Future Value Of A Complex Annuity: Mr. Smith, Age 80

future Value Of A Complex Annuity Mogul Mr Smith Age 80 Wants T

Determine the future value of a complex annuity and the required deposit amounts for Mr. Smith's retirement plan. Specifically, calculate the amount Mr. Smith will need at age 100 to fund his retirement withdrawals, and find the size of the annual deposits he must make into his retirement account to reach that goal, given the interest rate and repayment period. Additionally, evaluate the present value of three investment options with different cash flows, using a specified discount rate.

Paper For Above instruction

Retirement planning involves complex calculations of present and future values of annuities, especially when dealing with irregular cash flows and multiple investment options. This paper addresses the numerical methods and formulas necessary to determine the required savings and fund values in such scenarios, focusing on Mr. Smith's retirement case and three investment alternatives.

Future Value of a Complex Annuity and Corresponding Deposit Calculations

Mr. Smith, aged 80, aims to retire at age 100 and secure annual withdrawals of 1.2 billion at the beginning of each year for 8 years. To prepare for this, he plans to make 20 equal end-of-year deposits into an account earning 26% annually. To determine how much Mr. Smith needs at age 100 and the deposits required, we apply complex annuity formulas involving future value (FV), present value (PV), and annuity components.

Part A: Determining the Future Value Needed at Retirement

The goal is to find the amount Mr. Smith must have in his account at age 100 to fund his withdrawals. Since withdrawals occur at the beginning of each year, this constitutes an annuity due. The annual withdrawal is 1.2 billion, the withdrawal period is 8 years, and the account interest rate is 26%.

The formula for the future value of an annuity due (FV of withdrawals) is:

FV = P [(1 + r)^n - 1] (1 + r) / r

Where:

  • P = annual withdrawal = 1.2 billion
  • r = annual interest rate = 26% = 0.26
  • n = number of withdrawals = 8

Calculating the needed amount at age 100 involves summing the present value of these withdrawals at age 100, which is computed by discounting each payment back to age 100, or equivalently, by computing the present value of the annuity at age 100 considering withdrawals' timing.

Part B: Calculating the Required Annual Deposits

Next, we determine the size of the annual end-of-year deposits Mr. Smith must make over 20 years beginning at age 80 to accumulate the necessary amount at age 100. This is a future value of an ordinary annuity (series of equal deposits), which is expressed as:

FV = PMT * [(1 + r)^t - 1] / r

Where:

  • PMT = annual deposit (unknown)
  • r = 26% = 0.26
  • t = number of deposits = 20

Rearranged to solve for PMT:

PMT = FV * r / [(1 + r)^t - 1]

Once the FV required at age 100 (from Part A) is computed, substituting it in this formula yields the annual deposit amount needed.

Present Value of Investments with Different Cash Flows

Given three investment options with specified cash flows over a period and an annual discount rate of 22%, the present value (PV) of each investment is calculated by discounting each cash flow back to the present using the formula:

PV = Σ (Cash Flow in Year t) / (1 + r)^t

Here, sum over all cash flows, applying the discount rate of 22%. The calculation provides a clear comparison of each investment's current worth based on future cash flows, aiding investment decision-making.

Conclusion

Understanding and applying the formulas for future value, present value, and annuity calculations are essential for effective retirement planning and investment analysis. By using these formulas, individuals can optimize their savings strategies and evaluate different investment options based on quantitative metrics. Accurate computation ensures sufficient funding for retirement needs and informed investment choices, which are vital aspects of financial planning.

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