Economic Analysis - Cash Flow Analysis

Cash Flow

To illustrate the models for economic analysis, consider the following situation.

 A businessman is considering the purchase of an asset that has an initial cost of \$2000. The asset promises an annual return of \$600. It's operating cost is \$100 the first year, \$150 the second, and increases by \$50 in each subsequent year. The salvage value for the asset in 10 years is \$400, or 20% of its initial cost. The cash flow for this situation showing the individual components explicitly is in the figure below. If the businessman's minimal acceptable rate of return is 10%, should he invest in this project?

The cash flow shows the amounts expended or received in each period of the analysis. An arrow drawn downward is an expenditure and an arrow drawn upward is a receipt. To provide a general procedure for evaluation, we define the general cash outflow or cost at time t as , and the general cash inflow or income at time t as . The time index t takes integer values between 1 and n, where n is the time horizon of the analysis and the units of time measurement are years (some other time measure could be adopted). Cash flows within a year are accumulated and, for analysis purposes, are assumed to occur at the end of the year. The net inflow at time t is .

Time zero represents some arbitrary reference time and is usually identified as the present time. Most analyses must determine the appropriateness of some investment, and we will define P to be the value of the investment made at time zero.

Although the figure shows a single investment at time 0, a project may require investments at other times. Often time 0 represents the startup time a project and several outlays during a construction phase would be represented by negative cash flows at times with negative indices. Net cash flows may be positive or negative in any year.

Cash flow analysis assumes that all flows over the life of the project are known with certainty. The analysis will reduce the cash flow to a single quantity that measures the profitability of the project. That measure will be used to accept or reject the project.

Minimum Rate of Return

In order to combine cash flows that occur at different points in time into a single measure, it is necessary that some rate of return be specified. In most cases this quantity is determined by management and represents the minimum rate of return that an investment must earn if it is to be acceptable. The rate of return is similar to the rate that a bank pays to persons who invest their money with the bank. In the case of a production system, the investment is the design and installation costs, and the return is the stream of cash flows returned during operation. We call this the minimum acceptable rate of return (MARR) and use the notation i to represent it. For the example, i = 10%.

Present Worth Analysis

The present worth (or present worth cost) is a single measure that expresses the investment at time zero and all cash flows that occur over n periods as an equivalent amount at time zero. It is particularly appropriate when the cash flows associated with a single project or several competing alternatives vary over time.

The present worth of a single cash flow at time t is

When the numerator is positive, this could be interpreted as the amount of money that must be invested at time zero at interest rate i to allow a withdrawal at time t of the amount . If the numerator is negative, the present worth can be interpreted as the negative of the amount that can be borrowed at time zero at interest rate i such that the amount that must be repaid is . When operated upon in the manner of the equation above, the cash flow is said to have been discounted to time zero. The quantity

is called the discount factor for time t. When i is positive, the discount factor is always less than 1 for t greater than zero. The use of discounted cash flows assigns more worth to cash flows that occur closer to the present time than those that occur further in the future. It recognizes the "time value of money." Money available now is worth more than the same amount obtained at some time in the future.

The present worth of a series of cash flows occurring from times zero to n is the sum of the discounted cash flows.

NPW stands for the net present worth of the project.

When the NPW is positive, the investment results in a rate of return greater than the minimum rate i. In the absence of alternatives this would be an acceptable investment. When the present worth is zero, the investment returns exactly the minimum acceptable rate, and this too would indicate an acceptable investment. When the present worth is negative, the investment would not satisfy the requirement of the minimum rate, and should be rejected.

This site provides an add-in that performs present worth analyses for arbitrary cash flows. The figure below shows the form constructed by the add-in for the example.

The investment in row 9 as a negative number. The Salvage is the percentage of the investment that is returned at the end of the life of the asset (20% = 400/2000). The Start value indicates when the asset is put into service and the End value indicates the end of its life.

