It is now January 1, 2000, and you are offered the following deal: starting from

Question

It is now January 1, 2000, and you are offered the following deal: starting from year 2000, you will be receive each year the amount of dollars equal to the year, i.e., $2000, $2001, $2002, etc., until year 3000 inclusive. Payments occur at the year end and the interest rate remains at 10% throughout the life of the investment, what is the most you would pay for this investment? How much would you pay for this deal if the payments cover only the years from 2000 to 2050 inclusive? What can you conclude from the comparison?

Explanation / Answer

We have following formula for PV of a cash flow

PV = FV/ (1+r)^n

PV= 2000/(1.10)^1 + 2001/ (1.10)^2 + 2002/ (1.10)^3…….3000/ (1.10)^1001   ----(1)

Multiplying both sides by 1.1, we get:

PV x 1.10 = 2000 + 2001/1.10 +2002/ 1.10^2 …..3000/ 1.10^1000 --------(2)

Subtracting equation 1 from equation 2, we get:

PV x 0.10 = 2000 +1/1.10 + 1/1.10^2 ……1/1.10^1000 -3000/ 1.10^1001

The middle part is an annuity:

PV x 0.10 = 2000 +1 x PVIFA (10%, 1000) – 0

PV= 2010/ 0.10

PV = 20,100

So single payment for this series of payments would be 20,100.

Now, the payment will be restricted to 2050. So PV would be:

PV x 0.10 = 2000 + 1 x PVIFA (10%, 50) - 2050/ 1.10^51

                    = 2000+9.91-15.88

                    = 1994.03/0.1

                     = 19,940.30

Difference in PVs = 20100 -19,940.30

                                   = 159.70

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