THE VALUATION OF SHARES 415 controvery itself thus turns out to be an as T approaches infinity so that( 7)can empty one, the different expressions do be expressed as ha intrinsic intere highlighting different combinations of v(o)=Jim >1 variables they provide additional insights (1+p) (8) [X(t)-I()], open alternative lines of attack on some of the problems of empirical testing which we shall further abbreviate to Ⅱ. WHAT DOES THE MAR3Ay”V(h(+mx(0-(0).(9) In the literature on valuation one can The discounted cash flow approach nd at least the following four more or Consider now the so-called discounted less distinct approaches to the valuation of shares: (1)the discounted cash flow cash flow approach familiar in discus- approach; (2) the current earnings plus sions of capital budgeting. There, in val future in estment opportunities ap- uing any specific machine we discount at proach;(3)the stream of dividends ap- the market rate of interest the stream of proach; and(4)the stream of earnings cash receipts generated by the machine approach. To demonstrate that these ap- plus any scrap or terminal value of the proaches are, in fact, equivalent it will be machine; and minus the stream of cash helpful to begin by first going back to outlays for direct labor, materials,re- quation(5)and developing from it a pairs, and capital additions. The same valuation formula to serve as a point of approach, of course, can also be applied reference and comparison. Specifically, if to the firm as a whole which may be we assume, for simplicity, that the mar- thought of in this context as simply a ket rate of yield p()=p for all 4,then, large, composite machine. 5 This ap- setting t=0, we can rewrite(5)as 3 More general formulas in which p() is allowed V(0)-1+Ix(0)-I(0) (6)presented here merely by substituting the cum some product 1+pV(1) Since(5)holds for all t, setting t= 1 per- I1+p()]for(1+p)+1 mits us to express V(1)in terms of V (2) hich in turn can be expressed in terms tion that the remainder vanishes is of v(3)and so on up to any arbitrary only and is in no way essential to the argument substitutions. we obtain m of the two terms in(7), be finite, but this can always be safely assumed (1+)+X(0)-1(0) (7) mally taken in eacn mic theory when discussing te value of the assets of an enterprise, but much mor V(T) ty side. One of the few to apply the approach In general, the remainder term(1+)-.to the sha res as wellas the assets s Bodenorm in /I, V(r)can be expected to approach zero above 's it to derive a formula closely similar to his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES 415 controvery itself thus turns out to be an empty one, the different expressions do have some intrinsic interest since, by highlighting different combinations of variables they provide additional insights into the process of valuation and they open alternative lines of attack on some of the problems of empirical testing. II. WHAT DOES THE MARKET "REALLY" CAPITALIZE? In the literature on valuation one can find at least the following four more or less distinct approaches to the valuation of shares: (1) the discounted cash flow approach; (2) the current earnings plus future investment opportunities approach; (3) the stream of dividends approach; and (4) the stream of earnings approach. To demonstrate that these approaches are, in fact, equivalent it will be helpful to begin by first going back to equation (5) and developing from it a valuation formula to serve as a point of reference and comparison. Specifically, if we assume, for simplicity, that the market rate of yield p (t) = p for all t,3 then, setting t = 0, we can rewrite (5) as V (O) 1 IX (O)-I (0) ] + 1 +p ( (6) +-- V (1). Since (5) holds for all t, setting t = 1 permits us to express V(1) in terms of V(2) which in turn can be expressed in terms of V(3) and so on up to any arbitrary terminal period T. Carrying out these substitutions, we obtain T-1 V(O) = E(l+p)t+l[X(t)I(t)] +(1+p) V(T). In general, the remainder term (1 + P)-T. V(T) can be expected to approach zero as T approaches infinity4 so that (7) can be expressed as T-1 v (O) = rnim (8) X [X(t)-I(t)], which we shall further abbreviate to c 1 V(O) = 2 (1-+ I1 [X(t)-I(t)]. (9) t- (I+ P)t The discounted cash flow approach.- Consider now the so-called discounted cash flow approach familiar in discussions of capital budgeting. There, in valuing any specific machine we discount at the market rate of interest the stream of cash receipts generated by the machine; plus any scrap or terminal value of the machine; and minus the stream of cash outlays for direct labor, materials, repairs, and capital additions. The same approach, of course, can also be applied to the firm as a whole which may be thought of in this context as simply a large, composite machine.5 This ap- 3 More general formulas in which p(t) is allowed to vary with time can always be derived from those presented here merely by substituting the cumbersome product 1L [l+p(r)] for (1+p)t+' TO0 4 The assumption that the remainder vanishes is introduced for the sake of simplicity of exposition only and is in no way essential to the argument. What is essential, of course, is that V(O), i.e., the sum of the two terms in (7), be finite, but this can always be safely assumed in economic analysis. See below, n. 14. 5 This is, in fact, the approach to valuation normally taken in economic theory when discussing the value of the assets of an enterprise, but much more rarely applied, unfortunately, to the value of the liability side. One of the few to apply the approach to the shares as well as the assets is Bodenhorn in [1], who uses it to derive a formula closely similar to (9) above. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
