Risk Aversion and Stock Prices

Transcription

1 Rsk Averson and Stock Prces Ray C. Far revsed February 2003 Abstract Ths paper uses data on companes that have been n the S&P 500 ndex snce 1957 to examne whether rsk averson has decreased snce The evdence suggests that t has not. There s no evdence that more rsky companes have had larger ncreases n ther prce-earnngs ratos snce 1995 than less rsky companes. 1 Introducton It s clear that there has been a huge ncrease n the average prce-earnngs (PE) rato of U.S. stocks snce For example, the medan S&P 500 PE rato for s 26.41, whch compares to the medan of for Earnngs fell on average more than stock prces n 2001, and the S&P 500 PE rato Cowles Foundaton and Internatonal Center for Fnance, Yale Unversty, New Haven, CT Voce: ; Fax: ; emal: ray.far@yale.edu; webste: I am ndebted to John Cochrane and Jesse Shapro for helpful comments and to Alsa Levne for superb research assstance. 1 As dscussed n Secton 4, medans seem more approprate than means as measures of average PE ratos. For the S&P 500 PE rato, however, medans and means are close. For the mean s (versus 26.41), and for the mean s (versus 15.45). For the medan s and the mean s 17.33, both stll much lower than the medan and mean for Note that 1995 s not used n these calculatons; t s treated as a transton year. The S&P 500 PE rato s defned as the value of the S&P 500 stock prce ndex at the end of the year dvded by S&P 500 reported earnngs for that year.

2 for 2001 s 46.50! The PE rato for 2002 s The ratos for 2001 and 2002 are obvously hgh because earnngs are unusually low, but they are much hgher than exsted n prevous low-earnngs perods. For example, earnngs were low n 1991, and the PE rato for 1991 s There was, on the other hand, no correspondng large decrease n real long term nterest rates after The medan real AAA bond rate 2 for s.053, whch compares to the medan of.031 for and.057 for (The respectve means are.054,.037, and.059.) Why PE ratos have rsen so much snce 1995 wth lttle change n real long term nterest rates s a key queston n fnance. Does ths sgnal the end of the equty premum puzzle, about whch so much has been wrtten? 3 The possblty that s examned n ths paper s that the degree of rsk averson of the average nvestor fell n the last half of the 1990s. Ths could account at least n part for the ncrease n PE ratos relatve to real long term nterest rates. The paper uses data on companes that have been n the S&P 500 ndex snce 1957, whch s the frst year that the S&P ndex ncluded 500 companes. The data are dscussed n Secton 2, and the 65 companes that were used are lsted n Tables 1 and 2. The basc dea of the paper s the followng. Although the 65 companes are obvously sold establshed companes, they do dffer somewhat n rsk. The frst step (Secton 3) s to estmate the rsk of each company usng data from The real AAA bond rate used for these calculatons s the nomnal AAA bond rate mnus the percentage change n the GDP deflator over the prevous two years (at an annual rate). 3 See Kocherlakota (1996) and Segel and Thaler (1997) for revews of the lterature on the equty premum puzzle pror to the possble change n the premum n the last half of the 1990s. For more recent dscussons of a possbly fallng equty premum, see Segel (1999) and Jagannathan, McGrattan, and Scherbna (2000). For an nterestng set of results on the vews of fnancal economsts on the equty premum, see Welch (2000). 2

3 through Two measures of rsk are computed per company. The frst s the estmate of β from the CAPM model, and the second s a measure of the varablty of real earnngs growth. The second step (Secton 4) s to compute the change n each company s average PE rato for the perod before 1995 to the perod after Once ths s done, one can compare the changes n the average PE ratos across companes. If the degree of rsk averson of the average nvestor fell after 1995, one should expect the changes n the average PE ratos for the more rsky companes to be on average larger than the changes for the less rsky companes. 4 The results n Secton 5 show that ths s not the case. There s no evdence from these results that rsk averson has fallen. Other explanatons are needed for the large ncrease n PE ratos snce An advantage of usng companes that have been n the S&P 500 ndex for a long tme (n addton to data avalablty) s that these companes are less lkely than others to have changed n large ways snce The hypothess tested n ths paper s that the degree of rsk averson of nvestors has changed snce 1995, not the nherent rskness of companes. If the rskness of the companes has also changed, any dfferences found after 1995 mght be due to these changes rather than to changes n nvestors rsk averson. Note that survval bas s not a problem here. In fact, long run survval s good here because ths makes t more lkely that the rsk characterstcs of the frm have not changed. There would be selecton bas f frms were selected on the bass of how much ther PE ratos changed snce 1995, but ths s not the case. 4 A proof of the proposton that PE ratos ncrease more for more rsky companes when rsk averson falls s presented n the appendx for a partcular model. 3

