Pricing kernel monotonicity and term structure: Evidence from China

We present a pricing kernel that summarizes well the main features of the dynamics of interest rates and risk in postwar U.S. data and use it to uncover how the pricing kernel has moved with the short rate in this data. Our findings imply that standard monetary models miss an essential link between the central bank instrument and the economic activity that monetary policy is intended to affect, and thus we call for a new approach to monetary policy analysis. We sketch a new approach using an economic model based on our pricing kernel. The model incorporates the key relationships between policy and risk movements in an unconventional way: the central bank's policy changes are viewed as primarily intended to compensate for exogenous business cycle fluctuations in risk that threaten to push inflation off target. This model, while an improvement on standard models, is considered just a starting point for their revision. It leads to critical questions that researchers need to answer as they continue to revise their approach to monetary policy analysis. Copyright © 2009 by the National Bureau of Economic Research.

A set of elementary axioms for stochastic finance is presented wherein a prominent role is played by the state–price density, which in turn determines the stochastic dynamics of the interest–rate term structure. The fact that the state–price density is a potential implies the existence of an asymptotic random variable X ∞ with the property that its conditional variance is the state–price density. The Wiener chaos expansion technique can then be applied to X ∞ , thus enabling us to ‘parametrize’ the dynamics of the discount–bond system in terms of the deterministic coefficients of the chaos expansion. Using this method, we find that there is a natural map from the space of all admissible term–structure trajectories to the symmetric Fock space F naturally associated with the space of square-integrable random variables on the underlying probability space. An element of F is either coherent or incoherent, and a stochastic bond–price system is necessarily represented by an incoherent element of F . Making use of the linearity of F we derive simple analytic formulae for the bond–price system, the volatility structure, the short rate, and the risk premium associated with an arbitrary admissible term–structure model. Extensions to foreign–exchange markets and general asset systems are also developed.

Long-Term Risk with Stochastic Interest Rates

This article shows how to evaluate the performance of managed portfolios using stochastic discount factors (SDFs) from continuous-time term structure models. These models imply empirical factors that include time averages of the underlying state variables. The approach addresses a performance measurement bias, described by Goetzmann, Ingersoll, and Ivkovic (2000) and Ferson and Khang (2002), arising because fund managers may trade within the return measurement interval or hold positions in replicable options. The empirical factors contribute explanatory power in factor model regressions and reduce model pricing errors. We illustrate the approach on US government bond funds during 1986–2000.

By assuming that the stochastic discount factor (SDF) [Formula: see text] is a proper but unspecified function of state variables [Formula: see text], we show that this function [Formula: see text] must solve a simple second-order linear differential equation specified by state variables’ risk-neutral dynamics. Therefore, this assumption determines the most general possible SDFs and associated preferences, that are consistent with the given risk-neutral state dynamics and interest rate. A consistent SDF then implies the corresponding state dynamics in the data-generating measure. Our approach offers novel flexibilities to extend several popular asset pricing frameworks: affine and quadratic interest rate models, as well as models built on linearity-generating processes. We illustrate the approach with an international asset pricing model in which (i) interest rate has an affine dynamic term structure and (ii) the forward premium puzzle is consistent with consumption-risk rationales; the two asset pricing features previously deemed conceptually incompatible.

We estimate a discrete-time multivariate pricing kernel for the term structure of interest rates, using both yields and inflation rates. This gives a separate estimate of the real kernel and the nominal kernel, taking into account a relatively sophisticated dynamical structure and mutual interaction between the real and nominal side of the economy. Along with obtaining an estimate of the real term structure, we use the estimates to obtain a new perspective on how real and nominal influences interact to produce the observed term structure.

The asset pricing behavior is studied in an overlapping generations model that is dynamically inefficient because of inefficient aggregate risk sharing. The inefficiency is linked with the stochastic distribution of income over agents within a period. The introduction of a clearing house may eliminate the dynamic inefficiency, but depending on how the clearing house operates, households may only have partial insurance against income risk. Hence, the clearing house affects the marginal rate of substitution (MRS) across generations within a period and also the intertemporal marginal rate of substitution faced by a household, which is the stochastic discount factor (SDF) applied to income streams over time. The opportunity to insure against income risk is related to different concepts of Pareto optimality. The implications for asset prices and the market price of risk are derived. In particular, the matrix of contingent claim prices supporting a Pareto optimal allocation must have a dominant root no greater than unity. This is equivalent to restricting the matrix of within-period MRS to have a dominant root no greater than unity. The sequence of powers of the pricing matrix determines the time series properties of the SDF, asset prices and asset returns.

