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Wolman, Alexander L.. "Bond price premiums." Economic Quarterly. Federal Reserve Bank of Richmond. 2006. HighBeam Research. 23 Oct. 2018 <https://www.highbeam.com>.
Wolman, Alexander L.. "Bond price premiums." Economic Quarterly. 2006. HighBeam Research. (October 23, 2018). https://www.highbeam.com/doc/1G1-158577964.html
Wolman, Alexander L.. "Bond price premiums." Economic Quarterly. Federal Reserve Bank of Richmond. 2006. Retrieved October 23, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-158577964.html
This article provides a detailed introduction to consumption-based bond pricing theory, a special case of the consumption-based asset pricing theory associated with Robert Lucas (1978). To help make the theory more accessible to novices, we organize the article around the two famous interest rate decompositions associated with Irving Fisher. These complementary decompositions relate real or nominal long-term interest rates to expected future short-term interest rates (the expectations theory of the term structure), and relate short- or long-term nominal interest rates to the ex ante real interest rate and the expected inflation rate (the Fisher equation). According to consumption-based theory, the Fisherian relationships hold exactly only under certain restrictive conditions. We show what those conditions are, and we show that generalizations of the Fisherian relationships hold quite broadly in the consumption-based model.
The pure Fisherian relationships are shown to hold only as special cases of the relationship between individual preferences, future economic activity, and the returns on assets. Notable sufficient conditions for the pure expectations hypothesis are that households be neutral to risk and the price level behave like a random walk; the pure Fisher equation requires only risk neutrality. (1) In turn, long-term nominal bond prices may lie above or below the values dictated by the pure expectations hypothesis and the pure Fisher relationship--forward premiums and inflation-risk premiums may be positive or negative.
Interpreting bond prices of various maturities is an important challenge for the Federal Reserve. Nominal bond prices contain information about the public's expectations of inflation and of future short-term rates. And they contain information about the levels of short-term and long-term real interest rates. All these variables can be valuable signals to the Federal Reserve of the appropriateness of its policy. (2) However, extracting these signals requires an understanding of the potential limitations of the pure expectations hypothesis and the pure Fisher relationship. (3)
The article proceeds as follows. In Section 1 we provide a brief historical overview of the two interest rate decompositions. Section 2 lays out a modeling framework for thinking about bond price determination, and derives the basic bond pricing equations from which all else will follow. Section 3 derives the generalized expectations theory of the term structure and Section 4 derives the generalized Fisher equation. Section 5 combines the results of the previous two sections for a general discussion of the yield differential between short- and long-term bonds. Sections 2-5 provide a textbook treatment of bond pricing relationships. (4) Section 6 provides a selective review of applied research based on bond pricing theory. Section 7 concludes the article.
Although the usual statements of the expectations hypothesis and the Fisher equation are made in terms of interest rates, most of our derivations use zero-coupon bond prices. This is for analytical simplicity; working with bond prices is slightly easier, especially when the bonds are zero-coupon bonds. And given an expression for the price of a bond, one can always work out the corresponding interest rate.
1. BRIEF HISTORY OF INTEREST RATE DECOMPOSITIONS
The expectations hypothesis of the term structure and the Fisher equation both made early appearances in Irving Fisher's Appreciation and Interest (1896). (5) Chapter 2 of that work is devoted to a discussion of the equation, or "effect," that would later bear the author's name. The Fisher equation is typically thought of as relating "real" and "nominal" interest to the expected rate of inflation, but Fisher's analysis in Appreciation and Interest is more general. He relates the interest rates between two standards (for example real vs. nominal, or dollars vs. yen) to the relative rate of appreciation of the standards, as
1 + j = (1 + a)(1 + i), (1)
where i is the rate of interest in the appreciating standard, a is the rate of appreciation, and j is the rate of interest in the depreciating standard. In Fisher's words,
The rate of interest in the (relatively) depreciating standard is equal to the sum of three terms, viz., the rate of interest in the appreciating standard, the rate of appreciation itself, and the product of these two elements. (p. 9)
In our context, j is the nominal rate, i is the real rate, and a is the expected inflation rate.
In Chapter 5 and to some extent in Chapters 3 and 4, one can find the essence of the expectations hypothesis of the term structure. Most notably perhaps, on pages 28 and 29, Fisher writes,
A government bond, for instance, is a promise to pay a specific series of future sums, the price of the bond is the present value of this series and the "interest realized by the investor" as computed by actuaries is nothing more or less than the "average" rate of interest in the sense above defined.
By "'average' rate of interest in the sense above defined," Fisher means what we now understand to be the expected future path of short-term rates.
John Hicks (1939) and F. A. Lutz (1940) elaborated on Fisher's version of the expectations hypothesis in the 1930s and 1940s. Their versions of these interest rate decompositions continued to be based on reasoning regarding how returns among different assets should be related. Later, the development of consumption-based asset pricing theory (Lucas 1978) gave a formal foundation to Fisher's reasoning, while making clear that restrictive assumptions were needed for the Fisherian relationships to hold exactly. The discipline provided by consumption-based theory and the rise of rational expectations and dynamic equilibrium modeling in macroeconomics also led economists and finance theorists to de-emphasize certain elements of Fisher's theories regarding interest rates. For example, with respect to the expectations hypothesis, early versions were "usually understood to imply ... that interest rates on long-term securities will move less, on the average, than rates on short-term securities" (Wood 1964). It is now well understood that whether this will be true depends on the behavior of monetary policy and on the real shocks hitting the economy (Watson 1999). And with respect to the Fisher equation, prior to the consumption-based theory, researchers often emphasized not just the decomposition into real rates and expected inflation, but also the extent to which the real rate was invariant to changes in expected inflation (Mundell 1963). It is now understood that one cannot make general statements about this invariance, even though a version of the Fisher equation holds under general conditions.
2. MODELING FRAMEWORK
We use the modern theory of consumption-based asset pricing, first developed by Robert Lucas (1978), to study bond prices. For our purposes, the crucial elements in this theory are as follows. There is a representative consumer who has an infinite planning horizon, has a standard utility function (exhibiting risk aversion) over consumption each period, and discounts the utility from future consumption at a constant rate. (6) The consumer has a budget constraint which states that the sum of income from sales of real and financial assets (including income from maturing bonds), and income from other sources must not be exceeded by the sum of spending on current consumption, on purchases of real and financial assets, and on any other uses. With this framework, it is possible to price any asset. To do this, we use conditions describing individuals' optimal behavior. …
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