4 A number of people have suggested that at least some of the ncrease n PE ratos snce 1995 may be due to a fall n rsk averson. Shller (2000, p. 41) suggests that the rse of gamblng opportuntes may have led to changed atttudes toward rsk takng n other areas. Campbell and Cochrane (1999) have a model n whch rsk averson s lower n expansons than n recessons. Snce the perod between 1995 and 2000 was one of robust growth, ths model mples lower rsk averson n ths perod than otherwse. Glassman and Hassett (1999, p. 97) argue that n the last half of the 1990s people have been lowerng ther estmates of the overall rskness of stocks relatve to bonds, whch has drven up the prce of stocks. Whle ths s not necessarly a change n rsk averson, t s a change that ths paper tests. If there has been a decrease n nvestors estmates of the overall rskness of stocks (and not, say, also a decrease n rsk averson), t stll should be the case that more rsky companes have larger ncreases n ther PE ratos than less rsky ones. 2 Data on the 65 Companes A number of companes have been n the S&P 500 ndex snce the ncepton of the 500-company ndex n For ths paper 65 companes were chosen. These are companes for whch data exsted back to (or nearly back to) 1957 and whch were not affected by large mergers. The 65 companes are lsted n Tables 1 and 2 along wth varous varables for each company. The varables are explaned as the paper proceeds. The companes are ranked n the tables by the sze of ther β s, whch are estmated n the next secton. 4

5 Table 1 Results from the β Regressons p-val p-val Company β a SE β t α AP end beg 1 Alcan Inc TXU Corp Procter & Gamble Co PG&E Corp Phllps Petroleum Co AT&T Corp Mnnesota Mnng & Mfg Co Alcoa Inc Amercan Electrc Power Publc Servce Entrp Hercules Inc Ar Products & Chemcals I Brstol Myers Squbb Kmberly-Clark Corp Aetna Inc Wrgley (WM) Jr Co Hallburton Co Deere & Co Kroger Co Intl Busness Machnes Corp Caterpllar Inc Goodrch Corp General Mlls Inc Wnn-Dxe Stores Inc Henz (H J) Co Eastman Kodak Co Campbell Soup Co Phlp Morrs Cos Inc Southern Co Du Pont (E I) de Nemours Phelps Dodge Corp Pfzer Inc Hershey Foods Corp Ingersoll-Rand Co FPL Group Inc Ptney Bowes Inc Archer-Danels-Mdland Co Rockwell Intl Corp Dow Chemcal General Electrc Co Abbott Laboratores Merck & Co Penney (J C) Co Unon Pacfc Corp Scherng-Plough Pepsco Inc McGraw-Hll Companes Household Internatonal Inc Emerson Electrc Co General Motors Corp

6 Table 1 Results from the β Regressons p-val p-val Company β a SE β t α AP end beg 51 Colgate-Palmolve Co Eaton Corp Dana Corp Sears Roebuck & Co Cornng Inc General Dynamcs Corp Coca-Cola Co Boeng Co Ford Motor Co Peoples Energy Corp Goodyear Tre & Rubber Co May Department Stores Co ITT Industres Inc Raytheon Co Cooper Industres Inc Notes: β a = estmate of β from Secton 3 ( estmaton perod). SE β = estmated standard error of β a t α = t-statstc for estmate of the constant term. AP = Andrews-Ploberger statstc. p-val end = p-value for end-of-sample test. p-val beg = p-value for begnnng-of-sample test. 6

