Investing
Portfolio value with annual contributions
The lump sum compounds at the net return, and each annual contribution compounds for the years remaining. This is an ordinary annuity in years, not months.
The Formula
V = P x (1 + r_n)^n + A x ((1 + r_n)^n - 1) / r_n
The lump sum compounds at the net return, and each annual contribution compounds for the years remaining. This is an ordinary annuity in years, not months.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Initial investment | currency | Amount invested at the start. |
| r_n | Net return | percent | Gross return minus expense ratio. |
| n | Years | years | The investment horizon. |
| A | Annual contribution | currency | Amount added at the end of each year. |
| V | Final value | currency | The portfolio at the end of the horizon. |
What Each Symbol Means
Pcurrency
Initial investment
Amount invested at the start.
r_npercent
Net return
Gross return minus expense ratio.
nyears
Years
The investment horizon.
Acurrency
Annual contribution
Amount added at the end of each year.
Vcurrency
Final value
The portfolio at the end of the horizon.
See It Worked With Real Numbers
A formula on its own does not tell you which number goes where. The how to calculate etf costs page applies this equation to a concrete case, shows every intermediate step, and names the mistakes that produce a wrong answer even when the algebra is right.
Where This Formula Appears
This equation is the working core of How To Calculate ETF Costs, filed under Finance. Every number the guide produces comes from substituting into this one line.