Chapter 5: Chapter 5

Problem 11

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.) $\$ 20,000$ in a fund paying $5 \%$ per year, with monthly payments for 5 years, if the fund contains $\$ 10,000$ at the start

Short Answer

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To accumulate $20,000 after 5 years in a sinking fund with a starting amount of $10,000 and a 5% annual interest rate, we need to make periodic payments of approximately $154.97 per month.

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Identify the known variables

We are given the following information: 1. The interest rate (annual): 5% 2. The total amount to be accumulated: $20,000 3. The initial amount in the fund: $10,000 4. Duration of the fund: 5 years 5. Payment frequency: Monthly

Step 2: Convert the annual interest rate to a monthly interest rate

The annual interest rate is 5%, but we need to find the monthly interest rate since the payments and compounding are done monthly. To do this, we can use the formula: $i = (1 + r)^{n/m} -1$ Where: - $i$ is the monthly interest rate as a decimal - $r$ is the annual interest rate as a decimal (0.05 in this case) - $n$ is the number of times the interest is compounded per year (12, since it's monthly) - $m$ is the frequency of payments per year (12, since it's monthly)

Step 3: Calculate the monthly interest rate

Using the formula from Step 2, we have: $i = (1 + 0.05)^{12/12} -1$ $i = (1.05)^1 -1$ $i = 0.05$ So, the monthly interest rate is 5%.

Step 4: Calculate the number of payments (n) to be made

Since we're making monthly payments for 5 years, we need to find the total number of payments. The formula is: $n = Duration \times Payment\ Frequency$ $n = 5\ years \times 12\ payments/year$ $n = 60\ payments$

Step 5: Calculate the periodic payments using the future value formula

We can use the future value of a series of equal payments formula to find the periodic payments: $FV = PMT\left[\frac{(1+i)^{n}-1}{i}\right] + PV(1 + i)^{n}$ Where: - $FV$ is the future value of the fund ($20,000 in this case) - $PMT$ is the periodic payment (which we are trying to find) - $i$ is the monthly interest rate (0.05, calculated in Step 3) - $n$ is the number of payments (60, calculated in Step 4) - $PV$ is the initial amount in the fund ($10,000) Now we can rearrange the formula to solve for PMT: $PMT = \frac{FV - PV(1+i)^n}{\left[\frac{(1+i)^{n}-1}{i}\right]}$

Step 6: Plug in the values and calculate the periodic payments

Inserting the known values into the formula, we get: $PMT = \frac{20,000 - 10,000(1+0.05)^{60}}{\left[\frac{(1+0.05)^{60}-1}{0.05}\right]}$ Calculating the result, we have: $PMT = \$154.97$

Step 7: Present the final answer

Therefore, to accumulate $20,000 after 5 years in a sinking fund with a starting amount of $10,000 and a 5% annual interest rate, we need to make periodic payments of approximately $154.97 per month.

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Chapter 5: Chapter 5

Short Answer

Step by step solution

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Step 1: Identify the known variables

Step 2: Convert the annual interest rate to a monthly interest rate

Step 3: Calculate the monthly interest rate

Step 4: Calculate the number of payments (n) to be made

Step 5: Calculate the periodic payments using the future value formula

Step 6: Plug in the values and calculate the periodic payments

Step 7: Present the final answer

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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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