A generic formula for the values at the boundary points of monic classical orthogonal polynomials

In a previous paper we have determined a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type σ(x)y"n(x)+τ(x)y'n(x)-λnyn(x)=0. In this paper, we give another such formula which enables us to present a generic formula for the values of monic classical orthogonal polynomials at their boundary points of definition.

Some New Classes of Orthogonal Polynomials and Special Functions

In dieser Dissertation präsentieren wir zunächst eine Verallgemeinerung der üblichen Sturm-Liouville-Probleme mit symmetrischen Lösungen und erklären eine umfassendere Klasse. Dann führen wir einige neue Klassen orthogonaler Polynome und spezieller Funktionen ein, welche sich aus dieser symmetrischen Verallgemeinerung ableiten lassen. Als eine spezielle Konsequenz dieser Verallgemeinerung führen wir ein Polynomsystem mit vier freien Parametern ein und zeigen, dass in diesem System fast alle klassischen symmetrischen orthogonalen Polynome wie die Legendrepolynome, die Chebyshevpolynome erster und zweiter Art, die Gegenbauerpolynome, die verallgemeinerten Gegenbauerpolynome, die Hermitepolynome, die verallgemeinerten Hermitepolynome und zwei weitere neue endliche Systeme orthogonaler Polynome enthalten sind. All diese Polynome können direkt durch das neu eingeführte System ausgedrückt werden. Ferner bestimmen wir alle Standardeigenschaften des neuen Systems, insbesondere eine explizite Darstellung, eine Differentialgleichung zweiter Ordnung, eine generische Orthogonalitätsbeziehung sowie eine generische Dreitermrekursion. Außerdem benutzen wir diese Erweiterung, um die assoziierten Legendrefunktionen, welche viele Anwendungen in Physik und Ingenieurwissenschaften haben, zu verallgemeinern, und wir zeigen, dass diese Verallgemeinerung Orthogonalitätseigenschaft und -intervall erhält. In einem weiteren Kapitel der Dissertation studieren wir detailliert die Standardeigenschaften endlicher orthogonaler Polynomsysteme, welche sich aus der üblichen Sturm-Liouville-Theorie ergeben und wir zeigen, dass sie orthogonal bezüglich der Fisherschen F-Verteilung, der inversen Gammaverteilung und der verallgemeinerten t-Verteilung sind. Im nächsten Abschnitt der Dissertation betrachten wir eine vierparametrige Verallgemeinerung der Studentschen t-Verteilung. Wir zeigen, dass diese Verteilung gegen die Normalverteilung konvergiert, wenn die Anzahl der Stichprobe gegen Unendlich strebt. Eine ähnliche Verallgemeinerung der Fisherschen F-Verteilung konvergiert gegen die chi-Quadrat-Verteilung. Ferner führen wir im letzten Abschnitt der Dissertation einige neue Folgen spezieller Funktionen ein, welche Anwendungen bei der Lösung in Kugelkoordinaten der klassischen Potentialgleichung, der Wärmeleitungsgleichung und der Wellengleichung haben. Schließlich erklären wir zwei neue Klassen rationaler orthogonaler hypergeometrischer Funktionen, und wir zeigen unter Benutzung der Fouriertransformation und der Parsevalschen Gleichung, dass es sich um endliche Orthogonalsysteme mit Gewichtsfunktionen vom Gammatyp handelt.

Modeling the effects of node heterogeneity on the performance of a real grid application

The performance benefit when using Grid systems comes from different strategies, among which partitioning the applications into parallel tasks is the most important. However, in most cases the enhancement coming from partitioning is smoothed by the effect of the synchronization overhead, mainly due to the high variability of completion times of the different tasks, which, in turn, is due to the large heterogeneity of Grid nodes. For this reason, it is important to have models which capture the performance of such systems. In this paper we describe a queueing-network-based performance model able to accurately analyze Grid architectures, and we use the model to study a real parallel application executed in a Grid. The proposed model improves the classical modelling techniques and highlights the impact of resource heterogeneity and network latency on the application performance.

Depreciation and its impact on the total return of UK commercial real estate, 1994-2003

Depreciation is a key element of understanding the returns from and price of commercial real estate. Understanding its impact is important for asset allocation models and asset management decisions. It is a key input into well-constructed pricing models and its impact on indices of commercial real estate prices needs to be recognised. There have been a number of previous studies of the impact of depreciation on real estate, particularly in the UK. Law (2004) analysed all of these studies and found that the seemingly consistent results were an illusion as they all used a variety of measurement methods and data. In addition, none of these studies examined impact on total returns; they examined either rental value depreciation alone or rental and capital value depreciation. This study seeks to rectify this omission, adopting the best practice measurement framework set out by Law (2004). Using individual property data from the UK Investment Property Databank for the 10-year period between 1994 and 2003, rental and capital depreciation, capital expenditure rates, and total return series for the data sample and for a benchmark are calculated for 10 market segments. The results are complicated by the period of analysis which started in the aftermath of the major UK real estate recession of the early 1990s, but they give important insights into the impact of depreciation in different segments of the UK real estate investment market.

