Showing papers on "Efficient frontier published in 2001"

Optimal execution of portfolio transactions

Abstract: We consider the execution of portfolio transactions with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. For a simple linear cost model, we explicitly construct the efficient frontier in the space of time-dependent liquidation strategies, which have minimum expected cost for a given level of uncertainty. We may then select optimal strategies either by minimizing a quadratic utility function, or by minimizing Value at Risk. The latter choice leads to the concept of Liquidity-adjusted VAR, or L-VaR, that explicitly considers the best tradeoff between volatility risk and liquidation costs. ∗We thank Andrew Alford, Alix Baudin, Mark Carhart, Ray Iwanowski, and Giorgio De Santis (Goldman Sachs Asset Management), Robert Ferstenberg (ITG), Michael Weber (Merrill Lynch), Andrew Lo (Sloan School, MIT), and George Constaninides (Graduate School of Business, University of Chicago) for helpful conversations. This paper was begun while the first author was at the University of Chicago, and the second author was first at Morgan Stanley Dean Witter and then at Goldman Sachs Asset Management. †University of Toronto, Departments of Mathematics and Computer Science; almgren@math.toronto.edu ‡ICor Brokerage and Courant Institute of Mathematical Sciences; Neil.Chriss@ICorBroker.com

Dynamic Mean-Variance Portfolio Selection with No-Shorting Constraints

Abstract: This paper is concerned with mean-variance portfolio selection problems in continuous-time under the constraint that short-selling of stocks is prohibited. The problem is formulated as a stochastic optimal linear-quadratic (LQ) control problem. However, this LQ problem is not a conventional one in that the control (portfolio) is constrained to take nonnegative values due to the no-shorting restriction, and thereby the usual Riccati equation approach (involving a "completion of squares") does not apply directly. In addition, the corresponding Hamilton--Jacobi--Bellman (HJB) equation inherently has no smooth solution. To tackle these difficulties, a continuous function is constructed via two Riccati equations, and then it is shown that this function is a viscosity solution to the HJB equation. Solving these Riccati equations enables one to explicitly obtain the efficient frontier and efficient investment strategies for the original mean-variance problem. An example illustrating these results is also presented.

Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints

Abstract: We consider the mean-variance (M-V) model of Markowitz and the construction of the risk-return efficient frontier. We examine the effects of applying buy-in thresholds, cardinality constraints and transaction roundlot restrictions to the portfolio selection problem. Such discrete constraints are of practical importance but make the efficient frontier discontinuous. The resulting quadratic mixed-integer (QMIP) problems are NP-hard and therefore computing the entire efficient frontier is computationally challenging. We propose alternative approaches for computing this frontier and provide insight into its discontinuous structure. Computational results are reported for a set of benchmark test problems.

The Efficient Use of Conditioning Information in Portfolios

Abstract: We study the properties of unconditional minimum-variance portfolios in the presence of conditioning information. Such portfolios attain the smallest variance for a given mean among all possible portfolios formed using the conditioning information. We provide explicit solutions for n risky assets, either with or without a riskless asset. Our solutions provide insights into portfolio management problems and issues in conditional asset pricing.

Optimal Portfolios with Bounded Capital at Risk

Abstract: We consider some continuous-time Markowitz type portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the Capital-at-Risk. In a Black-Scholes setting we obtain closed form explicit solutions and compare their form and implications to those of the classical continuous-time mean-variance problem. We also consider more general price processes which allow for larger uctuations in the returns.

Conditional Value-at-Risk: Optimization Approach

Abstract: A new approach for optimization or hedging of a portfolio of finance instruments to reduce the risks of high losses is suggested and tested with several applications. As a measure of risk, Conditional Value-at-Risk (CVaR) is used. For several important cases, CVaR coincides with the expected shortfall (expected loss exceeding Values-at-Risk). However, generally, CVaR and the expected shortfall are different risk measures. CVaR is a coh erent risk measure both for continuous and discrete distributions. CVaR is a more consistent measure of risk than VaR. Portfolios with low CVaR also have low VaR because CVaR is greater than VaR. The approach is based on a new representation of the performance function, which allows simultaneous calculation of VaR and minimization of CVaR. It can be used in conjunction with analytical or scenario based optimization algorithms If the number of scenarios is fixed, the problem is reduced to a Linear Programming or Nonsmooth Optimization Problem. These techniques allow optimizing portfolios with large numbers of instruments. The approach is tested with two examples: (1) portfolio optimization and comparison with the Minimum Variance approach; (2) hedging of a portfolio of options. The suggested methodology can be used for optimizing of portfolios by investment companies, brokerage firms, mutual funds, and any businesses that evaluate risks. Although the approach is used for portfolio analysis, it is very general and can be applied to any financial or non-financial problems involving optimization of percentiles.

