Supplementary Material to Cash Flow, Consumption Risk, and the Cross-Section of Stock Returns

Transcription

1 Supplementary Materal to Cash Flow, Consumpton Rsk, and the Cross-Secton of Stock Returns Zh Da Mendoza College of Busness, Unversty of Notre Dame. E-mal:

2 Ths document contans supplementary materal to the paper ttled Cash Flow, Consumpton Rsk, and the Cross Secton of Stock Returns. It contans sx sectons. Secton A detals why the two-factor cash flow model captures the relaton between rsk premum and cash flow characterstcs n the smple economy dscussed n the paper. Secton B solves the rsk premum on an asset usng the usual return-based beta representaton, renforcng the ntuton behnd the two-factor cash flow model and also relatng the cash flow characterstcs drectly to the standard return-based consumpton beta. Secton C examnes a slghtly modfed smple economy where the cash flow of an asset s also exposed to long-run consumpton rsk and shows that the two-factor cash flow model s stll vald. Secton D shows that the emprcal long-run earnngs-based measures dentfy ther theoretcal counterparts up to scalng factors n the smple economy. Secton E examnes the performance of the cash flow models on ndustry portfolos. Secton F contrasts two related varables: cash flow duraton and the book-to-market rato. A. Two-factor Cash Flow Model as an Approxmaton Ths secton provdes the detals for Secton I.D n the paper and explans why the two-factor cash flow model captures the relaton between rsk premum and cash flow characterstcs n the smple economy dscussed n the paper. Proposton n the paper shows that the rsk premum on an equty strp wth a maturty n n the smple economy s RP (n) = ( + φ n λ ) + (γ ) ρ ] 2δ σ 2 ρ δ w. The rsk premum on a stock s just the value-weghted average of rsk prema of all equty strps and can be approxmated as ] E t R t+ Rf t w (n)rp (n). n= Consder a lnear approxmaton of the rsk premum on ndvdual equty strp RP (n) around some fxed maturty n : RP (n) RP (n ) + RP n (n )(n n ), 2

3 where RPn (n ) denotes the frst dervatve of RPn wth respect to n, evaluated at n ; the rsk premum on a stock then becomes ] ] E t R t+ Rf t RP (n ) + RPn (n ) w (n)n n. () n= Drect computaton shows that RP (n ) = a 0 + a λ, (2) RPn (n ) = a 2 λ, (3) w (n)n n a 3 zt, (4) n= a 0 = a = φ n + (γ ) ρ 2δ ρ δ a 2 = φ n log φ ] σ 2 w, ] σw, 2 + (γ ) ρ 2δ ρ δ + (γ ) ρ 2δ ρ δ ] σ 2 w. Gven a rsk-averse agent wth γ > and n >, t can easly be verfed that a 0 > 0, a > 0, and a 2 < 0. To understand a 3, defne a functon f(z t) as f(z t) = w (n)n. n= Consder a lnear approxmaton of f(z t) around z t = 0: f(z t) f(0) + f (0)z t. Choosng n = f(0), whch can be nterpreted as the Macaulay duraton of an asset wth zt = 0 (for example, the aggregate consumpton portfolo), then w (n)n n a 3 zt, n= a 3 = f (0). The cash flow covarance measure λ also enters w (n) through the convexty adjustment terms n A (n). Its mpact on w (n)n, however, s relatvely small. 3

