**Concept of Equivalence**

To compare alternatives that provide the same service over extended
periods of time when interest is involved, we must reduce them to an equivalent
basis that is dependent on: ---If two alternatives are economically equivalent,
then they are equally

desirable.

Equivalence factors are needed in engineering economy to make cash flows (CF) at different points in time comparable. For example, a cash payment that has to be made today cannot be compared directly to a cash flow that must be made in 5 years.

Since the time value of money changes according to:

1.The interest rate,

2.The amount of money involved,

3.The timing of receipt or payment,

4. The manner in which interest is compounded,

We need a way to reduce CF's at different times to an equivalent basis. Equivalence factors allow us to do so.

**Principles
of Equivalence**

- Equivalent cash flows have the same economic value at the same point in
time.

- Cash flows that are equivalent at one point in time are equivalent at any point in time.
- Conversion of a cash flow to its equivalent, at another point in time must
reflect the interest rate(s) in effect for each

period between the equivalent cash flows. - Equivalence between receipts and disbursements: the interest rate that
sets the receipts equivalent to the disbursements is

the actual interest rate (IRR). - Economic equivalence is established, in general, when we are indifferent between a future payment, or series of payments, and a present sum of money.

**Notation and Cash Flow Diagrams (CFDs)**

The following notation is utilized in formulas for compound interest calculations:

I = effective interest rate per interest period

N = number of compounding periods

P = present sum of money; the equivalent value of one or more cash flows at a reference point in time called present

F= future sum of money; the equivalent value of one or more cash flows at a reference point in time called future

A = end-of-period cash flows (or equivalent end-of-period values) in a uniform series continuing for a specified number of periods, starting at the end of the first period and continuing through the last period

1. The Horizontal line is a time scale, with progression of time moving from left to right. The period (e.g., year, quarter, month) labels can be applied to intervals of time rather than to points on the time scale.

2. The arrows signify cash flows and are placed at the end of the period. If a distinction needs to be made, downward arrows represent expenses (negative cash flows or cash outflows) and upward arrows represent receipts (positive cash flows or cash inflows).

3. The cash flow diagram is dependent on the point of view. The situations shown in the figure were based on the cash flows as seen by the lender. If the directions of all arrows had been reversed, the problem will have to be diagrammed from borrower's viewpoint.

**Cash Flow Diagram**