7 Table 2 Constructed Varables for the 65 Companes Company β a PE a PE b e a e b d a d b σ a 1 Alcan Inc TXU Corp Procter & Gamble Co PG&E Corp Phllps Petroleum Co AT&T Corp Mnnesota Mnng & Mfg Co Alcoa Inc Amercan Electrc Power Publc Servce Entrp Hercules Inc Ar Products & Chemcals I Brstol Myers Squbb Kmberly-Clark Corp Aetna Inc Wrgley (WM) Jr Co Hallburton Co Deere & Co Kroger Co Intl Busness Machnes Corp Caterpllar Inc Goodrch Corp General Mlls Inc Wnn-Dxe Stores Inc Henz (H J) Co Eastman Kodak Co Campbell Soup Co Phlp Morrs Cos Inc Southern Co Du Pont (E I) de Nemours Phelps Dodge Corp Pfzer Inc Hershey Foods Corp Ingersoll-Rand Co FPL Group Inc Ptney Bowes Inc Archer-Danels-Mdland Co Rockwell Intl Corp Dow Chemcal General Electrc Co Abbott Laboratores Merck & Co Penney (J C) Co Unon Pacfc Corp Scherng-Plough Pepsco Inc McGraw-Hll Companes Household Internatonal Inc Emerson Electrc Co General Motors Corp PE b1 PE b2 7

8 Table 2 (contnued) Constructed Varables for the 65 Companes Company β a PE a PE b e a e b d a d b σ a 53 Colgate-Palmolve Co Eaton Corp Dana Corp Sears Roebuck & Co Cornng Inc General Dynamcs Corp Coca-Cola Co Boeng Co Ford Motor Co Peoples Energy Corp Goodyear Tre & Rubber Co May Department Stores Co ITT Industres Inc Raytheon Co Cooper Industres Inc Mean of the Notes: β a = estmates of β from Secton 3 ( estmaton perod). PE a = medan PE rato PE b = medan PE rato e a = medan earnngs growth rate e b = medan earnngs growth rate d a = medan dvdend growth rate d b = medan dvdend growth rate PE b1 = predcted medan PE rato from equaton (3) n Table 3. [= exp( log PE b1 )] σ a = estmate of the varablty of the earnngs growth rate PE b2 = predcted medan PE rato from equaton (4) n Table 3. [= exp( log PE b2 )] PE b1 PE b2 8

9 For each company annual data were collected for on ts stock prce at the end of the year (Pt ), ts earnngs per share for the year (E t ), and ts dvdends per share for the year (Dt ). Adjustments were made for stock splts. The data were obtaned from the CRSP/COMPUSAT Merged Database from the webste of Wharton Research Data Servces. One company, Internatonal Paper Company (IP), was not used even though data exsted for all the years. Earnngs of IP for all fve years between 1996 and 2000 are very low, and the medan PE rato s for ths perod. Ths s not a sensble number, and f ths observaton were used for the emprcal work n Secton 5 t would be a huge outler. Rather than try to adjust the PE rato down n some way, the IP company was just not used. 3 Estmates of Rsk The β Regressons As noted n the Introducton, two measures of rsk are computed per company. The frst s β from the CAPM model. Let Pt m denote the value of the S&P 500 stock prce ndex at the end of year t, and let Dt m denote S&P 500 dvdends for year t. The market rate of return, Rt m, that s used for the β regressons s taken to be (P m t + D m t )/P m t 1. The rsk free rate, Rf t, s taken to be the one-year Treasury bll rate (average for the year). 5 5 Because of data lmtatons, the sx-month rate s used for 1958 (average for the year). The data were obtaned from the web ste of the Board of Governors of the Federal Reserve System. The bll rates are for the secondary market. 9

