risk. That is, there are no unintended risks, and all risks are compensated with additional expected returns. How can a portfolio manager construct such an efficient portfolio? We compare two approaches: (1) a rule-based system and (2) portfolio optimization. Building an efficient portfolio is a complex problem. To help simplify this complicated task, many portfolio managers use ad hoc, rule-based methods that partially control exposures to a small number of risk factors. For example, one common approach-called stratified sampling-ranks stocks within buckets formed on the basis of a few key risk factors, such as sector and size. The manager then invests more heavily in the highest-ranked stocks within each bucket, while keeping the portfolio's total weight in each bucket close to that of the benchmark. The resulting portfolio is close to neutral with respect to the identified risk factors (i.e., sector and size) while overweighting attractive stocks and underweighting unattractive stocks. Although stratified sampling may seem sensible, it is not very efficient. Numerous unintended risks can creep into the portfolio, such as an overweight in high-beta stocks, growth stocks, or stocks in certain subsectors. Nor does it allow the manager to explicitly consider trading costs or investment objectives in the portfolio construction problem. Portfolio optimization provides a much better method for balancing expected returns against different sources of risk, trade costs, and investor constraints. An optimizer uses computer algorithms to find the set of weights (or holdings) that maximize the portfolio's expected return (net of trade costs) for a given level of risk. It minimizes uncompensated sources of risk, including sector and style biases. Fortunately, despite the complex math, optimizers require only the various forecasts we've already described and developed in the prior section.11 To demonstrate the benefits of optimization, we compare two portfolios: one constructed using stratified sampling and the other constructed using an optimizer. The return and risk forecasts are from our CORE U.S. models.12 The benchmark and investment universe for both portfolios is the S&P 500 index. The stratified sampling (or rule-based) method divides stocks into eight buckets, two market-capitalization segments within each of four macro sectors. Within each bucket, stocks were ranked by expected return: Stocks in the bottom third were given a weight of "Mathematically, an optimizer solves a formula such as the following (where b denotes stock weights in the benchmark; w denotes the optimal stock weights in the final portfolio; a denotes the return forecasts; S denotes the covariance risk matrix; and F denotes the stocks' characteristics): Max w' a, subject to (w - b)' S (w-b) < (Target tracking error)2 \F'w-F'b\< = bounds on sector positions and other stock characteristics w > 0 (no short positions) E w = 1 (budget constraint that stock weights must sum to one) 12The CORE U.S. return model comprises six investment themes: profitability, valuation, earnings quality, momentum, management impact, and fundamental research. The CORE U.S. risk model is based on a factor structure that includes all of these investment themes, as well as other factors without expected returns (size, beta, etc.).