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 ConditionsTHE VALUATION OF SHARES 417 sets currently held, we get as an expres￾sion 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 defini￾tion 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 reveal￾ing 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 how￾ever 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 in￾vest 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