4 Fnally, t has to be verfed that a 3 = f (0) > 0. Drect calculaton of f (0) shows that f (0) > 0 n exp A (n) ] ( φ n ) exp A (n) ] > n exp A (n) ] ( φ n ) exp A (n) ] n exp A (n) ] exp A (n)] > n exp A (n) ] φ n exp A (n)] φ n. (5) Defne functon g(x) as g(x) = n exp A (n) ] exp(nx) exp A (n)] exp(nx). Then nequalty (5) s equvalent to g(0) > g(log φ), whch wll be true f g(x) s ncreasng n x or g (x) > 0. Drect calculaton of g (x) shows that g (x) > 0 n 2 a(n) > na(n) ] 2, (6) where a(n) = exp A (n) ] exp(nx) exp A (n)] exp(nx). Gven a(n) > 0, a(n) =, and h(n) = n 2 s a convex functon, nequalty (6) then follows from Jensen s nequalty. The ntuton behnd the postve a 3 s clear. The term z t can be nterpreted as the expected cash flow growth rate (relatve to aggregate consumpton growth). A hgher z t means that more cash flow wll occur n the future, thus ncreasng the present value-weghted tme as n w (n)n. Substtutng (2), (3), and (4) nto () gves us the two-factor cash flow model: E t R t+ Rf t ] γ0 + γ λ + γ 2 (z tλ ), γ 0 = a 0 > 0, γ = a > 0 and γ 2 = a 2 a 3 < 0. B. Beta Representaton n a Return Log-lnearzaton Framework In ths secton, I provde an alternatve dervaton of the rsk premum usng the usual return based beta-representaton framework to acheve three objectves: () to renforce the ntuton behnd the two-factor cash flow model; (2) to relate the cash flow characterstcs drectly to the standard return-based consumpton beta; and (3) to relate the cash flow characterstcs to the cash 4

5 flow rsk and dscount rate rsk. Under the usual assumpton that the return and the Stochastc Dscount Factor (SDF) are jontly log-normally dstrbuted (condtonally), the condtonal expected excess return of an asset can be expressed as 2 ] E t r σ 2 ] t+ rf t + t r t+ = cov t (r 2 t+, m t+ ). (7) I then use a log-lnear approxmaton to wrte the return on asset as where x t s the log prce to cash flow rato r t+ = κ 0 + κ x t+ x t + d t+ d t, κ exp(x ) = + exp(x ), κ 0 = log + exp(x ) ] κ x, ( ) log Pt D t at tme t and x s ts tme-seres average. Conjecture x t+ = a + b z t+ and use the cash flow process, consumpton growth process and the SDF specfed n the smple economy to evaluate the followng relaton: E t exp(mt+ + r t+) ] =. After collectng the terms on z t, we have φκ b b + ( φ) = 0, whch mples b = φ φκ. Then, 2 See Campbell (993), for example. rt+ ] = E t r t+ + β w t+ + κ φκ ε t+, where β = + κ φκ λ. 5

6 Equaton (7) can also be rewrtten usng a beta representaton n my economy: ] E t r σ 2 ] t+ rf t + t r t+ 2 = β λ c, (8) β = + κ φκ λ, λ c = + (γ ) ρ 2δ ρ δ ] σ 2 w, where β denotes the beta of asset, and λ c denotes consumpton rsk prema. The term κ = exp(x ) +exp(x ) s a log-lnearzaton constant where x s usually chosen to be equal to the average log prce-to-cash flow rato. The term β can be rewrtten usng x as β = + λ ( φ) exp(x )λ. (9) Snce the average log prce-to-cash-flow rato should be drectly related to cash flow duraton (see Proposton ), (9) also gves us the two-factor cash flow model. On the other hand, the usual practce of assumng a constant κ across all stocks effectvely elmnates the mpact of cash flow duraton when examnng cross-sectonal varaton n rsk prema. Campbell and Shller (988) decompose the return on an asset nto a component (N CF,t+ ) related to cash flow news and a component (N DR,t+ ) related to dscount rate news, wrtten as: rt+ ] E t r t+ = N CF,t+ N DR,t+, where (0) ( ) N CF,t+ = (E t+ E t ) κ j d t++j, j=0 ( ) N DR,t+ = (E t+ E t ) κ j r t++j. j= We may have the mpresson that N CF,t+ s related to cash flow covarance rsk and that N DR,t+ s related to cash flow duraton. By substtutng (0) nto (7), the beta can also be decomposed nto two parts: a cash flow beta (βcf ) and a dscount rate beta (β DR ) smlar to those n Campbell and Me (993) and Campbell, Polk, and Vuolteenaho (2003). Specfcally, β = βcf + β ( ) DR, κ βcf = λ φκ + ρ 2κ ρ κ, βdr = ρ 2κ ρ κ. 6

