Let's break down the problem step by step: ### **Given:** - **Option 1:** $100,000 today - **Option 2:** $300,000 in 13 years - **Discount rate (interest rate):** 11% per year - **Question:** What is the present value (PV) of $300,000 received in 13 years at 11% interest? --- ### **Step 1: Present Value Formula** \[ PV = \frac{FV}{(1 + r)^n} \] Where: - \(FV = \$300,000\) - \(r = 0.11\) - \(n = 13\) --- ### **Step 2: Calculate the Denominator** \[ (1 + 0.11)^{13} = (1.11)^{13} \] Let's calculate \((1.11)^{13}\): - \(1.11^1 = 1.11\) - \(1.11^2 = 1.2321\) - \(1.11^3 \approx 1.3676\) - \(1.11^4 \approx 1.5180\) - \(1.11^5 \approx 1.6849\) - \(1.11^6 \approx 1.8702\) - \(1.11^7 \approx 2.0759\) - \(1.11^8 \approx 2.3042\) - \(1.11^9 \approx 2.5577\) - \(1.11^{10} \approx 2.8391\) - \(1.11^{11} \approx 3.1514\) - \(1.11^{12} \approx 3.4981\) - \(1.11^{13} \approx 3.8829\) --- ### **Step 3: Compute Present Value** \[ PV = \frac{300,000}{3.8829} \approx 77,270.20 \] --- ## **Final Answer** **The present value of $300,000 received in 13 years at 11% is:** \[ \boxed{\$77,270.20} \] --- ### **Decision** Since $100,000 today is greater than $77,270.20, you should choose the $100,000 today.

**Introduction** Understanding the concept of present value is fundamental in financial decision-making, especially when comparing cash flows occurring at different points in time. Present value (PV) quantifies how much a future sum of money is worth today, considering a specific discount or interest rate. The core idea relies on the principle that money available today can be invested to earn interest, thus increasing its value over time. Conversely, future cash flows can be discounted back to their value in today's terms to facilitate comparison with immediate sums. In the context of the given problem, the key concepts involve calculating the present value of a future sum of money using a specified annual interest rate, often called the discount rate. This enables an investor to determine whether accepting a lump sum today or a larger sum in the future is more advantageous, by comparing their respective present values. **Explanation:** This introduction emphasizes the importance of present value calculations in evaluating investments and financial choices. It links the theoretical understanding of time value of money with practical decision-making, providing a foundation for analyzing the specific problem involving a choice between immediate and future sums of money. --- **Presentation of Relevant Formulas Required To Solve The Question** 1. **Present Value Formula** \[ PV = \frac{FV}{(1 + r)^n} \] - **Description:** This formula calculates the current worth (PV) of a future sum of money (FV) to be received after \(n\) years, discounted at an annual interest rate \(r\). - **Rationale:** The denominator \((1 + r)^n\) accounts for compounding over \(n\) years at rate \(r\), reflecting how the value of money grows over time when invested. By dividing the future value by this factor, the formula effectively 'discounts' the future sum back to its present value. 2. **Future Value Formula (for context, not directly needed here)** \[ FV = PV \times (1 + r)^n \] - **Description:** Calculates the amount of money accumulated after \(n\) years, given an initial principal \(PV\) invested at interest rate \(r\). - **Relevance:** Understanding this formula underscores the inverse relationship between present value and future value, reinforcing why discounting is necessary to compare sums across different time frames. **Explanation:** These formulas are essential tools for financial analysis, enabling the transformation of future sums into present terms. They underpin the calculations necessary to evaluate the attractiveness of different cash flow options based on an established rate of return. --- **A Detailed Step-by-Step Solution** **Step 1: Identify the relevant values** - Future value \(FV = \$300,000\) - Discount rate \(r = 0.11\) (11%) - Number of years \(n = 13\) *Explanation:* These parameters are directly provided or inferred from the problem. The initial step involves recognizing the variables needed for the present value calculation, ensuring clarity before proceeding. --- **Step 2: Calculate the discount factor \((1 + r)^n\)** \[ (1 + 0.11)^{13} = 1.11^{13} \] - Compute \(1.11^{13}\): \[ 1.11^{13} \approx 3.8829 \] *Supporting Statement:* The exponential calculation involves raising 1.11 to the 13th power, which accounts for compounding over 13 years at 11% annually. This factor indicates how much the future sum is 'discounted' to arrive at its present value. --- **Step 3: Calculate the present value** \[ PV = \frac{\$300,000}{3.8829} \approx \$77,270.20 \] *Explanation:* Dividing the future value by the discount factor converts the future sum into its present value. This step quantifies what the future amount of \$300,000 is worth today if invested at 11% annual return over 13 years. --- **Conclusion** The present value of \$300,000 to be received in 13 years at an 11% discount rate is approximately **\$77,270.20**. Comparing this to the immediate offer of \$100,000 reveals that the current sum offers a higher value in present terms. Therefore, based solely on the present value calculation, accepting \$100,000 today is the more advantageous choice. --- **Final summary:** - The present value of \$300,000 in 13 years at 11% interest rate is approximately **\$77,270.20** - Since \$100,000 today exceeds this amount, the best option is to take the immediate payment.