ROR calculation using a PW or AW relation

The ROR value is determined in a generically different way compared to the PW or AW value for a series of cash flows. For a moment, consider only the present worth relation for a cash flow series. Using the MARR, which is established independent of any particular project’s cash flows, a mathematical relation determines the PW value in actual monetary units, say, dollars or euros. For the ROR values calculated in this and later sections, only the cash flows themselves are used to determine an interest rate that balances the present worth relation. Therefore, ROR may be considered a relative measure, while PW and AW are absolute measures. Since the resulting interest rate depends only on the cash flows themselves, the correct term is internal rate of return (IROR); however, the term ROR is used interchangeably. Another definition of rate of return is based on our previous interpretations of PW and AW.

To determine the rate of return, develop the ROR equation using either a PW or AW relation, set it equal to 0, and solve for the interest rate. Alternatively, the present worth of cash outflows (costs and disbursements) PW_O may be equated to the present worth of cash inflows (revenues and savings) PW_I . That is, solve for i using either of the relations

The annual worth approach utilizes the AW values in the same fashion to solve for i .

The i value that makes these equations numerically correct is called i*. It is the root of the ROR relation. To determine if the investment project’s cash flow series is viable, compare i* with the established MARR.

Figure (6-2): Cash flow for which a value of i is to be determined.

The guideline is as follows:

The purpose of engineering economy calculations is equivalence in PW or AW terms for a stated i ≥ 0%. In rate of return calculations, the objective is to find the interest rate i* at which the cash flows are equivalent. The calculations are the reverse of those made in previous chapters, where the interest rate was known. For example, if you deposit $1000 now and are promised payments of $500 three years from now and $1500 five years from now, the rate of return relation using PW factors and Equation (6-1) is

The value of i* that makes the equality correct is to be determined (see Figure (6–2) ). If the $1000 is moved to the right side of Equation (6-3), we have the form 0 = PW.

The equation is solved for i * _ 16.9% by hand using trial and error or using a spreadsheet function. The rate of return will always be greater than zero if the total amount of cash inflow is greater than the total amount of outflow, when the time value of money is considered. Using i* = 16.9%, a graph similar to Figure (6–1) can be constructed. It will show that the unrecovered balances each year, starting with $ -1000 in year 1, are exactly recovered by the $500 and $1500 receipts in years 3 and 5.

It should be evident that rate of return relations are merely a rearrangement of a present worth equation. That is, if the above interest rate is known to be 16.9%, and it is used to find the present worth of $500 three years from now and $1500 five years from now, the PW relation is

This illustrates that rate of return and present worth equations are set up in exactly the same fashion. The only differences are what is given and what is sought.

There are several ways to determine i* once the PW relation is established: solution via trial and error by hand, using a programmable calculator, and solution by spreadsheet function. The spreadsheet is faster; the first helps in understanding how ROR computations work. We summarize two methods here and in Example (6-2).

i* Using Trial and Error The general procedure of using a PW-based equation is as follows:

1-      Draw a cash flow diagram.

2-      Set up the rate of return equation in the form of Equation (6-1).

3-      Select values of i by trial and error until the equation is balanced.

When the trial-and-error method is applied to determine i*, it is advantageous in step 3 to get fairly close to the correct answer on the first trial. If the cash flows are combined in such a manner that the income and disbursements can be represented by a single factor such as P/F or P/A , it is possible to look up the interest rate (in the tables) corresponding to the value of that factor for n years. The problem, then, is to combine the cash flows into the format of only one of the factors. This may be done through the following procedure:

1-      Convert all disbursements into either single amounts (P or F) or uniform amounts (A) by neglecting the time value of money. For example, if it is desired to convert an A to an F value, simply multiply the A by the number of years n. The scheme selected for movement of cash flows should be the one that minimizes the error caused by neglecting the time value of money. That is, if most of the cash flow is an A and a small amount is an F , convert the F to an A rather than the other way around.

2-      Convert all receipts to either single or uniform values.

3-      Having combined the disbursements and receipts so that a P/F, P/A, or A/F format applies, use the interest tables to find the approximate interest rate at which the P/F, P/A, or A/F value is satisfied. The rate obtained is a good estimate for the first trial.

It is important to recognize that this first-trial rate is only an estimate of the actual rate of return, because the time value of money is neglected. The procedure is illustrated in Example (6-2).

i* by Spreadsheet The fastest way to determine an i* value when there is a series of equal cash flows (A series) is to apply the RATE function. This is a powerful one-cell function, where it is acceptable to have a separate P value in year 0 and a separate F value in year n . The format is

When cash flows vary from year to year (period to period), the best way to find i* is to enter the net cash flows into contiguous cells (including any $0 amounts) and apply the IRR function in any cell. The format is

Where “guess” is the i value at which the function starts searching for i*. The PW-based procedure for sensitivity analysis and a graphical estimation of the i* value is as follows:

1-      Draw the cash flow diagram.

2-      Set up the ROR relation in the form of Equation (6-1), PW = 0.

3-      Enter the cash flows onto the spreadsheet in contiguous cells.

4-      Develop the IRR function to display i*.

5-      Use the NPV function to develop a PW graph (PW versus i values). This graphically shows the i* value at which PW = 0.