7 In my model, dscount rate news s drven by rsk-free rate dynamcs, whch are n turn drven by nnovatons n consumpton growth. Ths, together wth ψ equalng one, explans why the second terms n the cash flow beta and n the dscount rate beta offset each other. The cash flow covarance measure λ enters the expresson of β drectly whereas the cash flow duraton measure z t enters only ndrectly through κ. The mpact of the duraton on beta and expected return s therefore dffcult to examne. Although the return decomposton approach has many theoretcal advantages (e.g, economcally ntutve, allows for tme-varyng rsk premum), t does not allow us to see clearly the lnkage between rsk/return and fundamental cash flow characterstcs. Ths s why I choose to examne each equty strp separately n ths paper. C. Cash Flow Models wth Long-run Rsk In the paper, I model the cash flow covarance as the contemporaneous covarance between nnovatons n cash flow share and nnovatons n the aggregate consumpton growth. Ths smple specfcaton allows for both analytcal tractablty and easy economc nterpretaton. In ths secton, I model the cash flow covarance as the exposure of cash flow share to both long-run and short-run consumpton rsk and show that a smlar two-factor cash flow model can stll be derved n the smple economy. In the smple economy consdered n the paper, the log aggregate consumpton growth n the economy follows an ARMA(,) process: c t+ = µ c ( ρ ) + ρ c t + w t+ ρ 2 w t, w t N(0, σ 2 w). Defne x t = E t c t+ ] µ c. The ARMA(,) process can be rewrtten as c t+ = µ c + x t + w t+, x t+ = ρ x t + (ρ ρ 2 )w t+. As a result, x t, whch captures the condtonal expected consumpton growth rate, can be nterpreted as the long-run consumpton rsk. The term w t+, whch measures the contemporaneous nnovaton n consumpton growth, can be nterpreted as the short-run consumpton rsk. 7

8 I next model the cash flow growth rate on a stock (portfolo) as d t+ = c t+ + ( φ)z t + λ ( c t+ µ c ) + ε t+ = c t+ + ( φ)z t + λ (x t + w t+ ) + ε t+, z t+ = φz t λ (x t + w t+ ) ε t+. The only dfference between ths specfcaton and the specfcaton used n the paper s that cash flow covarance (λ) s now modeled as the exposure of cash flow share to both long-run and shortrun consumpton rsk, rather than the exposure to short-run consumpton rsk alone. To keep the algebra smple, I assume the cash flow share has the same exposure to both long-run and short-run consumpton rsk. Ths assumpton s also made mplctly n the co-ntegraton specfcaton of Bansal, Dttmar, and Lundblad (2002). Havng specfed the cash flow and aggregate consumpton growth process, I proceed to solve for the rsk premum expresson on an equty strp as a functon of ts maturty (n). The rsk premum s RP (n) = { + λ φ n + (ρ ρ 2 )A(n ) ]} + (γ ) ρ ] 2δ σ 2 ρ δ w, () where A(n) evolves accordng to the followng dfference equaton wth an ntal condton that A(0) = 0: φ n + ρ A(n ) = A(n). The rsk premum expresson s obtaned by followng the exact same procedure as descrbed n Appendx A of the paper and the detals are thus omtted. Due to the presence of the long-run rsk, the algebra s more complcated and an analytcal expresson cannot be obtaned. Compared to the equty strp rsk premum n the paper, the presence of long-run rsk results n one addtonal term (ρ ρ 2 )A(n ). It can be shown that for reasonable parameter values of φ and ρ (close to but smaller than one), A(n ) frst ncreases, then decreases n n. The term φ n, on the other hand, always decreases n n. A typcal long-run rsk model sets ρ to be close to one and slghtly greater than ρ 2. Such parameter choce allows consumpton growth to closely resemble an..d. process emprcally. At the same tme, the persstent expected consumpton growth rate leads to a larger rsk premum 8

