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Appendix to Limits to Arbitrage and Hedging : Evidence from Commodity Markets Viral
| Content Provider | Semantic Scholar |
|---|---|
| Author | Acharya, V. Lochstoer, Lars Ramadorai, Tarun |
| Copyright Year | 2013 |
| Abstract | In this Online Appendix, we solve a general equilibrium version of the model in the main text and show that the predictions of the model are robust to this extension. We also solve a general equilibrium model where the managerial costs of default are the motivation for firm hedging. We show that the implications of this calibrated model are qualitatively the same as in the model with risk averse managers. We also solve a model where default leads to supply disruptions, which alters the spot price dynamics and thus has implications for the futures risk premium. In this model higher default risk will tend decrease the futures risk premium as a supply disruption will benefit the long side of the futures contract. This is counter to our empirical results, which thus are consistent with the hedging story. Finally, we give some additional empirical results that were not included in the main paper and describe in more detail the micro data set used in the paper. ∗Acharya is at NYU-Stern, a Research Affi liate of CEPR and a Research Associate of the NBER. Lochstoer is at Columbia University. Ramadorai is at Said Business School, Oxford-Man Institute of Quantitative Finance, and CEPR. A part of this paper was completed while Ramadorai was visiting London Business School. Correspondence: Lars Lochstoer. E-mail: LL2609@columbia.edu. Mailing address: Uris Hall 405B, 3022 Broadway, New York, NY 10027. 1 General equilibrium version of main model In general equilibrium, the consumers’ consumption of other goods will typically be affected by the frictions in the commodity market. Thus, both the commodity ’demand’ shocks, Ct, and the marginal intertemporal rate of substitution will be affected when varying the frictions in the commodity market. Further, a general equilibrium model allow us to calibrate the model to gauge the likely magnitudes of the effect of the model’s frictions. We follow the same setup as in the partial equilibrium model given in the main paper, but now also solve the consumers’problem. Let consumers’preferences be given by: V = u (C0, Q0) + βE0 [u (C1, Q1)] , (1) where the felicity function is of the constant elasticity of substitution (CES) form: u(x, y) = 1 1− γ {( x(ε−1)/ε + ωy(ε−1)/ε )ε/(ε−1)}1−γ , (2) where ε is the intratemporal elasticity of substitution and γ is the level of relative risk-aversion. The standard intratemporal first order condition implies that the equilibrium commodity spot price St is given by: St = ω ( Ct Qt )1/ε , (3) as assumed earlier in the partial equilibrium version of the model. However, in the general equilibrium case we assume that the consumers own a Lucas tree producing the numeraire good At, as well as the commodity producing firms which produce the aggregate supply of the commodity Qt. However, the consumers must hire managers to manage the firms (their inventory and hedging decisions). The manager’s objective function is as in the partial equilibrium case (see Equation (3) in the main paper).1 Consumers can also invest in the commodity futures markets, but only through specialized funds who provide an aggregate number of contracts hs as the solution to the problem given in Equation (8) in the main paper. Denote the cost charged per contract by these funds as c. We assume the costs are incurred at time 0. The consumers’equilibrium consumption of other goods in the first period will equal C0 = A0 − c × h∗, where h∗ is the equilibrium open interest in the futures market, while in the last period C1 = A1. In equilibrium, consumers’net present value of a marginal investment in a commodity futures must be zero and so we have that c = E [Λ (S1 − F )]. Therefore, the aggregate loss due to intermediation is h∗E [Λ (S1 − F )]. Given the optimal position in futures contracts from Equation (9) in the main paper, we have that the equilibrium aggregate cost is E[Λ(S1−F )] γsσ 2 s 2 . The reason the consumers are willing to incur this cost is the utility gain from moving to more optimal Q0 and Q1 as the futures price affects the commodity producers’inventory decisions. In equilibrium, we have that E [Λ (S1 − F )] = γpγs γs+γp σsQ1 and so, substituting out σ 2 s, we have: Aggregate cost = c× h∗ = 1 γs ( γpγs γs + γp )2 ωQ 2(1−1/ε) 1 k, (4) We do not model the managers’consumption, but instead argue that this is a reasonable abstraction as there are very few managers relative to the total population and so their consumption is a minuscule component of aggregate consumption. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://pages.stern.nyu.edu/~sternfin/vacharya/public_html/pdfs/Online%20Appendix_Commodity_LimitsToHedging%20(1).pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