416 THE JOURNAL OF BUSINESS proach amounts to defining the value of investments in real assets that will yield the firm as more than the"normal"(market)rate of return. The latter opportunities, fre V(0)= 2(1+p) quently termed the“ good will’ of the (10) business, may arise, in practice, from any [(-0(4)1+①+p(T), of a number of circumstances (ranging all the way from special locational advan where R()represents the stream of cash tages to patents or other monopolistic outlays, or, abbreviating, as above, to To see how these opportunities affect the value of the business assume that in some future period t the firm invests I() v(0)f-0(1+p)i* [R(-0(01(11) dollars. Suppose, further, for simplicity, that starting in the period immediately But we also know, by definition, that following the investment of the funds [X(o-I(D]=[R(-0(0] since, X( the projects produce net profits at a con differs from R( and I( differs from stant rate of p*() per cent of I( )in each (and also by the depreciation expense if worth as of t of the (perpetual) stream of rather than gross profits and invest- the "good will "of the projects(i.e, the ment). Hence(11)is formally equivalent difference between worth and cost)will to(9), and the discounted cash flow ap- be proach is thus seen to be an implication I(PD-1(0)=I([P*(D) of the valuation principle for perfect The investment opportunities approach. ture"good will", as of now of this fu- markets given by equation(1) The present worth -Consider next the approach to valua- tion which would seem most natural I() (1+p)-(+), from the standpoint of an investor pro- posing to buy out and operate some al- and the present value of all such future ready-going concern. In estimating how opportunities is simply the sum much it would be worthwhile to pay for the privilege of operating the firm, the I() (1+p)-(+) to be paid is clearly not relevant, since the new owner can, Adding in the present value of the(uni- within wide limits, make the future divi- form perpetual)earnings, X(O), on the as dend stream whatever he pleases. For him the worth of the enterprise, as such The assumption that I() yields a uni is not restrictive in the present o will depend only on:(a) the"normal since it is always possible by rate of return he can earn by investing his capital in securities (i. e, the market the time shape of its actual ety p note also that the physical assets currently held by the the firm are behaving rationally, they will, o cogens rate of return); (b)the earning power of p"( is the auerage rate of return. If the i their cut-off criterion(cf. below p. 418) firm; and( ) the opportunities, if any, in this event we would that the firm offers for making additional mulas remain valid, however, even where"0<e his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
416 THE JOURNAL OF BUSINESS proach amounts to defining the value of the firm as T-1 V(O) = E t=O (0 P) (10) X [E (t-co() +(+p Tv (T), where IR(t) represents the stream of cash receipts and ()(t) of cash outlays, or, abbreviating, as above, to co v ( ?) = E 1+p teRW [st-(t I . ( 1 1) ,_O (1+p),+'(11 But we also know, by definition, that [X(t) -I(t)] = [IR(t) -()(t)] since, X(t) differs from IR(t) and 1(t) differs from CO(t) merely by the "cost of goods sold" (and also by the depreciation expense if we wish to interpret X(t) and I(t) as net rather than gross profits and investment). Hence (11) is formally equivalent to (9), and the discounted cash flow approach is thus seen to be an implication of the valuation principle for perfect markets given by equation (1). The investment opportunities approach. -Consider next the approach to valuation which would seem most natural from the standpoint of an investor proposing to buy out and operate some already-going concern. In estimating how much it would be worthwhile to pay for the privilege of operating the firm, the amount of dividends to be paid is clearly not relevant, since the new owner can, within wide limits, make the future dividend stream whatever he pleases. For him the worth of the enterprise, as such, will depend only on: (a) the "normal" rate of return he can earn by investing his capital in securities (i.e., the market rate of return); (b) the earning power of the physical assets currently held by the firm; and (c) the opportunities, if any, that the firm offers for making additional investments in real assets that will yield more than the "normal" (market) rate of return. The latter opportunities, frequently termed the "good will" of the business, may arise, in practice, from any of a number of circumstances (ranging all the way from special locational advantages to patents or other monopolistic advantages). To see how these opportunities affect the value of the business assume that in some future period I the firm invests 