AMDG
Chapter 4
On The Meaning Of (Economic) Life:
An Overview And Proposed Method Of Estimation
I.
Introduction
The concept of amortization, or decay, applies to all assets. This includes
intangible assets.
For tangible assets, economic obsolescence is relatively easy to observe. Tangible
assets wear out at a rate that can be observed at a physical level. In some cases,
this obsolescence occurs because of actual decay or entropy. In other cases,
physical assets obsolesce because their underlying functionality is displaced by a
competing functionality. In either case, obsolescence is an empirical fact that is
relatively easy to verify.
By contrast, intangible assets “decay” at a rate that is inherently more difficult to
observe. The reason for this is obvious. Intangible assets are, at essence, ideas.
They are non-physical in nature. This means that they can not be observed,
except indirectly through their effect on revenue and profits. If the assets
themselves can only be observed indirectly, it stands to reason that their rate of
change, or obsolescence, is even more removed from empirical observation.
It is largely for this reason that the concept of the “economic life” or “remaining
useful life” of intangible assets is one that is inherently fuzzy, frequently
confusing, and subject to numerous (often inconsistent) definitions. The result is
well known – valuations of intangible assets are commonly contested because of
disagreements relating to the economic life of the subject intangibles.
This chapter serves a twofold purpose. First, it attempts to summarize and
evaluate the meanings of “economic life” as applied to intangible assets. Second,
it proposes a new definition of economic life that is “endogenous” to the firm.
By this I mean that my proposed definition of “intangible asset life” is a function
only of objective financial data, rather than subjective determinations relating to
the “usage” and “productivity” of the intangible assets at issue.
The logic of my proposed approach to defining economic life can be summarized
as follows.
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1) First, recognize that today’s stock of intangible assets is the result of the
cumulative accretion of prior years of intangible asset investments (such
as R&D or marketing investments) by the firm. This is, at essence, a
recognition that, from an economic perspective, sunk intangible asset
investments are “capitalized” by the firm, even if this capitalization is not
always recognized by current accounting standards.
2) Second, define a prior year, or “layer,” of sunk intangible asset investment
as being “in service” today, if yields a return today.
3) Third, recognize that the economic life, N, of an intangible asset is
necessarily both the number of years over which a dollar of intangible
asset investment incurred today yields its required return in the future and
the number of layers, or prior years, of past investments that have
accumulated to form today’s stock of intangible assets. This is necessarily
true. For example, in a steady state, if today’s dollar of R&D investment
yields its required return over 5 years, then it must be the case that 5 years
of prior investments are supporting today’s intangible-related profit flow.
Therefore, if we know the number of layers of “in service” investments,
we know the time period over which today’s intangible asset investment
yields a return, and vice versa.
4) Fourth, recognize that there can only be a finite number of layers, or
previously sunk investments, in service today. We can see this by noting
that if this statement were not true, then the total return to currently in
service investments would be greater than residual profit – because it
would be infinite.1 In other words, some finite number, N, of prior years
of sunk investments will fully exhaust the firm’s per period residual profit
flow, simply because each layer requires (produces) a return if it is in
service, and an infinite economic life would imply infinite layers and
infinite profits.
5) Fifth, given #3 above (if we know the number of layers, we know the
future payoff period of today’s investments, and vice versa), by finding
the number of layers of prior investments, N, that exhausts the firm’s per
period residual profit, we have also found the economic life of today’s
investments in the same intangible. We have found an “endogenous”
definition of economic life – that is, an N that is entirely determined by the
firm’s profit flow and investment decisions.
It would be equal to I*r*N, where I is the intangible asset investment per year, r is the required
return on that investment given that it is a perpetuity, and N is the total number of prior years of
investments made by the firm. As N approaches ∞, so too does the profit flow attributable to the
firm’s sunk investments. This is obviously an absurd result.
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6) Sixth, recognize that this unique value for N is the only choice of economic
life that neither implies more, nor less, residual profit than actually exists.
Other definitions of economic life may arrive at an estimate of N that
corresponds precisely to the level of residual profit expected by the firm,
but will not do so necessarily.
This approach results in a valuation of existing intangible property that has four
very important characteristics. First, only this “endogenous” approach has the
advantage of ensuring that the choice of N is exactly consistent with the steady
state level of residual profit in the system. Importantly, I do not propose that this
is the only approach to estimating economic life that makes sense. Other
approaches, including subjective analyses of functional and economic
obsolescence, should be incorporated into any evaluation of intangible asset life.
However, there is only one N that exactly absorbs, or exhausts, the residual profit
in the system, given a rate of return and level of steady state intangible asset
investment.
Second, what this approach demonstrates is that the economic life of intangible
assets must be finite. N must be a finite number. As noted earlier, I show in this
paper that if N were not finite, the implication would be that the firm would
eventually have an infinite amount of residual profit per period (which is
obviously absurd).
Third, because the economic life, N, is chosen such that the ongoing intangible
asset investments are NPV=0,2 a buyer of intangible assets using this approach to
economic life (i.e., the party making post-transaction intangible asset
investments) is by definition NPV=0. This means that this approach is
fundamentally consistent with the IRS’ Investor Model framework, which simply
says that a buyer of any asset should be NPV=0 on the deal if the asset price is
correct.
Finally, this approach is very similar in principle to the valuation approach
outlined by Modigliani and Miller in their famous 1961 paper on optimal
dividend policy. Almost as an aside, Modigliani-Miller show that the value of
the firm is equal to the present value of a steady state flow of “normal” or
“competitive” returns, plus the present value of growth opportunities over a
The net present value of the intangible asset investments is zero – meaning that the present
value of the future benefits from the investments exactly equals the upfront cost of the
investments.
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“competitive advantage period.” My approach in this paper is quite
complementary, and provides a means of estimating the competitive advantage
period in the Modigliani-Miller model. This is important to grasp – the
endogenous approach to N that I develop in this paper will always result in
the unique N that produces a valuation of intangible assets that when summed
with the present value of “normal” or “routine” returns, equals enterprise
value. This is, in essence, a new way to value the firm and its component
parts.
This paper proceeds as follows. In Section II, I survey and evaluate the three
primary meanings of the term “economic life” that are in currency today. In
Section III, I describe an alternative approach to the question of intangible asset
life – the endogenous life approach. Section IV then examines the intangible
asset valuation and licensing implications of this approach, and demonstrates
that the endogenous approach to economic life is generally consistent with the
licensee-licensor profit split rule of thumb, as well as with approaches to
economic life that are currently used. Section V demonstrates that this approach
to economic life is consistent with, and in fact completes, the 1961 “competitive
advantage period” valuation model sketched out by Modigliani and Miller.
