Lesson Accumulate money using ordinary annuity; then spend money via sinking fund
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<H2>Accumulate money using ordinary annuity; then spend money via sinking fund</H2> <H3>Problem 1</H3>Grandma decides to put 1300 dollars every month into an account for you. She makes 23 monthly deposits, the last coming September 1, 2003 - the day you start college. She wants you to be able to withdraw money from this account at the beginning of each month, with the first withdrawal coming September 1, 2003 and the last coming June 1, 2008, (when you'll graduate). (Note: that makes 58 withdrawals total.) How much will you be able to withdraw each month if the account is earning a nominal interest rate of 9.3 percent compounded monthly? <B>Solution</B> <pre> Let' solve this problem into two parts: Part 1: Calculate the Accumulated Value of Grandma's Deposits 1. Monthly Deposit: $1300 2. Number of Deposits: 23 3. Interest Rate: 9.3% compounded monthly (effective rate r = 0.093/12 per month) Use the future value of an ordinary annuity formula: FV = {{{P * (((1 + r)^n - 1) / r)}}}, (1) where FV = Future Value P = Periodic Payment ($1300) r = Interest Rate per Period (0.093/12) n = Number of Periods (23) FV = {{{1300 * (((1 + 0.093/12)^23 - 1) / ((0.093/12)))}}} (2) Use MS Excel software in your computer. It is commonly considered as an appropriate tool for such calculations, since it provides the necessary precision. Copy and paste this formula into a MS Excel, and get the value of $32,592.78. Do not make intermediate rounding, since intermediate rounding are PROHIBITED in such calculation: they influence to the final result and lead to wrong answer. Part 2: Calculate the Monthly Withdrawal Amount 1. Accumulated Value (Present Value for Withdrawals): $32592.78 2. Number of Withdrawals: 58 3. Interest Rate: 9.3% monthly (effective compounding rate r = 0.093/12 per month) Use the present value of an annuity due formula since withdrawals are at the beginning of each month: PV = {{{M * ((1 - (1 + r)^(-n)) / r) * (1 + r)}}}, (3) where: PV = Present Value ($32592.78) M = Monthly Withdrawal Amount r = Interest Rate per Period (0.093/12) n = Number of Periods (58) Rearrange the formula (3) to solve for M: M = {{{((PV*r) / ((1 - (1 + r)^(-n)))) * (1/(1 + r))}}}, (4) M = {{{((32592.78*(0.093/12)) / ((1 - (1+0.093/12)^(-58)))) * (1/(1+0.093/12))}}}. (5) Again, use MS Excel software in you computer. It is commonly considered as an appropriate tool for such calculations, since it provides the necessary precision. I copied and pasted this formula into a MS Excel, and I got the value of $694.43. Do not make intermediate rounding, since intermediate rounding are PROHIBITED in such calculation: they influence to the final result and lead to wrong answer. <U>Answer</U>. 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