The return is entered in row 13 as a uniform series with value 600. The first payment time (at the end of year 1) is entered in the Start cell and the last payment time (at the end of year 10) is entered in the End cell. The uniform part of the operating cost is entered in row 14 as a negative number (-100). The operating cost is increasing by an amount \$50 per year. This is entered as a negative gradient (-50) in row 15. The Start cell indicates when the first non-zero payment occurs (2) and the End cell indicates when the last payment occurs. The Parameter for the gradient is the periodicity of the payments (1 means that it occurs annually).

The add-in automatically computes the present worth for each component of the cash flow in column S starting in row 9. The total NPW for the project is in cell Q2 (81.93). The fact that this is positive indicates that the project returns more than 10%. In fact, the computed entry in cell Q5 indicates that the project returns 11.13% on the initial investment of \$2000.

The form makes it easy to change features of the project and to experiment with alternative assumptions concerning the cash flow. For example if one changes the MARR to 15%, the NPW becomes a negative number indicating that the project is not acceptable if the investor's minimum rate were 15%.

The Economics add-in has many features available to evaluate single projects. The add-in presents cash flows in either tabular or graphical forms. It can analyze situations with inflation and/or taxes. See the add-in instructions for more details.

Net Annual Worth

The number computed in Q3 is the net annual worth (NAW) of the project. If the NPW amount (81.93) were used to buy an annuity with 10 end of the year payments, the annuity payments would be the NAW (13.33). The NAW will always have the same sign as the NPW, so it adds no new information regarding the acceptability of the project. The NAW measure is useful when alternatives with different lives are compared.

The annual worth is a single measure that expresses the investment at time zero and all cash flows that occur over n periods as a uniform annual amount. The annual equivalent of an investment at time zero, P, that obtains a salvage value S at time n is

The factor in the brackets is called the capital recovery factor. It determines a uniform amount that includes an allowance for the capital expended (PS) and the investment cost of the capital. The salvage value, since it is the part of the investment recovered at the end of the project, only contributes an investment cost (Si).

When the income and costs associated with a project are uniform over the time horizon with values I and C respectively, the uniform net annual worth of the project is

NAW = ICA.

When incomes and costs are not uniform, one first determines the NPW of the cash flows and then uses the capital recovery factor to convert that to a uniform annual equivalent.

This formula is used to calculate the NAW on the project form (Cell Q3 for the example).

The NAW can be used to evaluate a single project or compare alternatives. When NAW is positive, the project has sufficient profit to justify the investment given the MARR, while if NAW is negative the project is not justified.

Internal Rate or Return

A third way to evaluate a cash flow is by computing its internal rate of return (IRR). The IRR of a project is the value of the interest rate that causes the NPW and the NAW to be zero. To find the IRR we find the value of that solves the following equation.

This is a useful measure, because it is common to consider investments in terms of their rates of return. The fact that for some unusual cash flows there may be more than one value of that yields zero NPW, makes IRR difficult to interpret for those cases.

Except for very simple cash flows, there is no closed form equation for computing the IRR. Rather the procedure is to use trial and error to evaluate the NPW or NAW, whichever is more convenient, for different values of i, until the result becomes sufficiently close to zero.The add-in uses a binary search procedure. For the example the IRR has the value 11.13 as presented in cell Q5.

With the IRR computed it is easy to make decisions regarding a cash flow. The project is acceptable if its IRR is greater than the MARR and not acceptable if its IRR is less than the MARR.

Different Measures

We have described three measures for representing a cash flow with a single quantity, the NPW, the NAW and the IRR. If a project is judged acceptable with one of the measures, it will also be acceptable under both of the others. There is one additional measure the return on invested capital (RIC). For simple investments the IRR has only one value that causes the NPW to be zero, the IRR is equal to the RIC. For mixed investments, where some incomes may preceed some expenditures, the IRR and RIC may be different. The RIC is useful because it can have at most one value for a given cash flow while for some mixed cash flows the IRR may have more than one solution. It is discussed more fully on the IRR/RIC page in the Computations section.

Operations Management / Industrial Engineering
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by Paul A. Jensen