Brief Exercise 6-11: Chris Spear Invested $12,720 Today

Brief Exercise 6 11 Chris Spear Invested 12720today In A Fund That

Chris Spear invested $12,720 today in a fund that earns 6% compounded annually. To determine the future value of the investment after 3 years with annual compounding, we use the future value formula: FV = PV × (1 + r)^t, where PV is the present value, r is the annual interest rate, and t is the time in years. Using the given data, FV = $12,720 × (1 + 0.06)^3. Calculating this, FV ≈ $12,720 × 1.191016 ≈ $15,150. This matches the provided investment at 6% annual interest for 3 years.

For semiannual compounding, the interest rate per period is halved (6%/2 = 3%), and the number of periods doubles (2 × 3 = 6). The future value formula becomes FV = PV × (1 + r/n)^(nt), where n is the number of compounding periods per year. Substituting, FV = $12,720 × (1 + 0.03)^6 ≈ $12,720 × 1.194052 ≈ $15,182.25. Rounding to the nearest dollar, the future value is approximately $15,182.

Paper For Above instruction

Given the scenario of Chris Spear’s investment, we explore the impact of different compounding periods on the future value of the investment and extend this analysis to multiple financial situations, including present value calculations, interest rate determinations, and financial decision-making based on time value of money principles.

Calculating Future Value with Annual and Semiannual Compounding

Chris Spear’s initial investment of $12,720 at a 6% annual interest rate results in a future value of approximately $15,150 after three years when compounded annually. This calculation uses the classic compound interest formula FV = PV × (1 + r)^t. The annual interest rate remains constant, and the interest is compounded once per year, leading to predictable growth.

In contrast, semiannual compounding involves dividing the annual interest rate by two, resulting in 3% per half-year, and doubling the number of compounding periods over the same duration. This results in a slightly higher future value because interest accumulates more frequently. Using the formula FV = PV × (1 + r/n)^(nt), where n = 2, yields an approximate future value of $15,182.25 after three years, demonstrating the slight increase due to more frequent compounding.

Present Value and Investment Decisions

Transitioning from future value calculations, Tony Bautista’s need for $21,080 in two years, with a 12% annual interest rate, showcases the reverse operation: present value computation. The present value PV = FV / (1 + r)^t. Plugging in the figures, PV ≈ $21,080 / (1 + 0.12)^2 ≈ $21,080 / 1.2544 ≈ $16,805. This amount represents the required initial investment today to meet his future goal.

Furthermore, if Bautista’s investment earns interest quarterly instead of annually, the effective interest rate per quarter becomes 3% (12% / 4), and the total number of periods becomes 8 (4 quarters per year × 2 years). Using the formula for compound interest with periodic compounding, the present value becomes PV = FV / (1 + r/n)^(nt). The calculations yield a similar essential conclusion, with minor differences attributable to the more frequent compounding.

Interest Rate and Investment Growth

For Candice Willis, who aims to grow her $11,609 to $190,000 in 20 years, we can determine the required annual interest rate. The future value formula rearranged to solve for r: FV = PV × (1 + r)^t, or r = (FV / PV)^(1/t) – 1. Substituting, r ≈ (190,000 / 11,609)^(1/20) – 1 ≈ 15%. This indicates she must earn approximately 15% annually to reach her goal.

Accumulation of Funds and Time Value of Money

John Fillmore’s inheritance of $434,800, with the goal of purchasing a boat costing $333,900 in 7 years, illustrates the need to determine how much of his inheritance to invest at an 11% annual rate. Using the present value factor from tables, or the formula PV = FV / (1 + r)^t, the investment needed today is approximately $160,826. This ensures that, at 11% compounded annually, the fund will grow sufficiently to fund his retirement purchase.

Interest Computations and Investment Growth

Alan Jackson’s $47,000 investment at 10% for 10 years, with interest compounded differently, showcases the difference between simple and compound interest. Under simple interest, total interest is PV × r × t, leading to a total withdrawal of approximately $94,000. For compound interest, using the compounding formula, the final amounts over 10 years with annual and semiannual compounding are approximately $121,906 and $124,705 respectively, reflecting the effect of compounding frequency.

Interest Rate on Loan and Time Value of Money

LEW Company’s purchase of a machine costing $109,000 with a note payable of $141,158 in three years involves calculating the implied interest rate. Using the present value formula PV = FV / (1 + r)^t, the interest rate r ≈ 9%, consistent with the provided information.

Future Value and Investment Planning

Mike Finley’s goal to become a millionaire with a current balance of $299,246 earning 9% annually involves finding the time horizon. Using the future value formula FV = PV × (1 + r)^t, solving for t, yields approximately 14 years. Conversely, Sally Williams seeks to attain $1 million in 17 years from $371,365, requiring an annual interest rate of roughly 5%, calculated via the compound interest formula rearranged for r.

Discounting Future Receipts and Lease Accounting

The valuation of $300,000 received in the future at 10% discount rate indicates a present value of approximately $153,948, which implies a 7-year horizon using the discounting formula PV = FV / (1 + r)^t. Additionally, IFRS-compliant lease accounting requires recognizing the present value of lease payments, including security deposits with interest, resulting in a net receivable of approximately $64,844 after 10 years at 10% interest.

Assessing Zero-Interest Notes and Warranty Provisions

Barton Company’s sale involving a zero-interest note is valued using the market rate of 8%. The present value of the note’s face value is calculated to record the sale at its fair value, resulting in a discount recorded on the books. Jamison Company’s warranty expense estimate, based on historical defect and repair costs, amounts to approximately $198,000, representing the expected future warranty obligations.

Conclusion

The various scenarios demonstrate the critical role of the time value of money in personal and corporate finance. From calculating future and present values to determining interest rates and investment horizons, these principles underpin sound financial decision-making. Accurate application of compound interest formulas, discounting techniques, and understanding of interest compounding frequencies are essential skills for financial professionals and individuals alike, enabling optimal planning, investment growth, and risk management.

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