How this calculator works
A savings account has five quantities: the starting balance, the deposit added each period, the interest rate, the length of time and the closing balance. Fix any four and the fifth follows. This calculator solves for whichever one you pick, using the same compound-interest model in every direction. Interest is credited and deposits are added at the frequency you choose, so a monthly setting means twelve interest credits and twelve deposits a year. The rate is nominal: 5% compounded monthly is 5% / 12 each month, which is an effective annual rate slightly above 5%. When solving for the rate, the calculator searches between -99% and 200% a year and reports the annual nominal rate that makes the target balance come out exactly. A negative rate is possible when the target is below what the deposits alone add up to. The model assumes level deposits, a constant rate and no fees or tax.
Formula
Let i = annual rate / periods per year and n = periods. Balance = P(1+i)^n + D((1+i)^n - 1)/i, with P the starting amount and D the deposit. Starting amount needed = (G - D((1+i)^n - 1)/i) / (1+i)^n. Deposit needed = (G - P(1+i)^n) i / ((1+i)^n - 1). Time = ln((G + D/i)/(P + D/i)) / ln(1+i). At a zero rate: balance = P + Dn, deposit = (G - P)/n and time = (G - P)/D. The rate is found numerically because the equation has no closed form for i.
Worked example
Starting with $10,000 and adding $100 a month at 5% a year compounded monthly, after five years the balance is $19,634.20 from $16,000 deposited and $3,634.20 of interest. The same plan held yearly instead gives $13,315.38 after five deposits and five interest credits. To reach $20,000 in five years monthly, the deposit would need to be $105.38. To reach it with $100 a month, the time needed is 62.01 months at 5%.
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Frequently asked questions
What is the difference between nominal and effective rate?
The nominal rate is the quoted annual rate. The effective rate accounts for compounding within the year: 5% compounded monthly is about 5.12% effective. The calculator uses the nominal rate you enter and compounds it at the chosen frequency.
Why does solving for the rate give a negative answer?
If the deposits plus the starting amount add up to more than the target, the balance has to shrink for the target to be met. A negative nominal rate represents a loss, such as a fee or a falling investment.
Should deposits be at the start or the end of the period?
This calculator assumes the end of each period. Deposits at the start of each period earn one extra period of interest, so a real account that credits deposits immediately will do slightly better than the result shown.
Does this include tax on interest?
No. Interest in a taxable account is usually taxed as it is earned or when withdrawn, which lowers the effective return. Enter a rate net of expected tax if you want an after-tax projection.
Why is the daily option different from monthly?
Daily compounding credits interest 365 times a year on a slightly smaller periodic rate. At the same nominal rate, more frequent compounding gives a higher effective annual rate, so the balance is a little larger.