An arbitrage model of the term structure of interest rates

A formula for the price of default-free discount bonds of all maturities is found using a Black- Scholes type of arbitrage model which is based on the assumption that a portfolio of three default-free discount bonds of distinct maturities can be managed to be a perfect substitute for any other default-free discount bond. The formula relates the price of bonds to the real rate of interest, the anticipated rate of inflation and the equilibrium prices of interest rate and inflation risks. Bond prices are shown to be the expected value of the sure nominal proceeds of the bond discounted to the present at a random discount rate. It is shown that the unbiased expectations hypothesis is in general inconsistent with this model.     

The nominal duration of TIPS bonds

This paper studies the price responsiveness (effective duration) of U.S. government issued inflation-indexed bonds, known by the acronym TIPS (Treasury Inflation-Protected Securities), to changes in nominal interest rates, real interest rates, and expected inflation. Using the TIPS pricing formula derived by Laatsch and Klein [Q. Rev. Econ. Finance 43 (2002) 405], we first confirm that TIPS bonds have zero sensitivity to changes solely in expected inflation. By changes solely in expected inflation, we mean that the real rate remains unchanged and the nominal rate changes in accordance with the established Fisher [Publ. Am. Econ. Assoc. 11 (1896)] effect. We show that the first derivative of the TIPS price is zero whenever the real rate is held constant. Thus, the first partial derivative of the TIPS bond pricing formula with respect to expected inflation is zero and the first partial derivative of the TIPS bond price with respect to nominal rates is also zero, given, in each case, that we hold the real rate constant. We then temporarily shift the analysis to zero-coupon TIPS bonds and zero-coupon ordinary Treasury bonds. We prove that the nominal duration of zero-coupon TIPS bonds equals that of zero-coupon ordinary Treasury bonds when the real rate changes but expected inflation is held constant.However, if expected inflation changes and the change in the nominal rate does not yield a constant real rate, zero-coupon TIPS prices will change and they will change by a smaller percentage than will zero-coupon ordinary Treasury bonds. We analyze TIPS responsiveness to changes in nominal rates under such conditions. We derive an approximation to effective duration that demonstrates that the effective durations of various maturity zero-coupon TIPS bonds are approximately linear functions in time to maturity of the effective duration of the one-year zero-coupon TIPS bond, ceteris paribus.Nominal effective duration of TIPS bonds is certainly of interest to fixed income portfolio managers that might have a desire to include such bonds in their portfolio. After all, the greater portion of a typical fixed income portfolio is in traditional, noninflation protected bonds whose major risk exposure is to changes in nominal rates. To properly assess the role of TIPS bonds in the portfolio, portfolio managers need information as to how TIPS bonds respond to the changes in nominal rates that are driving the price behavior of the bulk of the portfolio's assets. Prior to concluding the paper, we demonstrate how portfolio managers can calculate the nominal durations of coupon TIPS bonds using the zero-coupon duration formula we derive.     

On pricing kernels and finite-state variable Heath Jarrow Morton models

Once a pricing kernel is established, bond prices and all other interest rate claims can be computed. Alternatively, the pricing kernel can be deduced from observed prices of bonds and selected interest rate claims. Examples of the former approach include the celebrated Cox, Ingersoll, and Ross (1985b) model and the more recent model of Constantinides (1992). Examples of the latter include the Black, Derman, and Toy (1990) model and the Heath, Jarrow, and Morton paradigm (1992) (hereafter HJM). In general, these latter models are not Markov. Fortunately, when suitable restrictions are imposed on the class of volatility structures of forward rates, then finite-state variable HJM models do emerge. This article provides a linkage between the finite-state variable HJM models, which use observables to induce a pricing kernel, and the alternative approach, which proceeds directly to price after a complete specification of a pricing kernel. Given such linkages, we are able to explicitly reveal the relationship between state-variable models, such as Cox, Ingersoll, and Ross, and the finite-state variable HJM models. In particular, our analysis identifies the unique map between the set of investor forecasts about future levels of the drift of the pricing kernel and the manner by which these forecasts are revised, to the shape of the term structure and its volatility. For an economy with square root innovations, the exact mapping is made transparent.     

