# Annuity

An annuity is a **finite** stream of cash flows of **identical magnitude** and **equal spacing in time**.

E.g., Savings, vehicle, home mortgage, auto lease, bond payments

![](https://956443069-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-M5-0RGr8Q9QLqT32rM-%2F-M5-0TBzyC8aF6M53_Dz%2F-M5-0_uLLW3usDYMdpPC%2FAnnuity.JPG?generation=1586990849328124\&alt=media)To compute the PV of an annuity, we could compute the PVs (at t=0) of the CFs individually and aggregate them, or we could use the formula below to directly compute the PV:

PV of Annuity = $$CF \* \frac{(1-(1+R)^{-T})}{R}$$

where the term multiplied to CF is called the **Annuity Factor**.

This formula assumes that the first cash flow occurs at t=1. If the first cash flow occurs today i.e. at t=0, then we must add CF to the above formula.

## Simple Example

```
How much do you have to save today to withdraw $100 at the end of each of the next 20 years if you can earn
5% per annum?
```

Here, CF=100, T=20, R=0.05

So, PV = $$100 \* \frac{(1-(1+0.05)^{-20})}{0.05}$$ = $1246.22

> **Sidenote**: A **mortgage** is an annuity where the borrowed amount is the present value of the annuity


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