weight in the S&P 500 benchmark; and stocks in the top third were given the remaining weight in proportion to their original benchmark weight. Table 23.3 presents some summary characteristics for the two portfolios. The optimized portfolio was designed to have the same predicted tracking error as the rule-based portfolio-namely, 2.8 percent. This immediately highlights one advantage of optimization: It can easily target a specific level of tracking error, while managers who use stratified sampling would need to design a completely different set of rules to hit a different tracking error objective. The optimized portfolio is also more efficient: It has a much higher expected alpha (3.4 percent versus 2.1 percent) and information ratio (1.22 versus 0.73) for the same level of risk. Further, risk is spread more broadly: The 10 riskiest positions in the rule-based portfolio consume 60 percent of the total risk budget, versus just 37.5 percent for the optimized portfolio. Also, more of the risk budget in the optimized portfolio is due to the factors that are expected to generate positive excess returns: 45.4 percent versus 23.2 percent. Finally, the forecast beta for the optimized portfolio is closer to 1.00, as unintended sources of risk (such as the market timing) are minimized. We can also show the benefits of portfolio optimization graphically. As stated previously, in an efficient portfolio without constraints, each stock's marginal contribution to risk should be proportional to its expected return. This means a plot of each stock's relative contribution to risk against its contribution to portfolio alpha should lie on a straight line. In practice, the plot is not a perfectly straight line, even for an optimized portfolio, because of portfolio constraints (e.g., no net short positions). As shown in Figure 23.2, the plot for the optimized portfolio falls much closer to the 45-degree line than the plot for the rule-based portfolio-making it significantly more efficient. Another benefit of optimizers is that they can efficiently account for transaction costs, constraints, selected restrictions, and other account guidelines, making it much easier to create customized client portfolios. Of course, when using an optimizer to construct efficient portfolios, reliable inputs are essential. Data errors that add noise to the return, risk, and transaction cost forecasts can lead to portfolios in which these forecast errors are maximized. Instead of picking stocks with the highest actual expected returns, or the lowest actual risks or transaction costs, the optimizer takes the biggest positions in the stocks with the largest errors, namely the stocks with the greatest overestimates of expected returns or the greatest underestimates of risks or transaction costs. A robust investment process will screen major data sources for outliers that can severely corrupt one's forecasts. Further, as described in TABLE 23.3 Summary Portfolio Characteristics Stratified Sampling Optimized Portfolio Tracking error 2.8% 2.8% Expected excess return 2.1% 3.4% Expected information ratio 0.73 1.22 Risk budget used by top 10 stocks 60.0% 37.5% Percent of risk from return factors 23.2% 45.4% Portfolio beta 1.03 1.01