SOME DEFINITIONS
Now let's see how these income and expense items fit into our cost benefit model. We've heard all our lives that we can't compare apples and oranges. In performing a CBA, the first step is to convert all oranges (and pears, bananas, etc.) into apples, however we choose to define those apples. There are several types of factors that we need to consider.
Present Values (PV): these are cash flows that occur now (or reasonably close to now). They include our startup expenses, as well as any other expenses and incomes (negative and positive cash flows) that occur at the beginning of the project, or close enough to the beginning that we consider any small delay to be trivial. All we need to describe these cash flows is the dollar amount involved.
Future Values (FV): these are the cash flows that occur some time in the future, naturally. Again, they can describe incomes or expenses, including our salvage costs and salvage incomes, as well as our non-recurring middle-of-project costs. To properly describe them we need three items: the dollar amount involved, the number of time units into the future, and the interest rate per time unit. These time units can be anything we choose: days, months, years, etc, but let's try to be consistent throughout our analysis. The interest rate will be selected based on a variety of soft factors (see below) and the time units we are using.
Annual Values (AV): these are our recurring cash flows that occur in equal amounts at equally spaced intervals. Calling them "annual" is just a convention; they might be monthly, daily, or any other interval. The AV concept is not strictly necessary, because we could define an AV just as a bunch of repeated FVs. However, since we tend to see a lot of them, it is worth using the AV definition as a short cut. To describe these, we need the repeating dollar amount, the spacing between payments, the total number of payments, and the interest rate.
That's it! Those are the various kinds of "fruit" that we need to reconcile into apples. To do our CBA, we need to convert all our PVs, FVs, and AVs into any one kind of cash flow. If we convert them all to present values, we call it a "present value analysis", which tells us what our project is worth in equivalent dollars right now. If we convert them all to future values, it is a "future value analysis", which tells us what our project is worth at some fixed point in the future. Likewise for the annual values. Which do we use? That is up to us, as long as we pick one that tells our story and makes our argument for us. For this article, we'll stick with present values for our "apple" definition. Alternately, we can do a rate of return analysis, which we'll discuss soon.
CONVERSION FORMULAS AND FACTORS
How do we do this conversion? A couple of formulas will take care of it. For our present value analysis, we want to convert everything to a PV. In that case, we need a formula to convert FVs to PVs and another formula to convert AVs to PVs:
PV = FV(1+ i)-n
PV = AV[((1 + i)n - 1) / (i(1 + i)n)]
where n is the number of time units and i is the interest rate per time unit.
Using these formulas, you can convert all of your expected future payments and incomes into their equivalent present values, thus "time compressing" all cash flows to the present day. Sum up the positives, subtract the negatives, and see if you are making money or losing it. Of course, there's still the issue of whether you are making enough money or not, but we'll get to that later.
An alternate method for converting cash flows is to use "compound interest tables" which are compiled and published. Banks use them all the time to calculate your loan payments. I use the tables at the back of "Engineering Economic Analysis" by Donald G. Newnan (1988, Engineering Press, Inc.), but many other equivalent publications exist. These tables give conversion factors for changing between PVs, FVs, and AVs by a simple multiplication. You'll get the same answer as if you had calculated from the formula, but the tables save time and are popular with non-techie types.
As a simple conversion example, let's say you expect to win the lottery and receive one million dollars per year for 20 years, and you want to know the present value of this windfall. This is clearly an AV, and we want to convert it to a PV. If we assume an interest rate of 10%, we would plug n = 20 and i = 0.10 into our AV to PV formula. Crunching through this will give you a present value of $8,514,000 for your winnings. Not as good as it sounded, is it?
As long as we can describe all of the cash flows identified for our project into our three formal constructs (PV, FV, and AV), we can convert them all into one total project present value. If the total value is positive, the project makes money. If it is negative, it loses money.
