Investment Theory and Risk Management, + Website

Visualizing Risk

In previous chapters, we analyzed market and interest rate risks and generated detailed risk-attribution statistics that decomposed and allocated total measured risk across the portfolio. In this chapter, I hope to provide a basis for thinking about risk and to develop a more rigorous set of risk management tools based upon that thinking. Let's begin by developing conceptually how we see risk geometrically.

Optimal portfolios are all conceptually related in that they provide a risk-minimizing allocation across assets—a set of weights that minimize portfolio risk. Risk, itself, is a weighted average of the individual asset risks as well as their covariances. For example, risk is essentially the sum w′Vw. We have already covered this ground. Covariances are estimated from historical returns that incorporate information about future expected returns and future risks in general; that is, we believe that markets are at least efficient aggregators of information. That is why we think covariances are informative. With this in mind, think of a two-asset portfolio again and the minimum variance portfolio. The minimum variance portfolio is the solution to the familiar quadratic form (usually subject to constraints):

For a two-asset portfolio, this problem can be visualized in three dimensions (x,y,z), where x and y are the two weights and z is the level of risk (dependent ...