As usual, investment projects will be represented by the corresponding expected cash flows. Since the future cash flows are uncertain, the results of decision making process are also uncertain. The detailed qualified analysis may reduce uncertainty, however.

We will deal with a set of competing projects. The decision maker may accept one or more projects and may even decide not to accept any. The projects are said to be mutually exclusive if at most one of the involved projects can be accepted. And they are said to be independent if an arbitrary number of the competing projects (including none of them) can be accepted.

There are two broad classes of investment projects that often arise in practise. In the first case, the investors use their own capital for the initial investment and they obtain incomes generated by the initial investment in successive periods. Such projects are characterized by negative payments in the initial period(s) and positive ones afterwards. Call them class I projects. In the second case, the investors take a loan at the beginning, make an investment, and then they acquire the benefits and also should pay back the loan. Such projects are characterized by positive payments in the initial period(s) and negative ones afterwards. Call them class II projects.

There is a variety of methods for decision making and we will mention only some of the principles. All the methods start with a careful analysis of the expected stream of payments including dividends, interest obtained or paid, salvage value of the assets at the end of the project's life, etc. The cost of capital (the valuation interest rate) should take into account the riskness (uncertainty) of the project.

3.7.1 Profitability Index

A simple indicator for a class I project CF = (CF0, CF\,..., OF?) is the profitability index defined by

This measure seems to be trivial but in fact it is, in some sense, equivalent to the measures based on the present value profile as we will see later. Among competing projects we select those with highest profitability indexes greater than one; we select none of them if all PF s are less than one.

Another simple and rough method is the payback method applied again to class I projects. It is based on the payback period that is the number of periods required to recover the initial outflows. Formally, let us keep assumptions of Theorem 3.4.1. For a class I project we have Aq < 0. Let k be the first index such that Ak > 0. Then the payback period is defined by

Here fc β 1 is the period just preceeding the full recovery, βAt-1 is the uncovered cost at the beginning of this period, and CFk (obviously positive) is the payment in the recovering period. If such a k does not exist, we set formally the payback period to infinity. Based on the payback method, we select the project(s) with the shortest payback period, or none of them if their payback periods all equal infinity.

A little better method based on this idea is the so called discounted payback method. Let i be a properly chosen project's cost of capital and define A^ β

Z{=o CFt/(l + i)*. Assume again < 0 and k the first index such that A^ > 0. Then the discounted payback period is defined as

If such a k does not exist we set formally the discounted payback period to infinity. The decisions based on the discounted payback method are the same as in case of the usual payback method.

3.7.2.1 Exercise and Problem. Analyze and try to prove the following conjecture. For a class I project of length T (At > 0), the discounted payback period approaches T as the interest rate approaches the internal rate of the project, i JRR.

Typically, for class I projects the present value is a decreasing (and often also convex) function of the valuation interest rate i and the opposite is true for class II projects; the present value is an increasing (and often concave) function of i. However, this is not the rule as shown in the following counterexample.

Consider an artificial cash flow CFZ = (-6,-10,-4,-8,-3,-5,18.5,18.5). The assumptions of Theorem 3.4.1 are fulfilled. The only IRR is 0.006372. But PV(CFz,i) is decreasing for i < 0.39 and increasing for i > 0.39.

Hence the investor should take care of the individual present value profile, i.e., the graph of the present value in dependence on the interest rate involved.

The leading rule is simple; for a given i accept the project if its present value at this interest rate i is positive:

For class I projects, the criterion of positive present value is equivalent to PI(CF,i) > 1. In case of independent projects we select all the projects with the positive present values at the given interest rate. If the projects are mutually exclusive we select that with the highest present value. If we investigate a set of projects which are mutually exclusive dependent on the valuation interest rate we should select the project that is determined by the upper envelope of the present value profiles.

For one project, the critical point is IRR. If PV is a decreasing function of i then we accept the project if the valuation interest rate is less than IRR and reject it otherwise. Analogously, if PV is an increasing function of i, we accept the project if the valuation interest rate is greater than IRR. For projects which do not possess a monotonous present value profile, we should perform a more careful analysis.

