You Need 28974 At The End Of Nine Years And Your Only Invest

You Need 28974 At The End Of Nine Years And Your Only Investment Ou

You need $28,974 at the end of nine years, and your only investment outlet is an 8 percent long-term certificate of deposit (compounded annually). With the certificate of deposit, you make an initial investment at the beginning of the first year.

a. What single payment could be made at the beginning of the first year to achieve this objective?

b. What amount could you pay at the end of each year annually for 10 years to achieve this same objective?

Paper For Above instruction

You Need 28974 At The End Of Nine Years And Your Only Investment Ou

Investment Calculation to Achieve Future Value

Investing a specific amount today or making annual payments to reach a target amount in the future is a common financial planning scenario. In this context, the goal is to accumulate $28,974 in nine years through an 8 percent annual compound interest rate, with two different investment strategies: a lump sum investment at the start and annual payments over a decade. This paper explores the calculations needed to determine the initial lump sum and the annual payment required to meet this future financial goal.

Problem Breakdown and Financial Concepts

The problem involves two distinct future value (FV) calculations based on different payment structures: a single initial investment (lump sum) and an annuity of yearly payments. Both are compounded annually at 8% interest. Understanding the core principles of compound interest and future value formulas is essential for solving the problem accurately.

Part A: Calculating the Present Lump Sum Investment

The goal here is to determine the amount of money that must be invested at the start of the period to reach $28,974 after nine years with compound interest. The future value of a single lump sum is given by the formula:

FV = PV × (1 + r)^n

Where:

  • FV = future value ($28,974)
  • PV = present value or initial lump sum investment (unknown)
  • r = annual interest rate (8% or 0.08)
  • n = number of years (9)

Rearranging the formula to solve for PV:

PV = FV / (1 + r)^n

Plugging in the values:

PV = 28,974 / (1 + 0.08)^9

PV = 28,974 / (1.08)^9

Calculating (1.08)^9:

(1.08)^9 ≈ 1.999005

Therefore,

PV ≈ 28,974 / 1.999005 ≈ 14,514.91

Thus, an initial investment of approximately $14,514.91 made at the beginning of the first year would grow to $28,974 in nine years at an 8% compounded annual interest rate.

Part B: Calculating the Annual Payments Over 10 Years

Alternatively, making equal annual payments over ten years to reach the same future value involves treating the payments as an ordinary annuity. The future value of an ordinary annuity (where payments are made at the end of each period) is calculated as:

FV = Pmt × [(1 + r)^n – 1] / r

Where:

  • Pmt = annual payment (unknown)
  • FV = future value ($28,974)
  • r = annual interest rate (0.08)
  • n = number of payments (10)

Rearranged to solve for Pmt:

Pmt = FV × r / [(1 + r)^n – 1]

Plugging in the values:

Pmt = 28,974 × 0.08 / [(1.08)^10 – 1]

Pmt = 2,317.92 / (2.158924 – 1)

Pmt = 2,317.92 / 1.158924 ≈ 2,000.80

Therefore, making annual payments of approximately $2,000.80 at the end of each year for ten years would also grow to about $28,974 by the end of the tenth year, assuming an 8% interest rate compounded annually.

Discussion and Implications

The calculations demonstrate how the timing and structure of investments impact the required amounts to reach specific financial goals. A lump sum investment today requires approximately $14,514.91 initially and will grow accordingly, while systematic annual payments of about $2,000.80 over ten years can also achieve the same savings goal, provided the interest rate remains constant and compounding is annual. Financial planning often involves choosing between these strategies based on available funds and income schedules, and understanding these formulas allows for more informed decision-making.

Conclusion

In summary, to meet a future savings goal of $28,974 in nine years at an 8% annual compound interest rate, an initial lump sum investment of approximately $14,514.91 is necessary. Alternatively, making annual end-of-year payments of roughly $2,000.80 over ten years can also achieve the same goal. Mastery of these formulas provides a valuable toolset for investors and financial planners in designing effective investment strategies tailored to individual circumstances and financial objectives.

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