Question 1: Sarah Wiggum Would Like To Make A Single Investm

Question 1 Sarah Wiggum Would Like To Make A Single Investment And H

Sarah Wiggum aims to accumulate $1.6 million by her retirement in 35 years. She is evaluating different investment scenarios based on varying annual returns. According to the information, when investing in a fund earning 3% annually, how much money must she invest today to reach her goal? Additionally, if she could achieve a 20% annual return, how soon could she then retire?

In financial planning, the core concept involved here is the present value (PV) of a future sum, which is calculated using the formula:

PV = FV / (1 + r)^n

Where FV is the future value, r is the annual interest rate, and n is the number of years. For the first scenario, calculating the initial investment for a 3% return:

PV = 1,600,000 / (1 + 0.03)^35

This calculation yields approximately $855,575.83. For the second scenario, to determine the number of years to reach the same goal at a 20% return, rearranging the formula to solve for n gives:

n = log(FV / PV) / log(1 + r)

Assuming the initial investment remains the same, or calculating the time to reach $1.6 million with an annual rate of 20%, yields approximately 18.8 years.

Paper For Above instruction

Financial decision-making involves understanding the relationship between present investments and future goals, especially through the lens of compound interest and the time value of money. Sarah Wiggum's case exemplifies how different annual returns significantly impact the amount she needs to invest today and the time horizon for her retirement savings. This analysis requires fundamental principles of present value calculations, which serve as the backbone for personal financial planning.

To determine the amount Sarah needs to invest now, one applies the present value formula. Given her goal of $1.6 million in 35 years and a fund with a 3% annual return, the present value can be computed as:

PV = 1,600,000 / (1 + 0.03)^35

PV ≈ $855,575.83

This means Sarah should invest approximately $855,576 today to reach her retirement goal at a 3% return. However, if she could secure a 20% annual return, her investment needs and time horizon change substantially. Solving for the number of years (n) at 20% interest involves logarithmic functions:

n = log(1,600,000 / PV) / log(1 + 0.20)

n ≈ 18.8 years

This demonstrates the dramatic effect higher returns have on the timeline to reach financial goals. The compound interest formula underscores the power of investment returns over time, emphasizing that higher yields significantly accelerate wealth accumulation.

Comparing these scenarios highlights the importance of identifying investments with favorable interest rates and understanding their impact on financial planning. Investors benefit from comprehending the time value of money, which informs decision-making, especially regarding choosing investment products, setting realistic retirement timelines, and assessing the sufficiency of savings contributions.

Furthermore, risk management is a critical part of this analysis. While higher returns are attractive, they often come with increased risk. Investors like Sarah must balance potential gains against their risk tolerance and investment horizon. Diversification, portfolio optimization, and careful asset allocation are strategies to mitigate the risks associated with high-yield investments.

In conclusion, financial planning hinges on the fundamental principles of present value and compound interest. Sarah Wiggum's scenario illustrates how different annual returns influence both her initial investment requirement and her retirement timeline. A thorough understanding of these concepts enables individuals to make informed decisions, optimize their investments, and achieve their financial goals efficiently.

References

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  • Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2019). Essentials of Corporate Finance. McGraw-Hill Education.
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