Derivatives: Markets, Valuation, and Risk Management

CHAPTER 3

Relation between Return and Risk

An important facet of valuation not yet discussed is the relation between expected return and risk. In the bond and stock valuation models discussed in Chapter 2, risk enters into the valuation formulas through the interest rate used to discount the expected cash flows to the present. In financial economics, the capital asset pricing model (CAPM)¹ provides the structural relation between expected return and risk. It relies on the assumption that individuals prefer more wealth to less wealth, but at a decreasing rate. Such individuals are risk averse, and risk aversion is the focus of the utility theory discussion in the first section. In the second section, we extend the discussion to show how such individuals allocate their wealth among securities. In the third section, we aggregate security demands across all individuals in the marketplace and identify the equilibrium expected return/risk relations for individual securities and security portfolios. Finally, in the fourth section, we apply the CAPM relations to evaluate portfolio performance.

UTILITY THEORY

In most financial economic models, individuals are assumed to be risk averse. Investors do not like risk but are willing to bear it if paid an adequate risk premium. Risk premiums arise from the nature of how an individual's satisfaction varies with wealth. Called a utility of wealth function, U(w) is the level of satisfaction (measured in units of utility) realized from having a level ...