Problem 76
Question
Use the table capabilities of your calculator to work Comparison of Two Accounts You have the choice of investing \(\$ 1000\) at an annual rate of \(7.5 \%\) compounded daily or \(7.75 \%\) compounded annually. Let \(Y_{1}\) represent the investment at \(7.5 \%\) compounded daily, and let \(Y_{2}\) represent the investment at \(7.75 \%\) compounded annually. Graph both \(Y_{1}\) and \(Y_{2}\), and observe the slight differences in the curves. Then use a table with \(Y_{3}=Y_{1}-Y_{2}\) to compare the graphs numerically. What is the difference between the returns for the investments after 1 year, 2 years, 5 years, 10 years, 20 years, 30 years, and 40 years? Why does the lower interest rate yield the greater return over time?
Step-by-Step Solution
Verified Answer
The difference grows with time, and the lower rate yields more due to daily compounding.
1Step 1: Understand the Formulas
For account 1, which is compounded daily, the formula is \(Y_1 = P \left(1 + \frac{r}{n} \right)^{nt}\), where \(P = 1000\), \(r = 0.075\), \(n = 365\), and \(t\) is the time in years. For account 2, which is compounded annually, the formula is \(Y_2 = P(1 + r)^t\), where \(P = 1000\), and \(r = 0.0775\).
2Step 2: Calculate Values Using Formulas
Calculate the values for both \(Y_1\) and \(Y_2\) using a calculator for different \(t\) values: 1, 2, 5, 10, 20, 30, and 40 years.
3Step 3: Populate the Table
Record the calculated values of \(Y_1\) and \(Y_2\) in a table. Also, calculate \(Y_3 = Y_1 - Y_2\) for each year to find the difference between the two investments.
4Step 4: Compare After 1 Year
Substitute \(t = 1\) in the formulas. For \(Y_1\), the result is approximately \(1077.78\), and for \(Y_2\), the result is approximately \(1077.50\). So, \(Y_3 = 0.28\).
5Step 5: Compare After 2 Years
Substitute \(t = 2\) in the formulas. Calculate \(Y_1\) and \(Y_2\); find \(Y_3 = Y_1 - Y_2\). The difference \(Y_3\) increases.
6Step 6: Compare After 5 Years
Substitute \(t = 5\) in the formulas. Calculate \(Y_1\) and \(Y_2\); find \(Y_3 = Y_1 - Y_2\). The difference \(Y_3\) continues to increase.
7Step 7: Compare After 10 Years
Substitute \(t = 10\) in the formulas. Calculate \(Y_1\) and \(Y_2\); find \(Y_3\) to observe that \(Y_3\) is larger.
8Step 8: Compare After 20, 30, and 40 Years
For each \(t = 20, 30, 40\), calculate \(Y_1\) and \(Y_2\). The difference \(Y_3\) grows significantly as time increases.
9Step 9: Explain Why Lower Rate Yields More
\(Y_1\), compounded daily, leverages compounding more frequently, enhancing the growth over time compared to \(Y_2\), which compounds annually. Hence, the effective annual interest rate is higher for \(Y_1\).
Key Concepts
Investment ComparisonInterest RateCompounding FrequencyFinancial Mathematics
Investment Comparison
When considering investment options, it's vital to understand how different interest rates and compounding frequencies impact potential returns. Investment comparison involves evaluating different financial products to see which offers the most benefit over time.
In the example where you have \(\$1000\) to invest, choices between different types of compounding can greatly affect your final returns. With one account offering \(7.5\%\), compounded daily, and another offering \(7.75\%\), compounded annually, it might seem intuitive to choose the higher interest rate.
However, it's important to realize that the frequency of compounding plays a crucial role in how investments grow over the years. By systematically comparing these investments over various time periods such as 1, 2, 5, 10, 20, 30, and 40 years, one can observe how the daily compounding can actually result in higher returns over the long term even with a lower interest rate.
In the example where you have \(\$1000\) to invest, choices between different types of compounding can greatly affect your final returns. With one account offering \(7.5\%\), compounded daily, and another offering \(7.75\%\), compounded annually, it might seem intuitive to choose the higher interest rate.
However, it's important to realize that the frequency of compounding plays a crucial role in how investments grow over the years. By systematically comparing these investments over various time periods such as 1, 2, 5, 10, 20, 30, and 40 years, one can observe how the daily compounding can actually result in higher returns over the long term even with a lower interest rate.
Interest Rate
An interest rate is essentially the cost of borrowing money or the reward for investing it. It's expressed as a percentage of the principal amount that is paid as interest over a specific period of time. In simple terms, it's what you earn from your money (or pay if you're borrowing) over some time.
However, just knowing the interest rate isn't enough. You also need to consider how frequently interest is compounded to understand the real return on investment. In our given example, two different interest rates illustrate that a seemingly higher rate (7.75%) compounded annually is less effective over time compared to a slightly lower rate (7.5%) compounded daily.
This is because the actual growth from interest depends not just on the rate itself but also on how often the principal earns that interest.
However, just knowing the interest rate isn't enough. You also need to consider how frequently interest is compounded to understand the real return on investment. In our given example, two different interest rates illustrate that a seemingly higher rate (7.75%) compounded annually is less effective over time compared to a slightly lower rate (7.5%) compounded daily.
This is because the actual growth from interest depends not just on the rate itself but also on how often the principal earns that interest.
Compounding Frequency
Compounding frequency refers to the number of times the interest is calculated and added to your balance in a year. The more frequently an investment compounds, the faster it grows. This is because each time compounding occurs, you earn interest on the new amount in your account, which includes the previous interest added.
In simpler words, frequent compounding can be likened to continuously rolling a snowball; every time you add more snow (or interest), the snowball (or balance) grows larger.
Using the exercise example, compounding daily at \(7.5\%\) can yield better long-term returns than compounding annually at \(7.75\%\), because you're adding interest much more frequently.
In simpler words, frequent compounding can be likened to continuously rolling a snowball; every time you add more snow (or interest), the snowball (or balance) grows larger.
- Daily compounding lets you add interest 365 times a year.
- Annual compounding only adds it once a year.
Using the exercise example, compounding daily at \(7.5\%\) can yield better long-term returns than compounding annually at \(7.75\%\), because you're adding interest much more frequently.
Financial Mathematics
Financial mathematics involves using mathematical techniques to solve problems related to finance. It's like the toolkit for understanding and working with financial concepts such as interest rates, compounding, and investments.
In our exercise, the mathematical formulas used to calculate compound interest are pivotal in revealing how different interest rates and compounding frequencies affect the monetary gains. For \(Y_1\), compounded daily, the formula is \([P(1 + \frac{r}{n})^{nt}]\) giving precise outcomes, whereas \(Y_2\) simply uses \(P(1+r)^t\).
Financial mathematics helps in accurately comparing the potential outcome of different investments and makes informed financial decisions possible, knowing exactly how much growth to expect in various scenarios.
In our exercise, the mathematical formulas used to calculate compound interest are pivotal in revealing how different interest rates and compounding frequencies affect the monetary gains. For \(Y_1\), compounded daily, the formula is \([P(1 + \frac{r}{n})^{nt}]\) giving precise outcomes, whereas \(Y_2\) simply uses \(P(1+r)^t\).
Financial mathematics helps in accurately comparing the potential outcome of different investments and makes informed financial decisions possible, knowing exactly how much growth to expect in various scenarios.
Other exercises in this chapter
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