The pricing kernel puzzle in forward looking data

See e.g. (Cochrane 2000, p. 50).

Dybvig (1988) and Beare (2011) provide another interesting view on the puzzle under the assumption of a complete market. Buying the market portfolio is the cheapest way to obtain the payoff of the market portfolio if and only if the pricing kernel is monotonically decreasing. Or stated differently, if the pricing kernel is locally increasing, there exists a portfolio of Arrow-Debreu securities that stochastically dominates the market return. See also Beare and Schmidt (2014a) for an empirical study on this issue.

For additional models of the pricing kernel puzzle, see Cuesdeanu and Jackwerth (2017).

B-splines are scaled truncated power functions, which can be used instead of polynomial splines (de Boor 1978, p. 96). Their theoretical characterization makes them favorable for the estimation procedure in Linn et al. (2017).

The methodology applied in Bliss and Panigirtzoglou (2004), Linn et al. (2017), as well as in our paper is forward looking in the sense that no historical return time series is used to explicitly compute the subjective density. Nevertheless, monthly realized returns are needed for calibrating the pricing kernel, see Sect. 2.3. Recovering the pricing kernel without using past returns but relying on more strict assumptions on the data generating process was pioneered by Ross (2015); see also the follow-up studies by Jackwerth and Menner (2017) and Jensen et al. (2017).

Using a different number of equally spaced values on the return axis does not change the results, see Sect. 5.2.

Consider also Figs. 1, 2, and 3 where such estimated pricing kernels are depicted. After scaling the pricing kernel, it can take on any value and is therefore not restricted by the initial value of 5.

Our method is robust to extending the pricing kernel below 0.8 and above 1.2 at the slopes of the last interior segments instead of horizontally. Moreover, the results do not change when fixing different basis points on the return axis, see Sect. 5.2.

Our formulation is equivalent to expressing the subjective density in terms of state prices \(\pi _t\) as the risk-free discount factor \(e^{-r_{f,t} \Delta t}\) cancels out when normalizing the subjective densities:

\( p_t=\frac{\pi _t}{m} \Big / \int _{0}^{\infty } \frac{\pi _t}{m} dR_t = \frac{e^{-r_{f,t} \Delta t} q_t}{m} \Big / \int _{0}^{\infty } \frac{ e^{-r_{f,t} \Delta t} q_t}{m} dR_t = \frac{q_t}{m} \Big / \int _{0}^{\infty } \frac{q_t}{m} dR_t \).

See Gneiting and Raftery (2007) for a comprehensive discussion of score functions.

A similar approach is used in Shackleton et al. (2010) for mixing forecasts from past returns and forward looking option prices.

We thank an anonymous referee for stressing this point.

See Beare and Moon (2015) for a detailed discussion. Intuitively, when simulating with a flat pricing kernel, the differences between the simulated restricted and unrestricted log scores are larger than the differences when simulating with a decreasing pricing kernel. Hence, the difference we obtain from the data has to be much larger in order to reject the null hypothesis, making the risk-neutral version of the test the most conservative way to test pricing kernel monotonicity.

See Cramer (1928).

Theoretically, one could also consider overlapping returns, e.g. monthly returns shifted by on day at a time. While the estimation of the unrestricted and restricted pricing kernel would be no issue in such setting, the simulation of the p values of the \(\Delta \) statistic is no longer straightforward. One would have to draw overlapping returns stemming from daily non-parametric densities with a monthly horizon. It is not clear at all how to do this without making restrictive assumptions on the data generating process.

The pricing kernel derived from an exponential utility function is given by \(m^{exp.}=\exp (-a \cdot R_t)\). Bliss and Panigirtzoglou (2004) identified \(a=6.33\) as the optimal value when looking at a monthly horizon and so we repeated our base procedure for \(a=2,4,6,8,\text { and }10\). Again, all p values decrease in a. All our tests, except the Berkowitz (2001) test, reject pricing kernel monotonicity at least at the \(10\%\) level.

In particular, Cuesdeanu (2016) argues that the tilde shaped pricing kernels in the canonical papers of Ait-Sahalia and Lo (2000), Jackwerth (2000), and Rosenberg and Engle (2002) can be explained by the sample period (characterized by a relatively low variance-risk-premium) in combination with the non-observability of out-of-the-money calls.

A model that could potentially explain w-shaped pricing kernels in returns is Bakshi et al. (2015). The authors describe an economy where investors are long and short volatility, and aggregation leads to a u-shaped pricing kernel in the volatility dimension. Since high (low) absolute returns are typically associated with high (low) volatility, a u-shaped pricing kernel in the volatility dimension could mechanically imply a w-shape in the return dimension.

The Berkowitz (2001) test might gain in power as the higher moments of the returns realized in this subsample are less extreme. Compared to the full sample, sample kurtosis drops from 7.18 to 3.74 and sample skewness is less negative: −1.24 in the full sample versus −0.66 in the low volatility subsample.

In case of linear interpolation, a pricing kernel is non-decreasing if and only if its values at the spline knots are non-increasing.

This GMM approach is equivalent to setting \(\widetilde{\Omega }_L\) to the identity matrix in Knüppel (2015).

Note that Linn et al. (2017) discuss the results for all numbers of moment conditions up to \(L=50\) in the working paper version only. In the published version of their paper they only show the results for \(L \in \{ 4, 5, 6, 7, 8, 9 \}\) and claim that the best pricing kernel obtains when the number of pricing kernel parameters equals to the number of moment conditions. They find that for the S&P 500, choosing \(L=7\) delivers the highest p values for the Berkowitz, Cramér van Mises, and Kolmogorov-Smirnov test statistics. Note that the p values in their Table 2 are upwards biased since fitting the moments of a distribution helps in fitting the distribution itself. Only the p values corresponding to the rows labelled ’No pricing kernel’ are unbiased since no pricing kernel optimization happened there.

The maximal realized return in our sample is at about 1.11.