# Present Value of Multiple Cash Flows
Most real decisions involve a series of cash flows, not a single amount. We classify these by (a) how long they last and (b) whether they are equal.
## 1. Finite period — cash flows that stop after `n₹ years
### A. Multiple Unequal cash flows
Discount each year's cash flow separately and add them up:
```
PV = CF₁/(1+r)¹ + CF₂/(1+r)² + CF₃/(1+r)³ + … + CFₙ/(1+r)ⁿ
```
### B. Multiple Equal cash flows (an annuity)
When every year's cash flow is the same, the sum collapses into a single factor — the Present Value Annuity Factor (PVAF):
```
PV = CF/(1+r)¹ + CF/(1+r)² + … + CF/(1+r)ⁿ
PV = Annual CF × PVAF(r%, n years)
```
PVAF is read from annuity tables; it saves you from discounting each year individually.
## 2. Infinite period — Perpetuity (cash flows that never end)
A perpetuity is a stream of cash flows expected to continue forever, with no end date.
### A. Equal cash flows forever
```
PV = Annual CF / Discount Rate
```
### B. Growing cash flows forever (growing perpetuity)
When the cash flow grows at a constant rate `g`:
```
PV = CF₁ / (Discount Rate − Growth Rate)
```
(CF₁ is next year's cash flow; this requires Discount Rate > Growth Rate.)
## 3. Timing of cash flows — when in the year do they arise?
The timing of cash flows materially changes the PV. Two cases:
| Type | When CF arises | Also called |
|---|---|---|
| Deferred Annuity | End of each year | CF in arrears |
| Annuity Due | Beginning of each year | CF in advance |
> Default rule: In the absence of information, always assume Deferred Annuity (cash flow at year-end).
### A. Deferred Annuity (equal CFs)
Same as the finite-equal-CF formula above:
```
PV = Annual CF × PVAF(r%, n years)
```
### B. Annuity Due (equal CFs)
Because each cash flow arrives one year earlier, it is discounted one period less. The adjustment:
```
PV = Annual CF × [1 + PVAF(r%, (n − 1) years)]
```
The first cash flow (received today) is taken at full value (₹1₹), and the remaining `n − 1₹ flows are discounted using PVAF for `n − 1₹ years.