Essays on Dual Risk Measures and the Asymptotic Term Structure

Abstract

This dissertation covers two distinct topics in the field of financial economics: risk measurement and the term structure of interest rates.<br /> The phenomenon of risk plays a ubiquitous role in finance and insurance as well as in economics, since it is involved in nearly all financial and economic activities. Risk is a key aspect in diverse situations, as the formation of investment decisions and the operation of financial markets. Conceptually decisions-making under risk depends on two distinct aspects: On the risk of the financial position, and on the subjective attitude towards risk of the investor. In brief, the literature captures the first aspect by various risk measures and the second by the concept of risk aversion. The first part of my thesis aims to connect both concepts by introducing the class of so-called dual risk measures which respect comparative risk aversion. This connection is achieved by an axiom, which dates back to the seminal work of Aumann and Serrano (2008). Roughly speaking, the axiom asserts that less risk averse agents accept riskier gambles. Chapter 1 clarifies for which positions dual risk measures exist. Appealingly these positions constitute the relevant case for a non-trivial decision under risk. In order to allow for applications Chapter 1 further derives numerical and closed-form solutions of dual measures by observing a connection to the theory of Laplace transforms. The primary result of Chapter 2 is a characterization of dual risk measures by a simple equivalent condition. This equivalence provides a representation theorem, which decomposes all dual risk measures. On the other hand it provides an easy construction method for these measures.<br /> Concerning the term structure of interest rates Chapter 3 focuses on a long-term perspective. The long-term behavior is essential for the valuation of long-term sensitive products, such as fixed-income securities, insurance and annuity contracts. For pricing and hedging of these instruments finance practitioners require a term structure for 100 years or more, whereas in most markets only 30 years are observable. Thus they rely on theoretical models, which extrapolate the evolution of the term structure beyond observable maturities. For the resulting limiting term structure Chapter 3 derives two results: under no arbitrage long zero-bond yields and long forward rates are monotonically increasing and equal to their minimal future value. Both results are inspired by Dybvig, Ingersoll and Ross (1996) and constrain the asymptotic maturity behavior of stochastic yield curves. They are fairly general and require only a minimal setting of buy-and-hold trading. Hence the results apply to various arbitrage-free term structure models and impose restrictions on their long-term behavior. These implications serve as a caution for modelers who specify an arbitrage-free term structure. Specifically, setting up the asymptotic yield or forward rate as a diffusion process or a process with systematic jumps necessarily imposes arbitrage opportunities.