Section VI discusses the relationship between realized rates of return to
intangible asset investments and discount rates for intangible asset-generated
profit. Section VII concludes.
II.
Intangible Asset Life – Three Common Definitions
While the assumption that intangible assets decay is a standard one, an accepted
definition of intangible asset life seems hard to come by. At present, there
appear to be three primary concepts of economic life in currency. These are as
follows.
1) Functional Obsolescence. Intangible assets have eroded in value when
they have ceased to function at a technical level. This definition is
commonly applied to software code or other technologies. For example, if
an average line of code, or a typical object in an object-oriented software
library, is no longer used in a software product after five years, then we
say that the economic life of software is five years.
2) Residual Profit Obsolescence. Intangible assets have eroded in value
when they are considered to no longer contribute to residual, or economic,
profit. That is, when an intangible asset no longer produces pricing
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power, and therefore to profit in excess of a company’s required return to
other functions and assets, that intangible asset has amortized.
3) Willingness To Pay. Intangible assets have eroded in value once a third
party would be unwilling to pay for access to the intangible. This
definition relies on the public good nature of intangible assets – that is, the
fact that intangible assets are goods that are non-excludable and non-rival
once they have leaked into the public domain. Since all intangibles
(knowledge goods) eventually leak into the public domain, all intangibles
eventually amortize to zero. As an example, the Pythagorean Theorem is
an intangible that is commonly used by engineers, but it is not
commercially transferable given that it has fully leaked into the public
domain – it is a public good.
There are advantages and disadvantages to each of these definitions. Beginning
with functional obsolescence, functional obsolescence has the advantage of being
relatively observable. That is, it is sometimes possible to clearly observe, at a
functional level, the rate at which an intangible asset ceases to be of use. Lines of
software code are the most commonly cited example of this – we can tell when a
line of code is replaced, and therefore is no longer functional.
However, functional obsolescence also has important conceptual disadvantages.
First, it does not apply to all asset types. Trademark assets, for example,
generally do not “functionally” obsolesce in a meaningful way. Under normal
conditions the only functional obsolescence that one can observe for trademark
assets is the discontinuation or replacement of a trademark asset. Second, and
perhaps more importantly, the concept of functional obsolescence may ignore the
“value echo,” or economic advantage, passed on to future intangible assets by
the current intangible assets. For example, even when we observe existing lines
of code being replaced by new ones, the new lines of code may incorporate
concepts or other programming devices that existed in the old code. Just as a
child’s DNA incorporates elements of the DNA of prior generations, lines of code
likely incorporate elements of logic from prior generations of software code. A
functional obsolescence concept has a difficult time capturing this.
The second definition of obsolescence, residual profit obsolescence, has the
distinct advantage that it emphasizes the economic profit, or profit in excess of a
required return, that results from the pricing power that intangible assets
generate. In other words, residual profit obsolescence “asks the right question,”
by focusing the analysis on pricing premia, or market power, that results from
successful investments in intangible assets.
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However, residual profit obsolescence suffers from a problem that is similar to
one of the problems faced by the functional obsolescence concept. Namely, this
approach to defining economic life requires that one take a stance regarding
whether or not current or expected future residual profit results only from
currently functional generation of intangible assets (i.e., lines of currently
functioning software code), or from current and prior generations of the asset. In
other words, residual profit obsolescence begs the same fundamental question as
functional obsolescence – namely, is the relationship between residual profit and
intangible assets limited to currently functioning intangibles or prior generations
that have a “value echo” reflected in today’s intangibles. Depending upon one’s
view of the way in which “current and old” intangibles map into residual profit,
this definition, like functional obsolescence, can produce widely varied estimates
of economic life. Said differently, while residual profit obsolescence focuses on
the right question (the timing of erosion in price premia and economic profit), the
subjectivity of the answer is of concern.
Finally, we come to “commercial transferability.” In a sense, commercial
transferability is the most objective of the three definitions of economic life. The
test here is a standard market test – are the intangible assets salable? Would a
buyer be willing to pay for the intangibles? The primary advantage of this
approach is that it sidesteps the question of whether or not current or previously
“functioning” generations of the intangibles matter. The question asked is,
simply, “for how long would a third party be willing to pay for your current
portfolio – however you define it – of intangible assets?” Equivalently, this
approach says “take this year’s investments, last year’s investments, the
investments of the year prior, and so on, and tell me how far back you go before
a third party would no long be willing to pay for the resulting intangible assets?”
Thus, in principle, the commercial transferability or willingness to pay definition
is objective. However, in practice, it is typically the case that the fact finding
necessary to implement this definition is very difficult to perform. By their
nature, intangible assets that generate price premia are unique, therefore markets
in the intangibles are not “thick,” or “liquid.” It is therefore difficult to estimate
how long a given layer of sunk intangible asset investment will be commercially
transferable, since there are very few market reference points to rely on. This
problem is particularly acute for marketing and customer-based intangibles.
What does it mean, exactly, to ask “how long would a third party be willing to
pay for your sunk marketing investments?” In my experience, the response
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when this question is asked of company marketing personnel is something along
the lines of “I’m not sure that the question makes sense.”
We thus arrive at the conclusion that most practitioners already understand
intuitively, there is no perfect way of thinking about economic life, or economic
obsolescence, for intangible assets. The most common definitions all have
drawbacks.
In the next section, I propose an alternative.
III.
A Proposed Definition
A.
Reasonable Criteria
Given the drawbacks of the definitions of economic obsolescence reviewed
above, it is worth asking the question – if one were to develop a definition of
economic life, what criteria might one require? Said differently, what would be
some of key characteristics of a meaningful and practical definition of economic
life?
Four desirable characteristics come to mind immediately. They are as follows.
1) Objective. Any definition of economic life should rest, to the greatest
degree possible, on objective data rather than subjective judgments.
2) Simple. Any definition of economic life should be straightforward, and
not rest on complex, or vague, concepts.
3) Intuitive. Any definition of economic life should be intuitive sensible. If,
for example, the definition produces short economic lives (rapid
obsolescence rates) under certain conditions, this should make sense
intuitively and be consistent with economic theory.
4) Finance-based. Any definition of economic life should be consistent with
basic concepts from finance theory – in particular, the theory of
investment decisions. In more precise terms, a definition of economic life
should require an analytical correspondence among three key variables:
a) levels of intangible asset investment, b) levels of economic or residual
profit, and c) the required rate of return on intangible asset investments.
What follows is a proposed definition of economic life that I believe meets these
four criteria. I also believe that this definition has significant conceptual
advantages over the three definitions in common currency.
