Lesson Regular withdrawals from an annuity payout account
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<H2>Regular withdrawals from an annuity payout account</H2> <H3>Problem 1</H3>You have $500,000 saved for retirement. Your account earns 5% interest and is compounded monthly. How much will you be able to pull out at the beginning of each month, if you want to be able to take withdrawals for 25 years? <B>Solution</B> <pre> Since withdrawals are made at the beginning of each month, use the general formula for an Annuity Due sinking fund A = {{{W*p*((1-p^(-n))/r)}}}. Here A is the initial amount at the account; W is the monthly withdrawn value at the beginning of each month; r is the nominal monthly percentage r = 0.1/12 presented as a decimal; p = 1 + r and n is the number of withdrawing periods (months, in this case). In this problem, W is the unknown; the monthly rate is r = 0.05/12 = 0.004166667, p = 1 + 0.004166667 = 1.004166667, the number of payment periods (= the number of months) is n = 25*12 = 300. So 500000 = {{{W*1.004166667*((1-1.004166667^(-300))/0.004166667)}}}. The factor {{{1.004166667*((1-1.004166667^(-300))/0.004166667)}}} is equal to 171.7727904; therefore W = {{{500000/171.7727904}}} = 2910.82 dollars. Thus you will be able to withdraw about $2910.82 every month (rounded to the closest lesser cent) during 25 years. </pre> \\\\\\\\\\\\\\\\\\\\\\\\ To make such complicated calculations as they are in this problem, you should have/use an appropriate calculator for such long formulas. Ideal choice is MS Excel, if you have it in your computer. Then you write a formula in a text editor, copy-paste it into an Excel work-sheet cell and click "Enter" - the result is ready in the next second. If you have no MS Excel in your computer, you may find similar free of charge calculators in the Internet. One such calculator is www.desmos.com/calculator It allows you to do the same thing: you write a formula in a text editor, copy-paste it into this calculator and click "Enter" - the result is ready in the next instance. <H3>Problem 2</H3>A sinking fund was established to provide quarterly payouts at the end of each quarter for 10 years. The initial amount deposited to the fund was $200,000. The account is compounded quarterly at 6.6% annually. What is the amount of the regular payout at the end of each quarter ? <B>Solution</B> <pre> For such sinking fund, the formula which connects the starting amount X and the quarterly payment Q is X = {{{Q*((1-p^(-n))/r)}}}. (1) In this formula, Q is the regular withdrawal per quarter; the factual quarterly compounding rate is r = {{{0.066/4}}}, p = {{{1 + 0.066/4}}}, and the number of payment periods is n = 10 years * 4 quarters = 40. So 200000 = {{{Q*((1-(1+0.66/4)^(-40))/(0.066/4))}}}. (2) The unknown is the value of quarterly payments Q. In this formula, we can calculate the factor (multiplier) {{{((1-(1+0.066/4)^(-40))/(0.066/4))}}} = 29.11257449. Then from formula (2) Q = {{{200000/29.11257449}}} = 6869.883668. We round it to the closest cent and get the <U>ANSWER</U>. The quarterly payment out is 6869.88 dollars. </pre> <H3>Problem 3</H3>Rajesh and Priya plan to retire at age 60 with a retirement income of $48,000 a year from their savings. Rather than pay themselves the whole amount at the beginning of each year, they have decided that withdrawing payments of $12,000 at the beginning of each quarter gives them the right balance of liquidity and maximized interest earnings. They feel they can safely earn an interest rate of 5.75%, compounded quarterly, on their money and they are budgeting based on the prediction that they will live until they are 90 years old. (a) How much money will they have to have saved by the time they are 60 and ready to begin their retirement, in order fulfill this plan? (b) If the same total calculated above was to be saved, but no interest earned whatsoever, how much would be available to live on each quarter? (c) If the full 30 years are lived and quarterly budget spent, how much money in total will have been utilized in retirement? (d) How much will have been earned in interest ? <B>Solution</B> <pre> This problem is about the starting amount of a sinking fund. The fund should provide the payments of $12,000 at the beginning of each quarter during 30 years. The fund is compounded quarterly at the annual interest rate of 5.75%. The general formula to calculate the starting amount of the account is X = {{{W*p*((1-p^(-n))/r)}}}. In this formula, W is the regular withdrawal per quarter, W = $12000; the factual quarterly compounding rate is r = 0.0575/4, p = 1 + r, and the number of payment periods is n = 30 years * 4 quarters = 120. So X = {{{12000*(1+0.0575/4)*((1-(1+0.0575/4)^(-120))/(0.0575/4))}}} = 694044.48 dollars. It is the <U>ANSWER</U> to the problem's question (a). The answer to question (b) is {{{694044.48/(30*4)}}} = 5783.70 dollars. It is the averaged available uniform withdrawal from $694044.48 at each quarter during 30 years - if compounding does not work. The answer to question (c) is 30*4*12000 = 1,440,000 dollars. It is the real total amount that the family withdraws/receives from this sinking fund during 30 years. The answer to question (d) is the difference $1,440,000 - $694,044.48 = $745955.52. It is the money that the sinking fund earns/accumulates in 30 years due to compounding. </pre> -------------------- In this site, there is a group of lessons associated with annuity saving plans and retirement plans. They are - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Ordinary-Annuity-saving-plans-and-geometric-progressions.lesson>Ordinary Annuity saving plans and geometric progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Annuity-due-saving-plans-and-geometric-progressions.lesson>Annuity Due saving plans and geometric progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problem-on-Ordinary-Annuity-saving-plans.lesson>Solved problems on Ordinary Annuity saving plans</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Withdrawing-a-certain-amount-of-money-periodically-from-a-compounded-saving-account.lesson>Withdrawing a certain amount of money periodically from a compounded saving account</A> (*) - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-Annuity-saving-plans.lesson>Miscellaneous problems on retirement plans</A> From these lessons, you can learn the subject and can see many similar solved problems. The closest lesson to your problem is marked (*) in the list. <H3>Problem 4</H3>You can purchase an investment that pays $75,000 at the beginning of each year for twenty years. Similar investments earn an 8% interest