10 The rate of return for company, Rt, s taken to be (P t + Dt )/P t 1, where Pt and Dt are defned n Secton 2. Observatons on R t are avalable begnnng n 1958 for all but three companes, where the begnnng year s 1960 for Aetna and 1963 for Household Internatonal and Unon Pacfc. For each of the 65 companes the followng regresson was run for the perod begnnng n 1958 (or later for the three) and endng n 1994: R t Rf t = β(r m t R f t ) + ɛ t, t = 1958,...,1994 The 65 estmates of β, denoted β a, are presented n Table 1, where the companes are ranked by the sze of the estmates. The estmated standard error of β a, denoted SE β, s also presented. The remanng four columns n Table 1 nclude the results of varous tests of the β regressons. The frst test s to add a constant term to each regresson. The CAPM model does not call for a constant term n the regresson, and so testng the hypothess that the constant term s zero s one test of the model. The t-statstc for the constant term estmate, denoted t α, s presented n the table for each of the 65 regressons. In only 7 of the 65 cases s the t-statstc greater than 2.0 n absolute value, and so the hypothess of a zero constant term s generally not rejected. The last three tests are stablty tests. The frst hypothess tested s that there s no structural break n the equaton between 1974 and The test due to Andrews and Ploberger (AP) (1994) was used. Ths test has the advantage that a sngle break pont does not have to be specfed, only a range of possble break ponts. Each regresson has 37 observatons, and the 5 percent crtcal value for 10

11 the AP statstc for ths number of observatons and one coeffcent s In Table 1 only 7 of the 65 values of AP are greater than 2.00, and so the stablty hypothess s generally not rejected. The next test s of the hypothess that there s no structural break near the end of the sample perod n The test due to Andrews (2002) was used. The p-values from ths test are presented n the table. Only 9 of the 65 p-values are less than.05, and so the end-of-sample stablty hypothess s generally not rejected. The fnal test s of the hypothess that there s no structural break near the begnnng of the sample perod n The Andrews (2002) test was also used for ths purpose. The p-values from ths test are presented n the table. Agan, only 9 of the 65 p-values are less than.05, and so the begnnng-of-sample stablty hypothess s generally not rejected. The overall results are thus farly supportve of the CAPM model for ths set of companes. The regressons are mostly stable, and most of the estmates of the constant term are not sgnfcant. On the negatve sde, most of the estmates of β are not sgnfcantly dfferent from 1.0, whch means that there s not much precson n the rankng of the β estmates. It wll be seen n Secton 5, however, that there s some evdence that the estmates of β are pckng up rsk dfferences across companes. 6 See Andrews and Ploberger (1994), Table I. 7 For Aetna the year was 1965, and for Household Internatonal and Unon Pacfc the year was

12 Varaton of Earnngs Another measure of the rsk of a company, not consstent wth the CAPM model, s the varaton of ts earnngs. Maybe the average nvestor looks only at a company s earnngs fluctuatons n judgng how rsky t s? The measure that was used s as follows. In the next secton the growth rates of each company s real earnngs are computed for These growth rates are ranked, and the medan, denoted e a, s computed. The varaton n the growth rate of earnngs, denoted σ a, s then taken from ths rankng to be the dfference between the value above whch 20 percent of the growth rates le and the value below whch 20 percent of the growth rates le. Ths range was used as the measure of varaton because t s not sensble to compute varances n the usual way due to extreme values at both ends of the rankng. The values of σ a of Table 2. are presented n the second-to-last column 4 Computng PE Ratos, Earnngs Growth, and Dvdend Growth Computng average PE ratos s problematc because earnngs can be very small or negatve. For the present calculatons the PE rato for a gven year was taken to be large (and postve) f earnngs for the year were negatve. The rato was taken to be large enough to put the observaton at the top when the observatons are ranked. The average PE rato was then taken to be the medan of the ranked observatons. Ths way of treatng negatve earnngs affects the calculaton of the average value 8 In some cases the frst year was later than