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I begin with the intuition behind the definition, and the process for arriving at an
estimate of economic life. I then move to the mathematics.
B.
The Basic Model
The logic of my proposed approach to defining economic life can be summarized
as follows.
1) First, recognize that today’s stock of intangible assets is the result of the
cumulative accretion of prior years of intangible asset investments (such
as R&D or marketing investments) by the firm. This is obvious –
intangibles, like any asset, are created by making investments.
2) Second, define a prior year, or “layer,” of sunk intangible asset investment
as being “in service” today, if yields a return today. “In service” layers are
like a first in – first out inventory. The investments come into service, live
for some period of time, and then amortize away.
3) Third, recognize that the economic life, N, of an intangible asset is
necessarily both the number of years over which a dollar of intangible
asset investment incurred today yields its required return in the future
(the life of a layer of sunk investment) and the number of layers, or prior
years, of past investments that have accumulated to form the intangible
asset. This is necessarily true. For example, if today’s dollar of R&D
investment yields its required return over 5 years, then it must be the case
that 5 years of prior investments are supporting today’s intangible asset.
Therefore, if we know the number of layers, we know the time period
over which today’s intangible asset investment yields a return (economic
life), and vice versa.
4) Fourth, consider that there can only be a finite number of layers, or
previously sunk investments, in service today. We can see this by noting
that if this statement were not true, then the total return to currently in
service investments would be greater than residual profit – because it
would eventually be infinite.3 In other words, some finite number, N, of
prior years of sunk investments will fully exhaust the firm’s per period
residual profit flow, simply because each layer requires (produces) a
It would be equal to I*r*N, where I is the intangible asset investment per year, r is the required
return on that investment given that it is a perpetuity, and N is the total number of prior years of
investments made by the firm. As N approaches ∞, so too does the profit flow attributable to the
firm’s sunk investments. This is obviously not possible.
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return if it is in service, and an infinite economic life would imply infinite
profits.
5) Fifth, given #3 above (if we know the number of layers, we know the
future payoff period of today’s investments, and vice versa), by finding
the number of layers of prior investments, N, that exhausts the firm’s per
period residual profit, we have also found the economic life of today’s
investments in the same intangible. We have found an “endogenous”
economic life – that is, an N that is entirely determined by the firm’s profit
flow and investment levels.
6) Sixth, recognize that this unique value for N is the only choice of economic
life that neither implies more, nor less, residual profit than actually exists.
Other definitions of economic life may arrive at the same N, but will not
do so necessarily.
It may be helpful to see this approach graphically before delving into the algebra.
The figure below provides a simplified depiction of how this approach works.
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The figure shows a firm that lives for nine periods. The key variables for the firm
are described in three separate panels, all with time on the horizontal axis and
dollars on the vertical axis, and all drawn to scale. In the top panel, we show
intangible asset investments of $3 per year – with each vertical segment or bar
representing a single year, and each box within that vertical segment
representing a dollar of intangible asset investment.
Directly below that, in the middle panel, we show “gross residual profit,” which
is residual profit earned before deduction of current period intangible asset
investments. In other words, gross residual profit is the payoff to prior period
sunk investments that are in service. The arrows pointing from the top panel to
the middle panel indicate that each layer of investment corresponds to a layer of
residual profit. This is important, because our approach solves for the economic
life, N, of each layer of investment that will result in total gross returns to the N
layers (in this case three) that exhausts the residual profit in the system. We see
in the figure, therefore, that gross residual profit is fully absorbed by the returns
in the middle panel.
This middle panel relies on two simplifying assumptions, which we will relax
when we develop this approach algebraically. First, the middle panel assumes
that pricing power (residual profit) begins to materialize immediately upon
sinking intangible asset investments – i.e., we assume no “gestation lag” for the
investments. Second, it assumes that the required rate of return to intangible
investments (i.e., the discount rate) is equal to zero. As a result, the gross returns
to each layer of intangible asset investment “ramp up” smoothly to the level of
$1, remain there for three years, and then ramp down smoothly. The shape of
the resulting returns has an area exactly equal to the R&D investment, and at our
assumed zero discount rate makes the intangible asset investments in the top
panel equal to exactly NPV=0.
Finally, in the bottom panel, we show “net” residual profit, which is equal to
gross residual profit minus investments made in the period. Net residual profit
is negative but rising in the early periods as the intangible asset investments
begin to accrete, flat at zero in the middle periods as the rate at which the
intangible asset investments accrete exactly offsets the rate at which they decay,
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and positive at the ending periods as investments end but the firm’s pricing
power remains while the assets amortize away.4
The key lesson from the graph is this. The firm has gross residual profit in each
period that results from its previously sunk intangible asset investments. If we
believe – and this is a key assumption – that we can estimate the rate of return to
the intangible asset investments, then there exists an N, which is both the number
of layers of sunk investments yielding a return and the time horizon over which
the intangible asset investment dollars yield that return, that fully exhausts gross
residual profit. Our goal, then, is simply to find N. In the diagram, N = 3.
Now let’s begin to generalize the approach shown in the diagram. First, define
the following variables.
N
=
I
Π
=
=
r
R
=
=
Economic life. The number of years over which an intangible asset
investment yields a return, and the number of “in service” layers.
Intangible asset investment (per year).
The annual future profit flow required in order for I to be NPV=0.
In other words, П is the profit flow per year that, over N years,
results in a present value of the return on one layer of investment to
equal the upfront cost of that investment (I).
The required rate of return on intangible asset investments.
The residual profit in the system that is attributable to pricing
power generated by the intangible asset investments. R is the
“gross” residual profit, before deduction of intangible asset
investment amortization. Note that R must be equal to N*П.
Now, the formula for the present value of a continuous even flow of П (in other
words we begin by assuming a zero growth rate), over N years, is:
(1)
𝑁
𝑃𝑉 = ∫0 П𝑒 −𝑟𝑇 𝑑𝑇,
where T represents time.
This can be integrated and shown to be equal to:
It bears noting that it is not necessary that the flat, or steady state, portion of the graph is zero.
The graph is drawn this way in part because of our simplifying assumption of a zero discount
rate. Of course, in “real life” the net residual profit will generally be positive.
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(2)
𝑒 −𝑟𝑁
𝑃𝑉 = П (
−𝑟
1
Π
− −𝑟) = 𝑟 (1 − 𝑒 −𝑟𝑁 ).
Recognizing that, for each layer of investment, we need to find a present value
(PV) of residual profit (Π) that is equal to I, we simply replace PV with I,
resulting in:
(3)
Π
𝐼 = 𝑟 (1 − 𝑒 −𝑟𝑁 ),
which is the formula equating the present value of profit flow, П, with upfront
investment, I.
