5.1 You are managing an endowment fund that is designed to provide perpetual annual payments for scholarships. The current value of the endowment is $2,000,000. 1. If the fund generates annual payments of $100,000, what is the implied interest rate? 2. The board of trustees wants to increase the annual scholarship payment by 3% each year. Assuming the same endowment value, what must the discount rate be to sustain this growth? 3. If the endowment receives an additional one-time donation of $500,000, how much can the annual payment increase (assuming no growth and the original interest rate)? — Solution: Part 1: Calculate the Implied Interest Rate The formula for the present value of a perpetuity is: PV = P r Rearranging to solve for r: r= P PV Substitute the given values: P = 100, 000, r= P V = 2, 000, 000 100, 000 = 0.05 2, 000, 000 Answer: The implied interest rate is 5%. — 1 Part 2: Calculate the Discount Rate for a Growing Perpetuity For a growing perpetuity, the formula is: PV = P r−g Rearranging to solve for r: r=g+ P PV Substitute the given values: P = 100, 000, P V = 2, 000, 000, g = 0.03 100, 000 2, 000, 000 r = 0.03 + 0.05 = 0.08 r = 0.03 + Answer: The discount rate must be 8% to sustain the 3% annual growth. — Part 3: Calculate the Increased Payment with Additional Donation The total value of the endowment after the donation is: P Vnew = P V + Donation = 2, 000, 000 + 500, 000 = 2, 500, 000 The payment is determined by: P = PV · r Using the original interest rate (r = 0.05): Pnew = 2, 500, 000 · 0.05 = 125, 000 The increase in the payment is: ∆P = Pnew − P = 125, 000 − 100, 000 = 25, 000 Answer: The annual payment can increase by $25,000, bringing the total annual payment to $125,000. — Summary of Results: 2 • Implied interest rate: 5%. • Discount rate for 3% growth: 8%. • Increased annual payment after donation: $25,000 (total payment: $125,000). 3