QUESTIONS

1. What are the most common constraints encountered in optimal portfolio allocation in practice?
2. State the standard definition of tracking error and discuss why other definitions of tracking error may be used.
3. How are transaction costs typically incorporated in portfolio allocation models?
4. Give an example of how the presence of taxes changes the concept of risk in portfolio optimization.
5. A limitation in the implementation of the mean-variance model for portfolio optimization is that one of the critical inputs in the model, the sample covariance matrix, is subject to considerable estimation risk. This can skew the optimizer towards suggesting extreme weights for some of the stocks in the portfolio and lead to poor performance. Explain what approaches can be used to deal with the problem.

¹ See Chapter 1 in John L. Maginn and Donald L. Tuttle (eds.), Managing Investment Portfolios: A Dynamic Process, 2nd ed. (New York: Warren, Gorham & Lamont, 1990).

² Harry M. Markowitz, “Portfolio Theory,” Journal of Finance 7, no. 1 (1952): 77–91.

³ Multiperiod portfolio optimization models are still rarely used in practice, not because the value of multiperiod modeling is questioned, but because such models are often too intractable from a computational perspective.

⁴ As the term intuitively implies, the ADV measures the total amount of a given asset traded in a day on average, where the average is taken over a prespecified time period.

⁵ R. Tyrell Rockafellar and Stanislav Uryasev, “Optimization of Conditional Value at Risk,” Journal of Risk 2, no. 3 (2000): 21–41.

⁶ Conditional Value-at-Risk measures the average loss that can happen with probability less than some small probability, that is, the average loss in the tail of the distribution of portfolio losses.

⁷ Another computationally tractable situation for minimizing CVaR is when the data are normally distributed. In that case, minimizing CVaR is equivalent to minimizing the standard deviation of the portfolio.

⁸ For a more detailed explanation of CVaR and a derivation of the optimization formulation, see Chapters 8 and 9 in Dessislava A. Pachamanova and Frank J. Fabozzi, Simulation and Optimization in Finance: Modeling with MATLAB, @RISK, and VBA (Hoboken, NJ: John Wiley & Sons, 2010).

⁹ Price movement risk costs are the costs resulting from the potential for a change in market price between the time the decision to trade is made and the time the trade is actually executed.

¹⁰ Market impact cost is the effect a trader has on the market price of an asset when it sells or buys the asset. It is the extent to which the price moves up or down in response to the trader's actions. For example, a trader who tries to sell a large number of shares of a particular stock may drive down the stock's market price.

¹¹ Versions of this model have been suggested in Gerald Pogue, “An Extension of the Markowitz Portfolio Selection Model to Include Variable Transactions Costs, Short Sales, Leverage Policies, and Taxes,” Journal of Finance 25, no. 5 (1970): 1005–1027; John Schreiner, “Portfolio Revision: A Turnover-Constrained Approach,” Financial Management 9, no. 1 (1980): 67–75; Christopher J. Adcock and Nigel Meade, “A Simple Algorithm to Incorporate Transaction Costs in Quadratic Optimization,” European Journal of Operational Research 79, no. 1 (1994): 85–94; Miguel S. Lobo, Maryam Fazel, and Stephen Boyd, “Portfolio Optimization with Linear and Fixed Transaction Costs and Bounds on Risk,” Annals of Operations Research 152, no. 1 (2007): 376–394; and John E. Mitchell and Stephen Braun, “Rebalancing an Investment Portfolio in the Presence of Convex Transaction Costs,” technical report, Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 2004.

¹² Here we are thinking of w_i as the portfolio weights, but in fact it may be more intuitive to think of the transaction costs as a percentage of amount traded. It is easy to go back and forth between portfolio weights and portfolio amounts by simply multiplying w_i by the total amount in the portfolio. In fact, we can switch the whole portfolio optimization formulation around, and write it in terms of allocation of dollars, instead of weights. We just need to replace the vector of weights w by a vector x of dollar holdings.

¹³ See, for example, Dimitris Bertsimas, Christopher Darnell, and Robert Soucy, “Portfolio Construction through Mixed-Integer Programming at Grantham, Mayo, Van Otterloo and Company,” Interfaces 29, no. 1 (1999): 49–66.