13 only n that the large values are put at the top before the medan s taken. For each company the medan was computed for the perod. For a few companes the earnngs data began after 1957, and for these companes the medan was computed for the perod consstng of the frst avalable observaton through The medan for company for ths perod wll be denoted PE a, where a denotes the (or slghtly shorter) perod. The medan for each company was also computed for the perod, whch meant rankng the fve yearly observatons and takng the thrd one. For one company, Cornng, three of the fve PE ratos were very large because of very low earnngs, and for Cornng the average PE rato was taken to be the second lowest rather than the thrd. The medan for company for ths perod wll be denoted PE b, where b denotes the perod. Both PEa and PE b are presented n Table 2. The last row n Table 2 presents the mean of the 65 observatons for each varable. The mean of PE a s 14.21, and the mean of PE b s There has thus been on average a large ncrease n the medan PE rato from before to after 1995 for these companes, whch s consstent wth the S&P 500 data dscussed n Secton 1. Four other varables per company were also computed: the medan growth rates of earnngs for the two perods, denoted e a and e b, and the medan growth rates of dvdends for the two perods, denoted d e and d b. Earnngs and dvdends from Secton 2 were frst deflated by the GDP deflator: ERt = Et /GDPD t and DRt = Dt /GDPD t, where GDP D t s the GDP deflator for year t, ER denotes real earnngs, and DR denotes real dvdends. The growth rate of real earnngs was 13

14 then computed as (ER t ER t 1 )/ER t 1 when ER t 1 was postve. When ER t 1 was zero or negatve, the growth rate was taken to be a large postve number f ERt >ERt 1 and a large negatve number f ER t <ERt 1. For each perod the growth rates were ranked and the medan of the ranked observatons was taken. 9 (As dscussed n the prevous secton, these growth rates of real earnngs for were used to compute σ a, the varablty measure.) The same procedure was followed for dvdends, where there are zero values for a few of the Dt but no negatve values. Agan, medans were computed for the perod up to 1994 and for the perod The four medan growth rates per company are presented n Table 2. It can be seen from the last row n Table 2 that on average earnngs growth was less after 1995 (mean of.044 versus.056) and dvdend growth was greater (mean of.040 versus.023). 5 The Cross Company Regressons Perod If was a perod n whch there were no large shfts n the rsk characterstcs of the 65 companes, then the estmates of β a or σ a may be reasonable approxmatons of the rskness of the companes. One would expect, other thngs beng equal, for more rsky companes to have on average lower PE ratos. If, 9 For the second perod, whch conssts of only fve observatons, ths procedure dd not result n sensble growth rates for fve companes (Boeng, Goodyear, Hallburton, ITT, and Phllps). For each of these fve companes total real earnngs were computed for and , and the growth rate (at an annual rate) between these two perods was used for e b. 14

15 therefore, ether β a or σ a s a good measure of rsk, t should have a negatve effect on PE a. One would also expect companes wth hgher average growth rates of earnngs and dvdends to have hgher average PE ratos, so that e a and d a should have postve effects on PE a. Usng the data n Table 2, log PE a was regressed on a constant, β a, ea, and d a for the 65 company observatons. Ths regresson s equaton (1) n Table 3. The coeffcent estmate for β a s negatve, as expected, and t has a t-statstc of 1.99, whch s sgnfcant at the 5 percent level for a one-taled test. The coeffcent estmates for the two growth rates are postve, as expected, although the estmate for earnngs growth only has a t-statstc of The sgnfcance of β a thus provdes further support for the CAPM model for ths set of companes. The results provde some evdence that the estmates of β are pckng up rsk dfferences across companes. If the estmates were not, they should not have a negatve effect on the average PE ratos. Regardng the other possble measure of rsk, σ a, log PEa was regressed on a constant, σ a, ea, and da for the 65 company observatons. Ths regresson s equaton (2) n Table 3. The coeffcent estmate for σ a s of the expected negatve sgn, but t only has a t-statstc of From ths regresson there s not much support for σ a beng a good measure of rsk Perod Equaton (1) n Table 3 can be used to predct what the average PE rato of a company should be n the perod f there were no structural breaks. Let ˆν a1 be the estmated error n equaton (1) for company. Ths error captures 15

