Discrete Compounding Cash Flow Calculator

Discrete Compounding: Key Formulas for Lump Sums and Annuities

This calculator uses discrete compounding per period. If you know the periodic rate \(i\) and the number of periods \(n\), a lump sum grows as \(F=P(1+i)^n\) and discounts back as \(P=\dfrac{F}{(1+i)^n}\). For a level annuity with payment \(A\), the ordinary annuity formulas are \(P=A\dfrac{1-(1+i)^{-n}}{i}\) and \(F=A\dfrac{(1+i)^n-1}{i}\). Annuity due (payments at the start) multiply these results by \((1+i)\).

Units & Good Practice

• Keep units consistent: if your cash flows are monthly, enter a monthly periodic rate and monthly periods.
• Enter rates as percentages (e.g., type “0.5” for 0.5% per period). The tool converts to decimals internally.
• Use the EAR/APR tab to convert a nominal APR with compounding frequency into an effective annual rate or vice versa.
• To solve for an unknown rate or number of periods, use the solver tab; it uses a robust bisection method.

Examples

If you deposit 1,000 for 12 months at 1% per month, \(F=1000\,(1.01)^{12}\approx 1126.83\). For a 36-payment ordinary annuity of 100 at 0.5% per month, \(P=100\cdot\frac{1-(1.005)^{-36}}{0.005}\approx 3362.03\). The equivalent annuity-due PV is \(3362.03\times1.005\approx 3378.84\).

Educational use only — not financial advice.

Understanding Discrete Compounding: Inputs, Outputs, and Best Practices

This calculator works with discrete compounding, where interest is applied a fixed number of times per period. It supports lump sums, level annuities (payments of the same amount each period), and quick conversions between nominal APR and effective annual rate (EAR). Everything runs privately in your browser.

Key Inputs

• Present Value (PV) — the value today. Enter a positive number for deposits/investments or the absolute value for loans.
• Future Value (FV) — the value at the end of n periods. Set to 0 for many loan-style problems when you want the balance to be fully repaid.
• Payment per period (A) — the level cash flow of an annuity. Choose timing: Ordinary (end of period) or Annuity Due (start of period).
• Rate per period (i) — interest as a percent per compounding period. Example: for a 12% APR compounded monthly, the periodic rate is \( i = 12\%/12 = 1\% \) per month.
• Number of periods (n) — count of compounding/payment periods. If you model monthly cash flows for 3 years, \( n = 36 \).
• APR ↔ EAR — use the EAR/APR tab to convert annual rates and derive the correct periodic rate for your scenario.

What the Calculator Computes

• Lump sums: Future Value \( F = P(1+i)^n \) or Present Value \( P = \frac{F}{(1+i)^n} \).
• Annuities (level payments): For an ordinary annuity, \( P = A \frac{1-(1+i)^{-n}}{i} \) and \( F = A \frac{(1+i)^n-1}{i} \). For annuity due, multiply those results by \( (1+i) \).
• Solvers: Find the unknown rate \( i \) or unknown number of periods \( n \) numerically from your PV/FV/payment inputs.
• APR/EAR tools: Convert between nominal APR with compounding frequency and the effective annual rate, and report the periodic rate.

Best Practices & Common Pitfalls

• Keep units consistent. If cash flows are monthly, use a monthly periodic rate and set \( n \) in months. Mixing annual rates with monthly periods leads to errors.
• Mind payment timing. Annuity due payments occur one period earlier and are worth more; the calculator applies the extra \( (1+i) \) factor automatically.
• Use EAR/APR correctly. Many disclosures quote APR but cash flows accrue at a periodic rate. Convert APR → EAR → periodic \( i \) as needed.
• Include fees if material. Upfront or recurring fees change your true PV/FV. Adjust PV (net of fees) or add fees as separate cash flows.
• Sign convention. Finance texts often use signs (outflows negative, inflows positive). For simplicity, this tool expects absolute values; just be consistent.
• Taxes & inflation. Results are in nominal terms. If you need real returns, deflate rates by expected inflation separately.

Percent and number formats differ by region (12,5% vs 12.5%). Enter rates as percent per period, and ensure your compounding frequency matches local conventions (e.g., monthly for consumer credit in many countries). Currency symbols are display-only; the math works with any currency.

Quick Example

Deposit 1,000 monthly for 36 months at 0.5% per month (ordinary annuity). Present value: \( P = 1000 \cdot \frac{1-(1.005)^{-36}}{0.005} \approx 32{,}881.11 \). The annuity-due PV is one period earlier: \( 32{,}881.11 \times 1.005 \approx 33{,}045.52 \).