University of Texas at Dallas
	School of Management

Finance 6301		Professor Day
Corporate Finance		Fall 1999

	Lecture 4:  Investment Decision Rules

Summary of Investment Decision Rules

	Most major corporations use two or more investment decision rules to evaluate requests for capital appropriations.  This note presents a short discussion of several of these rules.  However, since the payback rule and the average accounting rate of return rule are relatively simple and subject to a number of limitations, the balance of the discussion is devoted to a comparison of the internal rate of return criteria with the net present value rule.  However, prior to this comparison, each of these four investment decision rules is briefly summarized in terms of its strengths and weaknesses.

1.	Payback

	a)  No accounting for cash flows received subsequent to the cutoff date,
	b)  No explicit adjustment for either the timing or the uncertainty of future cash flows,
	c)  No an objective standard,
	d)  Useful for “small routine” expenditure decisions.

	The application of the payback rule is illustrated by the Table below. 

		Project	0	1	2	3	4

		A	<100>	20	 30	 50	100

		B	<100>	50	 30	 20	0

	Assuming that a firm applies the payback rule using a cutoff date of 3 years, projects A and B would be considered equally desirable.  However, note that for any positive discount rate project B has a negative net present value.  By contrast, project A has a positive net present value for most reasonable discount rates but has the same payback period as project B.  In other words, since the payback rule fails to consider cash flows occurring subsequent to the cutoff date, a company that relies solely on the payback rule would consider the two projects to be equally desirable.  While most companies do in fact supplement the payback rule with evaluations based either on the average accounting rate of return or some kind of internal rate of return criteria, the example illustrates the limitations of the payback rule.

2.	Average Accounting Rate of Return

	a)	Affected by arbitrary accounting standards for computing net income and book value,
	b)	No adjustment for time value of money,
	c)	No objective standard for setting cutoffs,
	d)	Not a rate of return in any meaningful economic sense.

3.	Internal Rate of Return

	a)	Subject to pitfalls associated with sequencing, timing, and scale problems,
	b)	Existence of multiple internal rates of return,
	c)	Misleading impression that the internal rate of return is a multiperiod rate of return,
	d)	Implicit adjustment for both the timing or the uncertainty of future cash flows,
	e)	When used properly, the IRR provides an objective standard for capital investment decisions,

	The internal rate of return is simply the discount rate that makes the net present value of the future cash flows from a capital investment project equal to zero.  In other words, the internal rate of return is simply the yield to maturity for a capital investment project.  The use of the internal rate of return rule in capital budgeting/project selection involves two distinct steps,

	a)	determine the internal rate of return (IRR) for the project,
	b)	accept any project offering an IRR greater than the firm's opportunity cost of capital (hurdle rate).

4.	Net Present Value (Discounted Cash Flow)

	a)	Explicit adjustment for both the timing or the uncertainty of future cash flows,
	b)	Objective standard of measurement,
	c)	Clear economic interpretation.

	In contrast to the payback rule, the net present value rule compares the present value of all of the future cash inflows (and outflows) from a project with the required investment, accepting any project for which the NPV is positive.  Ironically, the strength of the net present value rule is illustrated by two of the common justifications for use of the payback rule:  (1) the payback rule provides a simple means of adjusting for risk and (2) the payback rule provides valuable information to firms facing a “scarcity of capital” to finance new projects.  In theory, the net present value method provides a superior approach to adjusting for risk in that the appropriate premium for the risk of the project can be added directly to the discount rate used in the computation of NPV.  Further, to the extent that capital is plentiful at the firm's opportunity cost of capital, there is no reason to give priority to projects with short-run payoffs at the expense of projects having long-term payoffs.

Computation of the Internal Rate of Return

	It is important to note that the implicit role for the opportunity cost of capital in the application of the Internal Rate of Return Rule implies that, unlike Payback and Accounting Rate of Return, the IRR is based on an objective standard.  Further, because the IRR is a discount rate, the Internal Rate of Return Rule explicitly accounts for the time value of money.

