How the Calculators Work

Notation

• P is the deposit (principal).
• r is the nominal annual interest rate (APR), as a decimal: 4.00% is 0.04.
• APY is the annual percentage yield, as a decimal.
• n is the number of compounding periods per year.
• t is the time in years: months ÷ 12, or days ÷ 365.
• A is the amount withdrawn early, m a number of months and d a number of days of interest.

Balance and interest

From an interest rate (APR) and a compounding frequency:

FV = P × (1 + r/n)^n·t

From an APY, the compounding frequency is already built in:

FV = P × (1 + APY)^t

Interest is FV − P. The month-by-month schedule uses the same formula with t = k ÷ 12 for month k, plus a final part-month row when the term is set in days.

Compounding frequencies

Daily compounding uses n = 365, monthly n = 12, quarterly n = 4 and annually n = 1.

APY and APR

Interest rate to APY:

APY = (1 + r/n)ⁿ − 1

APY back to an interest rate:

r = n × ((1 + APY)^1/n − 1)

When you enter an APY, results use the APY balance formula and show the equivalent APR for the compounding you choose. When you enter an APR, results use the APR balance formula and show the APY.

APY from interest paid

This is the formula US institutions use for APY disclosures, where I is the interest paid over a number of days:

APY = (1 + I/P)^365/days − 1

Early withdrawal penalties

A penalty set in months of interest:

Penalty = A × r/12 × m

A penalty set in days of interest:

Penalty = A × r × d/365

If you enter an APY, it is first converted to an interest rate with the formula above. Interest earned so far uses the balance formula for the months held. For a full withdrawal, you would receive the deposit plus the interest earned so far, minus the penalty. For a partial withdrawal, the penalty is applied to the amount you take out, and you receive that amount minus the penalty; the rest stays in the CD and keeps earning at its rate. When the penalty is larger than the interest earned, the difference would come out of the deposit; some institutions do this, and the calculator shows it as a red line.

If the penalty exceeds the amount withdrawn, the result shows $0 received; we don't model penalties beyond that.

Interest paid out

When interest is paid out each month instead of being added to the CD, it does not compound:

I = P × r × t

The amount per month is P × r/12.

CD ladders

Each rung uses the APY balance formula for its own term. Total interest is the sum of each rung's interest, and the total returned is the sum of each rung's value at maturity. Rungs are held to maturity, with no reinvestment.

Blended APY = Σ(amount_i × APY_i) ÷ Σ amount_i

This is an amount-weighted simple average. It is a summary figure, not a yield.

Rounding and limits

Every calculation uses full double-precision numbers. Rounding happens only for display: money to the cent, rates to 3 decimal places on the page and 4 in the test cases.

• Deposit: $1 to $100,000,000.
• Rate: 0% to 30%; 0% is allowed.
• Term: 1 to 120 months, or 1 to 3,650 days.
• Penalty: 0 to 60 months, or 0 to 1,825 days, of interest.
• Months held must not be more than the term.

Input outside these limits shows a message next to the field instead of a result.

Test cases

Every number below is checked automatically against the same script the calculators run. Rates are example rates only. Tolerance is ±$0.005 on money and ±0.00005 on rates.

What we do not model

Taxes, fees, variable or step-up rates, reinvestment at maturity, and leap years (days are divided by 365). Add-on CDs with regular extra deposits are not covered.