Net Present Value Formula Explained with Mathematical Derivations
The NPV formula is the mathematical cornerstone of capital budgeting. Understand every term, summation symbol, and discounting variable.
Key Takeaways
- The general discrete formula: NPV = -I_0 + Σ [CF_t / (1 + r)^t].
- I_0 represents Year 0 initial cash outlay (not discounted).
- CF_t is net cash flow in year t, r is periodic discount rate, and t is time period.
- NPV can also be expressed for continuous compounding: NPV = -I_0 + Σ [CF_t * e^(-rt)].
1. The Mathematical Expression of Discrete NPV
The standard discrete NPV formula is written as:
$$\text{NPV} = -I_0 + \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t}$$
Where: - $I_0$ = Initial investment capital spent at Year 0 (Day 0) - $CF_t$ = Net cash flow received at period $t$ - $r$ = Annual discount rate (expressed as a decimal, e.g., 0.10 for 10%) - $t$ = Specific time period index ($t = 1, 2, \dots, n$) - $n$ = Total project duration or forecast horizon in years
2. Multi-Year Expansion Breakdown
Expanded across a 5-year capital project, the formula expands algebraically to:
$$\text{NPV} = -I_0 + \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \frac{CF_3}{(1+r)^3} + \frac{CF_4}{(1+r)^4} + \frac{CF_5}{(1+r)^5}$$
Notice that the denominator $(1+r)^t$ grows exponentially larger with each passing year. Consequently, a ₹10,00,000 cash flow received in Year 5 is worth far less in present purchasing power than an identical ₹10,00,000 received in Year 1.
3. Compounding Frequency Variations
When cash flows or compounding occur at different intervals, the formula adapts as follows:
| Compounding Frequency | Formula Representation | Effective Rate Adjustment | Standard Application |
|---|---|---|---|
| Annual (Standard) | NPV = -I₀ + Σ [CF_t / (1 + r)^t] | r = Annual Hurdle Rate | Corporate CapEx & Mergers |
| Monthly Compounding | NPV = -I₀ + Σ [CF_m / (1 + r/12)^m] | r_m = r / 12 | Commercial Leases & Mortgages |
| Quarterly Compounding | NPV = -I₀ + Σ [CF_q / (1 + r/4)^q] | r_q = r / 4 | Corporate Dividend Distributions |
| Continuous Discounting | NPV = -I₀ + Σ [CF_t · e^(-r·t)] | e^(-r·t) discount factor | Derivatives & Options Pricing |
Calculate Net Present Value Live
Run your cash flows through our free, real-time Net Present Value calculator.