	To illustrate the computation of the internal rate of return, consider the following example.  Suppose that a firm has the opportunity to invest $1000 today.  The firm expects that this investment will generate cash flows of $200 per year in perpetuity.  Since the IRR is the discount rate which makes the net present value of an investment project equal to zero, the IRR will be the solution to

		0	=     <$1000>     +     ,

which gives an internal rate of return of

		IRR	=     .

Although computation of the internal rate of return is straightforward in the case of an investment which generates perpetual future cash flows, the computation of the internal rate of return will generally require either a spreadsheet package or a sophisticated financial calculator.

Relation Between the IRR and the Net Present Value Rule

	The example above can be used to illustrate the relation between the internal rate of return and the net present value of a capital investment project.  Suppose that the firm has an opportunity cost of capital of 25 percent, which is greater than the internal rate of return for the project in question.  In this case, the net present value for the project will be

		NPV	=     <$1000>     +     ,

			=     < $200>  .

Therefore, whenever the firm's required rate of return is greater than the internal rate of return, the net present value of the project is less than zero.

	If the firm's required rate of return is less than the internal rate of return for the project, then the project will have a positive net present value.  To see this, assume that the firm has a required rate of return (opportunity cost of capital) of 10 percent.  In this case, the project has a net present value of

		NPV	=     <$1000>     +     ,

			=        $1000  ,

which shows that a project will have a positive net present value whenever the opportunity cost of capital is less than the internal rate of return.

		Pitfalls Associated with the Internal Rate of Return

	In addition to being difficult to compute and the possibility of multiple rates of return (noted by Ross, Westerfield and Jaffe), the IRR rule fails to give the correct answer to three classic investment decision problems.

Sequencing

	In some instances, the proper interpretation of the internal rate of return depends on the sequencing of cash inflows and outflows from a project.  To illustrate this problem, consider the following example for a firm whose opportunity cost of capital is 10 percent,

		Project	0	1	IRR	NPV @ 10%

		A	<$1000>	$1500	.50	$364
		B	 $1000	 <$1500>	.50	 <$364>

	Note that although both projects have the same internal rate of return, the net present value of A is positive while the net present value of B is negative.  The explanation for this seeming inconsistency between the internal rate of return rule and the net present value rule lies in the sequencing of the cash flows.  To resolve this problem, note that Project B should be interpreted as a loan since the cash inflow precedes the cash outflow.  When B is viewed as a loan, the internal rate of return of 50 percent represents the cost of obtaining capital.  Since the internal rate of return (i.e., the cost ) of the “loan” is greater than the opportunity cost of capital, we would reject Project B.  By contrast, Project A is a more conventional investment in the sense that the initial cash flow (at date 0) is negative, with positive future cash flow (at date 1).

Timing

	The Internal Rate of return rule is also unreliable for ranking mutually exclusive projects that

	1.  produce cash flows over different time horizons,
	2.  produce different patterns of cash flows over identical time horizons.

To see this, assume that a firm with a required rate of return of 10 percent is faced with the following mutually exclusive capital investment projects,

		Project	0	1	2	3	IRR	NPV @ 10%

		C	<$1000>	$1200	0	0	 .200	$91
		D	<$1000>	0	0	$1500	 .145	$127

	Although project C has a higher internal rate of return than project D, project D has a larger net present value given that the firm has a required rate of return of 10 percent.  In fact, it is straightforward to show that so long as the firm's required rate of return is less than 11.8 percent, project D will have a greater net present value than project C.  The inconsistency shown above can be avoided by relying on the net present value rule, which is of course based on the 'correct' opportunity cost of capital.