Solving for Π, we have:
(4)
𝑟𝐼
Π = (1−𝑒 −𝑟𝑁 ).
The expression on the right hand side of the equality is the profit required in
each of N future periods, as a function of the upfront investment level I, the
required rate of return (discount rate) r, and the economic life of that investment,
N.
As of yet, we have not solved for N. However, we can express the number of
layers, N, that exhausts the firm’s residual profit as R/Π. Substituting equation
(4) into R/Π, we have:
(4)
𝑅
𝑁=Π=
𝑅(1−𝑒 −𝑟𝑁 )
𝑟𝐼
.
With some rearranging, this leaves us with the following expression:
(6)
(1−𝑒 −𝑟𝑁 )
𝑁
−
𝑟𝐼
𝑅
= 0.
This is the equation that allows us to solve for N. That is, equation (6) says that
the economic life is that value for N that leaves the leftmost term,
𝑟𝐼
(1−𝑒 −𝑟𝑁 )
𝑁
, equal
to 𝑅 .
Unfortunately, the structure of equation (6) cannot easily be manipulated to
arrive at a closed form expression for N as a function of r, I, and R. However,
numeric (non-algebraic) solutions can be derived, and we can demonstrate that
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N is a unique number for every possible combination of r, I, and R. In other
words, we have found a formula that allows us to use a simple computer solver
(such as the one available in Microsoft Excel™) to find N, given the firm’s
intangible asset investment level, residual profit level, and an estimate of the
required rate of return to intangible asset investments.5, 6
Exhibit 2 displays the economic life, N, that results from equation (6). The
columns of the exhibit represent different levels of net residual profit – that is
residual profit, R, less investments incurred of I.7 Columns further to the right
provide higher net residual profit than columns further to the left. As an
example, the column that reads “100%” means that we have a firm with steady
state gross residual profit that is equal to two times the firm’s steady state
intangible asset investments (100% = R/I – 1). So, for example, this firm might
have R&D as a percentage of sales equal to 5 percent, and residual profit after
deduction of per period R&D equal to 5 percent of sales.
Correspondingly, the rows of Exhibit 2 provide for different required rates of
return to the intangible asset investment – that is, different rates at which the
required flows of profit, П, are discounted. The exhibit shows reasonable
discount rates, or required rates of return, ranging from 16 percent to 35 percent.
As I have discussed elsewhere, these rates are consistent with the economics and
finance literature regarding the required rate of return to intangible asset
investments.8
For a discussion of how to estimate the required rate of return to intangible asset investments,
see Reichert and Gray, Estimating the Required Return to Intangible Property Investments. Working
paper available upon request.
6 The application that we developed in Excel to solve for N is available upon request.
7 Note that net residual profit is residual profit as it is commonly defined in transfer pricing
matters.
8 Ibid. See also, Reichert, Technology’s Share of Operating Profit: What are the Implications of the
Empirical Economics Literature?
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The economic lives shown in Exhibit 2 should be relatively familiar to readers
that have some experience with intangible asset valuation. For example, at a
required rate of return to investment of 25 percent, and gross residual profit
equal to two times intangible asset investments, N is 6.4.
What, exactly, is happening in Exhibit 2? What is the intuition behind our
mathematical approach? First, it is important to recognize that this approach can
be thought of as “endogenous,” in the sense that it is determined by the firm’s
objective financial data, rather than by subjective judgments regarding the rate at
which intangible assets obsolesce. Economic life, N, is entirely determined by
just three variables: I, R, and r. Consistent with Exhibit 1, N is derived so that
the number of layers, or prior years of sunk investment, exactly “soaks up” or
exhausts the residual profit stream given that N is also the length of time over which
each of those prior layers live (i.e., N is the length of each layer). Equation (6), which
generates the values shown in Exhibit 2, is in essence picking layers of length N
𝑟𝐼
and height equal to П, or (1−𝑒 −𝑟𝑁 ), so that residual profit per period is entirely
exhausted by N*П – which is simply the number of layers times the required
profit per layer.
Exhibit 2 highlights several critically important relationships. First, economic life
(and therefore the number of “in service” layers) is a function of profitability. As
the firm’s level of residual profit increases, N also increases. Correspondingly, as
the firm’s level of net residual profit falls, so too does economic life. That is,
lower residual profit means that intangible asset investments are relatively more
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like period costs or factor inputs that yield a return equal to their cost in the same
period in which the costs are borne. In fact, while it is not shown in Exhibit 2,
equation (5) ensures that if the firm’s level of net residual profit is zero, then
economic life is zero. That is, if the firm earns no net residual profit, then
investments like R&D are simply period costs, rather than investments.
This is entirely intuitive. Certainly, if the firm conducts R&D, but no net residual
profit results from that R&D, then the “investment” that that R&D would seem
to represent is not an investment at all. Rather, the R&D must be a period cost
like any other – it exactly covers itself through revenue in the period incurred.
Put differently, the R&D does not result in a competitive advantage, so it is like
any other input.
Correspondingly, if a given intangible asset investment, I, produces significant
levels of gross and net residual profit, R, then one of two things (or both) is (are)
happening. Either the firm’s intangible asset investments yield a higher average
return, or more intangible asset investments have “accreted” to form the stock of
intangibles, or both. What Exhibit 2 shows is that all else equal – that is, at a given
required rate of return, r – as R increases the number of layers has to increase in
order to soak up the entire flow of R.
However, as r increases, N decreases, all else equal. This is for the obvious
reason that increases in r increase the “height” of each layer, П. This, in turn,
means that there are fewer of layers required to exhaust residual profit – hence N
falls.
Exhibit 2 also implies that N is slightly more elastic (sensitive) with respect to
increases in r than increases in net residual profit. Doubling r roughly halves N.
However, doubling net residual profit increases N by slightly less than two
times.
Finally, it bears noting that the values chosen for the rows in Exhibit 2 (the values
for r), and the values chosen for the columns (net residual profit) are consistent
with the values that we tend to see in practice. That is the range of values for r is
consistent with the economics literature on the return to intangible property
investments. Correspondingly, the range of values for R/I – 1 are consistent with
the levels of residual profit typically observed in companies with significant
intangible asset investments.
DRAFT
16
C.
Incorporating Growth into the Analysis
The model given above incorporates two simplifying assumption that can be
relaxed in order to enhance its realism. The first of these is that the model
assumes a steady state involving no growth. Relaxing this assumption is
straightforward.
