Let's break down the problem: **Given:** - Success ratio (probability of success, \( p \)) = 10% = 0.10 - Required portfolio rate of return (\( r_p \)) = 19.4% = 0.194 - Average length of loan (\( n \)) = 5 years - Loan amount (\( PV \)) = $101,000 **Question:** What is the **annual percentage rate (APR)** you would have to pay for this loan? --- ### Step 1: Required Return per Successful Loan Because only 10% of the loans succeed, successful loans must generate enough return to cover both the required overall portfolio return and the losses from failed loans. Formula: \[ \text{Required future value per successful loan} = \frac{(1 + r_p)^n \times \text{Loan Amount}}{p} \] --- ### Step 2: Plug in the Values \[ (1 + 0.194)^5 = 2.4247 \] \[ \text{Future value required} = \frac{2.4247 \times 101,000}{0.10} = \frac{244,894.7}{0.10} = 2,448,947 \] --- ### Step 3: Find the Required Annual Rate (\( r \)) Solve for \( r \) in the Future Value formula: \[ \text{Future Value} = \text{Loan Amount} \times (1 + r)^n \] \[ 2,448,947 = 101,000 \times (1 + r)^5 \] \[ \frac{2,448,947}{101,000} = (1 + r)^5 \] \[ 24.244 = (1 + r)^5 \] \[ 1 + r = (24.244)^{1/5} \] \[ 1 + r = 1.8921 \] \[ r = 1.8921 - 1 = 0.8921 \] --- ### Step 4: Convert to Percentage \[ \boxed{89.21\%} \] **Final Answer (rounded to four decimal places):** \[ \boxed{89.2100\%} \] --- **You would have to pay an annual percentage rate of 89.2100%.**

**1. Introduction** Understanding the concept of venture capital required rate of return involves recognizing the inherent risks and expected returns associated with early-stage investments. Venture capital investments typically entail high uncertainty and a significant probability of failure; therefore, investors demand a higher rate of return to compensate for these risks. The required rate of return reflects the minimum annual percentage gain an investor expects to earn to make the investment worthwhile, considering the potential for failure and the time value of money. In the context of a portfolio of loans or investments, the overall expected return must account for the success probability, the desired return on successful investments, and the average duration of the investment. When a venture capital firm like Blue Angel Investors seeks funding, the rate it must charge on its loans must align with its risk-adjusted return expectations to ensure profitability across its portfolio. This involves calculating an effective annual percentage rate (APR) that compensates for the risk of failure, time horizon, and targeted overall return. **Explanation:** This introduction is essential because it lays the foundation for understanding how risk and return interact in venture capital and lending scenarios. Recognizing the significance of the success probability, desired portfolio return, and investment duration allows for the formulation of the appropriate rate calculations. It prepares for analyzing how the required rate of return influences the interest rate that must be charged on a specific loan, enabling accurate financial decision-making within high-risk environments. --- **2. Presentation of Relevant Formulas Required To Solve The Question** - **Future Value (FV) of an investment:** \[ FV = PV \times (1 + r)^n \] *Where:* \(PV\) = Present value or initial loan amount \(r\) = annual interest rate (decimal form) \(n\) = number of years - **Adjustment for success probability in expected return:** To determine the total amount needed to compensate for the probability of success, the formula becomes: \[ \text{Adjusted Future Value} = \frac{FV_{desired}}{p} \] *Where:* \(p\) = success probability (decimal) - **Solving for the required annual interest rate (\(r\)) when future value is known:** \[ r = \left( \frac{FV}{PV} \right)^{1/n} - 1 \] *Rationale:* The first formula calculates how much an initial investment must grow over the period to meet the target return. The second adjusts for the success probability, recognizing that only a fraction of investments succeed, thus inflating the required future value to meet the overall portfolio return. The third formula isolates the annual interest rate necessary to grow the specific loan amount to the adjusted future value over the specified number of years. **Explanation:** These formulas are crucial because they translate the desired overall return and success probability into a specific annual rate that must be charged on individual loans. They underpin the mathematical process of deriving the interest rate that compensates for risk, time, and target returns, ensuring the calculations align with fundamental financial principles. --- **3. A Detailed Step-by-Step Solution** **Step 1: Determine the growth factor for the target portfolio return over 5 years** \[ (1 + r_{portfolio})^{n} = (1 + 0.194)^5 \] Calculate: \[ 1 + 0.194 = 1.194 \] \[ (1.194)^5 \approx 2.4247 \] *Explanation:* This step computes how much the initial investment must grow over 5 years to achieve a 19.4% annual return compounded annually. It forms the basis for understanding the total growth needed for the portfolio to meet its return target. --- **Step 2: Calculate the total future value required per successful loan** \[ FV_{required} = \frac{\text{Growth factor} \times \text{Loan amount}}{p} \] \[ FV_{required} = \frac{2.4247 \times 101,000}{0.10} = \frac{244,894.7}{0.10} = 2,448,947 \] *Explanation:* Since only 10% of loans succeed, the successful loans must collectively generate enough value to cover the portfolio's target return. Dividing by success probability inflates the required future value, ensuring that the expected successful outcomes meet the overall return goal. --- **Step 3: Solve for the annual interest rate \(r\)** \[ (1 + r)^n = \frac{FV_{required}}{PV} \] \[ (1 + r)^5 = \frac{2,448,947}{101,000} \approx 24.244 \] \[ 1 + r = (24.244)^{1/5} \] Calculate the fifth root: \[ (24.244)^{1/5} \approx 1.8921 \] \[ r = 1.8921 - 1 = 0.8921 \] Expressed as a percentage: \[ r = 89.21\% \] *Explanation:* Solving for \(r\) involves taking the fifth root of the total growth factor, which yields the annual rate needed to grow the original loan amount to the adjusted future value over five years. The high rate reflects the risk premium associated with the low success probability and the desired portfolio return. --- **Conclusion:** The annual percentage rate that must be charged on the loan to meet the venture capital firm’s return objectives, considering the success probability, is **89.2100%**. This rate compensates for the high risk of failure and the long-term growth needed to achieve the targeted 19.4% portfolio return over five years. --- **Summary:** By calculating the required growth to meet the portfolio's return, adjusting for the success probability, and solving for the annual interest rate, it is determined that an approximate APR of **89.2100%** is necessary for the loan. This comprehensive approach ensures that all risk, time, and return factors are incorporated into the interest rate calculation.