1(t) dollars. Suppose, further, for simplicity, that starting in the period immediately following the investment of the funds, the projects produce net profits at a constant rate of p*(t) per cent of I (t) in each period thereafter.6 Then the present worth as of t of the (perpetual) stream of profits generated will be I(t) p*(t)/p, and the "good will" of the projects (i.e., the difference between worth and cost) will be I(t)fP-22)-I(t) P* =1(t) [P (t) P P* The present worth as of now of this future "good will" is It P* ( ) p] (1 + p)-+ and the present value of all such future opportunities is simply the sum to P Adding in the present value of the (uniform perpetual) earnings, X(O), on the as- 8 The assumption that I(t) yields a uniform perpetuity is not restrictive in the present certainty context since it is always possible by means of simple, present-value calculations to find an equivalent uniform perpetuity for any project, whatever the time shape of its actual returns. Note also that p*(t) is the average rate of return. If the managers of the firm are behaving rationally, they will, of course, use p as their cut-off criterion (cf. below p. 418). In this event we would have p*(t) > p. The formulas remain valid, however, even where p*(t) < p. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES sets currently held, we get as an expres- The first expression is, of course sion for the value of the firm simply a geometric progression summing v(o)- X(O) to X(o/e, which is the first term of(12) +∑I() To simplify the second expression note (12) that it can be rewritten as p*(4) (1+p)-(4+) ∑({°*(∑(1+) To show that the same formula can be derived from(9) note first that our defini (1+p)-(4+1 tion of p*(t )implies the following relation between the X(t Evaluating the summation within the brackets gives x(1)=X(0)+p*(0)I(0) r()p*(t) (1+p) x()=X(t-1)+p*(t-1)I(t-1) (1+p)-(+ and by successive substitution X()=X(0)+∑p*(r)I() =∑r(2)*(4)-P(1+)-( hich is precisely the second term of Substituting the last expression for (12) X(t)in(⑨) yields la(12)has a number of reveal V(0)=[X(0)-I(0)](1+p)-1 widely used in discussions of valuation. 7 For one thing, it throws considerable light on the meaning of those much X(0)+∑p*(r)I(r) abused terms“ growth”and“ growth stocks. As can readily be seen from(12) I(4)(1+p)-((+1) a corporation does not become a"growth stockwith a high price-earnings ratio =X(0)∑(1+p)-4 I(0)(1+p)-1 glamor cae g orer s assets and earnings merely because ire growl (>p. For if p* ever large the growth in assets may be *(T)I(T)-I()the second term in(12)will be zero and X(1+p)-(+1) the firm's price-earnings ratio would not rise above a humdrum 1/p. The essence =X(0)∑(1+p)-4 of“ growth, ort, is not but the existence of opportunities to in- vest significant quantities of funds at ∑[∑?*(n)()-l(4-1) higher than"normal"rates of return f=1=0 i A valuation formula analogous to(12)though derived and interpreted X(1+p)(1+p)-(+1). is found in Bodenhorn(1]. Variants of(12)for certai special cases are discussed in Walter[201 his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES 417 sets currently held, we get as an expression for the value of the firm V(O) =(O) + E I (t) P t=O (12) xP* (t) - p +P XP()----( 1 + p)-(t+l). p To show that the same formula can be derived from (9) note first that our definition of p*(t) implies the following relation between the X(t): X (1) = X (O) + p* (O) I (O), .................... X (t) = X(t -1) +p* (t -1) I(t -1) and by successive substitution t-1 X (t) = X(O) + Yd p* X () Tr=O t=1,2 ...o . Substituting the last expression for X(t) in (9) yields V(O) = [X(O)-I(O)] (1 + p) +X X(O) +Ep*(r)I (r) t = =X(O)-(O +1p)-1 (1 t1-1 I ( ___ 0 t =1 T=O X ( + p)-t) CO =X(O) f, (I1+ p) -t t =1 + Y. *T) T-It1 t =1 T=O X (+ P) +5 {12 )(t The first expression is, of course, simply a geometric progression summing to X(O)/p, which is the first term of (12). To simplify the second expression note that it can be rewritten as 1:I (t) [p*t E ( 1+ P) -T tO0 T-=t+2 - ( 1 + p)(t+)] Evaluating the summation within the brackets gives E .1(t) , I (t) [p* (t)( + p) -(t+l + t00 - (1+p)-(t+1)] = I(t (t) ]* P +p -t) which is precisely the second term of (12). Formula (12) has a number of revealing features and deserves to be more widely used in discussions of valuation.7 For one thing, it throws considerable light on the meaning of those much abused terms "growth" and "growth stocks." As can readily be seen from (12), a corporation does not become a "growth stock" with a high price-earnings ratio merely because its assets and earnings are growing over time. To enter the glamor category, it is also necessary that p*(t) > p. For if p*(t) = p, then however large the growth in assets may be, the second term in (12) will be zero and the firm's price-earnings ratio would not rise above a humdrum i/p. The essence of "growth," in short, is not expansion, but the existence of opportunities to invest significant quantities of funds at higher than "normal" rates of return. 7A valuation formula analogous to (12) though derived and interpreted in a slightly different way is found in Bodenhorn [1]. Variants of (12) for certain special cases are discussed in Walter [201. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions