Let's break down the problem step by step: ### **Problem Recap** - **Success ratio**: 10% (Only 10% of investments succeed, 90% fail) - **Portfolio required rate of return**: 19.4% - **Loan period**: 5 years - **Loan amount**: $101,000 - **Find**: The annual percentage rate (APR) you must pay. --- ### **Step 1: Understanding the Venture Capital Math** Since only 10% of investments succeed, the successful investments must earn enough to cover all investments and provide the required portfolio return of 19.4%. Let \( r \) be the required annual return on your loan. #### **Future Value Needed** In 5 years, Blue Angel wants its portfolio to grow by 19.4% per year: \[ \text{Future Value (FV) of $101,000 in 5 years at 19.4%:} \] \[ FV = PV \times (1 + 0.194)^5 \] \[ FV = 101,000 \times (1.194)^5 \] \[ FV = 101,000 \times 2.4283 = 245,258.30 \] But only 10% of investments succeed, so each successful investment must return enough to cover all (including failed ones): \[ \text{Each successful investment must return:} \] \[ \text{Required FV per successful investment} = \frac{245,258.30}{0.10} = 2,452,583 \] --- ### **Step 2: Find Required Annual Rate (\( r \))** Set up the equation for the loan to grow from $101,000 to $2,452,583 in 5 years: \[ 101,000 \times (1 + r)^5 = 2,452,583 \] \[ (1 + r)^5 = \frac{2,452,583}{101,000} = 24.27 \] \[ 1 + r = (24.27)^{1/5} \] \[ 1 + r = 1.8921 \] \[ r = 1.8921 - 1 = 0.8921 \] \[ r = 89.21\% \] --- ### **Step 3: Final Answer** **The annual percentage rate for the loan is**: \[ \boxed{89.21\%} \] (Rounded to four decimal places: **89.2100%**)

**1. Introduction** Understanding the concepts of venture capital investments, expected returns, and risk-adjusted performance metrics is essential to evaluate the required rates of return on investments. Venture capital investments are characterized by high risk and potential high rewards, often involving a significant probability of failure. To compensate for the high risk, investors demand a higher rate of return compared to traditional investments. When assessing the appropriate rate to charge or expect on a venture loan, it is critical to consider the success ratio, the desired portfolio return, and the time horizon of the investment. In this context, the success ratio indicates the probability that an individual investment will succeed, contributing to overall portfolio performance. The venture capital required rate of return, in conjunction with the success ratio and the time horizon, informs the calculation of the necessary annual percentage rate (APR) for a specific loan. The challenge lies in adjusting the expected return to account for the low probability of success, ensuring that the overall portfolio achieves the targeted return. This introduction provides the foundation for analyzing the problem by linking key concepts such as success ratios, return requirements, and time value of money, which are integral to determining the appropriate loan rate. **Explanation:** The introductory concepts clarify the importance of success probability, target returns, and time horizon in financial decision-making related to high-risk investments. Recognizing that venture capital investments involve a low success rate but high potential payoff helps in understanding why the required rate of return must be adjusted accordingly. This foundational knowledge prepares for the subsequent application of mathematical formulas to derive the necessary annual percentage rate. --- **2. Presentation of Relevant Formulas Required To Solve The Question** **a) Future Value (FV) formula for compound interest:** \[ FV = PV \times (1 + r)^t \] - *Description:* Calculates the accumulated amount after \(t\) years when invested at an annual rate \(r\). - *Applicability:* Used to determine how much an initial investment will grow over the investment horizon at a given annual rate. **b) Adjustment for success ratio in expected return:** \[ \text{Required FV per successful investment} = \frac{\text{Target FV of the portfolio}}{\text{Success probability}} \] - *Description:* Since only a fraction of investments succeed, the expected returns from successful investments must be scaled up to compensate for failures. **c) Solving for the annual rate \(r\):** \[ (1 + r)^t = \frac{\text{Adjusted FV}}{PV} \] \[ r = \left(\frac{\text{Adjusted FV}}{PV}\right)^{1/t} - 1 \] - *Description:* Rearranged to find the annual rate \(r\) needed to grow the initial investment \(PV\) to the adjusted future value over \(t\) years. --- **Explanation:** These formulas facilitate translating the desired portfolio return into a per-investment required return, considering success probability and time horizon. The core concept involves understanding how compound interest impacts growth and how risk adjustment inflates the necessary returns to compensate for low success rates. Using these formulas ensures a precise calculation aligned with financial principles. --- **3. A Detailed Step-by-Step Solution** **Step 1: Calculate the future value of the portfolio at the desired return** Given: - Initial investment (PV) = $101,000 - Target portfolio return (annual) = 19.4% or 0.194 - Time horizon, \(t\) = 5 years Calculate the future value (FV) after 5 years: \[ FV = 101,000 \times (1 + 0.194)^5 \] \[ FV = 101,000 \times (1.194)^5 \] Calculate \((1.194)^5\): \[ (1.194)^5 \approx 2.4283 \] Thus: \[ FV = 101,000 \times 2.4283 \approx 245,258.30 \] **Explanation:** This step determines the amount Blue Angel Investors expects to have after 5 years to meet its target return, providing the future value benchmark for the investment. --- **Step 2: Adjust for the success probability** Given success ratio: - Probability of success = 10% or 0.10 Since only 10% of investments succeed, each successful investment must generate enough to cover the entire portfolio goal, scaled up by the inverse of success probability: \[ \text{Adjusted FV} = \frac{FV}{\text{Success probability}} = \frac{245,258.30}{0.10} = 2,452,583 \] **Explanation:** This adjustment accounts for the fact that only a fraction of investments will succeed; thus, successful investments need to produce a higher return to ensure the overall portfolio reaches the target. --- **Step 3: Calculate the required annual rate \(r\)** Using the compound interest formula: \[ (1 + r)^5 = \frac{\text{Adjusted FV}}{PV} = \frac{2,452,583}{101,000} \approx 24.27 \] Solve for \(r\): \[ 1 + r = (24.27)^{1/5} \] Calculate the 5th root: \[ (24.27)^{1/5} \approx 1.8921 \] Therefore: \[ r = 1.8921 - 1 = 0.8921 \] Expressed as a percentage: \[ r = 89.21\% \] **Explanation:** This step determines the annual percentage rate needed on the loan so that, after compounding over five years, the investment grows to the adjusted future value, accounting for the low success probability. --- **Final Answer:** The annual percentage rate required for the loan is **89.2100%**. --- **Conclusion:** By accounting for the low success probability, the targeted portfolio return, and the time horizon, it is evident that a substantially high annual rate—approximately 89.21%—must be charged to meet the desired investment objectives in a venture capital context characterized by high risk and low success probability.