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How do you calculate percent change when the starting amount is zero?

GlobalPublished 6 October 2026 · Written by BuyMeWhiskey Editorial Team · How we check facts

A percentage change from a starting amount of $0 is undefined because the standard formula divides by the starting amount. Report the absolute increase instead. For compound interest, state the principal, rate, compounding frequency and time period used in your formula or spreadsheet.

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Need to check a calculation? Start by writing down the original amount, final amount, rate and compounding frequency separately.

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Why percent change breaks at zero

The standard formula is (new value − old value) ÷ old value × 100. If the old value is $0, the denominator is zero, so the expression is undefined. There is no mathematically valid percentage change from $0 to a positive amount using the standard formula.

Do not replace zero with a made-up denominator

A common mistake is to say that a move from $0 to $100 is a 100% increase. It is not. You can say the amount increased by $100, or that it went from zero to $100, but the usual percentage-change measure cannot be used because the starting base is zero.

Compound interest uses a different formula

For a nominal annual rate compounded n times per year, the future value is commonly calculated as A = P(1 + r/n)^(nt), where P is principal, r is the annual rate as a decimal, n is the number of compounding periods per year and t is years. A problem that gives an effective annual rate should not be divided by 12; instead use the equivalent periodic rate.

Monthly compounding in a spreadsheet

In a spreadsheet, the direct formula can be written as `=P*(1+r/n)^(n*t)` using the appropriate cell references. For example, with principal in B2, annual rate in B3, compounding frequency in B4 and years in B5, `=B2*(1+B3/B4)^(B4*B5)` calculates the future value under the nominal-rate convention.

Converting an effective annual rate to monthly

When a rate is already an effective annual rate, the equivalent monthly rate is (1 + annual rate)^(1/12) − 1. Dividing an effective annual rate by 12 is a different assumption and can produce a slightly different result. Always check whether the quoted rate is nominal APR, annual effective yield or another contract-specific measure.

Daily interest with monthly compounding

Banks do not all calculate interest the same way. Some accounts accrue daily and credit monthly; others use another contractual convention. If the terms specify daily simple accrual at a stated annual rate, a common calculation is principal × annual rate ÷ day-count basis for each day, followed by the account's posting rule. Use the bank's disclosure rather than assuming every product uses the same day-count method.

Use formulas that match the product terms

For a real account, read the rate, APY or AER, compounding frequency and day-count language together. The displayed annual yield may already include compounding, so reverse-engineering it into a monthly rate without checking the definition can produce a wrong answer. In a school or work calculation, use exactly the convention stated in the question.

Why zero breaks the normal percentage-change formula

The usual percentage-change formula is (new value - old value) ÷ old value × 100. If the old value is zero, the denominator is zero, so the calculation cannot be completed. That is a mathematical issue, not a question of whether the increase is “infinite”.

From zero to a positive amount, a clearer statement is usually the absolute change: the value increased by the new amount. If a report requires a percentage, use a different measure that has a meaningful denominator, such as share of a total, or explain that the standard percentage change is undefined.

Monthly compound interest needs a stated rate convention

When an annual nominal rate is compounded monthly, the monthly rate is commonly represented by the annual nominal rate divided by the number of compounding periods. The future-value formula is A = P(1 + r/n)^(nt), where P is principal, r is the annual rate expressed as a decimal, n is the number of compounding periods per year and t is time in years.

Spreadsheet formulas are only as good as the inputs

In Google Sheets or Excel, put the principal, annual rate, number of compounding periods and term in separate cells when possible. That makes it easier to audit the calculation and change assumptions. If the bank quotes an effective annual rate rather than a nominal rate, do not blindly divide it by twelve; convert it using the rate definition and compounding convention supplied by the institution.

Daily interest and monthly compounding are not necessarily contradictory

Banks can calculate accrued interest using one frequency and credit it to an account using another. The account agreement determines the actual method. When reproducing a bank's calculation, use its stated annual percentage rate, daily balance method, compounding convention and posting dates instead of assuming that every savings product uses the same formula.

Worked example

Illustrative example. Start with $0 and end with $500. The absolute increase is $500, but the normal percent-change formula is ($500 - $0) ÷ $0 × 100, which is undefined because division by zero is not allowed. For a separate compound-interest example, suppose $10,000 earns an illustrative nominal annual rate of 4.8%, compounded monthly for one year. The future value is $10,000 × (1 + 0.048/12)^12 ≈ $10,490.70, so the illustrative interest is about $490.70. The two examples illustrate different issues: the first is a zero-denominator problem, while the second is a standard compound-growth calculation with a non-zero principal and a stated rate and compounding frequency. Write the assumptions down so another person can reproduce the result without guessing the rate convention.

What is the percentage increase from $0 to $100?#

The standard percentage-change formula is undefined because it divides by the starting value of zero. Report the absolute increase of $100 instead. Calling it a 100% increase incorrectly treats zero as though it were a valid percentage base. A spreadsheet should reproduce the stated formula and rate convention exactly.

Is going from zero to any amount an infinite percentage increase?#

Not under the standard percent-change formula. Division by zero is undefined, so “infinite percent” is not the correct mathematical result for ordinary reporting. Use the absolute difference or describe the change in terms of a ratio only when a separate, valid base exists.

How do I calculate monthly compound interest?#

For a nominal annual rate compounded monthly, use A = P(1 + r/12)^(12t). Put the annual rate in decimal form, the number of years in t, and use the account or question’s stated compounding convention. A spreadsheet should reproduce the stated formula and rate convention exactly.

Should I divide an annual interest rate by 12?#

Only when the stated rate is a nominal annual rate with monthly compounding under that convention. If the rate is an effective annual rate, the equivalent monthly rate is (1 + annual rate)^(1/12) − 1 instead. Check the terminology before doing the conversion.

What formula should I use in Google Sheets for compound interest?#

A simple direct formula is `=P*(1+r/n)^(n*t)`, with P, r, n and t replaced by cell references or values. For a real financial product, make sure the formula matches the product’s rate definition and compounding method. A spreadsheet should reproduce the stated formula and rate convention exactly.

Why can daily interest and monthly compounding look different?#

Interest may accrue daily but be credited to the account monthly, or the bank may use another contractual convention. The result depends on the stated annual rate, day-count basis and posting rules. Do not assume that “monthly compounding” means the same thing as “interest calculated monthly” in every product.

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Sources

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General information, not personal financial advice. Rules depend on your circumstances; check the official source or speak to a regulated adviser before acting.