Tracking error: Ex ante versus ex post measures

Abstract: Portfolio performance is usually evaluated against a prespecified benchmark portfolio. One most frequently used measure is tracking error (TE), sometimes defined as differences between portfolio returns and the benchmark portfolio returns. TE is simple and easy to calculate as well as a powerful tool in structuring and managing index funds. Two common sources of tracking errors come from the attempts to outperform the benchmark and the passive portfolio replication of the benchmark by a sampled portfolio.

Diversification Benefits of Emerging Markets Subject to Portfolio Constraints

Abstract: This paper examines the international diversification benefits subject to portfolio constraints --- in particular, constraints on short selling. We show that the international diversification benefits remain substantial for U.S. equity investors when they are prohibited from short selling in emerging markets. This result is robust to investment restrictions on non-native individuals. It is also unaffected by the fact that the U.S. equity index portfolio is not on the efficient frontier spanned by U.S. securities. The integration of world equity markets reduces, but does not eliminate, the diversification benefits of investing in emerging markets subject to short-sale constraints.

Portfolio Evaluation and Benchmark Selection: A Mathematical Programming Approach

Abstract: This paper illustrates a new approach to evaluating portfolios in the context of multiple performance measures. The approach is based upon linear programming techniques and identifies the n-dimensional efficient portfolio frontier. An illustrative example with commodity trading advisor (CTA) returns shows that benchmarks can be identified for each individual portfolio.

The Global Real Estate Crash: Evidence from an International Database

Abstract: The past decade has been a period of globalization in the world’s investment markets. Access to international investment has broadened dramatically as barriers to cross-border investment have lifted. During the 1980’s many equity and debt markets around the world performed well, and none more so than those in emerging markets. Willingness by U.S. investors to diversify beyond their borders was based, in part, upon the availability of statistical information about the risk and return characteristics of these markets, as well as upon the increasing use of optimization models to manage portfolio risk and return.

The Supported Solutions Used as a Genetic Information in a Population Heuristics

Abstract: Population heuristics present native abilities for solving optimization problems with multiple objectives. The convergence to the efficient frontier is improved when the population contains `a good genetic information'. In the context of combinatorial optimization problems with two objectives, the supported solutions are used to elaborate such information, defining a resolution principle in two phases. First the supported efficient solution set, or an approximation, is computed. Second this information is used to improve the performance of a population heuristic during the generation of the efficient frontier. This principle has been experimented on two classes of problems : the 1 || (Σ,Ci; Tmax) permutation scheduling problems, and the biobjective 0-1 knapsack problems. The motivations of this principle are developed. The numerical experiments are reported and discussed.

System and method for portfolio allocation

Abstract: The Portfolio Allocation System of the present invention is a comprehensive tool which accepts user specified scenarios describing selected aspects of future price evolution—and provides as output an efficient frontier for portfolio re-allocation—taking into account transaction costs and costs of carry (the cost associated with foregoing the risk free interest rate where applicable). The product uses intensive monte-carlo computations and is supported by a network of machines. Users can interact with the system over the internet or through modem—using a specially developed user interface.

Method of ascertaining an efficient frontier for tax-sensitive investors

Abstract: There are computerized processes for financial planning for individuals and groups whose financial portfolio would be subject to tax on certain events. But these processes do not take into account these taxes when optimizing investment decisions, since taxes levied on investment outcomes, typically on income and realized capital gains, may have an important impact on net portfolio results. This invention is a method for transforming the usual pretax information for calculation of an efficient frontier, unique to an investor's portfolio, in such a manner that any portfolio on the calculated frontier is efficient after incorporating the effect of taxes on the risk and expected return of each asset class permitted in the investor's portfolio. This invention addresses how this may be done and how certain facets of the process may be incorporated into a computer program or system so as to provide convenience to the potential user.

