Part 3: Value, Investment, and SEO Puzzles

Transcription

1 Part 3: Value, Investment, and SEO Puzzles Model of Zhang, L., 2005, The Value Premium, JF. Discrete time Operating leverage Asymmetric quadratic adjustment costs Counter-cyclical price of risk Algorithm of Krusell, P. and A. A. Smith Jr., 1998, Income and Wealth Heterogeneity in the Macroeconomy, JPE. Solves for equilibrium with heterogeneous agents and aggregate risks. Issue is how the cross-sectional distribution evolves in conjunction with the aggregate shock. Implications for the value premium. Implications for the capital investment and seasoned equity offering (SEO) puzzles. Zhang, JF, 2005: Analyze a single industry with price-taking firms. Constant elasticity demand: P t = Y η t. Output of firm i [0, 1] is Y it = e X t +Z it K α it with α < 1. X represents systematic risk and Z i represents idiosyncratic risk. Operating leverage: operating cash flow is P t Y it C for a constant C. Asymmetric costly adjustment: investment cost is { K θ + (I/K ) 2 if I/K > 0, h(i, K ) = I + K θ (I/K ) 2 if I/K < 0, for constants θ > θ +.

2 Stochastic Discount Factor and Risks M = SDF process, meaning M t+1 /M t is the date t SDF for pricing cash flows at t + 1. Firm i seeks to maximize E M t [π(x t, Z it, K it ) h(i t, K t )], t=0 Counter-cyclical price of risk: Assume log M t+1 = log β [γ 0 γ 1 (X t X)] X t+1, with X being an AR(1) process and γ 0, γ 1 > 0. Assume Z i are independent AR(1) processes with long run means of zero. Operating Leverage Value of firm is value without constant cost C minus value of consol bond paying C. Thus, firm value is levered, and leverage is higher when revenues P t Y it are low. Zhang emphasizes asymmetric costly adjustment and counter-cyclical price of risk. Kogan and Papanikolaou argue that asymmetric costly adjustment does little without operating leverage. Kogan, L. and D. Papanikolaou, 2012, Economic Activity of Firms and Asset Prices, Annual Review of Financial Economics Vol. 2.

3 State Variable The state of the economy at date t is defined by X t and {(Z it, K it ) i [0, 1]}. The names i of the firms are unimportant, so we can replace {(K it, Z it ) i [0, 1]} with the induced measure µ t on R R + : µ t (A) def = Leb {i [0, 1] (Z it, K it ) A}. Each firm s operating cash flow depends on (µ, x, z i, k i ): e x+z i k α i ( ) η e x+z k α dµ(z, k) C. R R + Fixed Point Each firm does dynamic programming, taking (µ, X) as an exogenous Markov process. Basic idea: Conjecture dynamics for µ: µ t+1 = g(µ t, X t ). Solve dynamic programming problems to compute optimal (µ t, X t, Z it, K it ) K i,t+1. The optimal policies yield a new map Find a fixed point ĝ = g. (µ t, X t ) µ t+1 def = ĝ(µ t, X t ).

4 Approximate State Variable In the fixed point map, replace µ with ν = (ν 1,..., ν n ), where def ν i = f i (z, k) dµ(k, z) R R + for functions f i. If we define the f i appropriately, then we can recover µ from a countable family ν 1, ν 2,.... For example, f 1 = 1 Ai for a countable basis A 1, A 2,... of the Borel σ field. We need operating cash flows to depend on (ν, x, z, k) in order to use ν in the dynamic programming. Define f 1 (z, k) = e z k α. Operating cash flow of firm i is e x+z i k α i e ηx ν η 1. Approximate Fixed Point Conjecture ν t+1 = g(ν t, X t ). Solve dynamic programming problems to compute optimal (ν t, X t, Z it, K it ) K i,t+1. The optimal policies yield a new map (ν t, X t ) ν t+1 def = ĝ(ν t, X t ). Find an approximate fixed point ĝ g. Increase the dimension of ν to better approximate µ and repeat.

5 Some More Details Look for the approximate fixed point within a parametric class, for example linear. Start with particular coefficients, defining g. To calculate ĝ: Numerically solve the dynamic programming problems using a finite grid for the state variables. Simulate the economy to produce a time series for (ν t, X t ) (discarding initial draws to obtain stationarity). Regress ν t+1 on (ν t, X t ) to obtain new approximate linear function ĝ. An approximate fixed point is obtained if ĝ g and if the R 2 in the regression is large. Law of Large Numbers In equilibrium of the original (not approximated) economy, dynamic programming yields (µ t, X t, Z it, K it ) K i,t+1 def = κ(µ t, X t, Z it, K it ). By the AR(1) assumption, Z i,t+1 = a + bz it + σε i,t+1 for independent standard normals ε i,t+1. For A = [z 0, z 1 ] [k 0, k 1 ], consider R R + 1 {(z,k) k0 κ(µ t,x t,z,k) k 1 } [ N ( ) ( )] z1 a bz z0 a bz N σ σ dµ t (z, k). This is µ t+1 (A) def = g(µ t, X t )(A) under some law of large numbers.

