Problem 11

Find the periodic payments necessary to accumulate the given amount in an annuity account. (Assume end-of-period deposits and compounding at the same intervals as deposits.) [HINT: See Quick Example 2.] $\$ \$ 20,000$ in a fund paying $5 \%$ per year, with monthly payments for 5 years, if the fund contains $\$ 10,000$ at the start

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To accumulate \$20,000 in the annuity account with a starting balance of \$10,000, periodic payments of approximately \$147.13 are necessary for the next 5 years, with monthly deposits and compounding interest at the same intervals.

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TABLE OF CONTENTS

Step 1: Calculate the interest rate per period

To find the interest rate per period, divide the annual interest rate by the number of periods in a year. Since there are 12 months in a year, the interest rate per month is: $r = \frac{5\%}{12} = \frac{0.05}{12}= 0.004167$

Step 2: Calculate the number of periods

Given that payments are made for 5 years with 12 payments each year, we have a total of: $n = 5 \times 12 = 60$

Step 3: Calculate the difference between the target amount and the starting amount

Since there is already $10,000 in the fund and we aim to reach $20,000, we are interested in finding out the payment needed to accumulate an additional $10,000: $FV' = \$20,000 - \$10,000 = \$10,000$

Step 4: Apply the FVOA formula and solve for P

We have all the necessary parameters, so we can now apply the FVOA formula to solve for the periodic payment P: $FV' = P\frac{(1+r)^n-1}{r}$ Substitute the values of FV', r, and n into the formula: $\$10,000 = P\frac{(1+0.004167)^{60}-1}{0.004167}$ Now solve for P using algebra: $P = \frac{\$10,000 \times 0.004167}{(1+0.004167)^{60}-1}$ $P = \frac{\$41.67}{(1.004167)^{60}-1}$ $P = \frac{\$41.67}{1.28323-1}$ $P = \frac{\$41.67}{0.28323}$ $P \approx \$147.13$

Step 5: Conclusion

To accumulate \$20,000 in the annuity account with a starting balance of \$10,000, periodic payments of approximately \$147.13 are necessary for the next 5 years, with monthly deposits and compounding interest at the same intervals.

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Chapter 2: Chapter 2

Short Answer

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Step 1: Calculate the interest rate per period

Step 2: Calculate the number of periods

Step 3: Calculate the difference between the target amount and the starting amount

Step 4: Apply the FVOA formula and solve for P

Step 5: Conclusion

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Chapter 3

Chapter 4

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Chapter 6

Chapter 7

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