General equilibrium pricing of CPI derivatives

We examine the issue of pricing forward futures and option contracts written on the Consumer Price Index (CPI), the change of which is a measure of inflation affecting the economy. Traditional approaches postulate an exogenous process for the price level and then derive CPI derivatives prices by standard arbitrage arguments. By contrast, we build the general equilibrium of a continuous time monetary economy that is affected by both real and nominal shocks. The price level and thus the inflation rate are found endogenously and solutions for the prices of CPI derivatives are obtained, which are in closed form in a specialized version of the economy.     

Consumption and Equilibrium Interest Rates in Stochastic Production Economies

In this paper, we analyze the behavior of equilibrium real interest rates in an identical consumer economy in which the preferences are represented by time additive logarithmic utility functions and production technologies are Cobb-Douglas with stochastic constant returns to scale. The following main results are established.

• (i) When there is no relative price uncertainty, it is shown that the equilibrium interest rate exhibits a mean reverting tendency. A nontrivial steady state distribution is found to exist for the equilibrium interest rate. The properties of the equilibrium interest rate are also derived and discussed.
• (ii) In a multigood economy, even with additive preferences across goods, the equilibrium interest rates depend explicitly on relative prices. The substitution possibilities in production technologies induce this result. This is in contrast to the findings of Richard and Sundaresan 11 who show that the analytical general equilibrium term structure of interest rates formula of Cox, Ingersoll, and Ross 5 is unaffected by the introduction of relative price uncertainty when the technologies are linear and hence involve no substitution.

Furthermore, we relate our results to those of Cox, Ingersoll, and Ross 5 , Breeden 3 , and Richard and Sundaresan 11 with special emphasis on stochastic production and realtive price uncertainty.     

The analysis and valuation of interest rate options

This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the Black-Scholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general, and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.     

政府债券、宏观效应与政策调节--基于规模和流动性双重视角的DSGE效应分析

本文基于政府债券规模和流动性的双重视角,构建动态随机一般均衡模型,通过理论分析和数值模拟,研究二者对实体经济发展和金融风险变量的影响。研究显示:政府债券规模的增加促进了投资,刺激了产出和劳动供给,但过度扩张对实体经济也带来"挤占投资"和物价上升、消费下降等负面效应,同时通过债券作为金融资产的特性向金融部门蔓延并形成金融风险集聚;政府债券流动性增强一定程度上刺激了投资,促进了物价稳定,但也存在"挤占消费"和引起经济波动等负面效应;而政府债券流动性的提升有利于缓释金融风险,对实体经济长期可持续发展有益,但流动性过高也会带来系统性金融风险集聚。本文从促进政府债券一级市场和二级市场协同发展、总量和结构合理匹配、政府债券与实体经济有机契合、政府债券流动性管理与金融供给侧密切衔接等方面提出政策建议。     

Optimal fiscal and monetary policy with sticky prices

In this paper I consider the role of state-contingent inflation as a fiscal shock absorber in an economy with nominal rigidities. I study the Ramsey equilibrium in a monetary model with distortionary taxation, nominal non-state-contingent debt, and sticky prices. With sticky prices, the Ramsey planner must balance the shock absorbing benefits of state-contingent inflation against the associated resource misallocation costs. For government spending processes resembling post-war experience, introducing sticky prices generates striking departures in optimal policy from the case with flexible prices. For even small degrees of price rigidity, optimal policy displays very little volatility in inflation. Tax rates display greater volatility compared to the model with flexible prices. With sticky prices, tax rates and real government debt exhibit behavior similar to a random walk. For government spending processes resembling periods of intermittent war and peace, optimal policy displays extreme inflation volatility even when the degree of price rigidity is large. As the variability in government spending increases, smoothing tax distortions across states of nature becomes increasingly important, and the shock absorber role of inflation is accentuated.     