9 and a potental soluton to the equty rsk premum puzzle. In ths case, ρ ρ 2 wll be very small and φ n + (ρ ρ 2 )A(n ), domnated by the frst term (φ n ), s lkely to be decreasng n n when n s not too small. Ths pattern has been confrmed n Fgure S. Place Fgure S about here Fgure S plots φ n, (ρ ρ 2 )A(n ), and ther sum separately as a functon of n (n number of months). The ARMA(,) parameters (ρ = and ρ 2 = 0.85) are taken from Bansal and Yaron (2000) and φ s chosen as 0.98 for the plot. As shown n the fgure, the value of φ n + (ρ ρ 2 )A(n ) peaks around year 2 (month 24), and s decreasng n n after that. As n the smple economy, for an equty strp wth nfnte maturty (n = ), the rsk premum becomes ] + (γ ) ρ 2δ ρ δ σw, 2 agan due to the mean-reverson n cash flow share, such that the mpact of cash flow covarance dmnshes wth maturty. A smlar two-factor cash flow model can be derved n ths economy where cash flow s also exposed to long-run rsk. Hgher cash flow covarance (λ ) should lead to hgher rsk premum as n (). Snce φ n + (ρ ρ 2 )A(n ) s decreasng n n after year 2, the nteracton between cash flow covarance and duraton wll be very smlar to that n the smple economy. When cash flow covarance (λ ) s postve, the rsk premum of an ndvdual equty strp generally decreases wth maturty. When ths happens, hgh duraton assets wll have lower returns snce a long-maturty cash flow wth lower return receves hgher present-value weght and the weghted average s lower. The reverse logc holds for negatve cash flow covarance, wth a hgher duraton leadng to a hgher return. Consequently, the product of cash flow covarance and duraton s negatvely related to the rsk premum on a stock. D. Theoretcal and Emprcal Cash Flow Characterstcs Ths secton proves that the emprcal long-run earnngs-based measures (Cov and Dur) dentfy ther theoretcal counterparts (λ and z) up to scalng factors n the smple economy. D. Cash Flow Duraton By defnton, ρ n s (t, n + ) = ρ n d (t, n + ) ρ n c t+n+. 9

10 Takng the expectaton at each portfolo formaton tme t on both sdes, { } E t ρ n s(t, n + ) ] = E t ρ n ( φ)z(t, n) = φ ρφ z t, cash flow duraton E z t ] can be dentfed (up to a scalng factor) by Dur = E Durt ] { = E = E Σ e t {E t Σ e] t = φ ρφ E z t], where Σ e t = ρ n e (t, n + ) and Σ c = ρ n c t+n+. t κ ρ ] } ξ t E t Σ c t D. Cash Flow Duraton To estmate the cash flow covarance λ, consder κ ρ ] } ξ t E t Σ c t ( ) ( cov ρ n s (t, n + ), ρ n w t+n+ = cov ρ n e ] ) (t, n + ) c t+n+, ρ n w t+n+. In my model specfcaton, the LHS s ( φρ) ( + ρ) λ σ 2 w. Therefore, by regressng ρ n e ] (t, n + ) c t+n+ on ρ n w t+n+, the regresson coeffcent Cov dentfes cash flow covarance (λ ) up to a scalng factor. E. Performance of the Cash Flow Models on Industry Portfolos Ths secton examnes the performance of the cash flow models on ndustry portfolos. Every June, I sort all stocks of ndustral frms (excludng fnancals and utltes) traded on NYSE, Amex, and NASDAQ nto ndustry portfolos accordng to a 7 Fama-French ndustry classfcaton. 3 The resultng 5 portfolos are: Food, Mnes(mnng and mnerals), Ol(ol and petroleum 3 I do not consder a fner ndustry classfcaton snce that would result n too few stocks n certan ndustry 0

