Asset-pricing Puzzles and Incomplete Markets

1. Asset-pricing Puzzles and Incomplete Markets Chris Telmer Presented by Jack Favilukis

2. Representative Agent Models • Every agent’s consumption growth is perfectly correlated, either by construction, or because markets are complete and agents trade away any idiosyncratic risk • 1=E[MRi] M=exp[δ-γΔc] for CRRA • Equity Premium Puzzle: E[ri]=γσi,Δc but σi,Δc is too small so need large γ • Risk Free Rate Puzzle: rf=δ+γE[Δc]-.5γ2σ2Δc but this will be to variable if γ is large

3. HJ Bounds & ramping up the SDF • E[MRe]=0  E[M]E[Re]=-Cov(M,Re)  |E[M]|*|E[Re]|=|corr(M,Re)|σeσm  σm/E[M]>E[Re]/σe=.15 • CRRA: SDF is not very volatile • Epstein-Zin: SDF also depends on expectations of future consumption because of recursive nature • Habit: SDF is now exp[δ-γΔc-γΔs], risk aversion is highest in bad times when consumption is close to habit

4. Heterogeneous Agent Models • 1=E[MjRi] for each agent j that trades asset i • Rather than adding terms to M, make consumption more volatile • Duffie & Constantinides: Lets pick Mi and a corresponding consumption stream for each agent so as to match any Ri • The assets available to agents must not span the space of idiosyncratic shocks to the wealth of agents, otherwise full insurance • There is evidence against full consumption insurance (i.e. due to moral hazard in insuring labor income)

5. The Model • Model is extension of Mankiw ’86, similar to Mehra-Prescott ‘85 • Aggregate Output: yt+1=λt+1yt • λ is a two state Markov process • Two ex-ante identical, CRRA agents: y1,t+y2,t=c1,t+c2,t • If λ is high yk,t=.5yt but if λ is low, with probability ½ yk,t=(1-γ)yt and yk’,t=γyt

6. Income Heterogeneity • When γ=½ all agents are identical, the further γ is away from ½ the more income heterogeneity there is • Let Qt=½ if λ is high, and (1-γ) or γ in the corresponding λ is low cases, then yk,t=Qtyt, yk’,t=(1-Qt)yt

7. The two extreme (and uninteresting) cases • Suppose markets were complete, each agent’s consumption would be perfectly correlated with aggregate, its easy to construct the SDF and price any asset; this problem is identical to Mehra-Prescott ‘85 (Table 1) • Suppose there was autarky, its easy to construct each agent’s SDF and price any asset according to that agent (Table 2) • Note: “easy” because bond holdings are not a state variable yet and there is only a single Euler equation to satisfy

8. Table 1: Complete Markets

9. Table 2: Autarky

10. Let there be trade! • Introduce a one period bond (with potential borrowing constraints) (i) 1≥E[Mk,t+1Rf] (Euler equation) (ii) yk,t+bk,t-1≥ ck,t+bk,t/Rf (Resource) (iii) b1,t+b2,t=0 (Equilibrium) (iv) bk,t ≥ b (Borrowing) • Whenever (iv) is not binding, (i) is binding • The relevant state variable is St=[Q1,t b1,t-1] and the solution is the functions bk(St) and Rf(St) such that (i)-(iv) hold

11. Solution Algorithm • Start with bT(ST)=0 and RfT(ST)=0 and plug into Euler Equations to solve for bT-1(ST-1), RfT-1(ST-1), keep iterating until convergence • At each step check if borrowing constraint is satisfied, if not, set quantity to borrowing constraint, set prices to the long agent’s shadow price

12. Table 3: Unconstrained Bond Trade

13. Table 4: Constrained Bond Trade

14. Takeaway • Very surprising (at least to me) that even with a single risk free asset the agents are able to insure themselves so well from very large idiosyncratic income shocks • Incomplete markets models are promising and relevant (i.e. only a small percentage of the population participates in the market, it is only their SDF’s that matter)