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### Rate of Return Analysis

#### Incremental analysis

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When comparing two or more alternatives, the alternative with the highest ROR is not necessarily the alternative that maximizes profit at the MARR, which is the appropriate goal. For example, suppose that an investor with at least \$1,000 available to invest is considering two mutually exclusive investment alternatives, X and Y, as follows:

Alternative X (ROR = 100%)

 EOY NCF (\$) 0 - 10 1 + 20

Alternative Y (ROR = 75%)

 EOY NCF (\$) 0 - 1000 1 + 1750

Despite having a lower ROR than alternative X, alternative Y is obviously the superior investment at any reasonable MARR because it generates more profit than X. Alternatives cannot be evaluated solely by comparing their individual ROR values. Comparisons must be made incrementally.

Example 1. Suppose that alternatives X and Y above are being considered by an investor with a MARR of 20%. The do-nothing alternative is rejected because the ROR of both X and Y is greater than 20%.

Next we set the default choice equal to X because it requires less investment than Y. If the incremental investment in Y over X is desirable, then we switch our selection from X to Y.

To determine if the incremental investment in Y over X is desirable, calculate the rate of return of the incremental net cash flow (Y - X). If the incremental ROR is greater than or equal to the MARR, then the more costly alternative (Y in this case) should be selected. Otherwise choose the lower cost alternative.

Form the incremental NCF by (more costly - less costly).

In this example, we form the incremental net cash flow (Y - X):

EOY 0: -1000 - (-10) = - 990

EOY 1: +1750 - (+ 20) = + 1730

 EOY Incremental NCF (\$) 0 - 990 1 + 1730

The rate of return of the incremental NCF is easily calculated in this example because the project duration is only one year:

Incremental ROR = ( 1,730 - 990) / 990 = 0.747 = 74.7%

Because the incremental ROR = 74.7% > MARR = 20%, the incremental investment in Y over X is desirable.

As was obvious from the data, the investor should select Y, the alternative with the lower ROR.

When three or more alternatives are under consideration, incremental ROR analysis is performed by a series of pairwise comparisons. At each step the current selection is compared to the least costly of the remaining alternatives under consideration. If the incremental ROR is greater than or equal to the MARR, then the more costly alternative is selected. Continue until all alternatives have been considered.

Example 2. Suppose that three mutually exclusive investment alternatives are being considered by an investor with a MARR of 10% and at least \$5k available to invest. The alternatives have equal lives. Partial economic data are given below. Which, if any, of these alternatives should the investor select?

 Alternative X Y Z Initial cost (\$k) 2 5 3 ROR (%) 18 12 16

Pairwise comparisons will be made in the following order based on initial cost: do-nothing, X, Z, Y.

Because at least one ROR > MARR, reject the do-nothing alternative. Thus the current selection is X.

Next compare X to Z. Suppose we find that the incremental ROR for (Z - X) is 13.4%. (There are insufficient data given to make this calculation.) Because the incremental ROR = 13.4% > MARR = 10%, we reject X and change the current selection to Z. Next we compare Z to Y. (If the incremental ROR had been less than 10% for Z - X, we would have rejected Z, kept X as the current selection, and next compared X to Y.)

Suppose that the incremental ROR for (Y - Z) is 8.8%. Because the incremental ROR < MARR, we keep Z as the current selection and reject Y.

All alternatives have now been considered; select Z.

### Rate of Return Analysis

#### Incremental analysis

Question 1

Question 2

Question 3

Question 1.

Calculate the ROR for the incremental net cash flow (X - Y), given the data below. Note that the appropriate increment is (X - Y), not (Y - X), because the initial cost of X exceeds the initial cost of Y.

 Alternative X Y Initial cost (\$k) 40 30 Uniform annual benefit (\$k) 12 10 Life (years) 8 8 Salvage value EOY 8 0 0 ROR (%) 24.95 28.98

Which of the selections below is correct?

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A ROR = 11.81%, the interest rate that is the solution to:
NPW = 0 = - 10 + 2 (P/A,i%,8)

B ROR = 28.98% - 24.95% = 4.03%

C ROR = 24.95% = ROR of X, because X is more costly

D ROR = 36.721%, the interest rate that is the solution to:
NPW = 0 = - 30 + 12 (P/A,i%,8)

Review topic

Question 2.

An investor with a MARR of 15% and at least \$40k to invest is using rate of return analysis to determine which, if either, of two mutually exclusive investment alternatives (X and Y) should be selected. Perform the analysis and make a recommendation.

 Alternative X Y Initial cost (\$k) 40 30 ROR (%) 22 26 Life (years) 8 8

Assume that the ROR of the incremental NCF (X - Y) is 10%.

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A Choose do-nothing

B Choose Y because the ROR of Y is greater than the ROR of X

C Choose Y because the incremental ROR < MARR

D Choose X because the incremental ROR > 0

Review topic

Question 3.

It is clear without calculating the ROR for the incremental investment (Y - X) that the incremental ROR will be negative, given the data below. Why?

 Alternative X Y (Y - X) Initial cost (\$) 20 30 10 Uniform annual benefit (\$) 10 12 2 Life (years) 4 4 4 Salvage value EOY 4 (\$) 0 0 0 IRR (%) 34.90% 21.86% < 0

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A Because ROR of X > ROR of Y

B Because there are no salvage values

C Because the \$10 incremental investment is never fully recovered.

D Because [ (ROR of X) - ( ROR of Y ) ] < MARR

Review topic 