(1) A factory costs $800,000. You think it will produce a net cash inflow of $170,000 a year for 10 years. (i) If the discount rate is 14%, is it a good idea to buy the factory for the $800,000 asking price? (ii) What would the maximum price be that you are willing to pay for the factory? What will the factory be worth after 5 years, in other words, what is the price that you can sell the factory for at the end of period 5?

Answer:

(i) Compare the PV of the costs, which is just the $800,000 with the PV of the expected future cash inflows. The future inflows are an annuity, so we can use the PV of an annuity formula:
 
PV = [170,000 / 0.14] * {1 - [1 / (1.14)¹⁰]} = $886,720, thus the profit you make equals $886,720 - $800,000 = $86,720. Yes, this is a good deal!

(ii) The maximum price you would be willing to pay for this factory is $886,720, because then you break even. If you pay more, you are making a loss.

(ii) Remember that any price, whether a selling or purchasing price, is determined by the discounted value of the future cash flows. Hence, at he end of 5 years, the factory’s value will be the present value of the five remaining $170,000 cash flows:
[$170,000 / 0.14] / [1 - (1 / 1.14)⁵] =  $583,610  (This is the price you will get when you sell in year 5 and answers the question.)

Answer:

The answer to this question is to find the PV of the stream of cash flows that this machine will generate:

Part I:
The first part is a growing annuity, with C = 50, g = 10%, r = 12%, t = 10, or,

PV₁ = 50 * {(1 / 0.02) - [(1 / 0.02) * (1.1 / 1.12)¹⁰]} = 50 * [50 - 41.756] = 412.20

Part II:
Figure out what the second growing (or rather, declining) annuity is. To avoid double counting of cash flows, we use the cash flow at t=11 as our first cash flow. Now we have a growing (declining) annuity with, C = $50 * (1.1)⁹ * (0.95) = $112
(We have had 9 years of growth at 10%, and then an additional year of negative growth of 5%), r = 12%, g = -5%, t = 9 (because in t=20 the machine is sold. We hence use the cash flows for years 11 through 19). Applying the formula for a growing annuity we have:

PV₂ = $112 {(1 / 0.12 - (-0.05)) - [(1 / 0.12 - (-0.05)) * ((1 + (-0.05)) / 1.12)⁹]} =

          $112 * {(1 / 0.17) - [(1 /0.17) * (0.95 / 1.12)⁹]} = 112 * {5.882-1.337} = $509.08

Part III:
Now we have to realize that the value of $509.08 is denoted in t = 10 dollars, and should thus be discounted further to t=0 dollars:

PV₂ (t = 0) = $509.08 / (1.12)¹⁰ = $163.91

Part IV:

Remember that we will another cash flow at t=20 from selling the machine. The present value of this cash flow is:

$500 / (1.12)²⁰ = $51.83

Part V:

Now we can add up all the t = 0 cash flows to find out what the maximum price is that we would be willing to pay for this machine:

 $412.20 + $163.91 + $51.83 = $627.94

Answer:

Let's compare the value of the machine in t = 20 if we keep it, to the resale value of $500. To do this, we are trying to figure out what the value of a growing (declining) perpetuity is, that has its first cash flow in year t=20 (the year we could also receive the $500 instead, if we sell the machine). But remember that to use the growing perpertuity formula to get a value in t = 20, we need to enter the cash flow in t = 21 in the formula, and then add the cash flow in year t=20.

So, what are the cash flows in t = 20 and t = 21 respectively? We know that in t = 11 the cash flow is equal to $112 (see above), and the value of the cash flow in t = 12 must be equal to $112 * 0.95 = $106.4, and for t = 13 we have $112 * (0.95)² = 101.08, etc. etc. Hence the cash flow in year t = 20 is equal to $112 * (0.95)⁹ = $70.59 and for t = 21 equal to $112 * (0.95)¹⁰ = $67.06

Using the formula for a growing (declining) perpetuity we have:

PV (t = 20) = $67.06 / (0.12 - (-0.05)) = $67.06 / 0.17 = $394.53, plus an additional $70.59 equals $465.12, therefore,yes, it is a good deal to receive the $500!