You just won the lottery! Your prize begins with a payment of $10,000, which you will receive one year from today. Every year forever after, the payments will increase by 4 percent annually. What is the present value of your prize at a discount rate of 6 percent per year?

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BlackTom AI · Accepted Answer

The problem asks for the present value of a perpetual growing annuity: a first payment of $10,000 one year from today, with payments growing at 4% per year thereafter, and a discount rate of 6% per year.
Option 1: $530,000. To evaluate, consider the standard formula for a growing perpetuity starting one year from now: PV = P1 / (r − g), where P1 is the first payment, r is the discount rate, and g is the growth rate. Plugging in P1 = 10,000, r = 0.06, g = 0.04 gives PV = 10,000 / (0.06 − 0.04) = 10,000 / 0.02 = 500,000. Since 500,000 ≠ 530,000, this option is not correct.
Option 2: $166,667. This value would correspond to a different ratio, perhaps if r − g were 0.06 − 0.10 or some misapplication. But the correct difference is 0.02, and with P1 = 10,000 the resulting PV is 500,000, so 166,667 is not supported by the standard formula.
Option 3: $555,000. This is close to the correct 500,000 but not equal. It might stem from rounding or a miscalculation of the growth/discount difference (0.06 − 0.04 = 0.02). However, using the precise formula yields 500,000, so 555,000 is incorrect.
Option 4: $500,000. This matches the result from the growing perpetuity formula PV = P1 / (r − g) with P1 = 10,000, r = 6% and g = 4%. Since 0.06 − 0.04 = 0.02 and 10,000 / 0.02 = 500,000, this is the correct computation for the given terms.
Option 5: $375,000. This value would imply a larger denominator or a smaller numerator than the actual setup. With the provided numbers, the precise calculation does not yield 375,000, so this option is incorrect.
In summary, the key concept is applying the growing perpetuity present value formula PV = P1 / (r − g) for a perpetual series that begins one year from now, ensuring that the discount rate exceeds the growth rate. The only option that aligns with this formula and the given P1, r, and g is $500,000.

You just won the lottery! Your prize begins with a payment of $10,000, which you will receive one year from today. Every year forever after, the payments will increase by 4 percent annually. What is the present value of your prize at a discount rate of 6 percent per year?

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