Method and apparatus for matching risk to return

Abstract: A system, apparatus, method, computer program code and means for matching a level of risk to an expected return for a financial product includes selecting a first and a second investment option and a duration. A risk and a corresponding return on investment for each of said investment options are calculated based on the duration. An efficient frontier is then calculated between the first and second investment options, where the efficient frontier defines a number of risks and corresponding expected returns on investment for the financial product.

Building a mean-downside risk portfolio frontier

Abstract: Publisher Summary This chapter presents an algorithm to minimize portfolios downside risk (DSR). Properties of the portfolio frontier, such as convexity, are demonstrated. Asset pricing relations are also derived, with and without a risk-free asset. Finally, a non-parametric approach is presented to calculate DSR, and an algorithm is illustrated to optimize it, and construct a non-parametric portfolio frontier.

Efficient frontier cutoff policies in credit portfolios

Abstract: Historically, account acquisition in scored retail credit and loan portfolios has focused on risk management in the sense of minimizing default losses We believe that acquisition policies should focus on a broader set of business measures that explicitly recognize tradeoffs between conflicting objectives of losses, volume and profit Typical business challenges are: ‘How do I maximize portfolio profit while keeping acceptance rate (volume, size) at acceptable levels?’ ‘How do I maximize profit without incurring default losses above a given level?’ ‘How do I minimize the risk of large loss exposures for a given market share?’ In this paper we are not concerned with which combination of objectives are appropriate, but rather focus on the cutoff policies that allow us to capture a number of different portfolio objectives When there are conflicting objectives we show that optimal policies yield meaningful tradeoffs and efficient frontiers and that optimal shadow prices allow us to develop risk-adjusted tradeoffs between profit and market share Some of the graphical solutions that we obtain are simple to derive and easy to understand without explicit mathematical formulations but even simple constraints may require formal use of non-linear programming techniques We concentrate on models and insights that yield decision strategies and cutoff policies rather than the techniques for developing good predictors

Measuring scale and scope economies in multiproduct banking? A stochastic frontier cost function approach

Abstract: Most studies on the economies of scale and scope in the banking industry assume no x-inefficiencies That is, banks are assumed to be always on their efficient frontier, which can in empirical studies confound scale and scope efficiencies with x-efficiencies The current paper employs a stochastic frontier cost function incorporating technical and allocative inefficiencies, as well as a system of share equations, to estimate scale and scope efficiencies Using data from Taiwan's banking industry, evidence is found that both scale and scope economies exist, and that the assumption of no x-inefficiencies results in underestimating such economies

Finding a maximum skewness portfolio

Abstract: Ways of finding a maximum skewness portfolio, with given return, variance and kurtosis, are presented. The methods take advantage of the special shape of the efficient portfolios manifold. Simpler solutions are obtained if the higher moments tensor has some particular structures. The problem of finding the optimal portoflio in a dynamic setting is also discussed. Areas where this portfolio is meaningful are outlined and an empirical application is fully developed.

Digital Portfolio Theory

Abstract: The Modern Portfolio Theory of Markowitz maximized portfolio expected return subject to holding total portfolio variance below a selected level. Digital Portfolio Theory is an extension of Modern Portfolio Theory, with the added dimension of memory. Digital Portfolio Theory decomposes the portfolio variance into independent components using the signal processing decomposition of variance. The risk or variance of each security's return process is represented by multiple periodic components. These periodic variance components are further decomposed into systematic and unsystematic parts relative to a reference index. The Digital Portfolio Theory model maximizes portfolio expected return subject to a set of linear constraints that control systematic, unsystematic, calendar and non-calendar variance. The paper formulates a single period, digital signal processing, portfolio selection model using cross-covariance constraints to describe covariance and autocorrelation characteristics. Expected calendar effects can be optimally arbitraged by controlling the memory or autocorrelation characteristics of the efficient portfolios. The Digital Portfolio Theory optimization model is compared to the Modern Portfolio Theory model and is used to find efficient portfolios with zero calendar risk for selected periods.

Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation

Abstract: The mean-variance formulation by Markowitz in the 1950s paved a foundation for modern portfolio selection analysis in a single period. This paper considers an analytical optimal solution to the mean-variance formulation in multiperiod portfolio selection. Specifically, analytical optimal portfolio policy and analytical expression of the mean-variance efficient frontier are derived in this paper for the multiperiod mean-variance formulation. An efficient algorithm is also proposed for finding an optimal portfolio policy to maximize a utility function of the expected value and the variance of the terminal wealth.

