Interest rate theory and geometry

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.

PurposeWhat causes the downward trend of real interest rates in major developed economies since the 1980s? What are the challenges of the near-zero interest and inflation rates for monetary policy? What can the policymakers learn from the latest developments in the monetary and interest rate theory? This paper aims to answer these questions by reviewing both basic principles of interest rate determination and recent academic and policy debates.Design/methodology/approachThe paper critically reviews the explanations for the downward trend of real interest rates in recent decades and monetary policy options in a near-zero interest rate environment.FindingsThe decline of real interest rates is likely an outcome of multiple technological, social and economic factors including diminished productivity growth, changing demographics, elevated tail-risk concerns, time-varying convenience yields of safe assets, increased global demand for safe assets, rising wealth and income inequality, falling relative price of capital, accommodative monetary policies, and changes in industry structure that alter the investment and saving behaviors of the corporate sector. The near-zero interest rate limits the space of central banks' response to economic crises. It also challenges some conventional wisdoms of monetary theory and sparks radically new ideas about monetary policy.Originality/valueThis survey differs from the existing work by taking a broader view of both economics and finance literature. It critically assesses the economic forces driving the global decline of real interest rates through the lens of basic principles and empirical evidence and discusses the merits and limitations of each proposed explanation. The study emphasizes the importance of a better understanding of economic forces driving diverging trends of corporate investment and saving behaviors. It also discusses the implications of the neo-Fisherism and the fiscal theory of price level for monetary policy in a low interest rate environment.

This paper provides cross-country empirical evidence on term premia. I construct a panel of zero-coupon nominal government bond yields spanning ten industrialized countries and nearly two decades. I hence compute forward rates and use two different methods to decompose these forward rates into expected future short-term interest rates and term premiums. The first method uses an affine term structure model with macroeconomic variables as unspanned risk factors; the second method uses surveys. I find that term premiums declined internationally over the sample period, especially in countries that apparently reduced inflation uncertainty by making substantial changes in their monetary policy frameworks. (JEL E13, E43, E52, G12, H63)

Invariance, observational equivalence, and identification: Some implications for the empirical performance of affine term structure models

Dai and Singleton (2002) and Duffee (2002) show that there is a tension in affine term structure models between matching the mean and the volatility of interest rates. This article examines whether this tension can be solved by an alternative parametrization of the price of risk. The empirical evidence suggests that, first, the examined parametrization is not sufficient to solve the mean-volatility tension. Second, the usual result in the estimation of affine models, indicating that some of the state variables are extremely persistent, may have been caused by the lack of flexibility in the parametrization of the price of risk. Term structure models have several uses, including pricing fixed-income derivatives, managing the risk of fixed-income portfolios, and detecting relationships between the term structure of interest rates and macrovariables such as inflation and consumption. To perform well in these tasks, term structure models must be numerically and econometrically tractable while matching the empirical properties of the term structure movements. At least two empirical properties of the term structure of interest rates have been well established by financial economists over the years [see Dai and Singleton (2003) for a survey]. First, the term premium, or the expected excess return of Treasury bonds, has a high time variability. Second, the volatility of interest rates is time varying. These two properties are so prominent in the data that they will be referred to as stylized facts. While these two stylized facts are very well established in the empirical literature, affine term structure models are thoroughly discussed in the theoretical literature. Affine models are those in which the yield of zero coupon bonds are affine functions of the model state variables. Classic

This paper provides a comparative evaluation of the behaviour of long-term sovereign yields in Czechia, Hungary and Poland from 2001 to 2019. An affine term structure model developed by Adrian, Crump and Moench (2013) is used as an empirical framework for the decomposition of the bond yields into term premium and risk-neutral components. We document a substantial compression in term premia which started in Central European economies around 2013 and played a decisive role in the changes that occurred in 10-year sovereign yields. This pattern, however, was more prevalent in Czechia and Poland than in Hungary. We show that long-term rates in all three economies remained higher than in Germany due to relatively large risk-neutral components. Nevertheless, cross-country correlations became increasingly dependent on term premium dynamics, both among Central European economies and between each of them and Germany. These results are robust to bias-correction in the baseline models and interpreted in the light of the general interest rates decline in the global economy. Potential policy implications are also discussed.