rate. How much will the payments be worth to you at the end of fifteen years? <B>Solution</B> <pre> Similar account at the end of fifteen years would be able to provide payments of $75,000 at the beginning of each year for 5 (five) years. So, the question is: find the amount, which, when invested at 8% annual interest rate, will provide payment (withdrawing) $75,000 at the beginning of each year for five years. +----------------------------------------------+ | It is a standard question/problem about | | a starting amount of a sinking fund. | +----------------------------------------------+ The general formula to calculate the starting amount at the account is X = {{{W*p*((1-p^(-n))/r)}}}. In this formula, W is the regular annual withdrawal, W = $75000; the factual annual compounding rate is r = 0.08, p = 1 + 0.08 = 1.08, and the number of payment periods is n = 5 years. So X = {{{75000*1.08*((1-1.08^(-5))/0.08)}}} = 323409.52 dollars. It is the <U>ANSWER</U> to the problem's question. </pre> <H3>Problem 5</H3>Calculate the purchase price of an annuity paying $200 at the end of each month for 10 years with an additional lump payment of $2000 on the same day as the last payment of $200, at 6.5% compounded monthly. <B>Solution</B> <pre> The sough purchase price is the sum of two amounts. First amount is the starting amount of an annuity that provides paying (from annuity to you) $200 per month for 10 years. Second amount is the starting amount which provides a lump payment (from the annuity to you) of $2000 in 10 years from now. To find first amount, use the formula X = {{{W*((1-p^(-n))/r)}}}. In this case, W = $200 is the monthly withdrawal; r is the effective monthly compounding rate r = 0.065/12; p = 1 + 0.065/12; n is the number of withdrawals (the same as the number of months, n = 10*12 = 120). So, X = {{{200*((1-(1+0.065/12)^(-120))/((0.065/12)))}}} = 17613.70 dollars for the first amount. To find the value of the second amount, Y, use this equation 2000 = {{{Y*(1+0.065/12)^(10*12)}}}. From this equation, Y = {{{2000/(1+0.065/12)^(10*12)}}} = 1045.924586, or 1045.92 dollars (rounded). Thus the <U>ANSWER</U> to the problem's question is this sum X + Y = 17613.70 + 1045.92 = 18659.62 dollars. </pre> My other lessons on Finance problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/Finance/Problems-on-simple-interest-accounts.lesson>Problems on simple interest accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Problems-on-discretely-compounded-accounts.lesson>Problems on discretely compounded accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Problems-on-continuously-compounded-accounts-.lesson>Problems on continuously compounded accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Find-future-value-of-an-Ordinary-Annuity.lesson>Find future value of an Ordinary Annuity</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Find-a-value-of-a-regular-deposits-for-an-Ordinary-Annuity.lesson>Find regular deposits for an Ordinary Annuity</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/How-long-will-it-takes-for-an-ordinary-annuity-to-get-an-assigned-value.lesson>How long will it take for an ordinary annuity to get an assigned value?</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Find-future-value-of-an-Annuity-Due-saving-plan.lesson>Find future value for an Annuity Due saving plan</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Ordinary-annuity-account-with-non-zero-initial-deposit-as-a-combined-total-of-two-accounts.lesson>Ordinary annuity account with non-zero initial deposit as a combined total of two accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Annual-depositing-and-semi-annual-compounding-in-ordinary-annuity-saving-plan.lesson>Annual depositing and semi-annual compounding in ordinary annuity saving plan</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/21-add-Regular-withdrawals-from-a-compounded-account.lesson>Variable withdrawals from a compounded account (sinking fund)</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Present-value-of-an-ordinary-annuity-saving-plan.lesson>Present value of an ordinary annuity cumulative saving plan</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Starting-value-of-a-sinking-fund.lesson>Problems on sinking funds</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Find-the-compounding-rate-of-an-ordinary-annuity.lesson>Find the compounding rate of an ordinary annuity</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Accumulate-money-using-ordinary-annuity-then-spend-money-via-sinking-fund.lesson>Accumulate money using ordinary annuity; then spend money via sinking fund</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Calculating-a-retirement-plan.lesson>Calculating a retirement plan</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Accumulating-money-via-ordinary-annuity-and-sending-simultaneously-via-sinking-fund.lesson>Accumulating money via ordinary annuity and spending simultaneously via sinking fund</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Loan-problems.lesson>Loan problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Mortgage-problems.lesson>Mortgage problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Amortizing-a-debt-on-a-credit-card.lesson>Amortizing a debt on a credit card</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/One-level-more-complicated-non-standard-problems-on-ordinary-annuity-plans.lesson>One level more complicated non-standard problems on ordinary annuity plans</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/One-level-more-complicated-problems-on-sinking-funds.lesson>One level more complicated problems on sinking funds</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/One-level-more-complicated-non-standard-problems-on-loans.lesson>One level more complicated non-standard problems on loans</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Using-Excel-to-find-the-principal-part-of-a-certain-loan-payment.lesson>Using Excel to find the principal part of a certain loan payment</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Finance-problems-solved-using-Excel.lesson>Using Excel to find the interest part of a certain loan payment</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Tricky-problems-on-present-values-of-annuities.lesson>Tricky problems on present values of annuities</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/OVERVIEW-of-my-lessons-on-Finance-section-in-this-site.lesson>OVERVIEW of my lessons on Finance section in this site</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I. Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
