## PV of Multiple Cash Flows — Finite Period
In finance, "cash flows for a finite period" means a series of inflows or outflows that occur over a specific, limited timeframe — as opposed to flows that continue indefinitely (a perpetuity).
Finite cash flows fall into two types:
```
Cash Flows for a Finite Period
├── Multiple Unequal CFs
└── Multiple Equal CFs (Annuity)
```
### A. Multiple Unequal Cash Flows
When the cash flow differs each year, discount each one separately and add them up:
$$PV = \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \frac{CF_3}{(1+r)^3} + \dots + \frac{CF_n}{(1+r)^n}$$
There is no shortcut here — every cash flow gets its own discount factor based on the year in which it arises.
### B. Multiple Equal Cash Flows (Annuity)
When the same amount recurs every year, the long form is:
$$PV = \frac{CF}{(1+r)^1} + \frac{CF}{(1+r)^2} + \dots + \frac{CF}{(1+r)^n}$$
Because the cash flow (CF) is constant, you can pull it out and use the Present Value Annuity Factor (PVAF):
$$PV = \text{Annual CF} \times PVAF(r\%, n\,\text{years})$$
The PVAF is simply the sum of the individual discount factors for years 1 to n. It is read directly from an annuity table, which makes equal-cash-flow problems far quicker than discounting year by year.
### Choosing the Right Approach
- Unequal CFs → discount each cash flow individually (use PVF table).
- Equal CFs → use the single-step PVAF shortcut (use PVAF/annuity table).