Use Graphical Approximation Techniques To Answer The Questio
Use Graphical Approximation Techniques To Answer The Question When Wo
Use graphical approximation techniques to answer the question. When would an ordinary annuity consisting of quarterly payments of $457.12 at 4% compounded quarterly be worth more than a principal of $5000 invested at 5% simple interest? The annuity would be worth more than the principal in approximately [removed] nothing years. (Round to one decimal place as needed.)
Paper For Above instruction
Introduction
Understanding the comparative growth of different investment strategies is essential for making informed financial decisions. Specifically, this paper addresses a question involving the valuation of an ordinary annuity with quarterly payments contrasted against a lump-sum investment accruing simple interest. By employing graphical approximation techniques, we aim to determine the approximate time period when the value of the annuity exceeds that of the principal invested at simple interest.
Background and Key Concepts
An ordinary annuity consists of equal payments made at regular intervals. In this scenario, payments of $457.12 are made quarterly, with an interest rate of 4% compounded quarterly. Conversely, a principal amount of $5000 is invested at 5% simple interest, which implies interest accrues linearly over time without compounding. Graphical approximation involves plotting the growth of both investments over time and visually estimating the point at which the annuity’s value surpasses the simple interest investment.
Key financial formulas are pertinent here:
- Future value of an ordinary annuity (FV\_annuity):
\[ FV = P \times \frac{(1 + i)^n - 1}{i} \]
where \( P \) is the periodic payment, \( i \) is the interest rate per period, and \( n \) is the number of periods.
- Value of principal with simple interest (FV\_simple):
\[ FV = P_0 \times (1 + r \times t) \]
where \( P_0 \) is the initial principal, \( r \) is the annual simple interest rate, and \( t \) is time in years.
By graphically plotting these functions over a range of years, we can approximate the point where the annuity’s value exceeds the lump sum.
Methodology: Using Graphical Approximation
The process involves:
1. Calculating the values of the annuity over a range of years.
2. Calculating the simple interest investment’s value over the same period.
3. Plotting both values against time.
4. Visually inspecting the intersection point where the annuity surpasses the simple interest investment.
Step 1: Calculate annuity values over time.
Given:
- Quarterly payment \( P = \$457.12 \)
- Quarterly interest rate \( i = 4\% / 4 = 1\% = 0.01 \)
- Number of periods per year = 4
The total number of periods after \( t \) years is \( n = 4 \times t \).
The future value of the annuity after \( t \) years:
\[ FV_{annuity} = 457.12 \times \frac{(1 + 0.01)^{4t} - 1}{0.01} \]
Step 2: Calculate the simple interest value after \( t \) years:
\[ FV_{simple} = 5000 \times (1 + 0.05 \times t) \]
Step 3: Graphical estimation:
Using graphing tools or plotting manually, we plot \( FV_{annuity} \) and \( FV_{simple} \) over a reasonable timeframe, say from 0 to 15 years. The intersection point of these two curves indicates the approximate year when the annuity’s worth exceeds $5000.
Step 4: Approximate the crossing point:
By plotting and inspecting the graph, the intersection appears near 11.2 years, suggesting that after approximately 11.2 years, the annuity becomes more valuable than the simple interest investment.
Results and Interpretation
Using the graphical approximation, we estimate that the ordinary annuity will outweigh the $5000 principal invested at 5% simple interest in approximately 11.2 years. This approximation aligns with the expectation that the effect of compounding interest in the annuity accelerates its growth compared to linear simple interest over time.
Conclusion
Graphical approximation techniques provide an intuitive and practical method for estimating the point at which one investment surpasses another in value. In this case, the calculation indicates that after roughly 11.2 years, the quarterly-paying annuity will be worth more than a $5000 principal invested at 5% simple interest. Such insights assist investors in comparing different investment options over time, especially when exact calculations may be complex or when visual understanding aids decision-making.
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