Module 06 Questions And Problems: Calculating Pv And Fv

Module 06 Questionsproblems Calculating Pv And Fv Using Online Calc

Calculate the future value (FV) of a $25,000 investment made today in a mutual fund expected to grow at an 8% annual rate over various periods: 3, 6, 9, and 12 years. Repeat the calculations assuming the rate increases to 10% for the same periods. Compare the values after 12 years with the different interest rates.

Determine the present value (PV) of a $120,000 investment received at the end of year 5 for discount rates of 3%, 6%, 9%, and 12%, respectively.

Calculate the future value of a series of annual $25,000 investments made at the end of each year, earning 5% over 35 years, assuming the investment is made immediately at the start of the period.

Paper For Above instruction

Financial mathematics plays a crucial role in personal and corporate decision-making, especially concerning investments. Understanding the concepts of present value (PV) and future value (FV) allows individuals and organizations to evaluate the worth of current investments and forecast the potential returns based on different interest rates and time horizons. This paper explores these core concepts through practical calculations, illustrating their significance in real-world financial scenarios.

Future Value Calculations of Lump-Sum Investment

The future value of an investment is the amount it will grow to over a specified period, considering a particular growth rate or interest rate. This calculation is fundamental for investors aiming to understand how their current investments will evolve. The formula for FV in case of compound interest is:

FV = PV × (1 + r)^t

where PV is the present value or initial deposit, r is the annual interest rate, and t is the time in years.

Applying this formula, consider a $25,000 initial deposit in a mutual fund with an 8% growth rate. For 3 years, the future value is:

FV = $25,000 × (1 + 0.08)^3 ≈ $25,000 × 1.2597 ≈ $31,493.50

Similarly, for 6, 9, and 12 years, the FV calculations are:

• 6 years: FV = $25,000 × (1.08)^6 ≈ $25,000 × 1.5869 ≈ $39,672.50
• 9 years: FV = $25,000 × (1.08)^9 ≈ $25,000 × 2.0066 ≈ $50,165
• 12 years: FV = $25,000 × (1.08)^12 ≈ $25,000 × 2.6137 ≈ $65,342.50

When the interest rate increases to 10%, the FV for the same periods are:

• 3 years: FV = $25,000 × (1.10)^3 ≈ $25,000 × 1.331 ≈ $33,275
• 6 years: FV = $25,000 × (1.10)^6 ≈ $25,000 × 1.772 ≈ $44,299
• 9 years: FV = $25,000 × (1.10)^9 ≈ $25,000 × 2.357 ≈ $58,935
• 12 years: FV = $25,000 × (1.10)^12 ≈ $25,000 × 3.138 ≈ $78,450

Comparison of these values after 12 years shows that higher interest rates significantly increase the accumulated value of the investment, emphasizing the importance of rate selection in long-term investing.

Present Value Calculations of a Future Sum

Present value is a fundamental concept that discounts future cash flows to their current worth, considering a specific discount rate. The PV formula is:

PV = FV / (1 + r)^t

For an investment of $120,000 to be received at the end of year 5, the PV under different discount rates is:

• 3%: PV = $120,000 / (1.03)^5 ≈ $120,000 / 1.1593 ≈ $103,448
• 6%: PV = $120,000 / (1.06)^5 ≈ $120,000 / 1.3382 ≈ $89,734
• 9%: PV = $120,000 / (1.09)^5 ≈ $120,000 / 1.5386 ≈ $77,967
• 12%: PV = $120,000 / (1.12)^5 ≈ $120,000 / 1.7623 ≈ $68,096

This analysis demonstrates how higher discount rates decrease the present value of a future sum, which is critical for valuation and decision-making related to investments and liabilities.

Future Value of an Annuity (Series of Equal Payments)

The calculation of the future value of an ordinary annuity, where payments are made periodically at the end of each period, uses the formula:

FV = P × [( (1 + r)^t ) - 1 ] / r

However, if payments are made at the beginning of each period (an annuity due), an additional multiplication by (1 + r) is necessary.

In this scenario, an obstetrician plans to invest $25,000 annually at 5% for 35 years, beginning immediately, which constitutes an annuity due. The FV is calculated as:

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

Substituting the values:

FV = $25,000 × [ ( (1.05)^35 - 1 ) / 0.05 ] × 1.05

Calculating:

( (1.05)^35 ≈ 6.3859 ), so:

FV = $25,000 × (6.3859 - 1) / 0.05 × 1.05 ≈ $25,000 × 5.3859 / 0.05 × 1.05 ≈ $25,000 × 107.7178 × 1.05 ≈ $25,000 × 113.1037 ≈ $2,827,593

Thus, the retirement account will be worth approximately $2.83 million after 35 years, emphasizing the power of consistent investing and compounding interest over the long term.

Conclusion

Understanding and accurately calculating the present and future values of investments are foundational for effective financial planning. The examples illustrated demonstrate how varying interest and discount rates impact outcomes significantly. Investors and financial managers must consider these factors carefully to optimize their strategies, whether making lump-sum investments, evaluating future liabilities, or planning for retirement. Leveraging online calculators aids in these computations, providing quick and precise insights to inform decision-making.

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