fundamental insight from portfolio theory. This insight was first suggested by Harry Markowitz (1952) and developed in his subsequent texts (1959 and 1987). Upon first reflection, this insight seems intuitive and not particularly remarkable. As we will see, however, getting it right in building portfolios is generally neither easy nor intuitive. The first complication is perhaps obvious. It is hard to quantify either expected returns or contributions to portfolio risk.11 Thus, balancing the two across different investments is especially difficult. Coming up with reasonable assumptions for expected returns is particularly problematic. Many investors focus on historical returns as a guide, but in this book we will emphasize an equilibrium approach to quantifying expected returns. We will return to this topic in Chapters 5 and 6. Here, we focus on measuring the contribution to portfolio risk, which, though still complex, is nonetheless more easily quantified. For an investor the risk that each investment adds to a portfolio depends on all of the investments in the portfolio, although in most cases in a way that is not obvious. The primary determinant of an investment's contribution to portfolio risk is not the risk of the investment itself, but rather the degree to which the value of that investment moves up and down with the values of the other investments in the portfolio. This degree to which these returns move together is measured by a statistical quantity called "covariance," which is itself a function of their correlation along with their volatilities. Covariance is simply the correlation times the volatilities of each return. Thus, returns that are independent have a zero covariance, while those that are highly correlated have a covariance that lies between the variances of the two returns. Very few investors have a good intuition about correlation, much less any practical way to measure or monitor the covariances in their portfolios. And to make things even more opaque, correlations cannot be observed directly, but rather are themselves inferred from statistics that are difficult to estimate and which are notoriously unstable.12 In fact, until very recently, even professional investment advisors did not have the tools or understanding to take covariances into account in their investment recommendations. It is only within the past few years that the wider availability of data and risk management technology has allowed the lessons of portfolio theory to be more widely applied. The key to optimal portfolio construction is to understand the sources of risk in the portfolio and to deploy risk effectively. Let's ignore for a moment the difficulties raised in the previous paragraph and suppose we could observe the correlations and volatilities of investment returns. We can achieve an increased return by recognizing situations in which adjusting the sizes of risk allocations would improve the 10In an optimal portfolio this ratio between expected return and the marginal contribution to portfolio risk of the next dollar invested should be the same for all assets in the portfolio. "Each of these topics will be the subject of later chapters. Equilibrium expected returns are discussed in Chapters 5 and 6, and deviations from equilibrium in Chapter 7. Estimating co-variance is the topic of Chapter 16. "Whether the unobserved underlying correlations themselves are unstable is a subtle question. The statistics used to measure correlations over short periods of time, which have estimation error, clearly are unstable.