Final answer:
The fair price for Brian to sell his corn dog stand today is $735,091, which is the present discounted value of a perpetuity that begins 6 years from now with a 7% discount rate.
Step-by-step explanation:
When calculating the present discounted value of an investment, we must account for the time it takes for cash flows to begin and the discount rate. In this case, the corn dog stand's first cash flow occurs 6 years from now at a rate of $77,000 annually, forever. This type of cash flow is known as a perpetuity. The formula we use for the present value of a perpetuity is PV = C / r, where C is the annual cash flow and r is the discount rate. However, given the time delay, we must first discount that perpetuity value back to the present, accounting for the 6-year delay.
To find the present value (PV) of the perpetuity starting in 6 years given the discount rate (r) of 7% (0.07), we use the formula: PV = C / r. Substituting $77,000 for C gives us PV = $77,000 / 0.07, which is equal to $1,100,000. This $1,100,000 is the value of the perpetuity in 6 years. We then must discount this value back to today, which can be done by dividing this amount by (1 + r)^n, where n is the number of years before the first cash flow.
Applying this, we get: Today's PV = $1,100,000 / (1 + 0.07)^6. This gives us a present value today of approximately $735,091. We round this to the nearest dollar to determine that the fair price for Brian to sell his corn dog stand today is $735,091.