Lesson One level more complicated problems on sinking funds
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<H2>One level more complicated problems on sinking funds</H2> <H3>Problem 1</H3>A sinking fund is designed to make 10 annual outpayments of $20,000 at the end of each year. Find the starting amount of this sinking fund, if the interest rate in the bank is 8% compounded quarterly. <B>Solution</B> <pre> It is a non-traditional sinking fund. The complication is that the outpayments are made annually, while compounding are made quarterly. So, we should construct an equivalent model, which will smoothly combine/treat these features. An account with the annual rate of 8% compounded quarterly works as the account with effective quarterly rate r = 0.08/4; hence, its the effective annual growth coefficient is t = {{{(1 + 0.08/4)^4}}} = {{{1.02^4}}} = 1.08243216, or an effective annual rate q = 0.08243216. Now this given sinking fund is equivalent to the ordinary annuity sinking fund with annual down payments of $20,000 and with effective annual rate q = 0.071859031 compounded yearly. Therefore, we can apply the standard formula for the starting value A of such sinking fund A = {{{W*((1-p^(-n))/r)}}}, where A is the starting value of a sinking fund; W is the annual withdrawal; n is the number of years; r is the annual rate (expressed as a decimal); and p = 1+r. In our problem A = {{{20000*((1-1.08243216^(-10))/0.08243216)}}} = 132741.78 (rounded). <U>ANSWER</U>. The starting value of this sinking fund is $132741.78. </pre> Thus, to solve the problem, we equivalently transformed the given saving plan into another saving plan, where deposits are synchronized with compounding. <H3>Problem 2</H3>An investment fund pays dividends at the end of each year for 10 years. The initial value of the fund was $200,000. What is the amount of the dividends if the fund is compounded quarterly at the annual rate of 6%? <B>Solution</B> <pre> In other terms, this investment fund is a sinking fund, which provides regular payouts annually and is compounded quarterly at the nominal compounding rate of 6% per year. The fund works in this mode during 10 years. After 10 years, the fund is empty. It is a non-traditional sinking fund. The complication is that the payouts are made annually, while compounding are made quarterly. So, we should construct an equivalent scheme, which will smoothly combine/treat these features. An account with 6% interest rate compounded quarterly works is equivalent to the account compounded annually with the effective rate of t = {{{(1+r/4)^4-1}}} = {{{(1+0.06/4)^4-1}}} = {{{1.015^4-1}}} = 0.061363551. +---------------------------------------------------------------------------+ | For such a scheme, the outpayments are synchronized with compounding, | | so we have a classic regular sinking fund. | +---------------------------------------------------------------------------+ Now we can apply a classic standard formula for sinking account, which connects the starting amount of the fund A with the annual outpayments W A = {{{W*((1-p^(-n))/t)}}}. (1) In this formula, A is the starting amount of the sinking fund; W is the regular annual outpayment values; the effective annual compounding rate is t = {{{(1+r/4)^4-1}}} = {{{(1+0.06/4)^4-1}}} = {{{1.015^4-1}}} = 0.061363551, p = 1 + t = 1 + 0.061363551 = 1.061363551, and the number of payment periods is n = 10 years. So, the equation (1) takes the form 200000 = {{{W*((1-1.061363551^(-40))/0.061363551)}}}. (2) The unknown is the value of annual payments W. In this formula, we can calculate the factor (multiplier) separately {{{((1-1.061363551^(-10))/0.061363551)}}} = 7.312772366. Then from formula (2) W = {{{200000/7.312772366}}} = 27349.41. We round it to the closest cent and get the <U>ANSWER</U>. The annual outpayment is 27349.41 dollars. </pre> Thus, to solve the problem, we synchronized compounding with outpayments and then applied a classic sinking fund formula to find the starting amount of the sinking fund. 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