Technical Chapter Two
Financial Analysis
Decisions regarding the choice of technology or process design require investment of capital. Therefore, these decisions utilize financial analysis of discounted cash flows or present values to determine the economic worth. In this chapter, several methods of financial analysis will be described, particularly as they relate to decisions in operations.
Typical operations decisions that require detailed financial analysis are
1. The purchase of new equipment or facilities
2. The replacement of existing equipment or facilities
An example of each of these decisions will be described later in the chapter.
TIME VALUE OF MONEY
In evaluating investments, we should consider the time value of money. We would rather have a dollar now than a dollar a year from now because we could invest the current dollar and earn a return on it for a year. Therefore, any future cash flows have less value to us than current cash flows. As a result, future cash flows must be discounted or reduced in value to their present values in order for future dollars to be comparable to present dollars.
Discounting of future cash flows is based on the idea of compound interest. If we have P dollars at the present time and invest it at an interest rate of i, the future value in 1 year will be
F1 = P + iP = P(1 + i)
In n years, at compound interest, the value of our P dollars will be
Fn = P(1 + i)n
This assumes that the interest is reinvested each year as it is earned.
If we divide the above equation by (1 + i)n, we will have
By turning the compound interest equation around, we see that the present value of an amount Fn paid in n years is simply P. We can, therefore, discount Fn to its present value by multiplying by the quantity 1/(1 + i)n. This quantity is known as the discount factor of the present value of $1 in year n. Values of the discount factor are tabulated in Appendix A. These factors can be used to convert any future cash flow to a present value amount.
For example, suppose an investment has an annual cash flow of $1000 after taxes for 5 years. The present value of this cash stream at 10 percent interest is $3790.
NYear / Return / / Present
Value
1 / $1000 / .909 / $ 909
2 / 1000 / .826 / 826
3 / 1000 / .751 / 751
4 / 1000 / .683 / 683
5 / 1000 / .621 / 621
$3790
In this case, each future cash amount was converted to a present value and the present values were then added. As a result, if we want to earn 10 percent on our money, we would be willing to invest $3790, the present value, now so as to get the future earnings of $1000 a year for 5 years.
In discounting future cash flows, it is also convenient to know the present value of a $1 annuity each year for n years. The annuity's present value is
Here we have discounted $1 each year back to the present time and added. The resulting values of P are tabulated in Appendix B for various interest rates and numbers of years.
We can solve the above problem directly by using the annuity's present values. For example, the present value of $1 a year for 5 years at 10 percent interest, from Appendix B, is 3.791. If $1000 is earned each year for 5 years, the present value is
P = $1000(3.791) = $3791
This is the same figure we arrived at above by adding the present values for each year (with the exception of round-off error in the last digit). The annuity table, therefore, saves time when uniform annual payments are present. The modern calculator and computer, however, may be even more convenient than the table.
In some investment problems, it is necessary to calculate an internal rate of return (IRR). The IRR is the interest rate which will just make the present investment equal to the future stream of earnings. In mathematical notation, suppose that an investment I has after-tax cash flows C1 in year 1, C2 in year 2, . . . , Cn in year n. The IRR is obtained by solving the following equation for i:
In general, the value of i is obtained by trial and error or iteration using the above equation. Suppose, for example, that we invest $5000 and earn $3000 in the first year, $2000 in the second year, and $2500 in the third year. Arbitrarily, assume i = 20 percent and solve the equation
Since $5335 is larger than the $5000 investment, we need a larger value of i to reduce the right-hand side. Assuming i = .30, we have
Since i = $4629 is smaller than the investment of $5000, the true value of i lies between 20 and 30 percent. Using linear interpolation, we can estimate that
Using i = .2475 in the equation to check the result, we have
Since this result, I = $4977, is slightly below $5000, we should reduce the interest rate a little more, perhaps to .24. By successive approximation, we will finally arrive at the interest rate to any desired degree of approximation.
In cases where the annual payments are equal, we can use the annuity tables (Appendix B) to find the IRR directly. For example, if the $5000 investment earns $2500 each year for 3 years, the following equation must be solved:
The annuity factor is therefore . From Appendix B, the interest rate for 3 years will be between 22 and 24 percent. The estimated figure, by interpolation, is 23.4 percent.
CHOOSING INVESTMENT PROJECTS
Now that we know how to calculate present values and internal rates of return, we can apply these ideas to choosing investment projects. Suppose there is a portfolio of investment alternatives. How should we choose among these alternatives or rank them in order of preference?
In general, there are three ways to make the choice; payback, present value, and IRR.
Payback. According to the payback method, a payback period for each investment is calculated as follows:
whereN = payback period in years
I = investment
S = salvage value
A = annual cash flow after tax
The investments in the portfolio are then ranked in order of their payback periods.
For example, suppose a $10,000 investment will earn $2000 a year after taxes, and there is no salvage value. The payback period for this investment is then 5 years.
