Amanda would receive an annual cash flow of $100,000 at the end of each year that will continue indefinitely. This cash flow is expected to grow at 5% per annum. Calculate the present value of these cash flows if the effective annual interest rate is 10%.

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BlackTom AI · Accepted Answer

To tackle this problem, first identify that we have a perpetuity that starts with a payment of 100,000 at the end of year 1 and grows at 5% per year indefinitely. The discount rate is 10% per year.

Option A: $666,666.67. This would correspond to the simple perpetuity formula without growth, i.e., 100,000 / 0.15 or some misapplication. Since the growth rate is positive (5%), we would not divide by 0.25 or 0.15 in this context, so this value is not correct for a growing perpetuity.

Option B: $675,902.38. This value is close to what one gets from a more nuanced calculation in certain irregular scenarios, but for a standard growing perpetuity with payments growing at 5% and discount rate 10%, the correct expression is different. This number does not align with the well-known formula for a growing perpetuity.

Option C: $2,000,000.00. This matches the standard formula for a growing perpetuity when the growth rate is less than the discount rate: PV = C1 / (i − g), where C1 is the first payment. Here C1 = 100,000, i = 0.10, g = 0.05, so PV = 100,000 / (0.10 − 0.05) = 100,000 / 0.05 = 2,000,000. This is the precise result for a perpetuity that grows at 5% per year starting with a 100,000 payment.

Option D: $1,000,000.00. This would be the present value if there were no growth (i = 10%) and the annual payment were 100,000 indefinitely, since 100,000 / 0.10 = 1,000,000. However, the problem states a 5% growth per year, so this ignores the growth factor and underestimates the value, making it incorrect in this context.

In summary, the only value that correctly incorporates both the perpetual nature and the 5% annual growth with a 10% discount rate is the growing perpetuity formula PV = C1 / (i − g). The given numbers align with i > g and yield 2,000,000, while the other options arise from applying incomplete or alternative formulas that do not match the problem's setup.

25400 Financial Literacy - Spring 2025 🔴 Practice Questions for In-class Quiz 1

Amanda would receive an annual cash flow of $100,000 at the end of each year that will continue indefinitely. This cash flow is expected to grow at 5% per annum. Calculate the present value of these cash flows if the effective annual interest rate is 10%.

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