The Master Formula
$$ROI = \left( \frac{\text{Current Value} - \text{Initial Cost}}{\text{Initial Cost}} \right) \times 100$$
Dimensional Analysis
The numerator is $(\text{Currency} - \text{Currency}) = \text{Currency}$. The denominator is also $\text{Currency}$. Therefore, ROI is a dimensionless quantity (a pure number or percentage). In Physics, like Strain or Refractive Index, dimensionless quantities represent ratios that define the "quality" or "property" of a system regardless of its scale.
Variations: Annualized ROI
If an investment lasts $t$ years, the simple ROI doesn't tell the whole story. We use the CAGR (Compound Annual Growth Rate) logic:$$Annualized ROI = \left[ (1 + ROI_{total})^{1/t} - 1 \right] \times 100$$This accounts for the time-value of money, much like comparing average power vs. total work done.
Shortcuts & Mnemonics
- The "One-Zero" Rule: If the Final Value is 10 times the Initial, the ROI is 900% (always $100 \times (\text{multiple} - 1)$).
- Sridharacharya-style Thinking: Always keep your "Initial" as the denominator. The "Base" never changes during the calculation.
- Mnemonic: "Gain over Drain" (Gain is the profit, Drain is the initial money spent).
Edge Cases
- Initial Cost = 0: The ROI becomes infinite ($\infty$). This happens in "Zero Investment" scenarios (like organic word-of-mouth growth).
- Final Value = 0: The ROI is -100%. You have lost everything.
- Negative Initial Cost: A mathematical anomaly in standard ROI, but in "Rebate" scenarios, it implies you were paid to take the asset.
