Lesson Problems on sinking funds
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<H2>Problems on sinking funds</H2> <H3>Problem 1</H3>Suppose a state lottery prize of $3 million is to be paid in 25 payments of $120,000 each at the end of each of the next 25 years. If the annual compounding rate at the bank is 11%, and the account is compounded annually, what starting amount should be deposited into the bank to provide uninterrupted outpayments for the prize ? <B>Solution</B> <pre> We have a sinking fund. Its initial value is unknown and we should find it. We know that the initial money is deposited once for 25 years at 11% annual rate, the outpayments are $120,000 at the end of each year and the account is compounded at the end of each year. Use the formula for the starting value of such a sinking fund A = {{{PMT*((1 - (1+r)^(-n))/r)}}}, where A is the starting value, PMT is the annual outpayment value, r is the annual rate, n is the number of payment/compounding (the same as the number of years, in this problem). With the given data, the formula for calculations is A = {{{120000*((1-(1+0.11)^(-25))/0.11)}}} = {{{120000*((1-1.11^(-25))/0.11)}}} = 1,010,609.36 dollars. <U>ANSWER</U>. The initial amount to be deposited into the bank for the prize is $1,010,609.36. This amount should be deposited initially, and it will provide no-failure payments of $120,000 at the end of each year during 25 years, under given conditions. </pre> <H3>Problem 2</H3>You just sold a house for $200,000. You can invest the money at 5% compounded semiannually. How much could you withdraw every 6 months, starting in 6 months, for the next 20 years? <B>Solution</B> <pre> This problem is about a sinking fund. The starting amount is A = $200,000. The fund is compounded semi-annually at the annual compounding rate r = 5%. You want to withdraw a regular amount at the end of each 6 months period during next 20 years. They want you determine the value of this regular withdraw amount W. Use the formula for a sinking fund A = {{{W*((1 - (1+r/m)^(-n))/((r/m)))}}}, (1) where A is the starting amount, W is the regular withdraw amount semi-annually, r is the annual compounding rate, m is the number of withdrawals per year (m= 2 in this problem), n is the total number of withdrawals/compounding (twice the number of years, in this problem), {{{r/m}}} is the effective rate of compounding per the 6 months period. With the given data, formula (1) takes the form 200000 = {{{W*((1-(1+0.05/2)^(-40))/((0.05/2)))}}} = {{{W*((1-1.025^(-40))/0.025)}}} = W*25.102775 dollars. From this equation, we find the semiannual withdraw value W = {{{200000/25.102775}}} = 7967.25 dollars. <U>ANSWER</U>. The semi-annual withdraw value is $7967.25. </pre> <H3>Problem 3</H3>A state lotto has a prize that pays $1,700 each week for 40 years. If the account can earn 3% interest on investments, how much money will they need to put into an account now to cover the weekly prize payments? <B>Solution</B> <pre> This problem is about the starting amount of a sinking fund, which pays out $1700 each week during 40 years, compounding weekly at 3% annual interest. The year is 365/7, or about 52 weeks, so I will use 52 weeks per year. Use the formula for the regular/(weekly) payment of a sinking fund IV = {{{PMT*((1 - (1+r)^(-n))/r)}}}, where IV stands for the initial value, PMT is the regular/(weekly) outpayment value, r is the annual rate, n is the number of payment/compounding (52 times the number of years, in this problem). With the given data, the formula for calculations is IV = {{{1700*((1-(1+0.03/52)^(-52*40))/(0.03/52))}}} = 2,058,841 dollars. <U>ANSWER</U>. The initial/starting value of the fund is about $2,058,841. This amount is deposited initially, and it provides no-failure payments of $1,700 at the end of each week during 40 years, under given conditions. Notice that the total outpayment in 40 years, under the given condition, is 40*52*1700 = 3,536,000 dollars. The difference $3,536,000 - $2,058,841 = $1,477,159 is the interest, which the fund earns in 40 years for the customer and pays out to the customer, together with the initial deposited amount. </pre> <H3>Problem 4</H3>A man will deposit with a trust company a sum just sufficient to provide his family with an annuity of 750 dollars per month for 15 years. How much does he deposit if the fund's annual compounding rate is 4.5% and compounding are made monthly ? <B>Solution</B> <pre> In this problem, they talk about the sinking fund, which is a monthly compounded account. It provides monthly outpayments of $750 every month during 15 years. The account is compounded monthly at the annual rate of 4.5%. They want you find the starting (= the original) amount of money in the fund. The total outpayment from the fund in 15 years is Total = 750*12*15 = 135000 dollars. To answer the problem's question, we should calculate the Present Value of this amount. The Present Value is PV = {{{135000/(1+0.045/12)^(12*15)}}} = 68822.95. Thus the problem is just solved. The initial amount in the fund must be $68,822.95. <U>ANSWER</U> Then the fund will be able to pay out $750 every month during 15 years. </pre> <H3>Problem 5</H3>A contract valued at $81,867.00 provides outpayment of $2,430.00 at the beginning of every 3 months. If interest is 6.5% compounded quarterly, calculate the term of this contract (in quarters and in years). <B>Solution</B> <pre> This problem is about a sinking fund compounded quarterly. The initial/starting value of the fund is $81,867. The fund makes outpayments of $2430 quarterly and is compounded quarterly at the annual rate of 6.5%. They want to know the term of this contract, i.e. the number of quarters. The feature of this fund is that it makes outpayments at the beginning of each quarter, while compounding is made at the end of each quarter. For such a fund, the formula connecting the starting value, the periodical regular outpayment value, the rate of compounding and the number of outpayments is A = {{{W*(1+r/m)*((1 - (1+r/m)^(-n))/((r/m)))}}}, (1) where A is the starting amount, W is the regular quarterly outpayment amount, r is the annual compounding rate, m is the number of withdrawals per year (m= 4 in this problem), n is the total number of outpayments/compounding (the number of quarters, in this problem), {{{r/m}}} is the effective rate of compounding per quarter (3 months) period. With the given data, formula (1) takes the form 81867 = {{{2430*(1+0.065/4)*((1-(1+0.065/4)^(-n))/((0.065/4)))}}}. (2) The unknown in this equation is 'n'. To find 'n', simplify equation (2) step by step {{{(81867/(2430*(1+0.065/4)))*(0.065/4)}}} = {{{1 - (1+0.065/4)^(-n)}}} 0.538710461 = {{{1 - (1+0.065/4)^(-n)}}} {{{(1+0.065/4)^(-n)}}} = 1 - 0.538710461 {{{1/(1+0.065/4)^n}}} = 0.461289539 {{{(1+0.065/4)^n}}} = 1/0.461289539 {{{(1+0.065/4)^n}}} = 2.167835851 At this point, take logarithm of both sides n*log(1+0.065/4) = log(2.167835851) Express n and then calculate its value n = {{{log((2.167835851))/log((1+0.065/4))}}} = 47.9999402. <U>ANSWER</U>. The number of quarters is 48. 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