First, the formula for the present value of a flow of П growing at an annual
growth rate of g, over N years, is:
𝑁
𝑁
𝑃𝑉 = ∫0 П𝑒 𝑔𝑇 𝑒 −𝑟𝑇 𝑑𝑇 = ∫0 П𝑒 (𝑔−𝑟)𝑇 𝑑𝑇.
(7)
As with equation (1), equation (7) can be integrated to form:
𝑒 (𝑔−𝑟)𝑁
𝑃𝑉 = П (
(8)
(𝑔−𝑟)
1
Π
− (𝑔−𝑟)) = −(𝑔−𝑟) (1 − 𝑒 (𝑔−𝑟)𝑁 ). 9
Once again, we replace PV with I, and solve for П:
−(𝑔−𝑟)𝐼
Π = (1−𝑒 (𝑔−𝑟)𝑁 ).
(9)
Given that N=R/П, we divide equation (9) into R, and with some manipulation
arrive at equation (10), which is analogous to equation (6).
(10)
(1−𝑒 (−𝑟+𝑔)𝑁 )
𝑁
−
(𝑟−𝑔)𝐼
𝑅
= 0.
Equation (10) can be solved just like equation (6) in order to find N. In fact, what
we find in equation (10) is that all we need to do to modify equation (6) in order
to find economic life is lower the discount rate r by subtracting g from it.10
Exhibit 3, below, is identical to Exhibit 2, except that Exhibit 3 assumes that the
firm is growing at 5 percent per year.
9
Note that the negative sign in front of the denominator in the term
obviously converts to
Π
, and that
(𝑟−𝑔)
Π
(𝑟−𝑔)
Π
(𝑟−𝑔)
(1 − 𝑒 (𝑔−𝑟)𝑁 ). Note then that the limit of
Π
−(𝑔−𝑟)
Π
(𝑟−𝑔)
(1 − 𝑒 (𝑔−𝑟)𝑁 ). This
(1 − 𝑒 (𝑔−𝑟)𝑁 ) as N−>∞ is
is the familiar formula for the present value of a continuously growing stream
of П, discounted at rate r.
10 This results from the mathematics of continuous time discounting.
DRAFT
17
As Exhibit 3 demonstrates, positive growth rates increase economic life, all else
equal. This makes intuitive sense, given that positive growth rates for the firm
mean steady state growth in R. In other words, the “height” of the layers of R
that must be absorbed by the flows of П that result from the firm’s intangible
asset investments is greater, all else equal, the higher is the firm’s growth rate.
This, in turn, implies that more “layers” of П are required to exhaust R.
Exhibit 4 is provided for the reader’s convenience. The exhibit shows the
difference between the economic lives shown in Exhibits 3 and 2, or put
differently the incremental addition to economic life, all else equal, from
increasing the growth rate from zero to five percent.
DRAFT
18
Exhibit 4 shows that, over the range of parameters assumed, economic life
increases by an average of around 2 years. For example, at a discount rate, r, of
25 percent and gross residual profit equal to two times investment, N increases
by 1.6 years.
D.
How Does a Gestation Lag Affect Life?
A “gestation lag” is defined as the average amount of time between the moment
at which intangible asset investments are made, and the moment at which these
investments come “into service.” A company’s investment gestation lag is
determined by technical and market considerations. In the case of R&D, for
example, the time period between incurring R&D investments and the moment
at which those investments are embedded into product features or production
processes is determined by the specific characteristics of the R&D process, the
production process, and other aspects of the firm’s operating cycle.
It is worth exploring the way in which a gestation lag affects our analysis.
Fortunately, this can be done very intuitively. In other words, we don’t need
complex math to understand how a positive gestation lag affects our
determination of N.
Imagine that we have a gestation lag, L, of one year. This means that at the end
of one year, the required total return to today’s investment, I is equal to Ier. In
other words, the opportunity cost of today’s investment (i.e., its required rate of
DRAFT
19
return) means that by the time that investment comes into service after a
gestation lag of one year, the intangible asset profit flow of П has to cover both
the original investment of I and “interest” on that investment at a continuously
compounded rate of r.
Similarly, after the gestation lag of one year has elapsed, at a constant growth
rate of g, the net residual profit margin RL/IL – 1 is equal to Reg/Ieg – 1 = R/I – 1.11
In other words, in a steady state, the residual profit margin is the same after one
year as it is today. This means that the same margin must be exhausted not by
the profit flow corresponding to I, but rather by a profit flow corresponding to
Ier, which is greater than I. Since larger levels of I mean larger per period flow of
intangible asset profit П, fewer “layers” are required to exhaust the residual
profit margin. In other words, economic life N falls as L increases.
This is an important finding, and differs from common practice. Typically,
practitioners view economic life and gestation lag as independent of one another.
However, what my analysis shows is that these two variables must be related.
Upon reflection, this result is intuitive. Because the residual profit stream is
finite, increasing the gestation lag must decrease economic life. Imagine that at
first the firm had a fixed residual profit margin and an economic life for its
intangible asset investments of N. N fully exhausts the residual profit margin.
Imagine then that we introduce a gestation lag, but keep all other attributes of
the firm the same. The longer time period that the investments must now wait
before realizing their return means that that return, which is П, is now larger.
This in turn means that the number of layers needed to exhaust residual profit
must fall (recall that N = R/П).
The reader can verify for himself or herself, using either equation (6) or equation
(10), that growing I and R by the same growth rate for a period of L, but
increasing the value of the sunk I by a factor of er, results in decrease in the value
of N. Exhibit 5 displays the values of N that result from assuming a growth rate
of 5 percent and a gestation lag of one year.
11
The superscript L denotes the flows of R and I one year hence.
DRAFT
20
Exhibit 6 demonstrates that, on the average, assuming a gestation lag of one year
decreases economic life, N, by roughly 75 percent of a year relative to the lives
shown in Exhibit 5.
E.
Reasonable Estimates of N
We know that N depends upon the magnitude of the net residual profit margin
as well as the realized rate of return to intangible asset investments. This means
DRAFT
21
that our estimate of N resides within a natural range, depending upon the
magnitudes of R, I, and r. It is therefore worth considering whether we can draw
any conclusions regarding expected, or typical, estimates of N, given what we
know about R, I, and r in the real world.
The economics literatures on returns to R&D and marketing investments come to
similar conclusions regarding the marginal productivity, or rate of return, to
these intangible asset investments. For example, Hall (2009) provides a
comprehensive survey of the results of over 50 empirical studies of the rate of
return to R&D investments. She finds that the interquartile range of rates of
return to R&D for the studies surveyed is 15 to 51 percent, the median estimate is
29 percent, and the average is 31 percent. The literature on the return to
marketing investments finds similar rates of return. These rates are consistent
with, though somewhat higher than, the values for r in Exhibits 2 through 6.
