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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

Expert verified
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.
See the step by step solution

## 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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