Case Application: Time Value Of Money - Richard E Mailed Me

Case Applicationtime Value Of Moneyrichard E Mailed Me That He And Mon

Richard E. emailed that he and Monica have differing opinions regarding the financial impact of his extra spending over the past 15 years. Richard calculated this as approximately $3,000 annually, believing that the total cost of $45,000 could be offset through investment. Conversely, Monica argued that the actual cost is significantly higher and requested an analysis of the situation. They were offered an investment opportunity of $20,000 that would grow to $70,000 over 20 years. They seek advice on whether to accept this investment. Additionally, Richard has the option to enroll in an employer-sponsored annuity costing $100,000 at age 65, which would provide annual payments of $8,000 over his estimated 17-year lifespan. The prevailing after-tax market rate of return on investments is 7 percent. Based on this scenario, the assignment involves analyzing their financial options and understanding the time value of money concepts involved.

Paper For Above instruction

The scenario presented by Richard and Monica highlights fundamental principles of personal finance, particularly the concept of the time value of money (TVM). Understanding TVM is crucial for making informed investment decisions, especially when evaluating whether to accept long-term investment opportunities or annuity options. This paper analyzes their situation by calculating the future value of the declared quarterly deficit, explaining the importance of compounding, assessing the return on the proposed investment, and evaluating the attractiveness of the annuity, ultimately illustrating the significance of TVM in personal financial planning.

1. Future Value of the $3,000-per-year deficit over 15 years

Richard estimated that the $3,000 annual deficit accrued over 15 years could be compensated if invested and compounded at a rate of 7 percent. To determine the future value of these annual deficits (which, when invested, would have grown), we employ the future value of an ordinary annuity formula, assuming annual contributions of $3,000 compounded yearly at a 7 percent rate. The formula is:

FV = P × [(1 + r)^n – 1] / r

where P is the annual payment ($3,000), r is the annual interest rate (0.07), and n is the number of years (15). Substituting the values:

FV = 3000 × [(1 + 0.07)^15 – 1] / 0.07

Calculating the compound factor:

(1.07)^15 ≈ 2.759

Thus:

FV ≈ 3000 × (2.759 – 1) / 0.07 ≈ 3000 × 1.759 / 0.07 ≈ 3000 × 25.114 ≈ $75,342

Therefore, if Richard had invested that $3,000 annually at 7% over 15 years, it would have accumulated approximately $75,342 by the end of the period.

2. Explanation of compounding and its impact on the cumulative amount

Compounding is the process where the returns or interest earned on an investment generate additional earnings over time. It effectively means earning interest on both the initial principal and the accumulated interest from previous periods. In the context of Richard's calculation, compounding allowed the invested annual deficits to grow exponentially, significantly increasing the accumulated amount over the 15-year period. The power of compounding accelerates wealth accumulation, especially over longer periods, by reinvesting interest earnings. This effect underscores why investing early and consistently can drastically enhance the growth of savings, highlighting the importance of the TVM concept in achieving long-term financial goals.

3. Return on the proposed $20,000 investment

The offer is to invest $20,000 today, which would grow to $70,000 in 20 years. To evaluate this, we calculate the compound annual growth rate (CAGR), which reflects the annualized return of the investment over the period:

CAGR = (Ending Value / Beginning Value)^(1/n) – 1

Plugging the values:

CAGR = (70,000 / 20,000)^(1/20) – 1 ≈ (3.5)^(0.05) – 1

Calculating the root:

3.5^0.05 ≈ e^{0.05 × ln(3.5)} ≈ e^{0.05 × 1.2528} ≈ e^{0.0626} ≈ 1.0646

Therefore:

CAGR ≈ 1.0646 – 1 ≈ 0.0646 or 6.46%

The expected annual return is approximately 6.46%, which is slightly below the target market rate of 7 percent. Factors influencing this return include transaction costs, taxes, market volatility, and the reliability of the projection. If we accept this investment, its risk profile and alignment with Richard and Monica's financial goals should be considered. Given the return slightly below the market rate, they should weigh whether the certainty of this investment compensates for the marginally lower return compared to market expectations or alternative investments with similar risk profiles.

4. Expected return and evaluation of the annuity

The annuity costs $100,000 at age 65 and provides annual payments of $8,000 over 17 years. To determine if this is a good deal, we calculate the present value (PV) of the annuity, using the market rate of 7 percent, and compare it to the cost. The PV of an ordinary annuity is given by:

PV = P × [1 – (1 + r)^–n] / r

where P = $8,000, r = 0.07, and n = 17:

PV = 8000 × [1 – (1 + 0.07)^–17] / 0.07

Calculating:

(1.07)^17 ≈ 3.276


(1.07)^–17 ≈ 1 / 3.276 ≈ 0.305

Thus:

PV ≈ 8000 × (1 – 0.305) / 0.07 ≈ 8000 × 0.695 / 0.07 ≈ 8000 × 9.929 ≈ $79,432

The present value of the annuity is roughly $79,432, which exceeds the cost of $100,000, implying that the annuity offers a lower value compared to its cost. Therefore, from a purely financial perspective, the annuity might not be an attractive investment unless Richard highly values the guaranteed income stream and risk mitigation. The expected return on this annuity, assessed by the internal rate of return (IRR), would be below the market rate, making it less appealing compared to other investment options that could yield higher returns.

5. Explaining the time value of money to Richard and Monica

The analysis of their financial options underscores the core principle of the time value of money—money available today is worth more than the same amount in the future because of its potential earning capacity. In practical terms, the future value of their investments is influenced by interest rates, compounding periods, and the duration of investments. As demonstrated, the $3,000 annual deficits, if invested at 7 percent, could accumulate over 15 years to a significant sum, illustrating how consistent saving and compound interest work together to grow wealth.

The evaluation of the $20,000 investment and the annuity further emphasize that accepting investments involves considering returns, risks, and the opportunity cost of funds. The marginally lower return on the proposed investment compared to the market rate indicates a trade-off between certainty and potential earnings. Similarly, calculating the present value of the annuity demonstrates how future income streams can be compared to their current costs, guiding decision-making in the context of personal financial goals.

Overall, understanding the TVM helps Richard and Monica make informed choices by quantifying how time, interest, and risk influence the value of money over the years. This comprehension enables them to evaluate which financial strategies align best with their long-term objectives, encouraging prudent financial planning and better resource management.

References

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