11 products), Clths(textles, apparel, and footware), Durbl(consumer durables), Chems(chemcals), Cnsum(drugs, soap, perfumes, and tobacco), Cnstr(constructon), Steel, FabPr(fabrcated products), Machn(machnery and busness equpment), Cars, Trans(transportaton), Rtal(retal stores) and Other. Table S presents varous portfolo characterstcs ncludng the portfolo book-to-market rato (BM), market equty (ME, measured n mllons) at formaton, and annual return durng the frst year after portfolo formaton. All values are tme-seres averages across a samplng perod from 964 to Place Table S about here I also drectly test the valdty of the AR() assumpton on the cash flow share for the 5 portfolos. I frst ft an AR() process for the cash flow share and compute the resduals. I then test whether these resduals volate the whte nose condton usng the Ljung-Box Q test. Both the Ljung-Box (LB) Q test statstcs and the assocated p-values are reported. In addton, I also test the statonarty of the cash flow share usng the Augmented Dckey-Fuller test wth a constant and a lag of one. The t-values are reported (** means the hypothess of a unt root can be rejected at the 99% confdence level and * means the hypothess can be rejected at the 95% confdence level). The cash flow share n year t s computed as the log of the rato between the portfolo cash flow (sum of common dvdend and common share repurchase) and aggregate consumpton durng year t. For seven out of the 5 ndustry portfolos, I am not able to reject the unt root hypothess, ndcatng that the key assumpton that cash flow share s mean-revertng mght not be very approprate for ndustry portfolos. As a result, the cash flow covarance defned n my paper mght not be measurng the true consumpton rsk on the ndustry portfolo. Table S also presents the cash flow covarance and duraton estmates for the 5 portfolos. As expected, a growng ndustry such as Cnsum s assocated wth a hgher cash flow duraton whle an ndustry wth lttle growth potental such as Steel has a lower cash flow duraton. I then nvestgate the performance of cash flow models on ndustry portfolos n a cross-sectonal analyss. The coeffcent or the rsk premum estmates on the cash flow models are obtaned from OLS regressons. However, the robust t-values are computed usng GMM standard errors that account for both cross-sectonal and tme-seres error correlatons wth the Newey-West formula of portfolos, renderng the estmaton of cash flow characterstcs very mprecse.

12 seven lags. The one-stage GMM estmaton s carred out by stackng moment condtons of both tme-seres regressons and cross-sectonal regressons. The results are presented n Table S2. The cash flow models do a reasonably good job n descrbng the cross-sectonal varaton of average excess returns on the 5 ndustry portfolos. The one-factor cash flow model wth only cash flow covarance has a R 2 of 42.5% (adjusted-r 2 s 38.%). The two-factor cash flow model wth only cash flow covarance and duraton has a R 2 of 59.8% (adjusted-r 2 s 53.%). Smlar to the fndngs n the paper, the ncluson of cash flow duraton mproves the R 2 by about 7%. Place Table S2 about here The rsk premum on cash flow covarance (Cov) s postve whle the rsk premum on the nteracton term (Cov Dur) s negatve, consstent wth the predcton of the theory. However, once the tme-seres estmaton errors on cash flow covarance and duraton are accounted for, both Cov and Cov Dur are assocated wth nsgnfcant rsk prema. Overall, the performance of the cash flow models on ndustry portfolos are qualtatvely smlar although the assocated statstcal sgnfcance s much weaker, possble due to ms-specfcaton of the cash flow share process, large tme-seres estmaton errors, and small sample sze n the cross-secton. F. Cash flow duraton vs. book-to-market rato Ths secton contrasts the cash flow duraton to the commonly studed book-to-market rato. Emprcally, book-to-market seems to be nversely related to cash flow duraton. Ths pattern should not surprse us. As Lntner (975) and Santa-Clara (2004) pont out, any measure of cash flow duraton wll be related to book-to-market smply as a result of accountng denttes. Makng use of the accountng clean surplus dentty and return-dvdend-prce relaton, Vuolteenaho (2002) shows that the log book-to-market rato (θ t ) can be approxmated as θ t = ρ j r t+j+ ρ j e t+j+, (2) j=0 j=0 where r denotes log returns. Therefore, an ncrease n future accountng earnngs that ncreases cash flow duraton measure wll at the same tme decrease the book-to-market rato. In turn, cash flow duraton s negatvely correlated the wth book-to-market. Lettau and Wachter (2007) study an economy n whch stocks are only dstngushed by the tmng of ther cash flows. In such an 2