Asset prices depend on two elements: the dynamics of the state variables and the pricing kernel. Traditional term structure models differ in factor dynamics. However, most of them imply a log-linear pricing kernel. We investigate empirically the role of factor dynamics and pricing kernel in pricing interest rate derivatives using a nonparametric approach. We find that interest rate cap prices are very sensitive to the specification of factor dynamics, especially when they are close to expiration. In addition, nonlinear log-pricing kernels improve the pricing of long-maturity caps, although significant pricing errors remain. Recent published studies document models that fit the London Inter-Bank offer Rate (LIBOR) and swap rates but do not price derivatives well, leading to the so-called ‘‘unspanned stochastic volatility puzzle.” Additional volatility factors seem to be needed to explain cap prices. However, the relative mispricing between interest rate caps and underlying LIBOR and swap rates could also potentially be due to mis-specifications of the parametric models used. Our paper provides evidence for unspanned stochastic volatility from a nonparametric perspective.

We consider the consumption and portfolio choice problem of a long-run investor when the term structure is affine and when the investor has access to nominal bonds and a stock portfolio. In the presence of unhedgeable inflation risk, there exist multiple pricing kernels that produce the same bond prices, but a unique pricing kernel equal to the marginal utility of the investor. We apply our method to a three-factor Gaussian model with a time-varying price of risk that captures the failure of the expectations hypothesis seen in the data. We extend this model to account for time-varying expected inflation, and estimate the model with both inflation and term structure data. The estimates imply that the bond portfolio for the long-run investor looks very different from the portfolio of a mean-variance optimizer. In particular, the desire to hedge changes in term premia generates large hedging demands for long-term bonds.

The Financial Risk Meter (FRM) employs Quantile-LASSO regression to identify systemic financial risk and dependencies among tail events across financial assets. This paper establishes, both theoretically and empirically, a meaningful economic relationship between the FRM index, derived from the penalization parameter in quantile LASSO regression, and the volatility of assets' pricing kernels, the attainable maximal Sharpe ratio, and market volatility. Despite the rapid growth of the crypto market and its increasing integration with traditional financial markets, there remains a dearth of risk measures in this space. FRM @ Crypto exhibits robust predictive capabilities in anticipating future market risk, potentially filling a critical void in this market.

We evaluate the asset pricing implications of a class of models in which risk sharing is imperfect because of limited enforcement of intertemporal contracts. These unconstrained households have lower consumption growth rates than all other households in the economy. We use household data on consumption growth from the U.S. Consumer Expenditure Survey to identify unconstrained households, to estimate the pricing kernel implied by these models and evaluate their performance in pricing aggregate risk. We find that for high values of the relative risk aversion coefficient, the limited enforcement pricing kernel generates a market price of risk that is substantially closer to the data than the one obtained using the standard complete markets asset pricing kernel.

Financial Contagion Risk and the Stochastic Discount Factor

Downside Risk Neutral Probabilities

Management Insights

A heat kernel approach is proposed for the development of a novel method for asset pricing over a finite time horizon. We work in an incomplete market setting and assume the existence of a pricing kernel that determines the prices of financial instruments. The pricing kernel is modeled by a weighted heat kernel driven by a multivariate Markov process. The heat kernel is chosen so as to provide enough freedom to ensure that the resulting model can be calibrated to appropriate data, e.g. to the initial term structure of bond prices. A class of models is presented for which the prices of bonds, caplets, and swaptions can be computed in closed form. The dynamical equations for the price processes are derived, and explicit formulae are obtained for the short rate of interest, the risk premium, and for the stochastic volatility of prices. Several of the closed-form models presented are driven by combinations of Markovian jump processes with different probability laws. Such models provide a basis for consistent applications in various market sectors, including equity markets, fixed-income markets, commodity markets, and insurance. The flexible multidimensional and multivariate structure on which the resulting price models are based lends itself well to the modeling of dependence across asset classes. As an illustration, the impact of spiraling debt, a typical feature of a financial crisis, is modeled explicitly, and the contagion effects can be readily observed in the dynamics of the associated asset returns.