The behavior of government of Canada real return bond returns

The Government of Canada first issued real return bonds, commonly referred to as index-linked bonds in the finance literature, in December 1991. This paper looks at how the prices of real return bonds should behave theoretically, and empirically examines the holding period returns of real return bonds between 1992 and 2003. This study finds that the real return bond holding period returns were positively related to changes in the year-over-year inflation rate but with a lagged effect, were negatively related to changes in nominal interest rates, but were unrelated to changes in either the stock exchange index or the value of the Canadian dollar.     

The ex ante real rate and inflation premium under a habit consumption model

This paper analyzes and quantifies ex ante components of bond yields – real rate of returns and risk premiums – from observed prices of nominal and indexed bonds in the United Kingdom from 1983 to 2000. The estimation uses an asset pricing framework based on a habit consumption model together with a joint formulation of consumption growth and inflation. Nominal yields carry a time-varying inflation premium that is significant throughout the period, increasing in the bond's maturity and contributing up to 25 basis points to yearly nominal yields. The analysis allows the extraction of the ex ante real rate from indexed bonds by properly taking into account both the incomplete indexation on these instruments and the inflation premium embedded in the nominal bonds.     

A theory of the nominal term structure of interest rates

A model of the nominal term structure of interest rates is developedthat has a positive and stationary process for the interestrate and delivers closed form expressions for the prices ofdiscount bonds and European options on bonds. Unlike the one-state-variableversion of the Cox, Ingersoll and Ross (1985) model this model- even in its one-state-variable version - allows the term premiumto change sign as a function of the state and the term to maturity,and also allows for shapes of the yield curve that are observedin the U.S. data but that are disallowed in the Cox, Ingersolland Ross model     

Futures and forward price differential and the effect of marking‐to‐market: Australian evidence

The objective of this paper is to examine the effects of marking‐to‐market of futures contracts on the price differential between futures and forward contracts based on the predictions of the Cox, Ingersoll and Ross (1981) (CIR) model. Cox et al ., (1981) derive a series of propositions with respect to the relationship between futures and forward prices and a set of testable implications. These are tested empirically in this paper using Australian data from November 1991 to June 1997. The results provide evidence of the presence of significant futures and forward price differences, where the futures price is consistently below the forward price. Only partial support is found for the Cox et al ., (1981) propositions, implying that the effect of marking‐to‐market is not able to fully account for the price differential. Therefore, it is not possible to rule out the influence of other institutional factors on the futures‐forward price difference.     

Bond price representations and the volatility of spot interest rates

A common approach to modeling the term structure of interest rates in a single-factor economy is to assume that the evolution of all bond prices can be described by the current level of the spot interest rate. This article investigates the restrictions that this assumption imposes. Specifically, we show that this Markovian restriction, together with the no-arbitrage requirement, curtails the relationship of forward rates and their volatilities relative to spot-rate volatilities. Among such Markovian models, only a few provide simple analytical relationships between bond prices and the spot interest rate. This article identifies the class of spot-rate volatility specifications that permit simple analytical linkages to be derived between bond prices and interest rates. Included in the class are the volatility structures used by Vasicek and by Cox, Ingersoll, and Ross. Surprisingly, no other volatility structures permit simple analytical representations.     

Optimal monetary policy with imperfect common knowledge

This paper determines optimal nominal demand policy in a flexible price economy in which firms pay limited attention to aggregate variables. Firms’ inattentiveness gives rise to idiosyncratic information errors and imperfect common knowledge about the shocks hitting the economy. This is shown to have strong implications for optimal nominal demand policy. In particular, if firms’ prices are strategic complements and economic shocks display little persistence, monetary policy has strong real effects, making it optimal to stabilize the output gap. Weak complementarities or sufficient shock persistence, however, cause price level stabilization to become increasingly optimal. With persistent shocks, optimal monetary policy shifts from output gap stabilization in initial periods following the shock to price level stabilization in later periods, potentially rationalizing the medium-term approach to price stability adopted by some central banks.     

