Comments on Anomalies by Lu Zhang

Transcription

1 Comments on Anomalies by Lu Zhang John H. Cochrane University of Chicago November 2, 2004

2 Q reminder X X V (K 0, {I t }) = E 0 M t D t = E 0 M t (θ t f( ) t=0 t=0 " 1+ α 2 Ã It!# I t ) s.t. +1 = (1 δ) + I t+1 ; K 0 =(1 δ)k 1 + I 0 FOC : V + V =0 I 0 K 0 1+α I 0 = V = marginalq K 0 K 0 X = E 0 M t (1 δ) t θ tf 0 ( )+ α t=0 2 Ã! 2 It constant returns to scale: V K = V K 1+α I t = V t = Q t = BE t ME t Implication: 1+α I t can substitute for V t, Q t, BE t ME t in any application.

3 Investment returns R t+1 = V t+1 + D t+1 V t =1+α I t V t K µ t 1+α I t+1 R t+1 = (1 δ) K + D t+1 t+1 +1 ³ 1+α I t µ 1 I t δ + D t+1 +(1+α) I t+1 α I t Stock return = investment return Ex post too. A first differenced version of q theory. Good: Emphasizes difference, where q theory works well. Good intuition for return anomalies. Bad: Can t do this with time to build, irreversible investment. There, stick to investment = marginal q

4 Qtheoryisprettygood! 1+α I t = V t 1. I/racks V/K, (and P/X); investment returns track stock returns. (a) Timeseries(seegraphs) (b) Cross section (see graphs) 2. Investment plans get timing better (Lamont). This suggests time-to plan is important for good empirical fits. (Of course).

5 4 3.5 I/BE ME/BE Investment/BE and ME/BE Real nonresidential fixed investment (BEA), ME/BE (Pastor/Veronesi), ME(CRSP)

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8 log(i/k) 0 I/K vs. M/B -- colors are industries log(mb) 20 Industry I/K and B/M, (Santos-Veronesi)

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10 Mining Food Apparel Paper Chemical Petroleum Construction Prim. Metals Fab. Metals Machinery Electric Eq. Transport Eq. Manufacturing Railroads Other Transport. Utilities Dept. Stores Retail Financial Other

11 Why do people think q is bad? 1+α I t = V t 1. Theory is rejected. No error! Predicts R 2 = 1! Errors are specification errors, unsurprisingly correlated with right hand variable, cashflow, etc. 2. Works better at high frequencies. Low frequency movement in p I, α, etc.? 3. Tried to line I up with interest rates. We now know that most variation in cost of capital is in the risk premium. 4. From 2, 3, found too high adjustment costs. Now more reasonable numbers.

12 Anomalies 1. Time series predictability, value/growth in the cross-section 1+α I t = V t = 1 E t α I t ln V t E t X j=1 X j=1 M t+j D t+j ρ j 1 ³ d t+j r t+j (d =ln(1+ D K )) (a) If E t r t+j varies (TS or XS) V t / varies V t / predicts returns r t+j = a + b V t + ε t+j (b) If E t d t+j varies V t / varies V t / predicts profitability d t+j = a + b V t + ε t+j (c) Time series: mostly effect a. Cross section: both a and b.

13 α I t ln V t E t X j=1 ρ j 1 ³ d t+j r t+j 2. What do V/K predictions of returns r and profits d have to do with I/K? (a) Zip. (b) I t / should also predict returns (TS, XS) and profitability (XS). i. This works pretty well. TS: Cochrane 1991 I/K forecasts returns. Lamont 2000: Investment plans work even better (see table). Direct XS estimate? ii. investment anomaly SEO anomaly (c) If α =0thenQ = 0. There must be some investment friction! (Investment is endogenous.)

14 r t+1 = a + bx t + ε x coeff. b s.e. I t 1 / (2.74) P t 1 (I t )/ (2.49) I t / (2.43) I t 1 /I t (0.29) P t 1 (I t )/I t (0.29) I t /I t (0.25) Lamont, Investment Plans and Stock Returns. P t 1 (I t ) is investment in year t plannedinyeart 1 Sample : it should work even better now!

15 Profitability Anomalies α I t ln V t E t X j ρ j ³ d t+j r t+j 1. Why should high E(d) have anything to do with high E(r)? High E(d) should just mean high V/K and high I/K and that s it. 2. Zhang s partial is d Holding investment (or V/K) contsant we should t+j {It } see high E(d) correlate with high E(r). High E(r) mustoffset the high E(d) if I/K and V/K are constant. Is that the empirical work?

16 An Explanation? The Q framework vs. the beta framework The Q framework vs. the SDF framework (beta = SDF!) Does the Q framework give a rational explanation for anomalies? No, alas. 1. Shiller might say: irrational exuberance raises V/K Firms act optimally, I/K is too high. It would be better if firms did not react. Q is how irrational prices cause bad allocations. 2. (JC: well, if people are rational at work, why irrational at home? But let s not argue about the coherence of behavioral stories.) 3. We can t really write a coherent endowment economy with fixed investment, prices must adjust.

17 Qdoesprovideaconnection to macroeconomic events. Market is not completely off on its own disconnected from economics. It s nice that one side of the market works well! Success of a (consumption) beta model would not be a full explanation either. Where do the betas come from? We don t live in an endowment economy. Only general equilibrium is an explanation alas

18 The Future 1. Quantitative explanation of puzzles. It would be lovely to see the same, low adjustment cost parameter in each case! 2. Time to build so we don t need to see only investment expenditures. Lamont and plans again. 3. Marginal q 6= average q. We will study price vs. present values in asset pricing, and investment vs. marginal present value. 4. Classic Q (Summers) had detailed and sophisticated calculation of taxes, replacement cost. Now we just use accounting book value. Worth improving X in V/X?

19 5. Wherearetheshocks? α I t log Ã! Vt X E t ρ j ³ d t+j r t+j j=1 1+α I t = 1 X E t Mt,t+j D t+j D t = θ t f( ) " 1+ α 2 Ã It (a) If technology/marginal productivity shocks, then shouldn t we see I/K and V/K forecasting profitability d but not returns r? (b) DoesthismeanthatmuchvariationinV/K, I/K must be due to preference shocks M, E(r)? (c) Not necessarily. E(d t+j ) c t risk aversion (e.g. habits) E(r t+j ) Then in an economy driven entirely by technology shocks, we see endogenous change in risk aversion, V/K, I/K forecast returns. Longahead D canresultinhighcurrent Er. (d) Similar, more complex mechanisms in the cross section. (Menzly, Santos and Veronesi, Kogan Zhang, Gourio, Gala, etc.)!# I t

20 6. Why no production-based asset pricing? Why do we lose the symmetry we hadinmicro? y t (s) =θ(s)f(k t 1 ) ThemodelswewritedownareLeontiefacross states. Thisisjustanaccident of history; we added shocks to f = f(k) from nonstochastic models. Firms do have actions to transform goods across states. We need a model of this! State 2 (shine) U= π( s) ucs [ ( )] + π( s) ucs [ ( )] How do we do this? ys ( ) = θ ( s ) f( k) t+ 1 t+ 1 t State 1 (rain)

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22 The algebra: Rt+1 s = V t+1 + D t+1 V t Rt+1 s = 1+α I t+1 K + D t+1 t α I t K t +1 = (1 δ) + I t+1 =(1 δ)+ I t à 1 I! t+1 Kt+1 =(1 δ) = (1 δ) µ 1 I t+1 +1