Busi 320 Comprehensive Problem 3 Use What You Have Learned

Busi 320 Comprehensive Problem 3use What You Have Learned About The Ti

Analyze each of the following decisions using the time value of money principle. Decide between multiple cash flow options based on their present value, computed at different interest rates. Then, evaluate different retirement saving strategies and determine the optimal approach for maximizing retirement savings, considering various contribution timelines, rates of return, and starting points.

Paper For Above instruction

The concept of the time value of money (TVM) is fundamental in financial decision-making, reflecting how the value of money changes over time due to potential earning capacity. Understanding how to evaluate options through present value (PV) calculations allows individuals to make informed choices that maximize their financial benefits. This paper provides a comprehensive analysis of two primary decision scenarios: selecting the most valuable cash flow set and determining optimal retirement saving strategies, using TVM principles.

Decision 1: Comparing Cash Flow Options

The first decision involves choosing among three cash flow options offered by a grandmother: a one-time gift of $10,000 today, an annuity of $1,600 annually for ten years, or a lump sum of $20,000 received after ten years. To determine the most valuable option, the present value of each needs to be calculated at different interest rates—4%, 7%, and 10%—over a ten-year period.

For Option A, receiving $10,000 today, the present value is straightforward; it equals $10,000 regardless of the discount rate since it is a cash flow received immediately. For Option B, receiving $1,600 annually, the PV is calculated using the present value of an annuity formula:

PV = P × [(1 - (1 + r)^-n) / r]

where P = annual payment ($1,600), r = interest rate, n = number of periods (10). At each rate, the PV of Option B decreases as the rate increases. Option C involves receiving $20,000 after ten years, which is a single future sum, discounted back to PV:

PV = FV / (1 + r)^n

where FV = $20,000. At each rate, the PV of Option C diminishes with increasing interest rates.

At 4%:

  • Option A: $10,000
  • Option B: PV = 1,600 × [(1 - (1 + 0.04)^-10) / 0.04] ≈ $13,656
  • Option C: PV = 20,000 / (1 + 0.04)^10 ≈ $13,675

Financial theory suggests selecting the option with the highest present value; hence, at 4%, Option C slightly edges out Option B.

Repeating similar calculations at 7% and 10% interest rates, the PVs decrease for Options B and C, but the relative advantages shift slightly depending on the rate. Generally, at higher rates, earlier cash flow options tend to be more favorable due to the higher discounting effect on future sums.

Decision 2: Retirement Planning Analysis

The second scenario involves Tom and Tricia planning for retirement with different contribution strategies. The key variables include the timing of contributions, the amount contributed annually ($2,400), the duration, and the expected annual rate of return (8%). They consider three strategies and a hypothetical situation to reverse-engineer their necessary contributions.

a) Saving nothing for 10 years, then contributing $2,400 annually for the next 35 years:

First, compute the future value of the 35-year annuity of $2,400 at 8% interest, which then acts as a lump sum at retirement:

FV = P × [( (1 + r)^n - 1) / r]

where P = $2,400, r = 0.08, n = 35. Plugging in figures yields:

FV ≈ $2,400 × [( (1 + 0.08)^35 - 1) / 0.08] ≈ $2,400 × 162.89 ≈ $391,005

Since they wait ten years to start saving, the accumulated amount at that point will also accrue interest over those early 10 years, compounded at 8%. Future value calculation includes growth during the initial period:

FV at 45 years = $391,005 × (1.08)^10 ≈ $391,005 × 2.1589 ≈ $844,072

b) Contributing $2,400 annually for 10 years, then doing nothing:

The future value after 10 years of annual contributions:

FV at year 10 = $2,400 × [( (1 + 0.08)^10 - 1) / 0.08] ≈ $2,400 × 14.486 ≈ $34,769

This amount then remains invested untouched for the remaining 35 years, accruing interest:

FV at retirement = $34,769 × (1.08)^35 ≈ $34,769 × 30.365 ≈ $1,055,812

c) Contributing $2,400 annually for all 45 years:

Calculating the future value of the entire 45-year annuity:

FV = $2,400 × [( (1 + 0.08)^45 - 1) / 0.08] ≈ $2,400 × 224.338 ≈ $537,211

d) To accumulate $925,000 at retirement after 45 years with total current contributions:

Assuming Tom and Tricia can contribute only for 20 years, and aiming for a future value of $925,000, the annual contribution P can be calculated:

$925,000 = P × [( (1 + 0.08)^20 - 1) / 0.08] × (1.08)^25

Calculating the bracketed term:

FV annuity factor = ( (1.08)^20 - 1) / 0.08 ≈ 51.725

Future value of 20-year contributions after 25 years of growth:

FV = P × 51.725 × (1.08)^25 ≈ P × 51.725 × 6.848 ≈ P × 354.565

Thus, P = $925,000 / 354.565 ≈ $2,607 per year.

Therefore, Tom and Tricia need to save approximately $2,607 annually for 20 years to reach their goal of $925,000 at retirement.

Conclusion

Financial theory emphasizes that selecting the best investment options depends heavily on discount rates and timing. The analyses demonstrate that lump-sum investments or early contributions tend to yield higher amounts due to the power of compounding. For retirement planning, starting early and making consistent regular contributions maximizes accumulated wealth, confirming the importance of time-aware investment strategies. Conversely, delaying savings diminishes potential growth, highlighting the benefits of early action and the significance of rate assumptions in financial planning.

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