Portfolio optimization

Konno and Yamazaki (1991) proposed a large-scale portfolio optimization model based on mean-absolute deviation (MAD). This serves as an alternative measure of risk to the standard Markowitz s MV approach, which models risk by the variance of the rate of return of a portfolio, leading to a nonlinear convex quadratic programming (QP) problem. Although both measures are almost equivalent from a mathematical point-of-view, they are substantially different computationally in a few perspectives, as highlighted by Konno and Wijayanayake (2002) and Konno and Koshizuka (2005). In practice, MAD is used due to its computationally-attractive linear property. 

Konno, H. and Wijayanayake, A. (2002) Portfolio optimization under D.C. transaction costs and minimal transaction unit constraints. Journal of Global Optimization, 22, 137. 

Konno, H. and Yamazaki, H. (1991) Mean-absolute deviation portfolio optimization model and its applications to Tokyo Stock Market. Management Science, 37, 519. 

Dow Epoxy DOW CHEMICAL COMPANY (THE) Dow Portfolio Optimization DOW CHEMICAL COMPANY (THE)... 

The probability of success is the product of the probability of technical success, the probability of regulatory success, and the probability of commercial success. The criteria for these probabilities of success need to be clearly defined and characterized so that future PMTs can translate the impact of project progress and decisions on the value of the projects in the portfolio (see section on portfolio optimization using sensitivity analysis). 

FIGURE 27.3 Relationship between project value and R D funds invested in their clinical development (project spend). The value of this hypothetical portfolio would be the cumulative value of its constituent projects. Project F clearly has the lowest expected value per "project spend." These low-value projects are usually considered as candidates for termination. In addition to termination as a possibility, effective companies evaluate the low-expected-value projects to identify the drivers that would lead to a significant increase in the project s expected value (see text, "Portfolio Optimization Using Sensitivity Analysis," and Figure 27.5). 

Sharpe ratio in portfolio optimization, although this assertion is only valid for symmetric distributions. This particular measure has not been used in two-stage stochastic engineering models to manage risk. This is not preferred because, as explained below, it is better to depart from single valued measures looking at the whole risk curve behavior instead. 

Because LBO and VC funds are illiquid, they are not typically included in MV portfolio optimizations. The reason is primarily time consistency of asset characteristics. That is, the risk associated with LBOs and VC funds is multiyear in nature. It cannot be equated directly with the risk of stocks. 

In an unadulterated world, MV analysis would simply be applied to after-tax return, risk, and correlations to produce tax-efficient portfolios. Regrettably, tax law is complex and there are a variety of alternative tax structures for holding financial assets. In addition, there are tax-free versions of some instruments such as municipal bonds. Further complicating the analysis is the fact that one must know the split between ordinary income and long-term gains to forecast MV model inputs. This muddles what would otherwise be a fairly straightforward portfolio-allocation problem and requires that tax structure subtleties be built into the MV framework to perform tax-efficient portfolio optimization. 

One additional application of MV analysis is to use the technique to reengineer implied market returns. This requires the problem be reformulated to select a set of returns given asset weights, risk, and correlations. The weights are derived from current market capitalizations for equities, bonds, and other assets. The presumption is that today s market values reflect the collective portfolio optimizations of all investors. Thus, the set of returns that minimizes risk is the market s forecast of future expected returns. This information can be compared with the user s own views as a reliability check. If the user s views differ significantly, they may need to be modified. Otherwise the user can establish the portfolio positions reflecting his or her outlook under the premise his or her forecasts are superior. 

While MV analysis stiU represents the current paradigm, other approaches to portfolio optimization exist and may eventually displace it. Value-at-risk simulation methodologies may ultimately prove more than tangential. Even so, for many practitioners there is stiU a long way to go before forecasting techniques, asset identification, md time horizon considerations are satisfactorily inte-... 

Construction of the variance-covariance matrix can be extremely burdensome and requires large amounts of data. A covariance matrix for n instruments must have n x n elements. For an index with 1,000 instruments, the corresponding variance-covariance matrix would contain 1,000,000 cells. This would make it less suitable for an automatic portfolio optimizer as the computational difficulty of inverting such a large matrix is great and prone to numerical errors. 