If we simply take the median and average estimates from Hall as a reasonable
estimate for the range of realized rates of return to intangible assets in general,
we can demonstrate that for almost any configuration R and I, the economic life
of the intangible assets must be well below 10 years. In fact, returning to Exhibits
2, 3, and 5, the empirical literature on rates of return to I suggests that we should
generally find ourselves in the bottom portion of these tables, with average
economic lives ranging from 5 to 7 years.
IV.
Using the Endogenous Life Approach to Value the Firm’s Intangible
Asset Inventory
We can assess the reasonableness of our endogenous life methodology by
examining whether or not its valuation and licensing implications are reasonable.
Specifically, we can ask: 1) whether or not the value of existing intangible assets
that results from this approach is reasonable as a share of enterprise value, and 2)
whether or not this approach has value implications that are consistent with
what we know about licensing (including empirical analyses of licensing
transactions).
Of course, in order to answer these questions, we have to first use the
endogenous life approach to value the firm’s intangible asset stock. The process
for doing so is straightforward, as follows.
DRAFT
22
1) Estimate N using the firm’s steady state financial forecasts for R and I, as
well as an estimate of r.12
2) Given N, R, r, and g, value the firm’s intangible asset stock, VIA, using the
formulae developed below.
3) Add the firm’s steady state financial forecast of routine profit, ПP, to the
forecast for net residual profit (R-I), to obtain total operating profit.13
4) Discount total operating profit to present value using the firm’s weighted
average cost of capital as the discount rate. The result is enterprise value,
EV.
5) Given the forecasts for net residual profit (R-I), routine profit (ПP), and our
estimate of r, solve for the discount rate on routine profits (𝑟̅ ) that results
in EV when the present value of net residual profit is added to the present
value of routine profit.14
6) Apply Test #1. Divide VIA, obtained in Step 2, by enterprise value,
obtained in Step 4. Examine VIA as a share of enterprise value and assess
its reasonableness in light of commonly accepted rules of thumb.
7) Apply Test #2. Using N, R, ПP, g, r, and 𝑟̅ , compute EV as per equation
(15). Compare this result to EV as computed in Step 4.
I first show how to derive the value of intangible assets, VIA, using this approach,
and then turn to the question of whether or not VIA is reasonable.
A.
Valuing the Firm’s Inventory of Intangible Assets
Under the approach outlined above, the formula for the value of the firm’s
intangible assets for g=0 is:
(11)
𝑁
𝑇
𝑁
𝑇
𝑉 𝐼𝐴 = ∫0 𝑅𝑒 −𝑟𝑇 (1 − 𝑁) 𝑑𝑇 = 𝑅 ∫0 𝑒 −𝑟𝑇 (1 − 𝑁) 𝑑𝑇.
Equation (11) simply says that the value of the firm’s intangible assets, VIA, is the
present value of the firm’s gross residual profit, R, over the period from T=0 to
There is a significant economics literature that estimates r for both R&D investments and
marketing investments. As I detail in another white paper, this literature demonstrates that r is
likely in the range of 20 to 40 percent. This means that, at the margin, the return to intangible
asset investments is 20 to 40 percent.
13 Technically, one should focus on cash flows rather than accrual-based accounting measures of
profit.
14 See Reichert and Gray, Estimating the Required Return to Intangible Property Investments.
12
DRAFT
23
T=N, given that the flow of R attributable to the in service intangible assets
amortizes over N years.15
Integrating equation (11) is a tedious process, given in the mathematical
appendix at the end of this paper. As demonstrated, after integration equation
(11) results in the following fundamental valuation equation:
(12)
𝑅
𝑉 𝐼𝐴 = 𝑁𝑟 2 (𝑒 −𝑟𝑁 + 𝑟𝑁 − 1).
As it turns out, equation (12) is a way to find the present value of any flow of
income that begins at level R and erodes evenly to zero (assuming a growth rate
of zero). While equation (12) may look odd, it is extremely useful. The value of
any evenly eroding asset can be derived using equation (12) by simply plugging
in parameters for the discount rate, r, the level of the profit flow being
discounted (in this case, gross residual profit, R), and life, N.
For example, if we assume an intangible asset that generates a gross residual
profit flow of $100, an economic life of 2 years, and a discount rate that
approaches zero, equation (12) produces a value of $100. This makes perfect
sense – we have a profit flow that begins at a level of $100 at time T=0, erodes to
$50 by T=1, and finally to zero by T=2. At a zero discount rate, the present value
is simply the area of the triangle of profit from T=0 to T=2, which can be found
1
simply by using the formula for the area of a triangle, or (2) ∗ 2 ∗ $100 = $100.
Similarly, if we assume the same $100 profit flow, a perpetual economic life, and
a discount rate of 10 percent, equation (12) produces a value of $1000 – which is
the value of a perpetuity ($1000 = $100 / 10%).16
In this case, equation (12) is discounting gross residual profit. As shown in
Exhibit 7, which is an extension of the graph shown in Exhibit 1, assuming an
intangible asset life of 3 years (N=3) and a discount rate of zero, equation 12 is
finding the present value of the area shown in the blue profit triangle that begins
at T=3 and ends at T=6. This is the profit area that includes the remaining
required returns to all intangible asset investments costs, I, sunk during the
period between T=0 and T=3.
15
𝑇
At T=N, the term(1 − ) = 0.
𝑁
In fact, one can show the limit of equation (12) as N -> ∞ is R/r, which is the formula for a
perpetuity.
16
DRAFT
24
Equations (11) and (12), as well as Exhibit 7, assume a zero growth rate. As
shown in the mathematical appendix, the formula for the case with growth is:
(13)
𝑅
𝑉 𝐼𝐴 = 𝑁(𝑟−𝑔)2 (𝑒 −(𝑟−𝑔)𝑁 + (𝑟 − 𝑔)𝑁 − 1).
Note that equation (13) is structurally identical to equation (12), but that we have
simply modified the “net discount rate” to account for growth.
B.
Are the Resulting Values Reasonable?
As noted earlier, a critical question is whether or not the intangible asset value,
VIA, is reasonable. As noted, we can assess the reasonableness of VIA by gauging
whether VIA is generally a reasonable share of enterprise value. That is, it is
commonly accepted that intangible assets constitute a share of enterprise value
residing within a given range, and we can test our approach by asking whether it
tends to also produce intangible asset values within that range.
