**Around and around: The expectations hypothesis**

We show how to construct models of the term structure of interest rates in which the expectations hypothesis holds. McCulloch (1993) presents such a model, thereby contradicting an assertion by Cox, Ingersoll, and Ross (1981), but his example is Gaussian and falls outside the class of finite-dimensional Markovian models. We generalize McCulloch's model in three ways: (*i*) We provide an arbitrage-free characterization of the unbiased expectations hypothesis in terms of forward rates; (*ii*) we extend this characterization to a whole class of expectations hypotheses; and (*iii*) we show how to construct finite-dimensional Markovian and non-Gaussian examples.

**Apparent arbitrage**

The doubling strategy does not produce an arbitrage in the space of signed measures equipped with the weak-* topology. (A signed measure represents a payout.) The absence of arbitrage opportunities is guaranteed by the existence of a valuation operator (a strictly positive continuous linear functional). Nevertheless, the sequence of signed measures generated by the doubling strategy produces what appears to be an arbitrage from the perspective of convergence almost everywhere of the corresponding sequence of Radon-Nikodym derivatives taken with respect to a fixed positive numeraire measure.

**Forces that shape the yield curve: Parts I and II**

The yield curve is shaped by (i) expectations of the future path of short-term interest rates and (ii) uncertainty about the path. Uncertainty affects the yield curve through two channels: (i) Investors attitudes toward risk as reflected in risk premia, and (ii) the nonlinear relation between yields and bond prices (known as convexity). The way in which these forces simultaneously work to shape the yield curve can be understood in terms of the conditions that guarantee the absence of arbitrage opportunities.

**Modeling the state-price deflator and the term structure of interest rates**

Generalizing the results of Kazemi (1992), we decompose the state-price deflator into the product of a positive martingale (the permanent component) and the inverse of the price of the very-long discount (VLD) bond (a trend-stationary process). We further decompose the permanent component into a *term-structure martingale* correlated with bond returns and a *neutrino factor* uncorrelated with bond returns. We analyze the implications of the constancy of one or both of these permanent components. When the term-structure martingale is constant, the risk premia for all bonds are determined by the covariance with the return on the VLD bond. When the neutrino factor is constant, risk premia for all assets are determined by the price of term structure risk. When both permanent components are constant, the state-price deflator is trend-stationary and all assets are priced by the VLD bond. Since exchange rates can be modeled as the ratio of two state-price deflators, expected exchange-rate depreciation depends on the ratio of the two neutrino factors. When neutrino factors are constant, an important source of variation is missing. We apply our analysis to Constantinides (1992) and Rogers (1997), and we provide a number of exponential-affine examples.

**Modeling negative autocorrelation in continuous time**

A continuous-time first-order autoregressive process cannot display negative autoregression at any sampling horizon, but a first-order system of two processes where only one is observed can produce a discretely sampled AR(1) with negative first-order autocorrelation.

**Consumption and asset prices with homothetic recursive preferences**

When preferences are homothetic, utility can be expressed in terms of current consumption and a variable that captures all information about future opportunities. We use this observation to express the differential equation that characterizes utility as a restriction on the information variable in terms of the dynamics of consumption. We derive the supporting price system and returns process and thereby characterize optimal consumption and portfolio decisions. We provide a fast and accurate numerical solution method and illustrate its use with a number of Markovian models. In addition, we provide insight by changing the numeraire from units of consumption to units of the consumption process. In terms of the new units, the wealth-consumption ratio (which is closely related to the information variable) is the value of a coupon bond and the existence of an infinite-horizon solution depends on the positivity of the asymptotic forward rate.

**Consumption and asset prices with recursive preferences: Continuous-time approximations to discrete-time models**

This paper presents tractable and efficient numerical solutions to general equilibrium models of asset prices and consumption where the representative agent has recursive preferences. It provides a discrete-time presentation of the approach of Fisher and Gilles (1999), treating continuous-time representations as approximations to discrete-time "truth." First, exact discrete-time solutions are derived, illustrating the following ideas: (*i*) The price-dividend ratio (such as the wealth-consumption ratio) is a perpetuity (the canonical infinitely-lived asset), the value of which is the sum of dividend-denominated bond prices, and (*ii*) the positivity of the dividend-denominated asymptotic forward rate is necessary and sufficient for the convergence of value function iteration for an important class of models. Next, continuous-time approximations are introduced. By assuming the size of the time step is small, first-order approximations in the stepsize provide the same analytical flexibility to discrete-time modeling as Ito's lemma provides in continuous time. Moreover, it is shown that differential equations provide an efficient platform for value function iteration. Last, continuous-time normalizations are adopted, providing an efficient solution method for recursive preferences.

