Suggested languages for you:

Americas

Europe

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

Expert verified
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. See the step by step solution ### Step by step solution ## Unlock all solutions Get unlimited access to millions of textbook solutions with Vaia Premium Over 22 million students worldwide already upgrade their learning with Vaia! ## 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.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features

## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner