Taylor Rules, McCallum Rules and the Term Structure of Interest Rates

Recent empirical research shows that a reasonable characterization of federal-funds-rate targeting behavior is that the change in the target rate depends on the maturity structure of interest rates and exhibits little dependence on lagged target rates. See, for example, Cochrane and Piazzesi (2002). The result echoes the policy rule used by McCallum (1994) to rationalize the empirical failure of the `expectations hypothesis' applied to the term- structure of interest rates. That is, rather than forward rates acting as unbiased predictors of future short rates, the historical evidence suggests that the correlation between forward rates and future short rates is surprisingly low. McCallum showed that a desire by the monetary authority to adjust short rates in response to exogenous shocks to the term premiums imbedded in long rates (i.e. "yield-curve smoothing"), along with a desire for smoothing interest rates across time, can generate term structures that account for the puzzling regression results of Fama and Bliss (1987). McCallum also clearly pointed out that this reduced-form approach to the policy rule, although naturally forward looking, needed to be studied further in the context of other response functions such as the now standard Taylor (1993) rule. We explore both the robustness of McCallum's result to endogenous models of the term premium and also its connections to the Taylor Rule. We model the term premium endogenously using two different models in the class of affine term structure models studied in Duffie and Kan (1996): a stochastic volatility model and a stochastic price-of- risk model. We then solve for equilibrium term structures in environments in which interest rate targeting follows a rule such as the one suggested by McCallum (i.e., the "McCallum Rule"). We demonstrate that McCallum's original result generalizes in a natural way to this broader class of models. To understand the connection to the Taylor Rule, we then consider two structural macroeconomic models which have reduced forms that correspond to the two affine models and provide a macroeconomic interpretation of abstract state variables (as in Ang and Piazzesi (2003)). Moreover, such structural models allow us to interpret the parameters of the term-structure model in terms of the parameters governing preferences, technologies, and policy rules. We show how a monetary policy rule will manifest itself in the equilibrium asset-pricing kernel and, hence, the equilibrium term structure. We then show how this policy can be implemented with an interest-rate targeting rule. This provides us with a set of restrictions under which the Taylor and McCallum Rules are equivalent in the sense if implementing the same monetary policy. We conclude with some numerical examples that explore the quantitative link between these two models of monetary policy.

Taylor Rules, McCallum Rules and the Term Structure of Interest Rates

term structure, monetary policy, Taylor rule

Stochastic volatility

Given the importance of return volatility on a number of practical financial management decisions, the efforts to provide good real- time estimates and forecasts of current and future volatility have been extensive. The main framework used in this context involves stochastic volatility models. In a broad sense, this model class includes GARCH, but we focus on a narrower set of specifications in which volatility follows its own random process, as is common in models originating within financial economics. The distinguishing feature of these specifications is that volatility, being inherently unobservable and subject to independent random shocks, is not measurable with respect to observable information. In what follows, we refer to these models as genuine stochastic volatility models. Much modern asset pricing theory is built on continuous- time models. The natural concept of volatility within this setting is that of genuine stochastic volatility. For example, stochastic-volatility (jump-) diffusions have provided a useful tool for a wide range of applications, including the pricing of options and other derivatives, the modeling of the term structure of risk-free interest rates, and the pricing of foreign currencies and defaultable bonds. The increased use of intraday transaction data for construction of so-called realized volatility measures provides additional impetus for considering genuine stochastic volatility models. As we demonstrate below, the realized volatility approach is closely associated with the continuous-time stochastic volatility framework of financial economics. There are some unique challenges in dealing with genuine stochastic volatility models. For example, volatility is truly latent and this feature complicates estimation and inference. Further, the presence of an additional state variable - volatility - renders the model less tractable from an analytic perspective. We examine how such challenges have been addressed through development of new estimation methods and imposition of model restrictions allowing for closed-form solutions while remaining consistent with the dominant empirical features of the data.Stochastic analysis

Examining the bond premium puzzle with a DSGE model

The basic inability of standard theoretical models to generate a sufficiently large and variable nominal bond risk premium has been termed the "bond premium puzzle." We show that the term premium on long-term bonds in the canonical dynamic stochastic general equilibrium (DSGE) model used in macroeconomics is far too small and stable relative to the data. We find that introducing long-memory habits in consumption as well as labor market frictions can help fit the term premium, but only by seriously distorting the DSGE model's ability to fit other macroeconomic variables, such as the real wage; therefore, the bond premium puzzle remains.Interest rates ; Econometric models

