I/EC
Costs
A Rapid Method for Finding W h e n profits are nomographs
constant,
avoid detailed
calculation
by Wishard H. Greist, Jr., Alan G. Bates, and James B. Weaver, Atlas Powder
• IME-CONSUMING calculations are sometimes mentioned as disadvan tages in using the profitability yard stick known as interest rate of re turn on investment. However, when profits are constant, nomographs like those in this article can be used in stead of detailed calculations. The nomographs provide a simplified means for determining profitability of relatively small investments, and for preliminary screening of proj ects in the feasibility stages. As is often the case, a simplified
Co.
nomograph can consider most but not all of the major variables. As more variables are introduced, nom ographs become more complex and their utility is diminished. Addi tional variables are introduced suc cessively below, until variations in all significant factors except profits can be considered. Simplified N o m o g r a p h
Figure 1 is a simple nomograph, based on sum of the years-digits depreciation, combined federal and
SUMMARY OF PUMP SELECTION ALTERNATIVES FIXED INVESTMENT, $
WORKING CAPITAL, $
52
A
INDUSTRIAL AND ENGINEERING CHEMISTRY
MAINTENANCE COST, $ YR.
state income taxes of 54%, and the following simplifying assumptions: 1. Constant annual profit before taxes and depreciation (P). 2. Project economic life equal to depreciable life (T). 3. Zero construction time—i.e., total investment expended at time zero. 4. Zero salvage value of fixed investment. 5. No start-up expense (S = 0). The nomograph allows determina tion of interest rate of return, R, as a function of Ρ and T, fixed in vestment (F), and working capital (HO. To demonstrate the use of the nomograph, consider the following alternatives for replacing a pump which has failed. Alternative A. Replace the exist ing pump with one of the same type which will cost $3000 and will have an economic and depreciable life of 10 years. It is estimated that the spare parts inventory (working capi tal in this case) will be $50 and the maintenance cost will be $300 per year. Alternative B. Install a different type of pump with the same life cost ing $3500 which has a spare parts inventory of $100 and a maintenance cost of $100 per year. Economic and depreciable life are again equal to 10 years.
Interest Rate of Return INTEREST RATE OF RETURN
START-UP CORRECTION FACTOR
TRIAL AND ERROR SOLUTION FOR INTEREST RATE OF RETURN
VOL. 53, NO. 3
·
MARCH
1961
53 A
COSTS To determine the interest rate of return for the $500 increment, the following steps are taken : 1. Calculate ΡΧ Τ 200 Χ 10 ., .... F ' -SOC" = 4 (s-le X) 100(1*' + /•?) 50 + 0 /·· = 500 =10 (scale Υ) 2. To find RΤ on the nomograph, connect scales X and Y and read scale Z. For this example, /?'/' = 210; therefore, R = 2 1 % per year. If 15% per year return is ac ceptable, the plant engineer knows he is justified in purchasing the more expensive pump (alternative B). Start-Up Expense If a project includes start-up ex pense (S) at time zero, the simplified nomograph can be used with one additional complexity. A factor is shown in Figure 2 which adjusts any initial expense outlay to an equivalent amount of working cap ital. This introduces a trial and error approach for the first time, but only two trials arc usually needed to obtain the correct answer. Assume in the example that an additional beforc-tax expense of $100 was required in case B. The following steps are taken to include the expensed outlay : 1. Find the / factor, which is 0.5 in this example for an RT value of 210 (obtained for the example excluding start-up expense), and calculate the/5 1 term. fS = (0.5)(100) = 50 2. Add the the original quirement of sum by the timate. 1 0 0 ^ . + /L)
F
=
result from step 1 to working capital re $50 and divide the fixed investment es
1OQ(50
For projects having construction times up to about 6 months, the use of the nomograph in Figure 1 will result in an error of less than 1 percentage point at returns between 10 and 20% per year. If the in vestment will be made over a period of 6 months or longer, a significant error will result. The more com plex nomograph presented in Figure 3 takes into consideration this vari able. The nomograph requires a trial and error approach, usually with 2 or 3 trials. The single nomo graph will not cover all project lives; Figure 3 is constructed to cover projects with a 10-year eco nomic and depreciable life. Other variables and simplifying assump tions are the same as those used in Figure 1. Figure 3 is more complex because of the introduction of construction time as a variable. However, the economic life (7") variable is re moved which allows the return to be read directly. Scale A corresponds to scale Y on Figure 1, and scale G to scale Z, except for the omission of the eco nomic life. Scales Β and C are in troduced in order to compound the investment value properly over the construction time to find its present value at time zero on scale D as a per cent of F. The use of Figure 3 is best demon strated by a sample problem. Assume Profit before taxes and déprécia- 2,000,000 tion (P), $/yr. Fixed investment (F), S 5,000,000 Working capital (W), S 500,000 Start-up expense, before taxes 250,000 (S), S Project life (T), yr. 10 Construction time, yr. 1 Procedure : 1. Calculate for the Ε scale.
the
ratio
4. Calculate the ratio for scale A.
(loo) l*+&
=
0 0 0 ) (500 + 250 QO X 0.50)
=
required
Scale E: (100) (Ç\ =
(3g) - « 2. With a straight edge between 40 on scale Ε and 100 on scale D, read the first trial rate of 24% per year on scale G. This would be the rate of return on an instantaneous investment (fixed only). 3. Using this first trial rate, RT = 240, an / factor of 0.50 can be read from Figure 2.
INDUSTRIAL A N D ENGINEERING CHEMISTRY
^
5. Find the point on the oneyear construction line (scale C) where the 24% return point would cross (scale B) ; line up this point with 12.5 on scale A and read 125 on scale D. 6. Connect scale D (125) with scale Ε (40) and read a second trial rate of about 17% per year on scale G. 7. Repeating steps C through Ε yields a result of 18% per year. Solving this problem by usual dis counting method yields a return of 17.8% per year. Although the approach may seem cumbersome at first, it becomes quite easy after a few practice tries. Figure 3 can also be used to solve for present worth at a given return without trial and error. The neces sary nomograph ratios for scales A and Ε are first determined. Then at a given return, the present value of the investment outlays is found by reading from scale A to the left side of scale D, while the present value of the cash income is found by reading from scale Ε to the right side of scale D. The sum of these present values is equal to the net present worth, as a per cent of F. More complex nomographs can be prepared to reflect certain prcsimplificd changing profit patterns. However, they are relatively un likely to occur, compared to con stant or extremely variable profit receipts. Therefore, nomographs which introduce varying profit pat terns are believed to be not worth presenting.
+ 50)
500 = 20 (scale Y)
3. Find a new RT value using the scale Y value of 20. Here, RT = 180. A second trial with S included still gives an RT value of essentially 180. In general, only two trials are re quired for an accurate result. The start-up expense has reduced the interest rate of return from 21 to 18%. 54 A
Extended Construction Period
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