Futures and forward price differential and the effect of marking‐to‐market: Australian evidence

The objective of this paper is to examine the effects of marking‐to‐market of futures contracts on the price differential between futures and forward contracts based on the predictions of the Cox, Ingersoll and Ross (1981) (CIR) model. Cox et al ., (1981) derive a series of propositions with respect to the relationship between futures and forward prices and a set of testable implications. These are tested empirically in this paper using Australian data from November 1991 to June 1997. The results provide evidence of the presence of significant futures and forward price differences, where the futures price is consistently below the forward price. Only partial support is found for the Cox et al ., (1981) propositions, implying that the effect of marking‐to‐market is not able to fully account for the price differential. Therefore, it is not possible to rule out the influence of other institutional factors on the futures‐forward price difference.     

Pricing Bonds and Bond Options with Default Risk

The pricing of bonds and bond options with default risk is analysed in the general equilibrium model of Cox, Ingersoll, and Ross (1985). This model is extended by means of an additional parameter in order to deal with financial and credit risk simultaneously. The estimation of such a parameter, which can be considered as the market equivalent of an agencies' bond rating, allows to extract from current quotes the market perceptions of firm's credit risk. The general pricing model for defaultable zero-coupon bond is first derived in a simple discrete-time setting and then in continuous-time. The availability of an integrated model allows for the pricing of default-free options written on defaultable bonds and of vulnerable options written either on default-free bonds or defaultable bonds. A comparison between our results and those given by Jarrow and Turnbull (1995) is also presented.     

A continuous time equilibrium model of forward prices and futures prices in a multigood economy

This paper is a theoretical investigation of equilibrium forward and futures prices. We construct a rational expectations model in continuous time of a multigood, identical consumer economy with constant stochastic returns to scale production. Using this model we find three main results. First, we find formulas for equilibrium forward, futures, discount bond, commodity bond and commodity option prices. Second, we show that a futures price is actually a forward price for the delivery of a random number of units of a good; the random number is the return earned from continuous reinvestment in instantaneously riskless bonds until maturity of the futures contract. Third, we find and interpret conditions under which normal backwardation or contango is found in forward or futures prices; these conditions reflect the usefulness of forward and futures contracts as consumption hedges.     

Are Indexed Bonds Really Inflation Proof? A Model of Real and Nominal Term Structures when Money has Real Effects

This paper studies the general behavior of the nominal and real term structures of interest rates in a general equilibrium framework. A central bank is introduced in the model as an agent facing a tradeoff between inflation and output and choosing a monetary policy variable. Prices and output are jointly determined in our model endogenously. Two multi-factor nominal and real term structure models are given as examples to illustrate the general model. In our economies, inflation indexed bonds are not completely inflation proof, but are still subject to the influence of inflation uncertainties. The models offer us an empirical framework that can be studied with indexed bond data and nominal bond data together in a single estimation.     

A MULTIFACTOR MODEL OF THE QUALITY OPTION IN TREASURY FUTURES CONTRACTS

The values of quality options in Treasury futures contracts are set relative to the prices of all coupon bonds in their respective deliverable sets. As a result, any model used to value the quality option should set its price relative to the set of observed bond prices. This requirement rules out the use of most simple equilibrium models that represent all bond prices in terms of a finite number of state variables. We use the two-factor Heath-Jarrow-Morton model, which permits claims to be priced relative to observable bond prices, to investigate the potential value of the quality option in Treasury bond and note futures. We show that the quality option has significantly more value in a two-factor interest rate economy than in a single-factor economy, and that ignoring it could lead to significant mispricing.