Portfolio Selection and the Analysis of Risk and Time Diversification

Abstract: This thesis is devoted to the analysis of three important issues in financial economics in general and portfolio selection in particular: the risk measure, estimation risk and time diversification. Besides a short introductory chapter the thesis consists of four empirical essays. In the second chapter, the effect of estimation risk on the efficient frontier in the lower partial moment framework is analyzed. A simulation approach is employed for the analysis of estimation risk in the MLPM-model because it can directly show the effect and magnitude of the estimation error on the portfolios. The results of the average difference between the actual and estimated portfolios show that the estimated portfolios are biased predictors of the actual portfolios in that they underestimate the risk in the portfolios and overestimate the portfolio mean returns. However, the estimates of the optimal portfolios can be improved. If our concern is the uncertainty in the optimal portfolio weights, then a bootstrap approach should be used to improve the optimizations since this approach produces the lowest root-mean squared errors in the study. In the third chapter, a downside risk approximation for calculating optimal portfolios in the discrete-time dynamic investment model is compared to the exact power function formulation that springs from the dynamic reinvestment model. The results show that the downside risk model approximates the dynamic model surprisingly well under both quarterly and annual revisions. However, the approximation seems to be correlated with the target rate of return in the downside risk formulation. In addition, the results suggest that the approximation perform best when the target rate of return is set high as compared to the mean returns of the basic assets. The fourth chapter analyzes whether or not mean-variance efficient portfolio weights for stocks and bills vary significantly with the investment horizon in a buy-and-hold strategy. Real returns from the U.S. asset market on a monthly basis from 1900 to 1997 were used in the analysis. As far as the question of estimation risk is concerned, the results showed that the estimation errors increased with the risk tolerance and with the investment horizon. However, the results in this study indicate that the optimal weights in stocks are not independent of the investment horizon, and that whether or not investors should tilt their portfolio weight towards or away from stocks in long horizon portfolios depends on the investor's risk aversion. The fifth chapter contains an analysis of whether the portfolio weights for stocks and bills, which are formed on the basis of direct expected utility maximization for a set of utility functions, vary significantly with the investment horizon. A non-parametric bootstrap approach is employed, which allows us to draw conclusions on whether or not differences between optimal portfolios are significant. Our analysis shows that the weights for stocks are significantly higher for long horizon investment as compared to the one-year horizon. We conclude that time diversification exists, and that the allocation decision seems to be independent of the utility function.

The Firm Under Uncertainty with General Risk-Averse Preferences: A State-Contingent Approach

Abstract: This paper summarizes and synthesizes recent developments in the state-contingent theory of production under uncertainty presented by Chambers and Quiggin (2000) with a particular focus on the case of generalized expected utility preferences. The problem of the risk-averse firm under price and production uncertainty is analyzed using a state-contingent production technology and general risk-averse preferences. The concept of an efficient frontier, which identifies all potentially optimal production plans for weakly risk-averse decisionmakers, is introduced and used to develop comparative static results. For constant absolute risky technologies, the efficient frontier is shown to correspond to a unique isocost contour.

Characterizing Efficient Portfolios

Abstract: We provide here a necessary and sufficient operational condition for determining whether a given portfolio is efficient in the sense of second-degree stochastic Dominance (SSD). This condition also enables one to find a direction for improving on an inefficient portfolio, in the sense that all risk averse investors would weakly prefer that change in the portfolio composition. This condition can be applied among others to resolve questions that have long been posed in the literature, concerning on whether the portfolios that are promoted by portfolio managers are in fact efficient in the above sense.