The payback method has several shortcomings. First, the length of the earning period of the investment is not taken into account. Two investments could have the same payback period but drastically different lifetimes. The second problem with the payback method is that it does not consider the time value of money. Thus different earnings streams are not evaluated differently. Finally, the above formula requires a constant annual cash flow. This assumption could easily be relaxed, however, by determining the time required for earnings to equal the investment.
Although the payback method has serious weaknesses, it is still quite popular because it gives a sense of time to recover the investment. Nevertheless, it is being replaced by the next two methods as ways to rank investment alternatives.
Net Present Value. Whenever a hurdle rate or cost of capital is specified for investment comparisons, the investments can be compared through the use of present values. The given hurdle rate is used as the “interest” rate, and all future cash flows are discounted to the present time. The net present value (NPV) is then computed as follows:
WhereI = investment required
Pj = present value of cash flow for year j
Whenever the net present value exceeds zero, the investment is worthwhile at the specified hurdle rate. If capital is limited, the investments can be ranked in terms of NPV from largest to smallest and funded in order of priority until capital is exhausted.
Internal Rate of Return. The IRR can also be used to rank investments and select those for funding from the portfolio. Figure T2.1 shows several investments ranked by IRR and the cost of capital as a function of the amount invested. Notice how the cost of capital increases when large amounts of investment are required. As a result, the IRR falls below the cost of capital for
alternatives E and F. In this case, alternatives A, B, C, and D should be funded because their IRR exceeds the cost of capital, and alternatives E and F should not be funded.
The NPV and IRR methods are the opposite of each other. With NPV, the``interest'' rate or cost of capital is used to compute NPVs; a positive NPV indicates a worthwhile investment. For IRR, the “interest” rate is not given but is computed and compared with the cost of capital. An IRR greater than the cost of capital is considered a worthwhile investment.
If two investments have equal lifetimes, both IRR and NPV methods will yield the same result. If investment lifetimes vary, however, these methods require additional assumptions to yield a correct answer. The additional assumptions must specify what is done after the lifetime of the shortest investment. Is the capital invested in a riskless investment, a like investment, a technologically superior alternative, or what? The answer to these questions will affect the ranking of the alternatives.
A series of applications of the above investment methods will be described next. In each case, not only the numerical analysis but also other factors in the decision will be discussed.
PURCHASE OF A NEW MACHINE
The operations department is considering the installation of a machine to reduce the labor required in one of its processes. The machine will cost $50,000 and have a 5-year life, with a salvage value of $10,000 at the end of 5 years. The pretax cash-flow savings in labor which will accrue over the cost of operating the machine is $11,000 per year. Assume a 50 percent tax bracket, straight-line depreciation, and a 10 percent investment tax credit. What is the NPV of the investment at a 15 percent hurdle rate after tax? What IRR does the investment provide?
In all investment problems first the cash flow must be determined on an annual basis. In this case, the annual cash flow is
Cash flow -- pretax$11,000
Depreciation 8,000
Net income$ 3,000
Additional taxes 1,500
Since the additional taxes paid are $1500 per year, the cash flow after tax is $9500 per year ($11,000--$1500). In the first year, there is an additional tax credit of $5000 (10 percent of $50,000). The net cash flow in the first year is therefore $14,500 = ($9500 + $5000). The after-tax cash flows - assuming that all of these occur at the end of the year - are shown in Figure T2.2. Here, investments and cash outflows are shown as negative numbers while cash inflows and salvage values are shown as positive numbers. It is always helpful to draw one of these cash-flow diagrams prior to making NPV or IRR calculations.
The NPV at 15 percent cost of capital is
Since the NPV is negative, the investment is not worthwhile at 15 percent.
The IRR is obtained by inserting i in place of .15 in the above equation and solving for an NPV = 0. Thus the IRR must satisfy the following equation:
Since the NPV at 15 percent was negative, we know that i < .15. As a trial value select i = .10 and plug into the right-hand side of the above equation. At i = .10, we have NPV = - $3232. Since NPV is still negative, try a smaller interest rate, say 5 percent, which yields an NPV of $3727. Since this NPV is positive, the interest rate must lie between 5 and 10 percent. By interpolation:
The IRR is thus estimated to be 7.7 percent.
In this case, there are other factors to be considered in the decision, such as a possible loss of flexibility after converting to the machine and more consistent quality due to the machine. Since the ROI is so low, these factors will probably not be overriding.
MACHINE REPLACEMENT
The second example is the well-known machine-replacement problem, where the decision is whether or not to replace a current machine with a new model. Suppose for the sake of this example that we have a 5-year-old motorcycle and are considering whether to replace it with a ``new'' one (only 2 years old). If the motorcycle is not replaced now, assume it will be driven for another 3 years. For each alternative, the following costs are given:
YearKeep Old Motorcycle
/ 1 / 2 / 3Maintenance / $ 200 / $250 / $ 300
Tires / 200 / --- / ---
Gas and oil / 600 / 600 / 600
Insurance and license / 150 / 125 / 100
Total / $1150 / $975 / $1000
Buy New Motorcycle
Maintenance / $ 50 / $150 / $ 200
Tires / --- / --- / ---
Gas and oil / 400 / 400 / 400
Insurance and license / 250 / 200 / 150
Total / $ 700 / $750 / $ 750
The new motorcycle is expected to get better gas mileage, as reflected in the above costs of gas and oil. The new motorcycle will have higher insurance and license costs but lower maintenance and tire costs, also as shown in the above numbers. The net result is that the new motorcycle will be less expensive to operate than the old one.
