Reference

The four return formulas, derived.

Simple ROI, CAGR, IRR, and time-weighted return all answer different questions about the same investment. Knowing which one to use is the first half of the work.

1. Simple total return

The crudest formulation. (Final value − initial investment) divided by initial investment.

ROI = (V_final − V_initial) / V_initial

$10,000 to $15,000 = +50 %. The figure ignores time entirely. Useful for headline reporting and for investments held less than a year. Misleading for everything else.

2. CAGR (Compound Annual Growth Rate)

The constant annual rate that takes initial value to final value over n years. The right number for buy-and-hold investments with no intermediate cashflows.

CAGR = (V_final / V_initial)^1/n − 1

Worked example. $10,000 → $15,000 over 5 years. CAGR = (15,000 / 10,000)^1/5 − 1 = (1.5)^0.2 − 1 ≈ 0.0845 = 8.45%.

Verify by compounding forward: 10,000 × (1.0845)⁵ ≈ 15,000. Round-trip clean.

3. IRR (Internal Rate of Return)

The CAGR equivalent for an investment with multiple cashflows. The rate that makes the net present value of the cashflow series equal to zero:

0 = Σ CF_t / (1 + r)^t

No closed-form solution exists in general. Solve numerically (Newton-Raphson is standard). The calculator on this site uses 80 iterations of Newton-Raphson, converging to within 1e-9 in typical cases.

Worked example. $10,000 initial deposit, $200/month for 5 years, final value $25,000. The cashflow series is [-10000, -200, -200, ..., -200, +24800] across 60 monthly periods. Solving for the monthly rate that zeroes the NPV gives ~0.69% per month, which annualises to ~8.6%.

4. TWR (Time-Weighted Return)

The return that strips out the effect of cashflow timing — the right measure of a manager's investment skill, because the manager doesn't control when the client adds or withdraws money.

TWR is computed by linking sub-period returns: divide the holding period into segments at each cashflow event, compute the return within each segment, and chain-link them:

TWR = (1 + r₁)(1 + r₂)...(1 + r_n) − 1

Where r_i is the segment return excluding the cashflow at the boundary. The calculator on this site does not compute TWR — it requires a full transaction history to identify segment boundaries, which is beyond a single-page tool. Institutional investors and fund managers use TWR for performance reporting; individual investors usually want IRR, which captures the timing of their personal cashflow decisions.

The IRR vs. TWR distinction matters in disputes. A fund manager who says “the fund returned 8 % per year” usually means TWR. A client who says “I earned 2 % per year” usually means IRR. Both can be correct simultaneously: the manager's investments did well, but the client added money near the top and withdrew near the bottom. IRR captures the client's experience; TWR captures the manager's skill.

The geometric vs. arithmetic mean trap

For multi-year returns, the geometric (compound) mean is correct; the arithmetic mean is wrong. A series of yearly returns of +50% then −33% has:

Arithmetic mean: (50 − 33) / 2 = +8.5%
Geometric mean: (1.50 × 0.67)^1/2 − 1 = 0.0%

The geometric mean is the truth: $100 → $150 → $100 is no return. The arithmetic mean systematically overstates compound performance, particularly for volatile portfolios. Always use geometric for multi-period reporting.

Sanity checks the calculator runs

No-contribution case: IRR engine and CAGR closed-form must agree to within 0.0001%. They are mathematically equivalent in this case.
Round-trip: the reported annualised rate, when compounded forward, must reproduce the user-entered final value.
Sign convention: CF₀ is negative (money out); intermediate contributions are negative; final value is positive (money in). Sign mismatches are caught at input parse.
Convergence: Newton-Raphson is allowed up to 80 iterations. Non-convergence triggers a fallback to the bisection method on a wider rate range.