Problem 1 Imagine that you have $\$ 100$... [FREE SOLUTION]

Chapter 14: Problem 1

Imagine that you have $\$ 100$ of ill-gotten gains stashed in an offshore bank account. Lest the IRS get too nosey, you plan to leave that account idle until your retirement in 45 years. a. If your bank pays you $3 \%$ annual interest, what will your account balance be upon retirement? b. If your bank pays you $6 \%$ interest, what will your account balance be upon retirement? c. Does doubling the interest rate double your accumulated balance at retirement? More than double it? Less? Explain your answer.

Short Answer

Expert verified

Doubling the interest rate leads to more than doubling the accumulated balance due to exponential growth from compounding.

Step by step solution

- Formula Selection for Compound Interest

To determine the future balance of a bank account with compound interest, we use the formula:\[ A = P (1 + \frac{r}{n})^{nt} \]where $ A $ is the amount of money accumulated after $ n $ years, including interest. $ P $ is the principal amount (initial money), $ r $ is the annual interest rate (decimal), $ n $ is the number of times that interest is compounded per year, and $ t $ is the time in years. In this problem, since there's only one compounding period per year, $ n = 1 $.

- Calculate with 3% Interest

For part (a), we substitute the values into the formula: $ P = 100 $, $ r = 0.03 $, $ n = 1 $, and $ t = 45 $.\[ A = 100 (1 + 0.03)^{{45}} \]Calculate $ 1.03^{45} $, then multiply by 100 to find $ A $.

- Calculation Results for 3% Interest

Calculate $ 1.03^{45} \approx 3.78593 $. Then,\[ A = 100 \times 3.78593 = 378.593 \]Your account balance after 45 years at $ 3\% $ interest will be approximately \$378.59.

- Calculate with 6% Interest

For part (b), substitute the values into the formula with $ r = 0.06 $.\[ A = 100 (1 + 0.06)^{45} \]Calculate $ 1.06^{45} $, then multiply by 100 to find $ A $.

- Calculation Results for 6% Interest

Calculate $ 1.06^{45} \approx 13.7648 $. Then,\[ A = 100 \times 13.7648 = 1376.48 \]Your account balance after 45 years at $ 6\% $ interest will be approximately \$1376.48.

- Analyze the Impact of Doubling Interest

In part (c), to see if doubling the interest rate doubles the accumulated balance, compare the balances from parts (a) and (b). Doubling $ r $ from $ 3\% $ to $ 6\% $ increases the balance from \$378.59 to \$1376.48. This growth is more than double. Hence, the effect of the compounding is exponential, and doubling the interest rate leads to more than double the accumulated balance.

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Key Concepts

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

Exponential Growth

To understand exponential growth, visualize how interest accumulates over time. It is not merely a steady increase. Instead, with each year, the balance grows larger. This growth is based on the new balance each year, not just the initial amount. For instance, a compound interest setup at a 3% interest rate will grow your balance not just by a straight 3% each year. Instead, interest is added to the balance, which includes previous interest. This repeated process results in exponential growth.

Effectively, as the years pass, the amount you gain each year increases. The formula utilized here is crucial for calculating exponential growth in compound interest:

Initial amount, or principal: $ P = \$100 $
Annual interest rate: $ r = 0.03 \text{ or } 0.06 $
Time in years: $ t = 45 $

Ultimately, the result of this growth showcases how time can amplify your savings in ways that simple interest cannot.

Interest Rate Impact

The interest rate plays a significant role in determining how much your balance will grow over time. A higher interest rate means your balance grows faster. Let's take an example from the exercise where the initial amount is $ \\(100 $.

If you have a 3% annual interest rate, after 45 years, your account will grow to roughly $ \$378.59 \). However, if the interest rate is doubled to 6%, the balance skyrockets to approximately $ \$1376.48 $. This increase is not linear because the interest rate's effect is compounded over time.

It's important to note that the impact of compounding interest implies that even small changes in the interest rate can have substantial long-term effects. A slight increase in the rate enhances the multiplier effect of compounding, causing the balance at the end to be more than just doubled.

Future Value Calculation

Calculating the future value of an account with compound interest involves using a specific formula:

$ A = P (1 + \frac{r}{n})^{nt} $
Where $ A $ is the future value, $ P $ is the principal amount or initial deposit,
$ r $ is the annual interest rate in decimal form, $ n $ is the compounding frequency (here, annual, so $ n = 1 $),
and $ t $ is the time the money is invested for in years.

In our context, to solve for $ A $ with interest rates of 3% and 6%, we treat compounding annually (which aligns with the problem scenario). So, plug the numbers into the formula, and you'll obtain the future value advantages after 45 years.

Comprehending how this calculation operates helps illustrate how investments of even small initial amounts can expand significantly with proper compounding. This understanding is crucial for making informed financial decisions and planning for future financial health.

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