The yield curve is shaped by (1) expectations of the future path of short-term interest rates and (2) uncertainty about the path. Uncertainty affects the yield curve through two channels: (1) investors’ attitudes toward risk as reflected in risk premia, and (2) 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.
The purpose of this paper is to provide an introduction to the modern theory of the term structure of interest rates using high-school algebra. In order to present the theory correctly, one must take uncertainty seriously. Nevertheless, the source of uncertainty can be modeled quite simply: All uncertainty is resolved by a single flip of a coin. In this setting, the author can rigorously present all three forces that shape the yield curve: expectations, risk aversion, and convexity. The analysis is organized around the conditions that guarantee the absence of arbitrage opportunities.
JEL classification: G12, A20
Key words: yield curve, convexity, absence of arbitrage, expectations
Some material is based on a memo written at the Federal Reserve Board co-authored with Christian Gilles. The observation that the taxable-tax-exempt spread is affected by convexity is due to Joel Lander. The authors thank Lucy Ackert, Christian Gilles, Frank King, Steve LeRoy, Saikat Nandi, Steve Smith, and Larry Wall for helpful comments. The views expressed here are the author’s and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the author’s responsibility.
Please address questions regarding content to Mark Fisher, Economic Advisor, Research Department, Federal Reserve Bank of Atlanta, 104 Marietta Street, N.W., Atlanta, Georgia 30303-2713, 404-498-8757, firstname.lastname@example.org.
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