Stoc k Valuation Based on EarningsThe PV approach to stock valuation discounts dividends Dt, not earningsEt:P0=XtDt(1 + r)t.If we were to discount earningsXtEt(1+r)tinstead, we would overestimatethe value of the stock by the PV of retained earningsXtEt−Dt(1+r)t.Of course, if the entire earnings are distribute d as dividends each year (Et=Dt), then P0=XtEt(1+r)t. In this case, the firm does not inve st any of its owncapital to gro w; it is thus plausible to assume that dividends=earnings stayconstant ov er time. Call this SCENARIO A. Then, applying the perpetuityformula,P0=D1r=E1r.Thus, in this case, the Price-Earnings ratio is given byP0E1=1r. (1)• Yardeni’s Fed Model is similar, as it says:P0E1=1r10,where r10is the interest rate on 10-year government bonds.SCENARIO B.Now assume that the firm retains some earnings, but inves ts them at the“requiredrateofreturn”r; in other words, the firm invests its e arnings in 0-NPV investments. Clearly, this cannot change the value of stock, that is: thePV of the dividends. Thu s, the formulasP0=E1r(2)andP0E1=1rstill apply. Of course, now both earnings and dividends will grow.1Example.E1= $10,r=0.1. (I am using alw ays per-share values).• In scenario A, P0=$100.1=$100.• In scenario B, applying the form ula (2), once again P0=$100.1= $100.Magically, this will be true ho wever mu ch the firm retains and whateverthe resulting dividend pattern is.For example, suppose the firm retains always 50% of its earnings, andpays out 50% as its dividends. Thus, D1=$5. Then one can show that ifretained earnings make a return of r =10% (as assumed), and if dividendsgrow at a constant rate g,g =0.5 × 0.1=0.05. (3)— Of course, the stock price must again be equal to the PV of futuredividends. Indeed, applying the Gordon growth formula, we getP0=D1r − g=$50.1 − 0.05= $100!• Equation (3) is a special case of the following formulag =(1− δ)π, (4)where δ =DtEtis the payout ratio (assumed constant over time), and 1−δ =Et−DtEtis the retention ratio,andπ is the return on retained earnings. Seesection 5.5 of the textbook for further explanation; equation (4) is thesame as equation (5.8) there. Assuming that the firm’s investments havezeroNPVisthesameasassumingthatπ = r.You don’t need to understand equation (4) fully, but you should under-stand the basic intuition behind it: earnings (and hence dividends) willgrow the more rapidly, the more the firm invests (here, for simplicity allinvestment is assumed to come from retained earnings), and the moreprofitable those in vestments are.2SCENARIO C:Suppose now that the firm keeps reinvesting its earnings in positiv e NPVprojects. Specifically, suppose that the return on its retained earnings is 12%ratherthan10%asinB.Thenitsstockprice will be accord ingly higher. Indeed,the constant growth rate will now beg =0.5 × 0.12 = 0.06,henceP0=D1r − g=$50.1 − 0.06=$125.The difference P0−E1r=$125− $100 = $25 is the Net Present Value of GrowthOpportunities (NPVGO).SCENARIO D:If the firm keeps making bad investments, the NPVGO may be negative.For example, if the return on its in vestments is only 8% (less than investor’srequired rate of return of 10%), it destroys value. Indeed, the constant growthrate will now beg =0.5 × 0.08 = 0.04,henceP0=D1r − g=$50.1 − 0.04=$83.33,yielding an NPVGO of $ − 16.67.• Scenarios C and D contain an important general lesson: earnings growthjustifies higher P/E-ratios only if it is based on genuinely profitable (i.e.positive NPV) investmen ts. A firm’s growth is howev er often profitneutral(zero NPV); for example, this is t ypically the case if the firm grows byacquiring other