While this also uses a variance-covariance matrix much like the full covariance method, the actual matrix is much more condensed. As an example, the matrix used in a 20-factor model would have a size of (20 X 20) 400 cells, which is moderate compared with the one-million-cell matrix mentioned previously for the full variance-covariance model. The advantages of using a multifactor model are that it easily allows for mapping a new issue into past data for similar bonds by looking at its descriptive characteristics, and it can be inverted for use in a portfolio optimizer without too much effort. The multifactor model is also more tolerant to pricing errors in individual securities since prices are averaged within each factor bucket. 

In his seminal paper published in 1952, Harry Markowitz formulated the framework for portfolio optimization, which helps an investor to maximize his utility depending on expected return and expected risk of the portfolio." The objective is to identify, on the basis of return and variance/covariance estimators, those portfolio weights that maximize the utility of the investor. The solutions for different investors depend on their individual risk preferences. In most cases, an investor achieves maximum utility when the return of his portfolio is maximal ... 

Martin L. Leibowitz and Roy D Henriksson, Portfolio Optimization with Shortfall Constraints A Confidence-Limit Approach to Managing Downside Risk, Financial Analysts Journal (March-April 1989), pp. 34-41. 

In order to answer the question what share should be taken up by corporate bonds in a portfolio, ex post simulations were run. The Markowitz approach of portfolio optimization is based on using expected returns. Since the question of determining the optimal fixed income portfolio is to be answered against the background of historical data, the return and variance/covariance estimators are replaced by their historical return means and variances/covariances respectively. These historical data are computed congruently to the relevant investment horizon. For a 3-year investment horizon, the return means and variances/covariances of assets are computed on the basis of 36 monthly returns. The same is, in analogy, done for a 5-year investment horizon on the basis of 60 monthly returns. Investment horizons of three, five, and 10 years are analyzed here. For the investment horizon of, for example, five years, the monthly data in the time window from February 1980 to January 1985 are used. 

Upon proving the mean reversion of the sample, the authors are now suggesting the implementation of the TRP ratio for the portfolio optimization purposes. 

This article focuses on the BL portfolio optimization. Others approaches will be used only as a benchmark which will enable us to assess the success of the augmented Black-Litterman s method. 

The common feature of all active portfolio optimization strategies is expectations about the factors that influence the performance of the class of assets ([5], p. 145). 

As shown with the minimum-variance method, solution of implementing traditional portfolio optimization is often expressed in highly concentrated portfolio. One alternative to overcome such difficulties is to use the equal weighting approach. This method is often considered as a naive diversification strategy which attempts to capture some of the potential gains from international diversification ([14], p. 229). Its major advantage is robustness as it does not require return or volatility forecasts, which is also one of the most important reasons for popularity of the EQW approach. Despite its simplicity and popularity, EQW certainly has some pitfalls. One of the most obvious is the fact that it does not account for volatilities and correlations between assets. 

In contrast to MPT, researchers of Goldman Sachs— Fisher Black and Robert Litterman—proposed a technique that tackles most of the problems, commonly associated to the classical portfolio optimization methods. They start the process of optimization with the assumption that investor chooses his optimum portfolio within a finite group of assets. In essence, the BL model turns the MPT on its head—it does not compute the optimal portfolio from the historical data, but rather assumes that a given portfolio in fact is the optimal one. This idea is backed by several researches which show that it is very difficult for investor to systematically outperform well-diversified benchmark. BL then derive the expected remrns for different positions in the portfolio. If investor agrees with the market assessment, benchmark becomes the optimal portfolio and the funds should be invested accordingly. On the other hand, if someone has different opinions about the expected returns of some of the stocks in the portfolio, the BL approach allows him to adjust the weights according to his projections. The result is the optimal portfolio, based on investor s individual assessment of market potential. 

The last stage of the BL portfolio optimization is application of the mean-variance approach, where we obtain the efficient portfolio weights based on previously calculated adjusted portfolio returns. ... 

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