6 References: Krusell-Smith and JEDC special issue den Haan, W.J., Assessing the accuracy of the aggregate law of motion in models with heterogeneous agents. JEDC. den Haan, W.J., Comparison of solutions to the incomplete markets model with aggregate uncertainty. JEDC. den Haan, W.J., Rendahl, P., Solving the incomplete markets model with aggregate uncertainty using explicit aggregation. JEDC. Kim, S., Kollmann, R., Kim, J., Solving the incomplete markets model with aggregate uncertainty using a perturbation method. JEDC. Krusell, P., Smith Jr., A.A., Income and wealth heterogeneity in the macroeconomy. JPE. Maliar, L., Maliar, S., Valli, F., Solving the incomplete markets model with aggregate uncertainty using the Krusell-Smith algorithm. JEDC. Reiter, M., 2010a. Solving heterogeneous-agent models by projection and perturbation. JEDC. Reiter, M., 2010b. Solving the incomplete markets economy with aggregate uncertainty by backward induction. JEDC. Young, E.R., Solving the incomplete markets model with aggregate uncertainty using the Krussell-Smith algorithm and non-stochastic simulations. JEDC. The Value Premium Panel A: Expected Value Premium Panel B: The V Expected Excess Return Benchmark Model 2 Book to Market Benchmark Model 2 Mode 0.01 Model x Figure 4. Time-varying spreads in expected excess return and in tween low-productivity (value) and high-productivity (growth) fir Source: Zhang, L., 2005, The Value Premium, JF 60, Horizontal axis: the spread in expected excess returns (Panel A) and the spread in book-to Aggregate productivity tweenx. firms Vertical with axis: lowspread idiosyncratic in expected productivity returns between andhigh firms with high idiosyn B-to-M and low B-to-M functions stocks. of aggregate Model 1: Symmetric productivity, adjustment x. As is costs evident and constant from Figure 2, sorting on f price of risk. Model z t, 2: inasymmetric the model is adjustment equivalent costs, to sorting constanton price book-to-market. of risk. In effect, Panel A expected value premium, and Panel B plots the time-varying spread in b Benchmark: Asymmetric adjustment costs, counter-cyclical price of risk.

7 Investment, New Issues and Risk Equity issues and capital investment are correlated. When a firm invests, it converts a growth option into assets in place, lowering risk. See Carlson, M., Fisher, A., and R. Giammarino, 2004, Corporate Investment and Asset Price Dynamics: Implications for the Cross-section of Returns, JF. Carlson, M., Fisher, A., and R. Giammarino, 2006, Corporate Investment and Asset Price Dynamics: Implications for SEO Event Studies and Long-Run Performance, JF. Other things equal, low-risk firms will invest more. So, high investment firms and firms that have made new issues should have lower average returns. Alternative story: market timing. Firms with over-priced stock issue it and subsequently have low returns. Li-Livdan-Zhang (RFS, 2009) Zhang (2005) model, but with costly external finance: When cash flow is positive, it is paid to shareholders. When cash flow is negative, it is raised with a fixed plus proportional cost. No cash holdings.

8 Source: Li, E. X. N., Livdan, D. and L. Zhang, 2009, Anomalies, RFS. Histogram (across simulations) of the mean return spread between the lowest quintile of capital investment (CI) firms and the highest quintile, where CI t = 3CE t 1 CE t 2 + CE t 3 + CE t 4 1, and CE is capital expenditure scaled by sales, following Titman, Wei, and Xie (2004). The Review of Financial Studies / v 22 n (A) (B) Figure 3 Empirical distributions for 300 the mean CI spread and the slopes and t-statistics of the Fama-MacBeth (1973) cross-sectional regressions of benchmark-adjusted returns on CI and CI DCF. 200 Panel A reports the mean CI spread 250 across 1000 simulations as well as its value in the real data, 16.9% per annum. CI denotes capital investment. The CI spread is the zero-investment portfolio that has a long position in the lowest 150 CI quintile and a short position 200in the highest CI quintile. In each simulation, we also run Fama-MacBeth (1973) cross-sectional regression: r a jt+1 =l 0t + l 1t CI jt + l 2t CI jt DCF jt + u jt+1, in which r a jt+1 is the benchmarkadjusted value-weighted return 150on individual stock j at month t. DCF is the dummy100 variable based on cash flow, measured as operating income scaled by total assets, measured in the model as π jt /k jt. If the cash flow of one firm is above the median 100cash flow of the year, its DCF equals one, and zero otherwise. Panel the slope B reports of ISSUE = 0.38 the joint empirical distribution of average the lslope 1t andof average ISSUE = l 2t Panel C reports the joint empirical 50 distribution of 50 their Fama-MacBeth t-statistics. (A) Mean CI spread; (B) Slopes of CI and CI DCF;(C)t-statistics of CI and CI DCF CI DCF slopes as well as that of their t-statistics in simulations. The panels (C) (D) show that the empirical estimates can be adequately explained by the model Specifically, Source: Li, E. the X. N., p-value Livdan, of D. the andempirical L. Zhang, CI 2009, and Anomalies, CI DCF RFS. slopes Histogram calculated with (across thesimulations) simulated of distribution 100 the averagein slope panel coefficient B is in the(the cross-sectional 350 p-value is calculated (Fama-MacBeth) regression the slope of ISSUE = by counting the number of simulations that have CI slopes higher than and have CI DCF slopes higher than 0.76 and dividing 250 r i,t+1 = a t + b t ISSUE it + ε i,t+1, this number by 1000.) Further, the p-value 60 of the t-statistics of the slopes in200 the data from the simulated distribution in panel C is where ISSUE it is 1 if the 40 firm conducted an SEO within the past 60 months and 0 Recent studies emphasize the importance for structural models 100 to explain the otherwise. failure of the CAPM 20 (e.g., Lettau and Wachter 2007 and Lewellen and Nagel 50 the slope of ISSUE = 0.60