Risk measures for income streams

Abstract: A new measure of risk is introduced for a sequence of random incomes adapted to some filtration. This measure is formulated as the optimal net present value of a stream of adaptively planned commitments for consumption. The new measure is calculated by solving a stochastic dynamic linear optimization problem which, for finite filtrations, reduces to a deterministic linear program. We analyze properties of the new measure by exploiting the convexity and duality structure of the stochastic dynamic linear problem. The measure depends on the full distribution of the income process (not only on its marginal distributions) as well as on the filtration, which is interpreted as the available information about the future. The features of the new approach are illustrated by numerical examples. Department of Statistics and Decision Support Systems, Universitaetsstrasse 5, University of Vienna, 1090 Wien-Vienna, Austria; e-mail: georg.pflug@univie.ac.at Department of Management Science and Information Systems and RUTCOR, Rutgers University, 94 Rockefeller Rd, Piscataway, NJ 08854, U.S.A.; e-mail rusz@rutcor.rutgers.edu 1 1 Motivation Since the seminal work of Markowitz it is well understood that consequences of economic activity with uncertain success must be judged in two different and well distinguished dimensions. The mean refers to the average result among a set of possible scenarios, while the risk dimension describes the possible variation of the results under varying scenarios. In the Markowitz model the risk is measured by the variance of the outcome (cf. [8, 9]). In the mean–risk setting the decision maker is faced with a two-objective situation: he/she wants to maximize the mean return and to minimize the risk at the same time. As for all multi-objective situations, there is in general no uniquely defined best decision, which is optimal in both dimensions and one has to seek for compromises. The set of solutions which are Pareto-efficient in the sense of these two objectives is called the mean–risk efficient frontier. In some models for optimal decision making the two dimensions are often mixed by introducing a nondecreasing concave utility function. Risk aversion, i.e. the degree of taking the risk dimension into account, can be modeled by the negative curvature of the utility function. However, it is highly desirable to clearly separate the two dimensions and to make the compromising strategy as transparent as possible, and the efficient frontier approach provides such a transparency. In the first step, the efficient frontier is calculated for a given decision problem and the non-dominated decisions are identified. In the second step, the compromise decision may be chosen among the efficient candidates. There is a vast literature on one-period decision models using several notions of measuring risk (see, e.g., [1, 7, 13, 14, 19]). In the multiperiod situation, however, most proposals focus on the risk contained in the terminal wealth (see [3, 11, 12] and the references therein). The purpose of this paper is to propose a risk measure for multiperiod models which incorporates the risk contained in intermediate incomes. Suppose that I1, . . . , IT is a stream of random incomes. A simple but inappropriate way of defining the multiperiod risk would be to look at the marginal variables separately and fabricate a combined risk measure as a combination of the univariate risk measures. The distinction can be made clear by advocating an example which goes back to Philippe Artzner. Suppose that a coin is thrown three times. In situation 1, a reward of 1 is paid if the coin shows more heads than tails. In situation 2, the same reward is paid if the last throw shows head. Do the two situations reflect the same risk for the decision maker? If the whole experiment is done in a few seconds, one is inclined to say ‘yes’. But suppose that the throw of the coin happens just once a year. Then in situation 1 the decision maker knows her income one year ahead, which is a clear advantage over situation 2. Thus situation 1 should turn out to be less risky than 2, although their income variables have identical marginals. We shall return to this example in section 7. To valuate the entire income stream and not just the terminal wealth appears to be appropriate in many models. For instance, pension funds promise a income streams to their clients. Since the rights emerging from a pension fund membership are not bequeathable, clients are not interested in the terminal wealth at some future moments of time, at which they may not be able to consume it. At least in Europe, pension funds are 2 only administrators and not owners of the funds. They are primarily interested in high management fees, which come from a large number of customers. Customers can only be attracted if a good income stream can be guaranteed. Thus it is in the own interest of a pension fund to keep an eye on the customer’s income stream process (see, e.g., [10, 17]). Besides that, the income stream risk must also be considered in other cases, where the primal investment is just made for the purpose of getting the income at later periods. Real options, or loans to companies are good examples here. The paper is organized as follows. After an introductory section about one-period measures, we introduce our concept of a multiperiod measure in section 3. Its properties are analyzed in section 4, and section 5 contains explicit linear programming models for the case of finitely many scenarios. In section 6 we consider mean–risk models for our measures and compare them to models based on the terminal wealth distribution. Illustrative examples are contained in section 7. All calculations can be done by standard linear programming packages. 2 The one-period case Let I be a random income variable defined on some probability space (Ω, F , P ). The risk contained in I is caused by the lack of information about its exact value. A variable, but predictable value of I is riskless. If a natural catastrophe, e.g. a flood, were completely predictable, there would be no risk and no company would insure against it. If a decision maker were clairvoyant, he/she would face no risk since he/she would see the future in a deterministic way and would be able to adapt to it. For us, normal humans, some but not all information about the future may be available. The amount of information available may be expressed in terms of some σ-algebra F ⊆ F . The extreme cases are the clairvoyant (F = F) and the totally uninformed (F = F0 = {Ω, ∅}). The ultimate goal of engaging in risky enterprises with uncertain income opportunities is consumption. Consumption, however, can only be realized after deciding about the amount one wants to commit for this purpose (to buy a house, a car etc.). Suppose that the decision maker decides to commit an amount a. In this case, he/she risks not achieving this decided target, since I may be less than a. However, he/she may insure against the shortfall event, i.e. the event that I 1. The costs for insurance decrease the possible consumption. If, on the other hand, some surplus is left after consumption, this surplus is discounted by a factor d < 1, since saving does not provide the same satisfaction as the consumption committed for. The Expected Net Present Value (ENPV) of the consumption and savings is therefore E(a + d[I − a] − q[I − a]−). A rational decision maker maximizes the ENPV with respect to the available information We use the notation [x] = max(x, 0) and [x]− = max(−x, 0).

