# 4. The Future Value and Present Value of a Series of Equal Cash Flows (Ordinary Annuities, Annuity Dues, and Perpetuities)

## e. calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;

What is an annuity?

An annuity is a finite set of sequential cashflows, all with the same value.

What is an ordinary (regular) annuity?

An ordinary annuity is just an annuity with payments at the end of the period instead of the beginning. I.e. its first cash flow occurs one period from now (indexed at t = 1).

What is the formula to calculate the future value of a regular annuity?

$$FV = A \times \frac{[(1 + r)^N - 1]}{r}$$

- A = annuity amount

- N = number of regular annuity payments

- r = interest rate per period

What is the formula for the present value of a regular annuity?

$$PV = A \times \frac{[1 - \frac{1}{(1 + r)^N}]}{r}$$

What is an annuity due?

An annuity due is similar to a regular annuity, except that the first payment (or cash received) occurs immediately (at period 0).

How do you use the BA II Plus Calculator to calculate the present value (PV) of an investment (annuity) that pays $1,000 per year for 10 years at 9% interest rate on your initial payment.

1. Clear the calculator `2ND CLR TVM`

2. `1000 PMT`

3. `9 I/Y`

4. `10 N`

5. `CPT PV`

= -$6,417.66

If you're using the BA II Plus to calculate an Annuity Due (payment/cash at beginning of period instead of end) what keys do you need to hit?

1. `2ND BGN (PMT)`

2. `2ND SET (ENTER)`

3. `2ND QUIT (CPT)`

- This sets the calculator to BEGIN mode instead of default END mode. Everything else is the same

- BE SURE TO SWITCH BACK TO END MODE WHEN DONE

What is a perpetuity?

A perpetuity is a perpetual annuity - an ordinary annuity that extends indefinitely

What is an example of a perpetuity?

A stock that pays constant dividends

How would you calculate the present value of a perpetuity if you know the interest rate you could invest that money at? E.g. what is the present value of a perpetuity that pays $450/yr which could be invested at 9%?

`450/.09`

= 5000

Basically, what amount of money would you need to invest at 9% in order to generate $450 every year forever?

##### Source:

- CFA