Assume that the new motorcycle will cost $5000 ($2000 with the trade-in) and it will be worth $3000 at the end of 3 years. Also assume that the old motorcycle is worth $3000 now and will be worth $1400 in 3 more years.
The cash-flow pattern for the difference between these two alternatives is shown in Figure T2.3. The new-motorcycle alternative requires a new investment of $2000 at time zero, including the trade-in. The new-motorcycle alternative will save $450 in the first year, $225 in the second year, and $250 in the third year. In addition, the salvage value of the new motorcycle will be $1600 ($3000 - $1400) more than the old motorcycle's salvage value at the end of 3 years. Since this is a personal decision, there will be no depreciation and no tax consequences.
As in the previous example, two questions can be asked: (1) is the new motorcycle worth the investment at some given rate of interest, perhaps 15 percent, and (2) what rate of interest does the investment earn? The first question can be answered by computing the net present value of the cash flow shown in Figure T2.3.
Since the net present value of the investment is negative, it is not worthwhile to buy the new motorcycle at 15 percent return on capital.
The IRR is the interest rate which makes the above net present value equal to zero. By iteration, we find that IRR = 9.6 percent.
In this example, there are several additional questions of interest. First, what is the value of capital to the motorcycle buyer? If the buyer takes the money out of a savings account, the after-tax value of capital might be about 3 or 4 percent. At this rate the investment would clearly be worthwhile, since the IRR is 9.6 percent. If the capital were diverted from other personal investments, which could earn, say, 15 percent after tax, then the investment would not be worthwhile.
The new-motorcycle buyer would also want to consider the intangible value of having a new motorcycle. The buyer might have fewer repair problems, there might be less squeaks and rattles, and there would be an aesthetic value to owning the new motorcycle. It would be difficult to put a dollar figure on such intangible benefits, and they would probably have to be incorporated subjectively in the decision. One way to evaluate these intangibles, however, is to compute their dollar value. If capital is worth 15 percent, the intangibles would have to be worth more than $222 (the negative amount of present value) to go ahead with the decision.
Similar problems are encountered in process technology choice when a current machine is being replaced by a new one. The new machine will require additional investment, but it will probably reduce annual operating costs. The new machine will also provide intangible benefits, as does the new motorcycle, such as fewer production disruptions through greater reliability.
The examples illustrate two different types of decisions on choice of process technology. The first example was a choice between an existing manual technology and a proposed new automated technology. The second example was one of technology replacement. In practice, there are other variations of these problems, but the basic principles still apply.
QUESTIONS
1. What are the advantages and disadvantages in using the payback method?
2. Under what conditions would you use the IRR or NPV method?
3. What problems are created by unequal investment lives? How are these problems handled?
4. Precisely how does depreciation affect cash flow?
5. Under what conditions would you choose an investment which does not meet the hurdle rate?
PROBLEMS
1. A factory is considering the installation of a new machine which will replace two workers who have been doing the job by manual methods. The combined wages and benefits of the two workers are $45,000 per year. The new machine will be run by a single operator paid $25,000 a year in wages and benefits; it will also require $5000 a year for maintenance and utilities expense. The machine will provide the same output as the manual method, but the investment and installation will be $60,000. For tax purposes, the machine can be depreciated over a 5-year period using a double declining balance. Assume a 50 percent tax rate, a 10 percent investment credit, and no salvage value.
a. Is the machine a good investment at a 15 percent interest rate? Use a 10-year machine life.
b. Calculate the IRR for this decision.
c. What other issues are important in this process technology decision?
2. A laundromat is considering replacing its washers and dryers. Two options are available. Option A: The new machines will cost $11,000 initially and $12,000 per year to operate. Expected life for taxes and operations purposes is 5 years. Option B: The new machines will cost $14,000 initially and $11,000 per year to operate. Expected life for taxes and operations purposes is 10 years.
a. Using straight-line depreciation, determine which option is the best at a 10 percent interest rate. Assume a 50 percent tax rate.
b. What is the break-even interest rate, where the two options have equal present values?
3. A machine has been used in production for 5 years and is being considered for replacement. At the present time the machine is fully depreciated but could be sold for $8000. The firm pays tax on half of all capital gains. The new machine being considered will cost $30,000 and will have a 5-year life for both tax and cash-flow considerations. The new machine will require one less operator at a savings of $22,000 per year in wages and benefits. Assume no salvage value, a 50 percent corporate income tax, and straight-line depreciation.