Duality for portfolio optimization with short sales

Abstract: We consider the classical Markowitz portfolio optimization problem with additional constraints representing so-called short sales. The two objectives of this multiobjective problem are the expected return and the variance of a portfolio combined by a number of risky securities. A multiobjective problem is established which is dual to this classical portfolio problem. Weak and strong duality assertions are verified. There we consider properly efficient solutions of the portfolio problem and Pareto-efficient solutions of the dual problem, respectively. The theoretical results are illustrated by means of an example representing the optimization problem for a portfolio containing some German blue chips.

Contractibility of the Efficient Frontier of Three-Dimensional Simply-Shaded Sets

Abstract: The aim of this paper is to study the geometrical and topological structure of the efficient frontier of simply-shaded sets in a three-dimensional Euclidean space with respect to the usual positive cone Our main result concerns the contractibility of the efficient frontier and refines a recent result of Daniilidis, Hadjisavvas, and Schaible (Ref 1) regarding the connectedness of the efficient outcome set for three-criteria optimization problems involving continuous semistrictly quasiconcave objective functions

La diversificazione del portafoglio delle famiglie italiane

Abstract: This paper performs an efficiency analysis of households portfolios based on the comparison of observed portfolios with the mean-variance frontier of assets returns. Data on household portfolios are drawn from the 2001 Centro Einaudi survey, a representative sample of the Italian population with at least a bank account. We find that most households' portfolios are extremely close to the efficient frontier once we explicitly take into account no short-selling constraints, while the null hypothesis of efficiency is rejected for all portfolios if we do not consider these constraints.

The relationship between size, diversification and risk

Abstract: Property portfolio diversification takes many forms, most of which can be associated with asset size. In other words larger property portfolios are assumed to have greater diversification potential than small portfolios. In addition, since greater diversification is generally associated with lower risk it is assumed that larger property portfolios will also have reduced return variability compared with smaller portfolios. If large property portfolios can simply be regarded as scaled-up, better-diversified versions of small property portfolios, then the greater a portfolio’s asset size, the lower its risk. This suggests a negative relationship between asset size and risk. However, if large property portfolios are not simply scaled-up versions of small portfolios, the relationship between asset size and risk may be unclear. For instance, if large portfolios hold riskier assets or pursue more volatile investment strategies, it may be that a positive relationship between asset size and risk would be observed, even if large property portfolios are more diversified. This paper tests the empirical relationship between property portfolio size, diversification and risk, in Institutional portfolios in the UK, during the period from 1989 to 1999 to determine which of these two characterisations is more appropriate.

Optimal Portfolios with Monitoring, Private Benefits of Control, and Budget Constraints

Abstract: This paper develops a model that combines features of portfolio theory and corporate governance research. The model analyzes the problem of how large investors form optimal equity portfolios when the return on each investment depends on the size the investment. The optimal solution counterweighs the benefits of higher returns through private benefits of control and monitoring of firm management with the costs of higher transaction prices and bearing diversifiable risk. The general model predicts that controlling investors will monitor more the firms in which they can appropriate more private benefits of control. Large investors are more likely to buy controlling blocks in smaller firms, firms with higher private benefits of control, and firms where the trading costs of buying large blocks are smaller. A special case is numerically solved using an integer-programming algorithm. The numerical analysis shows that investor utility is strictly increasing in available capital for purchasing controlling blocks. The optimal portfolios of large investors dominate the portfolios of small investors both in terms of expected return and risk. The classical result that the efficient frontier is the same for all investors regardless of their wealth does not hold in the setting of this paper.