Cointegration and Regime-Switching Risk Premia in the U.S. Term Structure of Interest Rates

To date the cointegrating properties and the regime-switching behavior of the term structure are two separate strands of the literature. This paper integrates these lines of research and introduces regime shifts into a cointegrated VAR model. We argue that the short run dynamics of the cointegrated model are likely to shift across regimes while the equilibrium relation implied by the expectations hypothesis of the term structure is robust to regime shifts. A Markov-switching VECM approach for U.S. data outperforms a linear VECM. Moreover, the regime shifts in the risk premium and the equilibrium adjustment reflect shifts in monetary policy.term structure, expectations hypothesis, cointegration, Markov-switching, monetary policy

Cointegration and Regime-Switching Risk Premia in the US Term Structure of Interest Rates

To date the cointegrating properties and the regime-switching behavior of the term structure are two separate strands of the literature. This paper integrates these lines of research and introduces regime shifts into a cointegrated VAR model. We argue that the short-run dynamics of the cointegrated model are likely to shift across regimes while the equilibrium relation implied by the expectations hypothesis of the term structure is robust to regime shifts. A Markov-switching VECM approach for U.S. data outperforms a linear VECM. We find significant shifts in risk premia and interest rate volatility. These regime shifts reflect changing inflation expectations and shifts in monetary policy, respectivelyterm structure, expectations hypothesis, cointegration, Markov-switching, monetary policy

Forward premium puzzle and term structure of interest rates: the case of New Zealand

Using monthly data for the United States dollar – New Zealand dollar exchange rate, this paper revisits the forward premium puzzle and applies a discrete no-arbitrage affine model of the term structure of interest rates to obtain historical estimates of the time-varying foreign exchange risk premium. The two-factor model is estimated via maximum likelihood for the period 1995-2006. The results of this study demonstrate that the modeled risk premium satisfies the required Fama’s conditions, and its inclusion in an extended GARCH(1,1) model is significant in explaining both the mean and the volatility of the exchange rate. However, consistently with the extant literature, the estimated risk premium does not preclude the presence of the forward premium anomaly. Lastly, out-of-sample forecasts of the exchange rate for different specifications and time periods reveal that predictions of the proposed model for the exchange rate are far from the accuracy of a simple random walk specification.

Cointegration and Regime-Switching Risk Premia in the U.S. Term Structure of Interest Rates

To date the cointegrating properties and the regime-switching behavior of the term structure are two separate strands of the literature. This paper integrates these lines of research and introduces regime shifts into a cointegrated VAR model. We argue that the short run dynamics of the cointegrated model are likely to shift across regimes while the equilibrium relation implied by the expectations hypothesis of the term structure is robust to regime shifts. A Markov-switching VECM approach for U.S. data outperforms a linear VECM. Moreover, the regime shifts in the risk premium and the equilibrium adjustment reflect shifts in monetary policyterm structure, expectations hypothesis, cointegration, Markov-switching, monetary policy

Dynamic pricing of general insurance in a competitive market

A model for general insurance pricing is developed which represents a stochastic generalisation of the discrete model proposed by Taylor (1986). This model determines the insurance premium based both on the breakeven premium and the competing premiums offered by the rest of the insurance market. The optimal premium is determined using stochastic optimal control theory for two objective functions in order to examine how the optimal premium strategy changes with the insurer’s objective. Each of these problems can be formulated in terms of a multi-dimensional Bellman equation. In the first problem the optimal insurance premium is calculated when the insurer maximises its expected terminal wealth. In the second, the premium is found if the insurer maximises the expected total discounted utility of wealth where the utility function is nonlinear in the wealth. The solution to both these problems is built-up from simpler optimisation problems. For the terminal wealth problem with constant loss-ratio the optimal premium strategy can be found analytically. For the total wealth problem the optimal relative premium is found to increase with the insurer’s risk aversion which leads to reduced market exposure and lower overall wealth generation

Realized volatility

Realized volatility is a nonparametric ex-post estimate of the return variation. The most obvious realized volatility measure is the sum of finely-sampled squared return realizations over a fixed time interval. In a frictionless market the estimate achieves consistency for the underlying quadratic return variation when returns are sampled at increasingly higher frequency. We begin with an account of how and why the procedure works in a simplified setting and then extend the discussion to a more general framework. Along the way we clarify how the realized volatility and quadratic return variation relate to the more commonly applied concept of conditional return variance. We then review a set of related and useful notions of return variation along with practical measurement issues (e.g., discretization error and microstructure noise) before briefly touching on the existing empirical applications.Stochastic analysis