Optimization of long-short portfolios through the use of fast algorithms takes advantage of models of covariance to simplify the equations that determine optimality. Fast algorithms exist for widely applied factor and scenario analysis for long-only portfolios. To allow their use in factor and scenario analysis for long-short portfolios, the concept of "trimability" is introduced. The conclusion is that the same fast algorithms that were designed for long-only portfolios can be used, virtually unchanged, for long-short portfolio optimization-provided the portfolio is trimable, which usually holds in practice.ong-short portfolios can take many forms, including market-neutral equity portfolios that have a zero market exposure and enhanced active equity portfolios that have a full market exposure, such as 120-20 portfolios (with 120 percent of capital long and 20 percent short). We describe a sufficient condition under which a portfolio optimization algorithm designed for long-only portfolios will find the correct longshort portfolio, even if the algorithm's use would violate certain assumptions made in the formulation of the long-only problem. 1 We refer to this condition as the "trimability condition." The trimability condition appears to be widely satisfied in practice.We also discuss the incorporation of practical and regulatory constraints into the optimization of long-short portfolios. A common assumption of some asset-pricing models is that one can sell a security short without limit and use the proceeds to buy securities long. This assumption is mathematically convenient, but it is unrealistic. In addition, actual constraints on long-short portfolios change over time and, at a given instant, vary from broker to broker and from client to client. The portfolio analyst charged with generating an efficient frontier must take these constraints into account. To our knowledge, all such constraints-whether imposed by regulators, brokers, or the investors themselves-are expressible as linear equalities or weak inequalities. Therefore, they can be incorporated into the general portfolio selection model.In the upcoming sections, we define the general mean-variance problem and outline some of the constraints on portfolio composition in the real world. We then show how the general meanvariance problem can be solved rapidly with a factor, scenario, or historical model by diagonalization of the covariance matrix. We next present the modeling of long-short portfolios and derive a condition under which these fast optimization techniques apply. And we illustrate the results.

A dividend discount model (DDM) is simply a formula for converting a stream of expected dividends into a present value, or price. DDMs add value in the investment process. The focus of this presentation is on how to use the DDM as a stock-picking tool. Figure 1 shows three different DDM formulations. The first represents the .generalized model, where projected dividends are discounted at a rate that corresponds to each time period in the future. While certainly comprehensive, this generalized form tends to be intractable, because the user must specify an infinite number of inputs.Under the assumptions that dividends grow at a constant rate G, and that the discount rate R is constant, the DDM reduces to the simple GordonShapiro model, as shown in Part 2 of Figure 1. But these assumptions are quite constraining, and the DDM is rendered less useful.Many practitioners use a three-stage DDM. This model is based on the assumption that the dividend stream of a company changes over the company's life cycle. Part 3 of Figure 1 illustrates a growth company. Dividends are assumed to grow quickly in the near term, followed by a slower, transitional rate of growth, and then level off at a slower growth rate during the mature stage.For the early growth stage, which is often assumed to last three to five years, explicit dividend estimates are made.

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