# Annuity Calculator

 Annuity Calculator Starting Principal: (\$) Growth Rate: (%) Years to Pay Out: Make payouts at: Beginning (annuity due) End (ordinary / immediate annuity) Results Annual Payout Amount: (\$)

## How to use the Annuity Calculator

1. Enter the “Starting Principal”;

2. Enter the “Growth Rate”;

3. Enter the “Years to Pay Out”;

4. Make payouts at Beginning (annuity due) or End (ordinary / immediate annuity);

5. Click on “Calculate” button to get the result for “annual payout amount”.

## Definition of Annuity

The goal of annuity is to make a starting balance pay a fixed annual amount over a given number of years. Annuities are used in retirement , as if 401(k), which is an employer-sponsored retirement plan that lets you defer taxes until you’re retired.

## Annuitized Payouts Are Important

Annuitized Payouts are the key to retirement accounts. You start with a lump sum at the start of retirement, and you want to invest at a set rate of return. Then, you can withdraw funds every year for income.

## Annuity Formula

How to find out the annual withdrawal amount without depleting the account early?
We’ll assume that the withdrawals are all equal to keep things simple, We write (w) for the annual withdrawal amount and (z) for (1 + r) to keep things neater.

The first few terms for the balance:

 Year Balance 1 P – w 2 (P – w)z – w 3 [(P – w)z – w]z – w

Multiplying the right sides out yields the pattern:

 Year Balance 1 P – w 2 Pz – w(1 + z) 3 Pz2 – w(1 + z + z2) Y PzY-1 – w(1 + z + z2 + . . . + zY-1)

The second part of the last line is w times the sum of a geometric series, which is:

1 + z + z2 + z3 + . . . + zn = [zn+1 – 1]/[z – 1]

Sometimes, we’ll want to do a series without that leading “1”, The result is:

z + z2 + z3 + . . . + zn = [zn+1 – z]/[z – 1]

So the annuity formula simplifies to:

Balance(Y) = PzY-1 – w[(zY – 1)/(z – 1)]

We’re assuming that P, r, and Y are all known and that we want to find w that makes the balance go to zero at time Y; so set Balance(Y) = 0 and solve for w, to get:

0 = PzY-1 – w[(zY – 1)/(z – 1)]
w = [PzY-1]/[(zY – 1)/(z – 1)]
w = [PzY-1(z – 1)]/[zY – 1]

Finally, write (z) out in terms of r, to get the annuity formula(The formula assumes payouts occur at the start of each year):

Annuity Formula 1: w   =   [ P(1 + r)Y-1r ] / [ (1 + r)Y – 1 ]

## Annuity Time Issues

Assuming you get payouts at the end of each year, which essentially means you get one more period of compounding before each payout, Solving for (w) thus gives you one more year of growth:

Annuity Formula 2: w   =   [ P(1 + r)Yr ] / [ (1 + r)Y – 1 ]

You can use our Annuity Calculator to see how much of a difference this makes between the end of the year and the beginning of the year.

## Present Value of an Annuity

You will get the formula for the present value of an annuity which is the starting principal you’ll need to achieve the payouts desired, if you solve either equation [Annuity Formula 1 ] or [Annuity Formula 2 ] for P.

Annuity Due: P   =   w [ (1 + r)Y – 1 ] / [ (1 + r)Y-1r ]
Ordinary / Immediate Annuity: P   =   w [ (1 + r)Y – 1 ] / [ (1 + r)Yr ]