How this calculator works
Cash flow growth is separate from investment return. The first payment remains the entered amount and growth begins with the second. 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. Inflation discounts the final nominal balance to starting purchasing power without changing the cash flow growth assumption.
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.
Worked example
Two year-end deposits starting at $100 and growing 10%, with a 10% investment return, become $100 x 1.1 + $110 = $220. At a 5% return and 10% cash flow growth, the same deposits become $105 + $110 = $215.
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Frequently asked questions
What happens when growth equals return?
The apparent zero denominator is removable. There are n identical future-value contributions after their growth is accounted for, giving n x (1+i)^(n-1) times the first payment.
Does growth start with the first payment?
No. The first payment is the stated amount. The second payment is multiplied by one periodic growth factor.
Can growing withdrawals exhaust the fund?
Yes. If a chosen term and withdrawals would make the balance negative, the calculator rejects that scenario. Solve the first withdrawal instead to fit a fixed term.