Chapter 5: Chapter 5

Problem 10

Find the periodic payments necessary to accumulate the amounts given in Exercises $7-12$ in a sinking fund. (Assume end-of-period deposits and compounding at the same intervals as deposits.) $\$ 100,000$ in a fund paying $7 \%$ per year, with quarterly payments for 20 years

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The periodic payment necessary to accumulate $100,000 in a sinking fund after 20 years, with a 7% annual interest rate and quarterly payments, is approximately $313.40 per quarter.

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Step 1: Calculating the periodic interest rate

Since there are quarterly payments, we have to divide the annual interest rate by the number of quarters in a year. The periodic interest rate is given by: $r = \dfrac{Annual Interest Rate}{Number of Quarters per Year}$ where $r$ is the periodic interest rate. $r = \dfrac{7 \%}{4}$ which can be written as a decimal: $r = \dfrac{0.07}{4}$ Calculating the value, we get: $r = 0.0175$

Step 2: Calculating the total number of periods

As there are quarterly payments for a period of 20 years, we need to find the total number of quarterly periods. To find the total number of periods, we can use the following formula: $Total\: Periods = Years * Number\: of\: Quarters\: per\: Year$ Plugging in the values: $Total\: Periods = 20 * 4$ $Total\: Periods = 80$ So, there are 80 total periods (quarters) over the 20 years.

Step 3: Using the sinking fund formula to find the periodic payments

Now, we can use the sinking fund formula to calculate the periodic payments required to accumulate the desired amount: $Periodic\: Payment = \dfrac{Total\: Amount * r}{(1 + r)^{Total\: Periods} - 1}$ Plugging in the values: $Periodic\: Payment = \dfrac{100,000 * 0.0175}{(1 + 0.0175)^{80} - 1}$ Calculating the result: $Periodic\: Payment = \dfrac{1,750}{(1.0175)^{80} - 1}$ $Periodic\: Payment = \dfrac{1,750}{5.5759}$ $Periodic\: Payment = 313.40$ The periodic payment necessary to accumulate $100,000 in the sinking fund after 20 years with a 7% annual interest rate and quarterly payments is approximately $313.40 per quarter.

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Chapter 5: Chapter 5

Find the periodic payments necessary to accumulate the amounts given in Exercises \(7-12\) in a sinking fund. (Assume end-of-period deposits and compounding at the same intervals as deposits.) \(\$ 100,000\) in a fund paying \(7 \%\) per year, with quarterly payments for 20 years

Short Answer

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Step 1: Calculating the periodic interest rate

Step 2: Calculating the total number of periods

Step 3: Using the sinking fund formula to find the periodic payments

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More chapters from the book ‘Finite Mathematics and Applied Calculus’

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

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