Scale

	The internal rate of return is difficult to use in selecting the optimal scale of investment.  To see this point, consider the following example taken from Ross, Westerfield and Jaffe.  A firm has the opportunity to produce a motion picture on a small scale (like the Brothers McMullen) or on a large scale (like Waterworld).  The cash inflows and outflows from the motion picture project are as described below.

		Project	0	1	IRR	NPV @ 25%

	Small Budget	<$10>	$40	3.00	$22
	Large Budget	<$25>	$65	1.60	$27

Note that although the small budget project has the greater internal rate of return, the large budget project has a greater net present value.  In other words, when there is a decision with regard to the optimal scale of a capital investment project, choosing the alternative with the highest internal rate of return does not maximize shareholder wealth.

	To see that the net present value rule produces the greatest level of shareholder wealth, consider the decision to escalate the scale of the project from a small budget picture to a large budget picture as an independent investment project.  The decision to increase the scale of the picture, which requires that the production company to spend an additional $15 (million) at date zero in return for additional revenues of $25 (million) at date one.  In other words, the incremental cash flows associated with the increase in scale are as shown below

		Project	0	1	IRR	NPV @ 25%

	Increase in Scale	<$15>	$25	.667	$5

	The analysis on the previous page shows that the increase in the scale of the project is a positive net present value project.  Alternatively, we could say that the scale of the project should be increased because the incremental cash flows have an internal rate of return greater than the opportunity cost of capital.

Multiple Internal Rates of Return

	One of the potential problems associated with using the internal rate of return is the fact that multiple solutions for the IRR may exist for projects which alternate between producing net cash inflows and net cash outflows.  In other words, any project for which the projected cash flows alternate (more than once) between producing positive cash flows and negative cash flows may have more than one internal rate of return.  As an example, consider the following example of a strip mining project.  For this project, the initial cash flow is negative as the mine is developed.  Although the cash flows from the project then become positive, the final set of cash flows are negative since recent environmental regulations require that the land be restored to something approximating its initial state.

		Project	0	1	2

	Strip Mining	 <60>	155	<100>

The internal rate of return for this project is the solution to

		0	=     <$60>    +    +    .

	You can easily verify that internal rates of return of both 25 percent and 33 1/3 percent satisfy the zero net present value condition given above.  Further, the net present value of the strip mining project will be positive for any discount rate between these two solutions.  While it may not be surprising that the net present value for the project is negative for discount rates in excess of 33 1/3 percent, you may be surprised to find that the net present value is also negative for any discount rate that is less than 25 percent.  This seeming paradox can be resolved by reflecting on the fact that a reduction in the discount rate below 25 percent increases the present value of the cash outflow in year two relative to the present value of the cash inflow in year one. 

	Unfortunately, your spreadsheet package may fail to warn you if there are multiple solutions for the internal rate of return.  Since these multiple solutions may arise in a number of practical situations, it is important to be aware of whether or not the occurrence of multiple solutions is indeed a possibility.   More importantly, if there is a projected future shift in a project's cash flow stream from net inflows to net outflows (assuming an initial shift from outflows to inflows), then it is important to know what your spread sheet package does when it encounters multiple solutions for the internal rate of return?  Some packages automatically report the positive internal rate of return whenever there is one positive solution and one negative solution.  Alternatively, when all solutions are positive, spreadsheet packages sometimes report the smaller of the positive solutions.  The important point to remember is that there is no intuitively correct internal rate of return when the pattern of incremental cash flows offers multiple solutions for the internal rate of return.

	A somewhat more intuitive explanation for the existence of multiple IRR's is provided by an application of the quadratic formula.  Assume that we have the opportunity to spend $35 (1000s) on a promotional campaign which will increase the usage of our main product over a two period horizon.  The marketing program will both increase usage of the product and accelerate consumption from year two into year one.  More precisely, if we choose not to aggressively market our product then cash flows will be $100 in year 1 and $200 in year 2.  By aggressively marketing the product, we can increase cash flow in year 1 to $190, but will decrease cash flow in year 2 to $150.  The incremental cash flows from this proposed investment are depicted by the time line below:

		Project	0	1	2

	Invest in Marketing	 <35>	190	150

	No Investment	 	100	200

	Incremental Cash Flow	<35>	90	<50>

	Given an opportunity cost of 12 percent, the net present value of the incremental cash flows from the marketing program is

		NPV	=     <$35>    +    +    

			=      $5.5  .