For two or more projects, the critical points are not only the IRR's of the individual projects but also their crossover rates. A crossover rate of two projects is such an interest rate for which the present values of the two projects are equal. Formally, let us consider two projects CFa and CFb- The crossover rate Iab is defined as a solution to the equation

Obviously, there may be more than one solution so that we must select that one with a reasonable economic interpretation. Since the present value is a linear function on the space of cash flows, we see that the crossover rate iab is in fact the internal rate of return determined by the difference between the two projects, IRRa-b-

In the neighborhood of the crossover rate the investor should take carc and carcfully study also the sensitivity of the present value profiles with respect to the interest rate. This is best done by looking on the duration and possibly on the convexity. Such an analysis will be better understood from the example.

3.7.3.2 Example. Let us consider five projects: (1) A: CFA = (-1000,300,500,200,100)

(3) C: CFc = (-851.18586,281.0005,170.39716,300,200)

Projects A, B, C, D are class I projects while E is a class II project. CFb represents the cash flow of a four years coupon bond purchased for the par value 1000 giving the holder yearly coupons of 47 with redemption value 1000. The present value profiles of these projects are shown in Figure 4. Visually the present value profiles of the projects A and C coincide. The payback periods for the first four projects are

PB(CFa) = 3.00 PB(CFb) = 3.82 PB(CFc) = 3.50 PB(CFd) = 4.83

and the discounted payback periods for two selected interest rates (i = 0.02, i = 0.04) are: PB(CFa, 0.02) = 3.40, PB(CFb, 0.02) = 3.89, PB(CFCl 0.02) = 3.70, PB(CFD) 0.02) = 4.99 and PB(CFa,QM) = 3.84, PB(CFb, 0.04) = 3.97, PB(CFc,QM) = 3.92, PB(CFd,QM) = +oo. In case of independent projects, based on the discounted payback method we accept projects A, B, C, D if i = 0 or i = 0.02. For i = 0.04 we accept A, B, C and reject D. If the projects are mutually exclusive, we accept only A for all three values of i.

Figure 4: Present values of 5 projects

Figure 4: Present values of 5 projects

The present value is a decreasing function of i for projects A, B, C, D, and an increasing function for project E, Thus the acceptance region depends on the corresponding IRR's:

IRRa = 0.0471 IRRb = 0.0470 IRRC = 0.0472 IRRD = 0.0208 IRRE = 0.0333.

Consider first the case of independent projects. We accept A, B, C, D for i < 0.021(= IRRd). For 0.021 < i < 0.033(= IRRE) we accept A, B, C. For 0.033 < i < 0.047 (approximately) we accept A, B, C, E; and we accept only E for i > 0.047.

Second, consider mutually independent projects A, C, E only. Since the projects A and C have almost identical present value profiles, we must look first at the difference of their present values. In Figure 5 we have plots of PV(CFc, i) β PV(CFA,i), D(CFc,i) - D(CFA,i), and C(CFc,i) - C(CFA,i). We see that PV(CFc,i) > PV(CFA,i) and that the difference is negligible. We also have PV{CFA,0.02) = PV{CFc,0.02) and D(CFA,0.02) = D(CFc,0.02). Since the convexities fulfil the inequality C(CFc,i) > C(CFA,i) we can decide in favor of project C against A. Further, the crossover rate for projects C and E is IRRq-e = 0.0391. Thus to summarize, for i < 0.0391, we accept C and for i > 0.0391 we accept E, among the candidates A, C, E. If we consider all the five projects, then we obviously select B for i < IRRb-b = 0.041 and E for greater values of i.

Suppose that the cash flow in question depends also on another variable or parameter y say, CF = CF(y). For decision making, an important measure is the value of y = y(i) such that the present value for a given interest rate i is zero. Call this value the internal value of the cash flow and denote it by HIV. (HIV has been introduced in [83] but we admit that such a simple indicator might have been known before.) Mathematically, HIV is defined implicitly by the relation

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