16 Table 3 Regressons Across Companes Dependent Varable s log PE a Explanatory Varables Eq. cnst β a e a d a σ a SE R 2 (1) (22.79) (-1.99) (1.46) (3.33) (2) (41.44) (1.36) (2.26) (-1.01) (3) log PE b1 = β a +.839e b db +ˆν a1 (4) log PE b2 = σ a +.805e b db +ˆν a2 Notes: t-statstcs are n parentheses. # obs. = 65. ˆν a1 ˆν a2 = estmated error n equaton (1) for company. = estmated error n equaton (2) for company. See Table 2 for the other notaton. all the effects on a company s average PE rato that are not pcked up by ts β, growth rate of earnngs, and growth rate of dvdends. If t s assumed for each company that β a and ˆν a1 have not changed (no structural breaks n ths sense), but that perceved earnngs growth and dvdend growth have changed (from e a to e b and from d a to d b ), then the predcton of the average PE rato for s as lsted n equaton (3) n Table 3. Ths calculaton uses the coeffcents estmated for the perod, but the values of earnngs growth and dvdend growth. The predctons of PE b from equaton (3) are presented n Table 2, where PE b1 = exp( log PE b1 ). It s clear from Table 2 that these predctons, whch 16

17 are the predcted average PE ratos for , are close to the actual average PE ratos for Because β a and ˆν a1 are used n equaton (3), the only reason log PE a and log PE b1 dffer for a gven company s because the growth rates of earnngs and dvdends dffer between the two perods. The net effect of these dfferences s n general not large,.e., log PE a and close. log PE b1 are n general Note that the βs have not been reestmated for the predctons: β a s used n equaton (3). It would not have been practcal to reestmate the βs because fve observatons per company s not enough to get trustworthy estmates. More to the pont, however, as dscussed above, the analyss n ths paper s based on the assumpton that the rsk characterstcs of the companes have not changed,.e., that a company s β has not changed. Equaton (2) n Table 3 can also be used to predct what the average PE rato of a company should be n the perod f there were no structural breaks. Ths s done n equaton (4), where ˆν a2 s the estmated error n equaton (2) for company. Ths predcton, of course, uses as the measure of rsk σ a nstead of β a. The predctons of PEb from equaton (4) are also presented n Table 2, where t s agan clear that these predctons are close to the actual average PE ratos for The man nterest of ths paper s to examne the dfference between the actual average PE rato for (PE b ) and the predcted average under the assumpton of no structural changes ( PE b1 or PE b2 ). Table 2 shows that on average ths dfference s large and postve. Now, f the ncrease n the average PE ratos s due to a fall n nvestors rsk averson, more rsky companes should have had 17

18 larger ncreases. (See the appendx for a proof of ths for a partcular model.) The tests are presented n Table 4. Row 1 s a regresson of log PE b log PE b1 on a constant and β a. If there has been a decrease n rsk averson, the coeffcent estmate of β a should be postve and sgnfcant. The estmate s postve but not sgnfcant, wth a t-statstc of There s thus lttle evdence that hgh β companes had on average more of a non predcted ncrease n ther PE ratos than dd low β companes. The regresson n row 1 n Table 4 uses the predctons of log PE b from equaton (3) n Table 3. Any msspecfcaton n equaton (3) wll affect the results n row 1 n Table 4. A smpler test, whch does not depend on equaton (3), s to regress the actual log change n the average PE ratos, log PE b log PE a, on a constant and β a. From the perspectve of equaton (3), the assumpton s beng made that e b = e a and d b = d a. In other words, the assumpton s that the perceved growth rates of earnngs and dvdends have not changed from to (as well as β a and ˆν a1 not changng). Ths regresson s n row 2 n Table 4, where the coeffcent estmate for β a s stll not sgnfcant. The man concluson s thus not senstve to whether or not the predctons from equaton (3) are used. The above analyss can be repeated wth σ a n place of β a and log PE b2 n place of log PE b1. Ths s done n rows 3 and 4 of Table 4. The coeffcent estmate of σ a n both rows s negatve and sgnfcant. These results say that f we take σ a as measurng rsk, the least rsky companes have had the largest ncrease n ther PE ratos. Ths, of course, s opposte to what would be the case f rsk averson has fallen. It s unclear, however, how much weght should be put on ths result gven that σ a s not sgnfcant n equaton (2) n Table 3, but at the least there s 18