	Since the project has an NPV of zero when the cash flows are discounted at the internal rate of return, the internal rate of return for the project is the solution to

		0	=     <$35>    +    +    .

	Note that if we multiply the expression above by (1 + IRR), we have

		<35> (1 + IRR)+    90 (1 + IRR)    +    <50>     =     0  ,

which is a quadratic equation in (1 + IRR).  Therefore, in this special case, we can solve for the IRR using the quadratic formula, which states that if x is the solution to

		a x+    b x    +    c     =     0  ,

then x can be found from the quadratic formula

		x	=      .

	The quadratic formula shows that there are two solutions to any quadratic equation, one where we add the term under the square root sign and one where we subtract it.  In our problem, 'a' is <35>, 'b' is 90, and 'c' is <50>.  Therefore, we have 

		1 + IRR	=      

which implies that both 75.4 percent and -18.8 percent are legitimate internal rates of return for the proposed investment project.  Unfortunately, neither of these internal rates of return appears to be a particularly relevant indicator as to the economic worth of the project.  While the quadratic formula is of little use in determining the IRR for more general streams of cash flows, the +/- nature of the formula motivates the possibility that cash flow streams which shift from negative to positive and back again will generally have two internal rates of return.  More importantly, the example shows situations may arise where none of the multiple solutions to the zero net present value relation can be meaningfully compared to the firm's hurdle rate/opportunity cost of capital.

Modified Internal Rate of Return

	The internal rate of return is sometimes mistakenly interpreted as a multiperiod rate of return.  Since the internal rate of return is the discount rate which makes the net present value of an investment project equal to zero, nothing more and nothing less, this interpretation is incorrect.  To clarify this point, consider the following stream of cash flows,

		Project	0	1	2

	 Primary	 <$2663>	$2000	$2000

It is straightforward to show these cash flows have an IRR of 32 percent, 

		0	=     <$2663>    +    +    .

	Note that 32 percent is not the rate at which the initial investment grows over the two period investment horizon.  In fact, it is impossible to determine this rate without an explicit assumption as to the rate at which cash flows received prior to the end of the investment horizon (i.e., $2000 received at date one) are reinvested.  However, given a reinvestment assumption (concerning the rate at which intermediate cash flows can be reinvested) it is possible to compute the rate at which the initial investment grows over the investment horizon.  This rate is referred to as either the reinvested rate of return or the modified internal rate of return.

	To illustrate the computation of the reinvested rate of return, assume the $2000 received at date one can be reinvested at 10 percent.  Then the inflow of $2000 at date one is implicitly netted out against a $2000 outflow ( investment) in exchange for an additional $2200 (principal plus interest) to be received at date two.  Given this reinvestment assumption, the cash flow stream becomes

		Project	0	1	2

	 Primary	 <$2663>	$2000	$2000

	 Reinvestment	 	 <$2000>	$2200

	Net Cash Flow	 <$2663>	$0	$4200

	If interim cash flows are reinvested at 10 percent, the initial investment of $2663 will grow to $4200 when the project terminates in two years.  The reinvested (modified internal) rate of return is simply the rate at which the initial investment of $2663 must grow in order to equal the terminal value of $4200 at the end of the investment horizon.  In other words, given our reinvestment assumption (which determines the terminal value), the reinvested rate of return is the solution to 

		Investment  (1 + RRR)	=     Terminal Value  .

Therefore,  the reinvested rate of return is equal to

			RRR	=     -     1

which represents a reinvested rate of return of 25.6 percent.