Present Value of an Annuity Due

To calculate the present value of the cash streams offered by the networking company, we will use the present value formulas for both an annuity due and a regular annuity. Present Value of an Annuity Due Definition: An annuity due is an annuity where payments are made at the beginning of each period. In this case, you receive $500,000 today (which is already in present value terms) and then $500,000 at the beginning of each of the next six years. Payment (PMT): $500,000 Number of periods (n): 6 Interest rate (r): 6% or 0.06 The present value (PV) of an annuity due can be calculated using the formula: [ PV = PMT \times \left(1 + r\right) \times \left(1 - \left(1 + r\right)^{-n}\right) / r ] However, since you also receive $500,000 today as the initial payment, we will add that to the present value of the annuity due. Calculation Steps: 1. Calculate the present value of the annuity due: [ PV = 500,000 \times \left(1 + 0.06\right) \times \left(1 - \left(1 + 0.06\right)^{-6}\right) / 0.06 ] Calculating the components step by step: - (1 + r = 1 + 0.06 = 1.06) - ((1 + r)^{-n} = (1.06)^{-6} \approx 0.7054) - (1 - (1 + r)^{-n} = 1 - 0.7054 \approx 0.2946) Now substituting into the formula: [ PV = 500,000 \times 1.06 \times \frac{0.2946}{0.06} ] Calculating: - First calculate the fraction: (\frac{0.2946}{0.06} \approx 4.91) Then: [ PV = 500,000 \times 1.06 \times 4.91 \approx 2,605,300 ] 2. Add the initial payment: The total present value of the cash stream is: [ Total PV = Initial Payment + PV of Annuity ] [ Total PV = 500,000 + 2,605,300 \approx 3,105,300 ] Present Value of an Annuity In this scenario, you will not receive any payments until the first sale in two years. Therefore, the cash stream consists of six payments of $500,000 starting at the end of year two. Payment (PMT): $500,000 Number of periods (n): 6 Interest rate (r): 6% or 0.06 The present value (PV) of a regular annuity can be calculated using the formula: [ PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r ] However, since these payments start in year two, we will first calculate the present value at year one and then discount it back to today. Calculation Steps: 1. Calculate the present value of the annuity starting in year two: [ PV = 500,000 \times \left(1 - (1 + 0.06)^{-6}\right) / 0.06 ] Calculating the components step by step: - ( (1 + r)^{-n} = (1 + 0.06)^{-6} \approx 0.7054) - (1 - (1 + r)^{-n} = 1 - 0.7054 \approx 0.2946) Now substituting into the formula: [ PV = 500,000 \times \frac{0.2946}{0.06} \approx 500,000 \times 4.91 \approx 2,455,000 ] 2. Discount this back to today: Since this PV is at the end of year one, we need to discount it back one year: [ PV_{today} = PV_{at,year,one} / (1 + r) ] [ PV_{today} = 2,455,000 / (1 + 0.06) \approx 2,455,000 / 1.06 \approx 2,316,415 ] Summary - Present Value of Cash Stream with Immediate Payment: - Total PV: $3,105,300 - Present Value of Cash Stream Starting in Two Years: - PV Today: $2,316,415 These calculations provide a clear financial picture for both scenarios regarding your software deal with the networking company.    

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