19 Table 4 Regressons Across Companes Dependent Explanatory Varables Varable cnst β a σ a SE R 2 1 log PE b log PE b (0.40) (1.18) 2 log PE b log PE a (1.82) (0.44) 3 log PE b log PE b (7.34) (-3.12) 4 log PE b log PE a (8.97) (-2.92) Notes: t-statstcs are n parentheses. # obs. = 65. See Table 3 for the computaton of See Table 2 for the other notaton. log PE b1 and log PE b2. no support for the hypothess that rsk averson has fallen. 6 Concluson A remarkable feature of the data for the 65 companes s on average the large ncrease n the medan PE rato from to Ths ncrease s not explaned by hgher earnngs or dvdend growth n , snce the predcted PE ratos from equatons (3) and (4) n Table 3 are much lower on average than the actual ratos. (Earnngs growth was n fact on average lower n than earler, although dvdend growth was hgher.) The man pont of ths paper s to show that larger ncreases n PE ratos dd not occur for the more rsky companes. Ths s contrary to what one would expect f 19

20 there were a fall n the degree of rsk averson of the average nvestor after Some other explanaton s needed for the large average PE ncreases. The results n ths paper may have mplcatons for the future growth of stock prces. Snce the degree of rsk averson does not appear to have fallen, the reason for the large PE ncreases may be due to somethng less fundamental and permanent. If, for example, they have been due to unrealstcally large expectatons of future earnngs or dvdends, the PE ncreases are less lkely to last than f they have been due to a fall n rsk averson. 20

21 Appendx Consder a partcular company. Let P = prce per share, E = earnngs per share, D = dvdends per share, g = the growth rate of dvdends, r = the rsk free rate, γ = the coeffcent of rsk averson, β = the rsk of the company, and σ 2 = the varance of consumpton changes. γ, β, and σ 2 are non negatve; a decrease n γ s a fall n rsk averson; and a decrease n β s a fall n the rsk of the company. Followng Cochrane (2001), p. 19, the expected return on the company s stock s r + βγσ 2. It s assumed that r g>0 and thus that r + βγσ 2 g>0 snce β, γ, and σ 2 are non negatve. If r s not greater than g, then the PE rato below s not defned for all values of βγσ 2. Usng the Gordon model, the PE rato s P E = D ( ) 1 E r + βγσ 2 g (1) or n log form log P E = log D E log(r + βγσ2 g) (2) Takng the dervatve of equaton (11) wth respect to γ yelds z = log(p /E) γ = βσ 2 r + βγσ 2 g > 0 (3) The dervatve of z wth respect to β s z β = σ 2 (r g) (r + βγσ 2 g) 2 > 0 (4) Thus when γ falls, the ncrease n log(p /E) s larger the larger s β. 21

22 References [1] Andrews, Donald W.K., 2002, End-of-Sample Instablty Tests, Cowles Foundaton Dscusson Paper No. 1369, May. [2] Andrews, Donald W.K., and W. Ploberger, 1994, Optmal Tests When a Nusance Parameter s Present Only Under the Alternatve, Econometrca, 62, [3] Campbell, John Y., and John H. Cochrane, 1999, By Force of Habt: A Consumpton-Based Explanaton of Aggregate Stock Market Behavor, Journal of Poltcal Economy, 107, [4] Cochrane, John, H., 2001, Asset Prcng, Prnceton: Prnceton Unversty Press. [5] Glassman, James K., and Kevn A. Hassett, 1999, DOW 36,000, New York: Three Rvers Press. [6] Jagannathan, Rav, Ellen R. McGrattan, and Anna Scherbna, 2000, The Declnng U.S. Equty Premum, Federal Reserve Bank of Mnneapols Quarterly Revew, 24 (Fall), [7] Kocherlakota, Narayana R., 1996, The Equty Premum: It s Stll a Puzzle, Journal of Economc Lterature, 34 (March), [8] Shller, Robert J., 2000, Irratonal Exuberance, Prnceton, New Jersey: Prnceton Unversty Press. [9] Segel, Jeremy J., 1999, The Shrnkng Equty Premum: Hstorcal Facts and Future Forecasts, Journal of Portfolo Management, 26 (Fall), [10] Segel, Jeremy J., and Rchard H. Thaler, 1997, The Equty Premum Puzzle, Journal of Economc Perspectves, 11 (Wnter), [11] Welch, Ivo, 2000, Vews of Fnancal Economsts on the Equty Premum and on Professonal Controverses, Journal of Busness, 73 (October),