13 economy, they show that stocks wth cash flows weghted more to the future (hgh duraton) have low prce ratos (book-to-market rato, for example) and earn low return. Therefore, cash flow duraton can potentally explan value premum. My results, on the topcal level, seem to support ther hypothess snce value stocks ndeed have lower duraton than growth stocks. However, I would requre further analyss to answer a more nterestng queston: can the cash flow duraton alone explan value premum? If the cash flow duraton alone perfectly explans value premum, we would expect further sortng on book-to-market to generate no spread n returns once we control for cash flow duraton. To control for cash flow duraton, I frst sort all stocks accordng to the ex-ante cash flow duraton measure Dur t nto three groups: Low Duraton, Medum Duraton, and Hgh Duraton. Wthn each group, I further sort stocks accordng to BM nto three subgroups. To make sure that such portfolo constructon s mplementable, at each year, I reestmate duraton usng data from 965 through the current year, so the duraton measure Dur t s only computed usng nformaton avalable at year t. For ths reason, I start my portfolo constructon at year 975. Table S3 contans the results of the double sort. Snce BM and duraton are negatvely correlated, sortng on BM wthn each duraton group wll lkely nduce a spread n cash flow duratons. Ths s partcularly true for stocks n Low Duraton groups n whch low BM stocks have a cash flow duraton measure of.28 but hgh BM stocks have a cash flow duraton measure of only In contrast, the spread s much smaller for stocks n Medum and Hgh Duraton groups. Therefore, f cash flow duraton alone explans the value premum, I should expect that further sortng on BM generates no sgnfcant spread on returns for these stocks wth smlar cash flow duraton. Ths s not the case. Value stocks stll earn much hgher returns than growth stocks n the same cash flow duraton group. Ths fndng s not necessarly nconsstent wth a duraton-based explanaton of value premum f we nterpret prce-based BM as a less nosy measure of cash flow duraton. However, under the hypothess that duraton rsk alone explans value premum, we wouldn t expect the return spread nduced by the second sort on BM to be systematcally related to cash flow covarance. Ths s not what we fnd n the data. Table S3 shows that return spread can be explaned by the cash flow covarance rsk value stocks have ndeed hgher cash flow covarance rsk than growth stocks. Ths last fndng suggests that cash flow covarance rather than duraton s more mportant n explanng value premum. Place Table S3 about here 3

14 REFERENCES Bansal, Rav, Robert Dttmar, and Chrstan Lundblad, 2002, Consumpton, dvdends, and the crosssecton of equty returns, Workng Paper, Duke Unversty. Bansal, Rav, and Amr Yaron, 2000, Rsk for the long run: A potental resoluton of asset prcng puzzles, Workng Paper, NBER. Campbell, John, Chrstopher Polk, and Tuomo Vuolteenaho,, 2003, Growth or glamour, Workng Paper, Harvard Unversty and Northwestern Unversty. Campbell, John, 993, Intertemporal asset prcng wthout consumpton data, Amercan Economc Revew 83, Campbell, John, and Janpng Me, 993, Where do betas come from? asset prce dynamcs and the sources of systematc rsk, Revew of Fnancal Studes 6, Campbell, John, and Robert Shller, 988, The dvdend-prce rato and expectatons of future dvdends and dscount factors, Revew of Fnancal Studes, Da, Zh, 2006, Three essays on asset prcng, PhD dssertaton, Northwestern Unversty. Lettau, Martn, and Jessca Wachter, 2007, Why s long-horzon equty less rsky? A duraton-based explanaton of the value premum, Journal of Fnance 62, Lntner, John, 975, Inflaton and securty returns, Journal of Fnance 30, Santa-Clara, Pedro, 2004, Dscusson of Impled equty duraton: A new measure of equty rsk, Revew of Accountng Studes 9, Vuolteenaho, Tuomo, 2002, What drves frm-level stock returns, Journal of Fnance 57,

