Testing monotonicity of pricing kernels

Abstract

International audienceThe behaviour of market agents has been extensively covered in the liter- ature. Risk averse behaviour, described by Von Neumann and Morgenstern (Theory of games and economic behavior. Princeton University Press, Princeton, 1944) via a concave utility function, is considered to be a cornerstone of classical economics. Agents prefer a fixed profit over an uncertain choice with the same expected value, however, lately there has been a lot of discussion about the empirical evidence of such risk averse behaviour. Some authors have shown that there are regions where market utility functions are locally convex. In this paper we construct a test to verify uncertainty about the concavity of agents’ utility function by testing the monotonicity of empirical pricing kernels (EPKs). A monotonically decreasing EPK corresponds to a concave utility function while a not monotonically decreasing EPK means non- averse pattern on one or more intervals of the utility function. We investigate the EPKs for German DAX data for the years 2000, 2002 and 2004 and find evidence of non- concave utility functions: the null hypothesis of a monotonically decreasing pricing kernel is rejected for the data under consideration. The test is based on approximations of spacings through exponential random variables. In a simulation we investigate its performance and calculate the critical values (surface)

Reference-Dependent Preferences and the Empirical Pricing Kernel Puzzle*

“…Typically, we find either a hump-shaped PK or a U-shaped PK. Statistical significance of the local nonmonotonicity of the EPKs was initially investigated by Golubev, Ha ¨rdle, and Timofeev (2014), who test the local concavity of the utility function implied by the DAX 30 index options and the historical index returns. In a subsequent paper, Ha ¨rdle, Okhrin, and Wang (2015) build uniform confidence bands for the PK estimates and test if a decreasing PK implied by the Black-Scholes model can be fitted within the confidence corridor.…”

Volatility-dependent probability weighting and the dynamics of the pricing kernel puzzle

“…Among others, the puzzle has been confirmed by Ait-Sahalia and Lo (2000) and Rosenberg and Engle (2002). Moreover, Beare and Schmidt (2016) and Golubev et al (2014) perform statistical tests and reject pricing kernel monotonicity for the S&P 500 and the German DAX, respectively. 4 In the recent past, several studies proposed possible solutions to the pricing kernel puzzle.…”

Distributional Replication

“…See, in particular, Figure 2 in [15], in which the state price density is an increasing function of the market return for monthly return levels between approximately −3% and 3%, and decreasing elsewhere. Other empirical studies of the relationship between the state price density and market returns have largely confirmed that it is often nonmonotone [16][17][18][19][20][21][22]. See also [23] for a discussion of the relevance of such nonmonotonicity for constructing density forecasts of market returns.…”