holding period, total return before costs and taxes is
250−200+4200=27%,\frac{250-200+4}{200}=27\%,
while price return remains 25%25\%. This distinction between income and price gains is explained by FINRA; the Equities primer develops the calculation (full lesson requires membership; public preview available).

A change in an account balance also includes deposits and withdrawals. Depositing USD 200 into a USD 200 account doubles its balance without earning an investment profit. Measuring performance requires separating external cash flows and their timing from investment gains. Likewise, a raw price fall caused by a stock split is not an economic loss: compare consistently adjusted prices or the value of the complete holding.

Check the baseline before interpreting the sign

  • Zero: a move from zero has no defined percentage change under this formula. Report the absolute change instead.
  • Negative: a loss shrinking from USD 20 to USD 10 means a reported profit moves from −20-20 to −10-10. The formula gives 10/(−20)=−50%10/(-20)=-50\%, even though the result improved. State that the loss narrowed by USD 10 or by 50% of its previous magnitude; do not present the signed-baseline ratio as an ordinary growth rate.
  • Already a percentage: a rate moving from 5% to 6% rises by one percentage point, or by 20% relative to the old rate. Those are different descriptions of the same change.

Always state the interval. A percentage change over a holding period is not automatically an annual rate.

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