Joan Robinson on Keynes's Liquidity Preference Function and the Rate of Interest and Her Failure to Grasp the Connection between the General Theory, the Liquidity Preference Function, Uncertainty, and the Weight of the Evidence from the a Treatise on Probability

This paper develops a method to approximate arbitrage-free bond yields within a term structure model in which the short rate follows a Gaussian process censored at zero (a shadow-rate model as proposed by Black, 1995). The censoring ensures that model-implied yields are constrained to be positive, but it also introduces non-linearity that renders standard bond pricing formulas inapplicable. In particular, yields are not linear functions of the underlying state vector as they are in affine term structure models (see Piazzesi, 2010). Existing approaches towards computing yields in shadow-rate models suffer from high computational burden or low accuracy. In contrast, I show that the technique proposed in this paper is sufficiently fast for single-step estimation of a three-factor shadow-rate term structure model, and sufficiently accurate to evaluate yields to within approximately half a basis point.

In affine term structure models (ATSM) the stochastic Jacobian under the forward measure plays a crucial role for pricing, as discussed in Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). Their approach leads to a deterministic integro-differential equation which, apparently, has the advantage of by-passing the solution to the Riccati ODE obtained by the standard Feynman-Kac argument. In the generic multi-dimensional case, we find a procedure to reduce such integro-differential equation to a non linear matrix ODE. We prove that its solution does necessarily require the solution of the vector Riccati ODE. This result is obtained proving an extension of the celebrated Radon Lemma, which allows us to highlight a deep relation between the geometry of the Riccati flow and the stochastic calculus of variations for an ATSM.

An affine term structure model with Fed chairs’ speeches

In econometric applications of the term structure, affine models are among the most used ones. Nevertheless, even presenting a closed form characteristic function, its estimation procedure still presents many points to be understood and difficulties to be removed. In this note, we address one of these points. Suppose we estimate an affine dynamic term structure model, and also apply principal component analysis to the interest rate database available. A very plausible question would inquire about the relation (if any) between the principal components obtained assuming no dynamic restrictions, and the dynamic factors estimated using the proposed term structure model. We answer this question when estimating a standard affine model using zero coupon data. We show that each principal component can be approximated by a linear transformation of the dynamic factors. Although simple, this is an important step to the understanding of the mechanics of dynamic affine term structure models. A numerical example using U.S. zero data illustrates the result

The present value of an annuity is the current value of the future payments from an annuity, given a specified rate of discount. The Future value of an annuity is the estimated value of current asset at specific time in the future based on an assumed growth rate. The future and present values of annuity were mathematically generated on the basis of interest rate theory. This was anchored on the fact that every compound interest problem involves the annual rate and the rate per compounding period. The derivations of both present and future values of annuity were established with real life applications. It was also shown that as the frequency of compounding periods increases, the compound amount behavior tends to exponential growth rate. It was deduced that effective rate of interest is a function of nominal rates and compounding periods.

Adaptive dynamic Nelson–Siegel term structure model with applications

One of the most salient debates concerning microcredit pertains to the unexpectedly high rates of interest charged on microloans - rates which frequently are deemed usurious by outside parties. Microcredit is supposed to be to the advantage of poor borrowers, but at the same time microfinance institutions face comparably high costs. This chapter seeks to give theoretical input into the debate on fairness in microcredit interest rates. By drawing on both contemporary discussions in the industry as well as abstract philosophical perspectives on fairness, we classify and theorise four conceptions about fairness in interest rates and pinpoint their limitations. We finally argue for a combination of consequentialist and rights-based concerns which seeks to balance the needs of clients and institutions. We also draw out some practical implications of this view for both microfinance institutions and governments.

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.