Payout Annuities

Payout annuities are typically used after retirement. With a payout annuity, you start with money in the account, and pull money out of the account on a regular basis. Any remaining money in the account earns interest. After a fixed amount of time, the account will end up empty. Perhaps you have saved $500,000 for retirement, and want to take money out of the account each month to live on. You want the money to last you 20 years. This is a payout annuity. The formula is derived in a similar way as we did for savings annuities. The details are omitted here.

Payout Annuity Formula

$P_{0}={\frac {d\left(1-\left(1+{\frac {r}{k}}\right)^{-Nk}\right)}{\left({\frac {r}{k}}\right)}}$

P₀ is the balance in the account at the beginning (starting amount, or principal).

d is the regular withdrawal (the amount you take out each year, each month, etc).

r is the annual interest rate (in decimal form. Example: 5% = 0.05)

k is the number of compounding periods in one year.

N is the number of years we plan to take withdrawals.

Like with annuities, the compounding frequency is not always explicitly given, but is determined by how often you take the withdrawals.

When do you use this

Payout annuities assume that you take money from the account on a regular schedule (every month, year, quarter, etc.) and let the rest sit there earning interest.

• Compound interest: One deposit
• Annuity: Many deposits.
• Payout Annuity: Many withdrawals

Example

After retiring, you want to be able to take $1000 every month for a total of 20 years from your retirement account. The account earns 6% interest. How much will you need in your account when you retire?

In this example,

d = $1000 (the monthly withdrawal)

r = 0.06 (6% annual rate)

k = 12 (since we’re doing monthly withdrawals, we’ll compound monthly)

N = 20 (since were taking withdrawals for 20 years)

We’re looking for P₀; how much money needs to be in the account at the beginning.

Putting this into the equation:

${\begin{aligned}&{{P}_{0}}={\frac {1000\left(1-{{\left(1+{\frac {0.06}{12}}\right)}^{-20(12)}}\right)}{\left({\frac {0.06}{12}}\right)}}\\&{{P}_{0}}={\frac {1000\times \left(1-{{\left(1.005\right)}^{-240}}\right)}{\left(0.005\right)}}\\&{{P}_{0}}={\frac {1000\times \left(1-0.302\right)}{\left(0.005\right)}}=\$139,600\\\end{aligned}}$

You will need to have $139,600 in your account when you retire.

Notice that you withdrew a total of $240,000 ($1000 a month for 240 months). The difference between what you pulled out and what you started with is the interest earned. In this case it is $240,000 – $139,600 = $100,400 in interest.

You know you will have $500,000 in your account when you retire. You want to be able to take monthly withdrawals from the account for a total of 30 years. Your retirement account earns 8% interest. How much will you be able to withdraw each month?

In this example,

We’re looking for d.

r = 0.08 (8% annual rate)

k = 12 (since we’re withdrawing monthly)

N = 30 (30 years)

P₀ = $500,000 (we are beginning with $500,000)

In this case, we’re going to have to set up the equation, and solve for d.

${\begin{aligned}&500,000={\frac {d\left(1-{{\left(1+{\frac {0.08}{12}}\right)}^{-30(12)}}\right)}{\left({\frac {0.08}{12}}\right)}}\\&500,000={\frac {d\left(1-{{\left(1.00667\right)}^{-360}}\right)}{\left(0.00667\right)}}\\&500,000=d(136.232)\\&d={\frac {500,000}{136.232}}=\$3670.21\\\end{aligned}}$

You would be able to withdraw $3,670.21 each month for 30 years.

Licensing

Content obtained and/or adapted from:

• Payout Annuities, Lumen Learning under a CC BY-SA license