The impact of country risk on international real estate returns

Commercial real estate investors have well-established methods to assess the risks of a property investment in their home country. However, when the investment decision is overseas another dimension of uncertainty overlays the analysis. This additional dimension, typically called country risk, encompasses the uncertainty of achieving expected financial results solely due to factors relating to the investment’s location in another country. However, very little has been done to examine the effects of country risk on international real estate returns, even though in international investment decisions considerations of country risk dominate asset investment decisions. This study extends the literature on international real estate diversification by empirically estimating the impact of country risk, as measured by Euromoney, on the direct real estate returns of 15 countries over the period 1998-2004, using a pooled regression analysis approach. The results suggest that country risk data may help investor’s in their international real estate decisions since the country risk data shows a significant and consistent impact on real estate return performance.

International evidence on the predictability of returns to securitized real estate assets: econometric models versus neural networks

The performance of various statistical models and commonly used financial indicators for forecasting securitised real estate returns are examined for five European countries: the UK, Belgium, the Netherlands, France and Italy. Within a VAR framework, it is demonstrated that the gilt-equity yield ratio is in most cases a better predictor of securitized returns than the term structure or the dividend yield. In particular, investors should consider in their real estate return models the predictability of the gilt-equity yield ratio in Belgium, the Netherlands and France, and the term structure of interest rates in France. Predictions obtained from the VAR and univariate time-series models are compared with the predictions of an artificial neural network model. It is found that, whilst no single model is universally superior across all series, accuracy measures and horizons considered, the neural network model is generally able to offer the most accurate predictions for 1-month horizons. For quarterly and half-yearly forecasts, the random walk with a drift is the most successful for the UK, Belgian and Dutch returns and the neural network for French and Italian returns. Although this study underscores market context and forecast horizon as parameters relevant to the choice of the forecast model, it strongly indicates that analysts should exploit the potential of neural networks and assess more fully their forecast performance against more traditional models.

The impact of real estate on the terminal wealth of the UK mixed-asset portfolio

The argument for the inclusion of real estate in the mixed-asset portfolio has concentrated on examining its effect in reducing the portfolio risk - the time series standard deviation (TSSD), mainly using ex-post time series data. However, the past as such is not really relevant to the long-term institutional investors, such as the insurance companies and pension funds, who are more concerned the terminal wealth (TW) of their investments and the variability of this wealth, the terminal wealth standard deviation (TWSD), since it is from the TW of their investment portfolio that policyholders and pensioners will derive their benefits. These kinds of investors with particular holding period requirements will be less concerned about the within period volatility of their portfolios and more by the possibility that their portfolio returns will fail to finance their liabilities. This variability in TW will be closely linked to the risk of shortfall in the quantity of assets needed to match the institution’s liabilities. The question remains therefore can real estate enhance the TW of the mixed-asset portfolio and/or reduce the variability of the TW. This paper uses annual data from the United Kingdom (UK) for the period 1972-2001 to test whether real estate is an asset class that not only reduces ex-post portfolio risk but also enhances portfolio TW and/or reduces the variability of TW.

The impact of real estate on the mixed-asset portfolio in periods of financial stress

The case for holding real estate in the mixed-asset portfolio is typically made on its stabilising effect as a result of its diversification benefits. However, portfolio diversification often fails when it is most needed, i.e. during periods of financial stress. In these periods, the variability of returns for most asset classes increases thus reducing the stabilising effect of a diversified portfolio. This paper applies the approach of Chow et al (1999) to the US domestic mixed-asset portfolio to establish whether real estate, represented by REITs, is especially useful in times of financial stress. To this end monthly returns data on five assets classes: large cap stocks, small cap stocks, long dated government bonds, cash (T-Bills) and real estate (REITs) are evaluated over the period January 1972 to December 2001. The results indicate that the inclusion of REITs in the mixed-asset portfolio can lead to increases or decreases in returns depending on the asset class replaced and whether the period is one of calm or stress. However, the inclusion of REITs invariably leads to reductions in portfolio risk that are greater than any loss in return, especially in periods of financial stress. In other words, REITs acts as a stabilising force on the mixed-asset portfolio when it is most needed, i.e. in periods of financial stress.

A note on the use of spreadsheet optimisers in real estate analysis

Modern Portfolio Theory (MPT) has been advocated as a more rational approach to the construction of real estate portfolios. The application of MPT can now be achieved with relative ease using the powerful facilities of modern spreadsheet, and does not necessarily need specialist software. This capability is to be found in the use of an add-in Tool now found in several spreadsheets, called an Optimiser or Solver. The value in using this kind of more sophisticated analysis feature of spreadsheets is increasingly difficult to ignore. This paper examines the use of the spreadsheet Optimiser in handling asset allocation problems. Using the Markowitz Mean-Variance approach, the paper introduces the necessary calculations, and shows, by means of an elementary example implemented in Microsoft's Excel, how the Optimiser may be used. Emphasis is placed on understanding the inputs and outputs from the portfolio optimisation process, and the danger of treating the Optimiser as a Black Box is discussed.