How this calculator works
The final balance is fixed at zero so the result values only the recurring receipts. Choose the payment amount, term, return or opening balance to solve. The nominal return is converted to the selected payment interval before valuing cash flows. The first cash flow is the entered amount; later cash flows grow by the equivalent periodic growth rate. Choose beginning or end timing and yearly summaries or individual payment rows. Enter a whole number of payment periods. Time solving finds the first payment period reaching the goal, adjusting the last cash flow to finish there; this mode supports non-negative returns and cash flow growth up to a 100-year horizon. Rate solving searches nominal annual returns from -99% to 200%. Dates use a 365.25-day model year and are illustrative rather than bank settlement dates.
Formula
Let j be the nominal annual return, m the compounding count and q payments per year. i = (1 + j/m)^(m/q) - 1, or exp(j/q) - 1 for continuous compounding. g = (1 + annual cash flow growth)^(1/q) - 1. For n payments, A = ((1+i)^n - (1+g)^n)/(i-g), with A = n(1+i)^(n-1) when i=g. Multiply A by (1+i) for beginning payments. Closing balance = opening balance x (1+i)^n + signed first payment x A. Here closing balance is zero. Opening balance = payment x A / (1+i)^n.
Worked example
At 10% annually, two year-end receipts of $110 have a present value of $100 + $90.909091 = $190.909091. If those receipts occur at each year beginning, the value is $110 + $100 = $210. At zero return, three $100 receipts are worth $300.
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Frequently asked questions
Does an annuity due change the rate?
It changes when money moves. Beginning payments gain or lose one extra interval of return compared with otherwise identical end payments.
Can I use a zero return?
Yes. With no cash flow growth, each payment simply adds to or subtracts from the balance. Equal interest and cash flow growth use a finite limiting formula.
Are the projected returns guaranteed?
The arithmetic uses a steady return assumption. Market changes, tax, fees, guarantees and account restrictions must be considered separately.