Lars Peter Hansen Prize Lecture: Uncertainty Outside and Inside Economic Models
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410 The Nobel Prizes for what alternative? Thus a complementary approach is to characterize proper- ties of the family of s’s that do satisfy (4). These solutions might well violate the parametric restriction. The interesting challenge is how to characterize the family of SDFs that solve (4) in useful ways. Here I follow a general approach that is essentially the same as that in Almeida and Garcia (2013). I choose this approach both because of its flexibilty and because it includes many interesting special cases used in empiri- cal analysis. Consider a family of convex functions ϕ defined on the positive real numbers: 18
φ(r) = 1 θ(1+θ) (r) 1 + θ −1 ⎡⎣ ⎤⎦ (5) for alternative choices of the parameter θ. The specification θ = 1 is commonly used in empirical practice, in which case ϕ is quadratic. We shall look for lower bounds on the
φ
Es ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ by solving the convex optimization problem: 19
λ = inf s >0
φ
⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ subject to E s ′
− ′
q [ ] = 0. (6)
φ
Es ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ ≥ λ.
Notice that E φ s Es ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ hence λ are nonnegative by Jensen’s Inequality be- cause ϕ is convex and ϕ (1) = 0. When θ = 1, 18 This functional form is familiar from economists’ use of power utility (in which case we use –ϕ to obtain a concave function), from statisticians’ use of F-divergence measures between two probability densities, the Box-Cox transformation, and the applications in the work of Cressie and Read (1984). 19
Notice that the expectation is also an affine transformation of the moment generating function for log s. 6490_Book.indb 410 11/4/14 2:30 PM Uncertainty Outside and Inside Economic Models 411
2E φ s Es ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ is the ratio of the standard deviation of s to its mean and 2λ is the greatest lower bound on this ratio. From the work of Ross (1978) and Harrison and Kreps (1979), arbitrage considerations imply the economically interesting restriction s > 0 with prob- ability one. To guarantee a solution to optimization problem (6), however, it is sometimes convenient to include s’s that are zero with positive probability. Since the aim is to produce bounds, this augmentation can be justified for mathemati- cal and computational convenience. Although this problem optimizes over an infinite-dimensional family of random variables s, the dual problem that opti- mizes over the Lagrange multipliers associated with the pricing constraint (4) is often quite tractable. See Hansen et al. (1995) for further discussion. Inputs into this calculation are contained in the pair (y, q) and a hypothetical mean Es. If we have time series data on the price of a unit payoff at date t + l, Es can be inferred by averaging the date t prices over time. If not, by changing Es we can trace out a frontier of solutions. An initial example of this is found in Hansen and Jagannathan (1991) where we constructed mean-standard devia- tion tradeoffs for SDFs by setting θ = 1. 20,21
While a quadratic specification of ϕ (θ = 1) has been the most common one used in empirical practice, other approaches have been suggested. For instance, Snow (1991) considers larger moments by setting θ to integer values greater than one. Alternatively, setting θ = 0 yields
φ
Es ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ = E s log s − log Es ( )
⎤⎦ Es ,
20 This literature was initiated by a discussion in Shiller (1982) and my comment on that discussion in Hansen (1982a). Shiller argued why a volatility bound on the SDF is of interest, and he constructed an initial bound. In my comment, I showed how to sharpen the volatility bound, but without exploiting that s > 0. Neither Shiller nor I explored mean-standard deviation tradeoffs that are central in Hansen and Jagannathan (1991). In effect, I constructed one point on the frontier characterized in Hansen and Jagannathan (1991).
21 When θ is one, the function φ continues to be well defined and convex for negative real numbers. As noted in Hansen and Jagannathan (1991), if the negative choices of s are allowed in the optimization problem (which weakens the bound), there are quasi- analytical formulas for the minimization problems with simple links to Sharpe ratios commonly used in empirical finance. 6490_Book.indb 411 11/4/14 2:30 PM Yüklə 1,76 Mb. Dostları ilə paylaş:
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