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Chapter 2: Problem 17

Fareshta and Ahmad want to save to help send their child to college. Their plan is to put aside \(\$ 50\) every week. Suppose they deposit that money into an account that pays \(3.5 \%\) APR compounded weekly. a. How much money will be in the account in 18 years? (assume 52 weeks in a year) b. What minimum initial lump sum deposit would they need to make today to have the same balance in 18 years if they weren't putting aside the \(\$ 50\) per week?

Short Answer

Expert verified
$62,971.53 in 18 years; $36,298.16 needed today as a lump sum.

Step by step solution

01

Determine the number of weeks

Since the account is compounded weekly and Fareshta and Ahmad plan to save for 18 years, we first determine the total number of weeks. Calculating this gives: \[ n = 18 ext{ years} \times 52 ext{ weeks per year} = 936 ext{ weeks} \] This is the total number of weeks over which deposits will be made.
02

Calculate the equivalent weekly interest rate

The annual percentage rate (APR) is 3.5%. We need to convert this to a weekly interest rate since the compound interest is applied weekly. The formula to convert an annual rate to a weekly rate is: \[ r = \left( \frac{3.5\%}{100} \right) \,/\, 52 \approx 0.0006731 \] This is the equivalent weekly interest rate.
03

Apply the Future Value of Annuity Formula

To find the future value of the weekly deposits made into the account, we use the Future Value of an Annuity formula for regular weekly deposits. The formula is: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where \( P \) is the weekly deposit \( ( \(50) \), \( r \) is the weekly interest rate, and \( n \) is the number of weeks. Substituting the known values gives: \[ FV = 50 \times \frac{(1 + 0.0006731)^{936} - 1}{0.0006731} \approx \\) 62,971.53 \] This is the amount that will be in the account after 18 years of regular weekly deposits.
04

Determine present value of the future amount

To find what initial lump sum deposit they would need to achieve the same future value without making additional deposits requires the Present Value formula. The formula is: \[ PV = \frac{FV}{(1 + r)^n} \] Using the future value we found earlier, the weekly interest rate, and number of weeks: \[ PV = \frac{62971.53}{(1 + 0.0006731)^{936}} \approx \$36,298.16 \] This is the lump sum amount they need today to have the same future balance in 18 years without making weekly deposits.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interest Rate Conversion
When working with financial calculations over different compounding periods, it's crucial to convert the interest rate into a form that matches the compounding frequency. In our example, the interest rate given was an annual percentage rate (APR) of 3.5%, but our compounding and payment period is weekly. Converting this APR to a weekly interest rate makes all subsequent calculations much easier and more accurate. To do this, you divide the annual rate by the number of periods in a year鈥攊n this case, weeks. So, the annual rate of 3.5% translates into a weekly rate of approximately 0.0006731. This conversion allows us to reflect the incremental growth of savings more precisely, given the compounding effect of interest over shorter intervals. Remember, proper conversion ensures that you're comparing like with like, leading to more truthful financial forecasting.
Compound Interest
Compound interest plays a pivotal role in growing savings, especially over long periods. Unlike simple interest, which is calculated once on the principal amount, compound interest takes into account the accumulated interest on both the initial principal and the past interest. So, in the case of Fareshta and Ahmad, where they are depositing money weekly, their savings not only grow from their deposits but also from the interest earned on yesterday's interest. This is why compound interest is often referred to as "interest on interest," and it's what causes the exponential growth of invested money over time. The beauty of compound interest is maximized with frequent compounding periods. In our scenario, each week offers a new opportunity to earn interest on a slightly larger amount, owing to the interest that was applied the previous week. As time passes, this compounding effect becomes more pronounced, significantly increasing the future value of the deposited funds.
Present Value Calculation
The concept of present value (PV) is fundamental in assessing how much future money is worth today. Essentially, it tells us that a specific amount of cash today has different value compared to the same amount in the future due to its earning potential. In the scenario of Fareshta and Ahmad, if they want to know how much they need to deposit as a lump sum today to match the future value of their planned savings, they apply the present value formula. This formula helps in determining what amount, invested today at a given interest rate, would equal their accumulated savings after 18 years. Calculating this involves using the future value found from their deposits and interest and discounting it back to the present using the compound interest formula. The resulting value, approximately $36,298.16, shows them the amount they'd need today to achieve their future financial goal without making the weekly deposits. The Present Value concept is not only helpful in savings but is crucial for making informed decisions in various financial planning and investment scenarios.

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Most popular questions from this chapter

Pat loves painting. With tax, the total spent is about \(\$ 46\) each month on supplies. Once a week (52 weeks per year) the art class cost Pat \(\$ 15.75 .\) How much is Pat spending on painting in a year?

Imagine that at the start of a certain year, you will deposit \(\$ 1000.00\) into a savings account, and then you will leave the account alone. Each year after the opening deposit, the amount in the account will grow to be \(103 \%\) of its previous year's balance. a. After two years, the account balance will have experienced two growth amounts of \(103 \%\). You can find this account balance amount here, with the spreadsheet computation \(=1000 *(103 \%) *\) (103\%). Perform this computation in a spreadsheet and write the balance that you find. b. Now enter the spreadsheet computation \(=1000 *(103 \%)^{\wedge} 2 .\) Notice that the result here, which involves using a power, gives the same answer as you found in part (a). Comparing the two spreadsheet computations: Explain why they give the same result. c. Using the pattern in part (b) above, and carefully choosing the power: Compute the balance that will be in the account fifteen full years after the account was originally opened. (Round to the nearest cent) d. (Challenge) Make a spreadsheet that shows the account balance each individual year for 30 years. From the date of the opening deposit: What minimum number of full years will you have to wait, until the balance finally exceeds twice its opening deposit amount? (Use cell references, the fill down feature, and dollar formatting) e. (Challenge) Imagine the opening balance of the account was \(\$ 5000.00\) instead of \(\$ 1000.00\) (and everything else about the account stays the same). Make a similar spreadsheet as you did in part (d), and using this spreadsheet, find the minimum number of full years you will have to wait this time, until the balance finally exceeds twice its opening deposit amount. How does this answer compare with your answer in part (d)? Do you think your answer would be the same here, for any positive opening balance you may choose for the account?

For twelve full years, and into an account that pays \(3.5 \%\) APR compounded quarterly: Yanhong will either pay \(\$ 1500\) at the end of each calendar quarter, or, deposit a single lump sum that will give the same future value amount. a. If Yanhong chooses the single lump sum option, then how much will Yanhong need to deposit? b. If Yanhong needs to have earned \(\$ 100,000\) in this account at the end of the twelve years, then the quarterly deposit amount will need to be increased. What would the new quarterly deposit amount need to be? c. (Challenge): If Yanhong will make quarterly deposits into this account for the twelve years, but also has \(\$ 8,000\) to additionally deposit into this account right away: What would the new quarterly deposit amount need to be, so that the total balance after twelve years is \(\$ 100,000 ?\)

You want to buy a \(\$ 25,000\) car. The company is offering an interest rate of \(2 \%\) APR for 48 months (4 years). What will your monthly payments be?

Lynn bought a \(\$ 300,000\) house, paying \(10 \%\) down, and financing the rest at \(6.5 \%\) APR for 30 years. a. Find her monthly payments. b. How much interest will she pay over the life of the loan? c. What percentage of her total payment was interest?
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