In his An Elementary Introduction to Mathematical Finance, 3rd Edition book, pg. 55, Sheldon Ross has a question -

A company needs a certain type of machine for the next five years. They presently own such a machine, which is now worth 6,000 (dollars) but will lose 2,000 in value in each of the next three years, after which it will be worthless and unuseable. The (beginning-of-the-year) value of its yearly operating cost is 9,000, with this amount expected to increase by 2,000 in each subsequent year that it is used. A new machine can be purchased at the beginning of any year for a fixed cost of 22,000. The lifetime of a new machine is six years, and its value decreases by 3,000 in each of its first two years of use and then by 4,000 in each following year. The operating cost of a new machine is 6,000, plus an additional 1,000 every following year.

He does Present Value Analysis on cash flows for alternate scenarios, that is, whether a new machine is bought at year 1, 2, 3.. His cash flow for year 1 purchase seems incorrect though. The flow is (in 1000 dollars)

22, 7, 8, 9, 10, −4

It says 22 for year 1 purchase of the new machine - but if the company is buying in the beginning of year 1, shouldn't they pay 22K + 6K = 28K, that is, including the operating costs? The year 2 above shows 7K for operating expense, which looks correct.

Any ideas?


1 Answer 1


22k seems fine to me, assuming a new machine is bought on the first year.

Indeed, you face a cost of 22k (sell price of new machine) plus 6k (operating cost of new machine), but you also earn 6k by selling your current machine at its face value (since it won't last 5 years as required and it will only lose value from now on, better sell it asap).

  • $\begingroup$ Ah - seem to have missed that one, the selling of the old machine at year 1. $\endgroup$
    – BBSysDyn
    Apr 27, 2016 at 8:00
  • $\begingroup$ @user616 After a quick read through, this is in fact mentioned by the author (see p.55 sentence about a 'replaced machine') when he unravels the cash flows associated to buying a new machine at the beginning of year 3. $\endgroup$
    – Quantuple
    Apr 27, 2016 at 8:09

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