Exhibit 8 is constructed in a manner similar to Exhibits 2 through 6, with r shown
in the rows and the net residual profit margin given in the columns. It bears
noting that in order to construct Exhibit 8, one must assume a given ratio of net
residual profit (R-I) to routine profit (ПP). The exhibit given below assumes a
ratio of 1:1 for these values.
DRAFT
25
Exhibit 8
Intangible Asset Value (VIA) As A Percentage Of Enterprise Value (EV)
Assuming That Net Residual Profit = ПP
Exhibit 8 is quite interesting. What it demonstrates is that, assuming that net
residual profit and routine profit are of equal magnitudes, the identifiable
intangible asset share of enterprise value is consistent with generally accepted
shares of enterprise value, and with the licensor-licensee rule of thumb. The
values shown range from 17.1 percent to 37.6 percent, and the average is almost
exactly 25 percent.
V.
Valuing the Enterprise – Completing the Modigliani-Miller Model
In 1961, Modigliani and Miller published an article in the Journal of Business
entitled Dividend Policy, Growth, and the Value of Shares. In that article, they
outlined three basic enterprise valuation techniques, and showed that all three
approaches are analytically equivalent. Two of these are in common currency
today: the present value of dividends approach, and the present value of free
cash flow approach. The third, which they called the “investment opportunities
approach,” is far less commonly employed. Importantly, the investment
opportunities approach was only sketched out in their paper.
The idea behind the investment opportunities approach is that the value of the
firm is equal to its steady state “normal,” or “competitive,” level of profit, plus
the value of economic profit over a “competitive advantage period.” The
competitive advantage period is the time horizon over which the firm expects to
DRAFT
26
retain its ability to price at a premium given the stock of intangible assets in
place. In terms that “value investors” will understand, it is the time period over
which the firm’s “competitive moat” will last. In terms used thus far in this
paper, the competitive advantage period sounds very similar to our concept of
intangible asset life, N.
Importantly, in the Modigliani-Miller paper, the competitive advantage period is
subjectively determined. In other words, Modigliani and Miller provide no
guidance regarding how to pin down the competitive advantage period, and
offer no examination of the relationship between the firm’s other financial
variables and the competitive advantage period. Given this, and given that we
can now use our endogenously determined estimate of N to value the firm’s
intangible assets over their economic life, it is worth examining the relationship
between enterprise value developed using commonly employed methods (i.e.,
discounted free cash flows) and the sum VIA + PV(ПP), where PV(ПP) is the
present value of routine profits. If our math thus far has been correct, we expect
that the enterprise value of the firm should equal VIA + PV(ПP). In other words,
the value of the firm should equal the value of its non-routine intangible assets
plus the value of is expected routine profit flow.
The fact that VIA + PV(ПP) = EV is implicit, although at first hard to see, in Exhibits
1 and 7. Exhibit 1 shows that the net residual profit must begin at negative levels
(economic losses are borne) while the firm’s intangible asset stock ramps up to its
steady state level (periods 1 through 3 in the exhibit). Then, during the steady
state period (4 through 6) the firm realizes zero net residual profit. Finally, in the
last three periods (7 through 9), the firm realizes positive net residual profit. In
present value terms, the payoff during the firm’s last three periods exactly
compensates the losses borne during the first three. This means that at time zero
the firm’s net residual profit is, in present value, equal to zero. However, at any
time after period 3 the present value of the firm’s net residual profit is equal to
the gross residual profit payoff in periods 7 through 9. Exhibit 7, in turn,
demonstrates that this gross residual profit payoff in the last three periods is the
value of the firm’s intangible asset stock at any time after period 3. Put
differently, in a steady state the present value of net residual profit is equal to
VIA. And, since enterprise value is simply the present value of routine profit plus
the present value of net residual profit, then enterprise value must also equal the
present value of routine profit plus VIA.
In the case of a zero growth rate, if VIA + PV(ПP) = EV, then we can express the
value of the enterprise as:
DRAFT
27
(14)
𝑅
𝐸𝑉 = 𝑉 𝐼𝐴 + 𝑃𝑉(П𝑃 ) = 𝑁𝑟 2 (𝑒 −𝑟𝑁 + 𝑟𝑁 − 1) +
П𝑃
𝑟̅
,
where EV is enterprise value, 𝑟̅ is the discount rate applicable to routine profits,
and all other variables are defined as before. Again, N is generated by using a
solver program to find the unique value for N that makes equation (6) equal to
zero.
Similarly, in the case of a positive growth rate, the value of the firm is:
(15)
П𝑃
𝑅
𝐸𝑉 = 𝑉 𝐼𝐴 + 𝑃𝑉(П𝑃 ) = 𝑁(𝑟−𝑔)2 (𝑒 −(𝑟−𝑔)𝑁 + (𝑟 − 𝑔)𝑁 − 1) + (𝑟̅ −𝑔),
where all variables are defined as before.
As noted earlier, my approach differs from Modigliani-Miller in two important
ways. First, formulas (14) and (15) rely on an endogenous definition of economic
life, rather than a subjectively determined competitive advantage period.
Second, my approach explicitly draws the analyst’s attention to the firm’s
intangible asset investments, routine profits, and residual profit streams,
whereas Modigliani-Miller focuses only on net residual profit (the difference
between the firm’s realized return on invested capital and the firm’s WACC).
Exhibit 9, below, simply shows that the value for EV obtained using equation (15)
as a percentage of EV obtained by discounting the sum of net residual profit and
routine profit using the firm’s weighted average cost of capital. As the exhibit
demonstrates, the result under my approach is identical to the value obtained
using a traditional discounted cash flow method. We have, in fact, found a new
way to value the enterprise.
DRAFT
28
Exhibit 9
Intangible Asset Value (VIA) Plus PV(ПP) As A Percentage Of Enterprise Value (EV)
VI.
Allowing for Differences Between the Residual Profit Discount Rate
and the Rate of Return to Intangible Asset Investments
Thus far, I have assumed that the rate of return, r, to intangible asset
investments, I, is the same rate at which one should discount the realized returns,
or gross residual profit. However, it may be (although is not necessarily) the
case that the realized returns to the firm’s intangible asset investments should be
discounted at a rate that differs from the realized rate of return, r.
The reason that this may be appropriate is simple. The discount rate at which
any flow of profit should be discounted is a function of the systematic risk of that
profit flow. And, since intangible asset investment decisions may not be made in
a perfectly competitive environment (indeed, gaining breathing space from the
competition is the reason for the intangible asset investments), there is no reason
to expect that the systematic risk of (or required return to) the firm’s gross
residual profit flow will be the same as the realized return.