15 φ n mth (ρ -ρ 2 )A(n-) mth 3 φ n- +(ρ -ρ 2 )A(n-) mth Fgure S. Equty strp rsk premum terms as functons of maturtes. Ths fgure plots varous terms as functons of maturtes (n) n the equty strp rsk premum expresson. The top plot corresponds to φ n. The mddle plot corresponds to (ρ ρ 2 )A(n ). The bottom plot corresponds to ther sum. The parameters (at the monthly frequency) used n ths fgure are: ρ = 0.965, ρ 2 = 0.85, and φ =

16 Table S Characterstcs of Industry Portfolos Every June, I sort all stocks of ndustral frms (excludng fnancals and utltes) traded on NYSE, AMEX, and NASDAQ nto ndustry portfolos accordng to the 7 Fama-French ndustry classfcaton. The resultng 5 portfolos are: Food, Mnes(mnng and mnerals), Ol(ol and petroleum products), Clths(textles, apparel, and footware), Durbl(consumer durables), Chems(chemcals), Cnsum(drugs, soap, perfumes, and tobacco), Cnstr(constructon), Steel, FabPr(fabrcated products), Machn(machnery and busness equpment), Cars, Trans(transportaton), Rtal(retal stores), and Other. Ths table presents varous portfolo characterstcs. The portfolo book-to-market rato (BM), market equty (ME, measured n mllons) at formaton, annual return durng the frst year after portfolo formaton are reported. All values are tme-seres averages across a samplng perod from 964 to I also drectly test the valdty of the AR() assumpton on the cash flow share for the 5 portfolos. I frst ft an AR() process for the cash flow share and compute the resduals. I then test whether these resduals volate the whte nose condton usng the Ljung-Box Q test. Both the Ljung-Box (LB) Q test statstcs and the assocated p-values are reported. In addton, I also test the statonarty of the cash flow share usng the Augmented Dckey-Fuller test wth a constant and a lag of one. The t-values are reported (** means the hypothess of a unt root can be rejected at the 99% confdence level and * means the hypothess can be rejected at the 95% confdence level). The cash flow share n year t s computed as the log of the rato between the portfolo cash flow (sum of common dvdend and common share repurchase) and aggregate consumpton durng year t. Fnally, the cash flow covarance (Cov) and average duraton (Dur) for each portfolo are also reported. Food Mnes Ol Clths Durbl Chems Cnsum Cnstr Steel FabPr Machn Cars Trans Rtal Other nobs BM ME Return LB Q test stat p-value ADF t-value -6.2** -5.7** ** ** -6.00** ** ** ** Dur Cov

17 Table S2 Performance of Cash Flow Models on Industry Portfolos Ths table reports the results of cross-sectonal regressons of average excess returns on the 5 portfolos on cash flow duraton and covarance measures. The coeffcent estmates are obtaned from OLS regressons. However, the robust t-values are computed usng GMM standard errors whch account for both cross-sectonal and tme-seres error correlatons, wth Newey-West formula of seven lags. The one-stage GMM estmaton s carred out by stackng moment condtons of both tme-seres regressons and cross-sectonal regressons. Fnally, both R 2 s adjusted-r 2 s of the regressons are reported. The samplng perod s from 964 to 995. ntercept Cov Dur Cov R 2 / adj R 2 One factor: Coeffcent Robust t-value Two Factors: Coeffcent Robust t-value Table S3 Duraton and BM-sorted Portfolos Each year from 975 to 996, I sort all stocks frst nto three groups accordng to a rollng-wndow ex-ante cash flow duraton measure, and wthn each group, I further sort stocks nto three subgroups accordng to ther book-to-market rato. The book-to-market-rato, annual excess returns, pont estmates of cash flow duraton, and covarance are reported n the table. BM Excess Return growth value growth value Low Dur Med Dur Hgh Dur Dur Cov growth value growth value Low Dur Med Dur Hgh Dur