¹⁴ As we explained earlier, this constraint can be written in an equivalent, more optimization solver-friendly form, namely,

¹⁵ The computation of the tax basis is different for stocks and bonds. For bonds, there are special tax rules, and the original price is not the tax basis.

¹⁶ The exact rates vary depending on the current version of the tax code, but the main idea behind the preferential treatment of long-term gains to short-term gains is to encourage long-term capital investments and fund entrepreneurial activity.

¹⁷ See David M. Stein, “Measuring and Evaluating Portfolio Performance after Taxes,” Journal of Portfolio Management 24, no. 2 (1998): 117–124.

¹⁸ See Apelfeld, Roberto, Gordon Fowler and James Gordon, “Tax-Aware Equity Investing,” Journal of Portfolio Management 22, no. 2 (1996): 18–28. These authors show that a manager can outperform on an after-tax basis with high turnover as well, as long as the turnover does not result in net capital gains taxes. (There are other issues with high turnover, however, such as higher transaction costs that may result in a lower overall portfolio return.)

¹⁹ Dividends are taxed as regular income, that is, at a higher rate than capital gains, so minimizing the portfolio dividend yield should theoretically result in a lower tax burden for the investor.

²⁰ See Apelfeld, Fowler, and Gordon, “Tax-Aware Equity Investing.”

²¹ Dan DiBartolomeo, “Recent Advances in Management of Taxable Portfolios,” working paper, Northfield Information Services, 2000.

²² See George Constantinides, “Capital Market Equilibrium with Personal Taxes,” Econometrica 51, no. 3 (1983): 611–636; Robert M. Dammon and Chester S. Spatt, “The Optimal Trading and Pricing of Securities with Asymmetric Capital Gains Taxes and Transaction Costs,” Review of Financial Studies 9, no. 3 (1996): 921–952; Robert M. Dammon, Chester S. Spatt, and Harold H. Zhang, “Optimal Consumption and Investment with Capital Gains Taxes,” Review of Financial Studies 14, no. 3 (2001): 583–617; and Robert M. Dammon, Chester S. Spatt, and Harold H. Zhang, “Optimal Asset Location and Allocation with Taxable and Tax-Deferred Investing,” Journal of Finance 59, no. 3 (2004): 999–1037.

²³ The Securities and Exchange Commission (SEC) in general prohibits cross-trading but does provide exemptions if prior to the execution of the cross trade the asset manager can demonstrate to the SEC that a particular cross trade benefits both parties. Similarly, Section 406(b)(3) of the Employee Retirement Income Security Act of 1974 (ERISA) forbids cross-trading, but there is new cross-trading exemption in Section 408(b)(19) adopted in the Pension Protection Act of 2006.

²⁴ Arlen Khodadadi, Reha Tutuncu, and Peter Zangari, “Optimization and Quantitative Investment Management,” Journal of Asset Management 7, no. 2 (2006): 83–92.

²⁵ The iterative procedure is known to converge to the equilibrium, however, under special conditions. See Colm O'Cinneide, Bernd Scherer, and Xiaodong Xu, “Pooling Trades in a Quantitative Investment Process,” Journal of Portfolio Management 32, no. 3 (2006): 33–43.

²⁶ O'Cinneide, Scherer, and Xu, “Pooling Trades in a Quantitative Investment Process.”

²⁷ The issue of considering transaction costs in multi-account optimization has been discussed by others as well. See, for example, Bertsimas, Darnell, and Soucy, “Portfolio Construction through Mixed-Integer Programming at Grantham, Mayo, Van Otterloo and Company.”

²⁸ As we mentioned earlier in this chapter, realistic transaction costs are in fact described by nonlinear functions, because costs per share traded typically increase with the size of the trade due to market impact.

²⁹ For example, if asset i is a euro-pound forward, then a trade in that asset can also be implemented as a euro-dollar forward plus a dollar-forward, so there will be two additional assets in the aggregate trade vector t.

³⁰ Note that η_k,i equals 1 if w_k,i is the actual dollar holdings.

³¹ Note that, similarly to , we could introduce additional sell variables , but this is not necessary. By expressing aggregate sales through aggregate buys and total trades, we reduce the dimension of the optimization problem, because there are fewer decision variables. This would make a difference for the speed of obtaining a solution, especially in the case of large portfolios and complicated representation of transaction costs.

