As we saw on the previous page in an annuity, you start with nothing, put money into an account on a regular basis, and end up with money in your account. On this page we will learn about a variation called a Payout Annuity<\/strong>.<\/p>\r\n 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.<\/p>\r\n A payout annuity<\/strong> allows an individual to receive regular payments, usually on a monthly or yearly basis, for a predetermined period of time.<\/p>\r\n<\/div>\r\n<\/section>\r\n Unlike a savings annuity, which requires the individual to make regular payments into the annuity, a payout annuity only requires a single lump sum investment. The individual can choose the length of the payout period, which can be for a set number of years or for the rest of their life. Once the payout period has ended, the annuity is depleted and no further payments are made.<\/p>\r\n<\/section>\r\n Payout annuities are typically used after retirement. Perhaps you have saved [latex]$500,000[\/latex] for retirement, and want to take money out of the account each month to live on. You want the money to last you [latex]20[\/latex] 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.<\/p>\r\n <\/p>\r\n When using formulas in application, or memorizing them for tests, it is helpful to note the similarities and differences in the formulas so you don't mix them up. Compare the formulas for savings annuities versus payout annuities.<\/p>\r\n <\/p>\r\n Like with annuities, the compounding frequency is not always explicitly given, but is determined by how often you take the withdrawals.<\/p>\r\n When do you use this?<\/strong><\/p>\r\n 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.<\/p>\r\n With these problems, you need to raise numbers to negative powers.\u00a0 Most calculators have a separate button for negating a number that is different than the subtraction button.\u00a0 Some calculators label this ([latex]-[\/latex]) , some with [latex]+\/-[\/latex] .\u00a0 The button is often near the [latex]=[\/latex] key or the decimal point.<\/p>\r\n If your calculator displays operations on it (typically a calculator with multiline display), to calculate [latex]1.005^{-240}[\/latex] you'd type something like:\u00a0 [latex]1.005 [\\wedge] [(-)] 240[\/latex]<\/p>\r\n If your calculator only shows one value at a time, then usually you hit the ([latex]-[\/latex]) key after a number to negate it, so you'd hit: [latex]1.005 [y^{x}] 240 [(-)] =[\/latex]<\/p>\r\n Give it a try - you should get [latex]1.005^{-240}=0.302096[\/latex]<\/p>\r\n<\/section>\r\n <\/p>\r\n We\u2019re looking for [latex]P_0[\/latex]\u00a0how much money needs to be in the account at the beginning.<\/p>\r\n Putting this into the equation:<\/p>\r\n [latex]\\begin{align}&{{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{align}[\/latex]<\/p>\r\n You will need to have [latex]$139,600[\/latex] in your account when you retire.<\/p>\r\n Notice that you withdrew a total of [latex]$240,000[\/latex] ([latex]$1000[\/latex] a month for [latex]240[\/latex] months). The difference between what you pulled out and what you started with is the interest earned. In this case it is [latex]$240,000 - $139,600 = $100,400[\/latex] in interest.<\/p>\r\n View more about this problem in this video.<\/p>\r\n [embed]https:\/\/youtu.be\/HK2eRFH6-0U[\/embed]<\/p>\r\npayout annuities<\/h3>\r\n
payout annuity formula<\/h3>\r\n
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\r\n[reveal-answer q=\"261541\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"261541\"]In this example,\r\n\r\n\r\n\r\n
\r\n [latex]d = $1000[\/latex]<\/td>\r\n the monthly withdrawal<\/td>\r\n<\/tr>\r\n \r\n [latex]r= 0.06[\/latex]<\/td>\r\n [latex]6\\%[\/latex] annual rate<\/td>\r\n<\/tr>\r\n \r\n [latex]n = 12[\/latex]<\/td>\r\n since we\u2019re doing monthly withdrawals, we\u2019ll compound monthly<\/td>\r\n<\/tr>\r\n \r\n [latex]t = 20[\/latex]<\/td>\r\n \u00a0since were taking withdrawals for [latex]20[\/latex] years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n