In this section, we will describe how the basic FV formula is developed by making extensive use of present value constructs and building on the foun- dation of the DDM.
We begin by defining the tangible value (TV) to be the economic book value associated with the earnings stream that can be derived from the current business without the addition of further capital. A valuable simplification is obtained by using a fixed annual payment Eto create a figurative annuity that has the same present value (PV) as the literal earnings stream. If k is the appropriated risk-adjusted discount rate (usually taken as equivalent to the COC—the cost of equity capital), then
Moreover, if Bis the book value, then the earnings Ecan be expressed as a return on this book value,
E=rB
so that
With the value derived from all current business embedded in the TV, we can now turn to future prospects as the remaining source of firm value.
There appears to be an almost congenital human need to view all growth as a smooth, consistent, and readily extrapolatable process. This compulsion may be rooted in the understandable desire for order and pre- dictability in confronting future developments that are intrinsically uncer- tain. However, this forced smoothing of growth prospects can lead to a number of fundamental errors.
In actuality, new projects deliver a sequence of returns over time, consisting of investment inflows in the early years followed by a subse- quent pattern of positive returns. A simple fixed franchise spread is far too simplistic to capture such a complex return pattern over time. How- ever, in most cases, the project’s present value contribution can be prox-
TV=rB k TV= E
ied by an appropriately chosen fixed return Rper dollar invested that is received year after year in perpetuity. The project’s net return pattern can thus be rendered equivalent (in PV terms) to a level annuity based on the dollar amount invested at a fixed spread (R–k) over the COC k. Piling these heroic assumptions one upon another, all of a firm’s fran- chise projects can be compressed into an “average” annual spread per dollar of funding that reaches from the current time into perpetuity. (Ac- tually, this hierarchy of assumptions is not that far afield from the com- mon corporate practice of requiring new projects to provide some “hurdle rate” above the COC.)
With this perpetual spread model as a gauge of value-added return, the next step is to size the totality of the firm’s franchise projects (i.e., to determine how many dollars could be invested at the given franchise spread). In the standard DDM, a growth rate is chosen, and the size and timing of future projects are implicitly set by the level of earnings avail- able for reinvestment. We sought to move beyond this standard ap- proach for several reasons, starting with our desire not to be constrained by the smooth-growth hypothesis. Another compelling rationale is the emergence of modern information systems and global capital markets. Carried to an (admittedly theoretical) extreme, an efficient financial market should be able to provide capital to any worthwhile project— any project that enjoys a positive franchise spread. Thus, in this hypo- thetical limit, it would be the opportunities for franchise investment that would be the scarce resource, rather than the capital required to fund them.
These growth opportunities might arise in some irregular fashion over the course of time. However, a project’s size may be normalized by com- puting the PV of the total dollar amount that could be invested over time in each such opportunity. The sum of these PVs could then be viewed as equivalent to a single dollar amount that, if invested today, would act as a surrogate for all such future opportunities. This concept of the PV of all fu- ture growth prospects has the virtue of considerable generality. No longer are we restricted to smooth compounded growth at some fixed rate. Virtu- ally any pattern of future opportunities—no matter how erratic—could be modeled through this PV equivalence.
The PV of all growth opportunities will generally sum to a massive dollar value. To make this term intuitive—and estimable—it seemed rea- sonable to represent it as some multiple G of the current book value (B), that is, PV of investable opportunities = G×B.
At this point, we have a PV of future investable dollars in terms of an equivalent single investment today. At the same time, we found that a pat- tern of franchise returns over time could be represented by an annual fran-
chise spread of (R –k) per dollar invested. Consequently, the product of this franchise spread and the PV of investable opportunities corresponds to an annual dollar return—above the COC—that could be earned in per- petuity from the full panoply of the firm’s positive growth prospects. Thus, the firm’s added value from its growth prospects, the FV, takes on the form of a perpetual stream of “net-net” profits with annual payments, (R–k)(G×B).
We now basically have the firm’s theoretical value expressed in terms of two level annuities—a current earnings stream Efor the TV and a flow of net-net profits from new projects for the FV. But the PV of a level dollar annuity is perhaps the most basic equation in finance: One simply divides the annual payment by the discount rate k(i.e., the COC). For the TV, we have earlier found
and now for the FV,
When the FV is added to the TV, we obtain a theoretical valuation Pfor the firm’s current business and all its foreseeable future value-additive projects,
From the very beginning, our intent was to use this valuation result as the numerator in a P/E ratio, with the current economic earnings as the de- nominator. But recall that the firm’s earnings E can be represented as the product (r×B) of the book value Band the current return on equity r. At this point, we have the serendipitous result that dividing by the earnings leads to an even simpler formulation for the FV,
P E k k R k G B = + = + − × TV FV 1 ( )( ) FV= − × 1 k (R k G B)( ) TV= E k
where , a compact expression that we came to call the “fran- chise factor.”
The FV model thus became very easily stated, especially in (P/E) terms,
This development may look straightforward (and even perhaps rather ob- vious), but the actual route traveled was quite torturous, with many false starts and blind alleys. When Stan and I finally stumbled (quite the appro- priate word) on this expression, it was definitely a eureka moment that neither of us are likely to forget. It was during a Sunday phone conversa- tion, and we suddenly realized that with this definition of the franchise factor FF we had a very general and elegant expression for the theoretical P/E ratio.
Little did we realize the extent to which we had opened the door to myriad further questions about equity valuation. Some of those other is- sues were later addressed by Stan and/or myself, sometimes in conjunction with other colleagues.