**A simple model of the failure of the term structure of interest rates**

This paper presents a simple exponential-affine model of the term structure that replicates the failure of the expectations hypothesis quite well. The model has two Gaussian state variables. Under the equivalent martingale measure, both state variables (the short rate and its stochastic mean) determine the shape of the yield curve, while under the physical measure the short rate is Markovian. Thus instantaneous forward rates have a classical errors-in-variables structure: They are moved by two variables (that are independent under the physical measure), but the short-term interest rate is not moved by one of them. This model shows that stochastic volatility is not required to model the failure of the expectations hypothesis. The key is stochastic risk premia.

**The absence of arbitrage and general equilibrium in continuous time: An overview**

We review the absence-of-arbitrage restrictions in a Brownian motion setting, relying on the existence of a *state-price deflator*. We show how to find useful representations by either changing the numeraire or by changing the *deflator-asset* and its associated equivalent martingale measure. We examine the relation between utility and the state-price deflator, and the absence-of-arbitrage condition for a perpetuity is applied to finding (*i*) the optimal relationship between consumption and wealth and (*ii*) the nominal interest rate.

**Risk-neutral pricing and risk neutrality: A tale of two equivalent martingale measures**

Two properties of asset prices have often been associated with risk-neutrality: (a) the expected return on all assets equals the risk-free rate (the expected-return property) and (b) the value of an asset equals the present value of the expectation (under the physical measure) of its payoff (the certainty-equivalent property). In this paper, we show the following: (i) the two properties are equivalent when interest rates are deterministic but mutually exclusive when interest rates are stochastic, (ii) an economy of risk-neutral investors who consume only at a single point of time in the future will support the certainty-equivalent property, (iii) the distribution that can be uncovered using options prices is associated with the certainty-equivalent property, (iv) the distribution associated with the expected-return property cannot be uncovered using option prices when interest rates are stochastic, although they can be uncovered from futures prices on options that expire on the delivery date. We provide an empirical investigation of the relationship between these measures in the context of multi-factor models of the term structure.

**Estimating exponential-affine models of the term structure**

We show how to estimate any model of the term structure of interest rates in the affine-exponential class, which includes the Vasieck (1977), Cox, Ingersoll, and Ross (1985), and Longstaff and Schwartz (1992) models, among many others. For most models in this class, analytical expressions for both bond prices and the conditional distribution for the state variables are not available. However, there are (*a*) fast and accurate numerical solutions to the bond-price partial differential equation and (*b*) closed-form expressions for the first two conditional moments for the state variables. We show how to construct a quasi-maximum likelihood estimator using (*a*) and (*b*) based on the maximum likelihood estimator of Chen and Scott (1993). We discuss extensions to other estimation techniques.

**The term structure of repo spreads**

We characterize the term structure of repo spreads in terms of forward prices and forward rates. The analysis builds on Duffie (1996), who showed that expected future overnight repo spreads are capitalized into a security's price. We show how to calculate correctly the forward price of a bond on special; we derive an absence-of-arbitrage restriction on the drift of the process for forward repo term spreads in the spirit of Heath, Jarrow, and Morton (1992); and we characterize an extension of the unbiased expectations hypothesis in terms of that restriction. Finally, we describe the average behavior of the term structure of repo spreads for U.S. Treasury securities over the auction cycle, and we examine empirically the unbiased expectations hypothesis of the term structure of repo spreads.

**Term premia in exponential-affine models of the term structure**

We derive expressions for various term premia in the context of the class of affine-exponential models of the term structure. In addition, we show how regression tests of the expectations hypothesis can be understood in terms of these models. In particular, we derive expressions for the regression coefficients in terms of the parameters of the models. Moreover, these expressions can used to define a GMM estimator of the parameters in a given non-Gaussian affine-exponential model.

**The term structure of tax-exempt spreads: The effect of convexity**

When taxable yields are tied to tax-exempt yields at the short end of the yield curve, the volatility of the before-tax short-term interest rate will be greater than the volatility of the after-tax short-term interest rate, and the relative shapes of the two term structures will be affected by this relationship. In particular, the yield on a default-free tax-exempt zero-coupon bond will be greater than the "after-tax yield" on a default-free taxable zero-coupon bond. Estimates of the size and variability of this effect are provided.

**Fitting the term structure of interest rates with smoothing splines**

We describe a technique for fitting the term structure of interest rates using smoothing splines, which incorporate a "roughness" penalty. An increase in the penalty reduces the effective number of parameters. We use generalized cross validation to choose adaptively the penalty and hence the effective number of parameters. We show how our technique can be used to spline an arbitrary transformation of the discount function, using a B-spline bases. Our Monte Carlo simulations and estimation results suggest that fitting a smoothing spline to the forward rate curve using generalized cross validation produces the best results.

**Happy-hour economics**

Seasonal increases in demand may cause price to fall in markets with travel or search cost. These costs limit the effective substitutability among products and brands. When individual demand increases, the benefits from additional travel or search increase. Increased willingness to travel or search can translate into more elastic demand curves facing monopolistic competitors, in which case — as long as marginal cost does not increase too quickly — price falls.

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