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Percentage point

Measure the difference between percentage levels, convert it to basis points or decimal units, and distinguish it from a relative percentage change.

A percentage point is the unit for the arithmetic difference between two percentage levels. A rate rising from 3% to 4% increases by 1 percentage point. This answers how far apart the rates are. A percentage change answers how large that increase is relative to the starting rate.
The distinction matters when reading a loan quote, a deposit rate or an inflation release: “up 1%” can describe two different calculations. Use “up 1 percentage point” for the difference between 3% and 4%. A 1% relative increase from 3% would instead give 3.03%.

Subtract levels; divide for relative change

Let p0p_0 and p1p_1 be the numbers before the percent signs, so a 3% rate has p0=3p_0=3. Let r0=p0/100r_0=p_0/100 and r1=p1/100r_1=p_1/100 be decimal rates. Then

Change in percentage points=p1−p0=100(r1−r0).\text{Change in percentage points}=p_1-p_0=100(r_1-r_0).
A basis point is one hundredth of a percentage point. Therefore
1 percentage point=100 basis points=0.01 in decimal-rate units.1\text{ percentage point}=100\text{ basis points}=0.01\text{ in decimal-rate units}.

For the move from 3% to 4%:

DescriptionCalculationResult
Percentage-point change4−34-3+1 percentage point
Basis-point change100(4−3)100(4-3)+100 basis points
Decimal-rate change0.04−0.030.04-0.03+0.01
Relative percentage change100(0.04−0.03)/0.03100(0.04-0.03)/0.03Approximately +33.33%

Both the point change and the relative change are valid; they measure different things. The relative calculation divides by the original rate. Reversing the move gives a decrease of 1 percentage point; always state the direction or retain the sign.

What the same rate move means in cash

Suppose a hypothetical deposit of USD 10,000 earns simple interest for exactly one year. Compare a 3% annual rate with a 4% annual rate on the same principal, before fees or tax, with no reinvestment:

I0=10,000(0.03)=300 USD,I1=10,000(0.04)=400 USD,I1−I0=10,000(0.01)=100 USD.\begin{aligned} I_0&=10{,}000(0.03)=300\text{ USD},\\ I_1&=10{,}000(0.04)=400\text{ USD},\\ I_1-I_0&=10{,}000(0.01)=100\text{ USD}. \end{aligned}

The extra USD 100 is 33.33% of the original USD 300 interest, rounded to two decimals. It is 1% of the USD 10,000 principal. Choosing the denominator changes the question; neither denominator changes the 1 percentage point rate difference.

A hypothetical annual simple rate rises from 3% to 4%: plus 1 percentage point, plus 100 basis points, or a 33.33% relative increase. On USD 10,000 for one year, interest rises from USD 300 to USD 400.
A hypothetical annual simple rate rises from 3% to 4%: plus 1 percentage point, plus 100 basis points, or a 33.33% relative increase. On USD 10,000 for one year, interest rises from USD 300 to USD 400.
Original hypothetical example. Subtraction measures the rate gap; division by the old rate measures relative growth. Relative percentages are rounded to two decimals.
This cash calculation relies on the stated simple-interest period. A percentage-point difference alone does not specify compounding, day count or cash income. Compare rates on the same quotation basis. For the underlying conversion and unit checks, see Arithmetic, Algebra, Equations and Functions (membership required for the full lesson; public preview available).

Zero and negative starting rates

Point differences remain useful when a relative growth rate becomes awkward:

  • From 0% to 1%: the increase is 1 percentage point, or 100 basis points. The relative percentage change is undefined because its denominator is zero.
  • From −0.50% to −0.25%: the rate rises by 0.25 percentage points, or 25 basis points. Applying the signed relative-change formula gives (−0.0025+0.005)/(−0.005)=−0.5( -0.0025+0.005)/(-0.005)=-0.5, or −50%. Calling that a “50% fall in the rate” would obscure that the rate moved upward toward zero. Report the endpoints and the positive point change instead.

Keep the underlying quantity consistent too. A percentage-point change in volatility compares two volatility levels; it is not a percentage-point change in variance. Variance is volatility squared and uses different units.

Sources

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