It may therefore be appropriate in some cases to use a combination of r and 𝑟̅ for
discounting residual and routine profit flows that differs from the combination
that is implied by the realized rate of return to intangible asset investments.
Importantly, this should not be confused with changing the realized rate of
DRAFT
29
return must be used to develop N. The realized rate of return should be in the
range suggested by the economics literatures on returns, at the margin, to R&D
and marketing investments.
VII.
Conclusions
This paper has two primary findings. First, there exists a unique intangible asset
life, N, that is endogenous to the firm’s steady state financial position. In other
words, N can be found objectively from the firm’s investment-related variables,
rather than using the subjective methods and definitions of intangible asset life
that are commonly employed today.
I noted at the beginning of Section III that there are four obvious criteria for any
definition of intangible asset economic life. These are: 1) objectivity rather than
subjectivity, 2) simplicity, 3) intuitiveness, and 4) consistency with basic finance
concepts. The approach that we have developed for determining economic life,
in my view, meets all four criteria. Clearly, the approach does not rely on
subjective estimates of the amortization rate of intangible assets. Thus, criterion
1 seems easily satisfied. Further, as we have shown in the foregoing formulas,
exhibits, and discussion, criteria 3 and 4 (intuitive and consistent with finance
theory) are certainly satisfied. Finally, as for criterion 2 (simplicity), it must be
admitted that this approach has more than a hint of complexity that at first
glance other approaches to intangible asset life do not. However, recognizing
that simplicity should be defined in such a way as to include the absence of
logical inconsistencies that must be rationalized away, this approach certainly
meets criterion 2. That is, as noted earlier, only the “endogenous” approach to N
necessarily exhausts the firm’s residual profit flow. In other words, only the
endogenous approach to N is necessarily logically consistent with the firm’s
configuration of R, I, and r.
The second main finding of this paper is that once N is obtained, a new method
for valuing the firm becomes available. While I detail the advantages of this new
approach in a separate paper, for our purposes here suffice to say that this
approach to enterprise valuation has several distinct advantages. First, it
provides an analytically complete way of expressing what Modigliani and Miller
recognized forty years ago – that the firm’s value is equal to a steady state
normal profit stream plus the value of excess profits earned over an expected
competitive advantage period. I say “analytically complete” because my
approach fully endogenizes, or “pins down” the competitive advantage period
that Modigliani and Miller envisioned. Second, this approach to firm valuation
DRAFT
30
allows the analyst tremendous flexibility in defining intangible asset investments
versus routine assets and activities, and provides a valuation of the firm in terms
of its routine and non-routine components. This is of great advantage in transfer
pricing contexts, as well as allocations of the purchase price of an acquired firm.
DRAFT
31
Mathematical Appendix
Equation (11), now re-written as equation (A-1), is:
𝑁
𝑁
𝑇
𝑇
(A-1) 𝑉 𝐼𝐴 = ∫0 𝑅𝑒 −𝑟𝑇 (1 − 𝑁) 𝑑𝑇 = 𝑅 ∫0 𝑒 −𝑟𝑇 (1 − 𝑁) 𝑑𝑇.
This can be decomposed into the sum of (or in this case difference between) two
integrals, as follows.
𝑁
𝑁
1
(A-2) 𝑉 𝐼𝐴 = 𝑅 (∫0 𝑒 −𝑟𝑇 𝑑𝑇 − 𝑁 ∫0 𝑇𝑒 −𝑟𝑇 𝑑𝑇).
Several manipulations then lead us to our result in equation (12). These are as
follows.
(A-3) 𝑉
𝐼𝐴
= 𝑅(
𝑒 −𝑟𝑇
𝑁
1
(𝑟𝑇+1)𝑒 −𝑟𝑇
| +𝑁(
𝑒 −𝑟𝑁
(A-4) 𝑉 𝐼𝐴 = 𝑅 ((
−𝑟
1
1
− −𝑟) + 𝑁 (
| )),
0
(𝑟𝑁+1)𝑒 −𝑟𝑁
𝑟2
𝑒 −𝑟𝑁 −1
𝑅(𝑟𝑁+1)𝑒 −𝑟𝑁 −𝑅
𝑟
𝑁𝑟 2
(A-5) 𝑉 𝐼𝐴 = −𝑅 (
)+(
),
𝑅(𝑒 −𝑟𝑁 (𝑟𝑁+1)−1)
𝑟
𝑁𝑟 2
)+(
𝑅(𝑒 −𝑟𝑁 (𝑟𝑁+1)−1)
𝑁𝑟 2
𝑅(𝑒 −𝑟𝑁 −1)
)−(
𝑟
1
− 𝑟 2 )),
𝑒 −𝑟𝑁 −1
(A-6) 𝑉 𝐼𝐴 = −𝑅 (
(A-7) 𝑉 𝐼𝐴 = (
𝑟2
−𝑟 0
𝑁
),
),
𝑅
(A-8) 𝑉 𝐼𝐴 = 𝑁𝑟 2 (𝑒 −𝑟𝑁 (𝑟𝑁 + 1) − 1 − 𝑟𝑁(𝑒 −𝑟𝑁 − 1)),
(A-9) 𝑉 𝐼𝐴 =
𝑅
𝑁𝑟 2
(𝑟𝑁𝑒 −𝑟𝑁 + 𝑒 −𝑟𝑁 − 1 − 𝑟𝑁𝑒 −𝑟𝑁 + 𝑟𝑁),
𝑅
(A-10) 𝑉 𝐼𝐴 = 𝑁𝑟 2 (𝑒 −𝑟𝑁 + 𝑟𝑁 − 1),
which is the result given in equation (12).
In the case of growth, we begin with a very similar integral, and precisely the
same manipulations apply. Specifically, we begin with:
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32
𝑁
𝑇
𝑁
𝑇
(A-11) 𝑉 𝐼𝐴 = ∫0 𝑅𝑒 −(𝑟−𝑔)𝑇 (1 − 𝑁) 𝑑𝑇 = 𝑅 ∫0 𝑒 −(𝑟−𝑔)𝑇 (1 − 𝑁) 𝑑𝑇.
Noting that the term (r-g) can simply be re-written as r’, we see immediately that
all of the manipulations given in equations (A-2) through (A-10) apply to (A-11)
as well, once we substitute r’ for (r-g). This means that our result in the growth
case must be:
𝑅
(A-12) 𝑉 𝐼𝐴 = 𝑁(𝑟−𝑔)2 (𝑒 −(𝑟−𝑔)𝑁 + (𝑟 − 𝑔)𝑁 − 1).
DRAFT