³² Note that γ = 1 defines linear transaction costs. For linear transaction costs, multi-account optimization produces the same allocation as single-account optimization, because linear transaction costs assume that an increased aggregate amount of trading does not have an impact on prices.

³³ The information ratio is the ratio of (annualized) portfolio residual return (alpha) to (annualized) portfolio residual risk, where risk is defined as standard deviation.

³⁴ For further details, see Chapters 6, 7, and 8 in Frank J. Fabozzi, Petter N. Kolm, Dessislava A. Pachamanova, and Sergio Focardi, Robust Portfolio Optimization and Management (Hoboken, NJ: John Wiley & Sons, 2007).

³⁵ Philippe Jorion, “Bayes-Stein Estimator for Portfolio Analysis,” Journal of Financial and Quantitative Analysis 21, no. 3 (1986): 279–292.

³⁶ See Chapter 8 (p. 217) in Fabozzi, Kolm, Pachamanova, and Focardi, Robust Portfolio Optimization and Management.

³⁷ See, for example, Oliver Ledoit and Michael Wolf, “Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection,” Journal of Empirical Finance 10, no. 5 (2003): 603–621.

³⁸ For an overview of such models, see David Disatnik and Simon Benninga, “Shrinking the Covariance Matrix—Simpler is Better,” Journal of Portfolio Management 33, no. 4 (2007): 56–63.

³⁹ For a step-by-step description of the Black-Litterman model, see Chapter 8 in Fabozzi, Kolm, Pachamanova, and Focardi, Robust Portfolio Optimization and Management.

⁴⁰ See Fisher Black and Robert Litterman, “Global Portfolio Optimization,” Financial Analysts Journal 48, no. 5 (1992): 28–43.

⁴¹ See Richard O. Michaud, Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation (Oxford: Oxford University Press, 1998); Jorion, “Bayes-Stein Estimator for Portfolio Analysis”; and Bernd Scherer, “Portfolio Resampling: Review and Critique,” Financial Analysts Journal 58, no. 6 (2002): 98–109.

⁴² For derivation, see, for example, Chapter 12 in Fabozzi, Kolm, Pachamanova, and Focardi, Robust Portfolio Optimization and Management or Chapter 9 in Pachamanova and Fabozzi, Simulation and Optimization in Finance.

⁴³ See Jyh-Huei Lee, Dan Stefek, and Alexander Zhelenyak, “Robust Portfolio Optimization—A Closer Look,” report, MSCI Barra Research Insights, June 2006.

⁴⁴ See Robert Stubbs and Pamela Vance, “Computing Return Estimation Error Matrices for Robust Optimization,” report, Axioma, 2005.

⁴⁵ For a more in-depth coverage of the topic of estimating input parameters for robust optimization formulations, see Chapter 12 in Fabozzi, Kolm, Pachamanova, and Focardi Robust Portfolio Optimization and Management.

⁴⁶ See, for example, Donald Goldfarb and Garud Iyengar, “Robust Portfolio Selection Problems,” Mathematics of Operations Research 28, no. 1 (2003): 1–38; and Karthik Natarajan, Dessislava Pachamanova, and Melvyn Sim, “Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization,” Management Science 54, no. 3 (2008): 573–585.

⁴⁷ See Aharon Ben-Tal, Tamar Margalit, and Arkadi Nemirovski, Robust Modeling of Multi-Stage Portfolio Problems”,” in High-Performance Optimization, edited by H. Frenk, K. Roos, T. Terlaky, and S. Zhang (Dordrecht: Kluwer Academic Publishers, 2000), pp. 303–328; and Dimitris Bertsimas and Dessislava Pachamanova, “Robust Multiperiod Portfolio Management with Transaction Costs,” Computers and Operations Research 35, no. 1, special issue on Applications of Operations Research in Finance (2008): 3–17.

⁴⁸ For further details, see Fabozzi, Kolm, Pachamanova, and Focardi (2007).

⁴⁹ See Sebastian Ceria and Robert Stubbs, “Incorporating Estimation Errors into Portfolio Selection: Robust Portfolio Construction,” Journal of Asset Management 7, no. 2 (2006): 109–127.

⁵⁰ See Lee, Stefek, and Zhelenyak. “Robust Portfolio Optimization—A Closer Look.”