Computing Markowitz efficient frontiers using a spreadsheet optimiser

Markowitz showed that assets can be combined to produce an 'Efficient' portfolio that will give the highest level of portfolio return for any level of portfolio risk, as measured by the variance or standard deviation. These portfolios can then be connected to generate what is termed an 'Efficient Frontier' (EF). In this paper we discuss the calculation of the Efficient Frontier for combinations of assets, again using the spreadsheet Optimiser. To illustrate the derivation of the Efficient Frontier, we use the data from the Investment Property Databank Long Term Index of Investment Returns for the period 1971 to 1993. Many investors might require a certain specific level of holding or a restriction on holdings in at least some of the assets. Such additional constraints may be readily incorporated into the model to generate a constrained EF with upper and/or lower bounds. This can then be compared with the unconstrained EF to see whether the reduction in return is acceptable. To see the effect that these additional constraints may have, we adopt a fairly typical pension fund profile, with no more than 20% of the total held in Property. The paper shows that it is now relatively easy to use the Optimiser available in at least one spreadsheet (EXCEL) to calculate efficient portfolios for various levels of risk and return, both constrained and unconstrained, so as to be able to generate any number of Efficient Frontiers.

Modal coupling in linear control systems using robust eigenstructure assignment

Eigenvalue assignment methods are used widely in the design of control and state-estimation systems. The corresponding eigenvectors can be selected to ensure robustness. For specific applications, eigenstructure assignment can also be applied to achieve more general performance criteria. In this paper a new output feedback design approach using robust eigenstructure assignment to achieve prescribed mode input and output coupling is described. A minimisation technique is developed to improve both the mode coupling and the robustness of the system, whilst allowing the precision of the eigenvalue placement to be relaxed. An application to the design of an automatic flight control system is demonstrated.

Robust pole assignment in linear state feedback

Numerical methods are described for determining robust, or well-conditioned, solutions to the problem of pole assignment by state feedback. The solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized. It is shown that for these solutions, upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized. A measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed-loop poles, and hence the suitability of the given poles for assignment.

Robust eigenstructure assignment in quadratic matrix polynomials: nonsingular case

Feedback design for a second-order control system leads to an eigenstructure assignment problem for a quadratic matrix polynomial. It is desirable that the feedback controller not only assigns specified eigenvalues to the second-order closed loop system but also that the system is robust, or insensitive to perturbations. We derive here new sensitivity measures, or condition numbers, for the eigenvalues of the quadratic matrix polynomial and define a measure of the robustness of the corresponding system. We then show that the robustness of the quadratic inverse eigenvalue problem can be achieved by solving a generalized linear eigenvalue assignment problem subject to structured perturbations. Numerically reliable methods for solving the structured generalized linear problem are developed that take advantage of the special properties of the system in order to minimize the computational work required. In this part of the work we treat the case where the leading coefficient matrix in the quadratic polynomial is nonsingular, which ensures that the polynomial is regular. In a second part, we will examine the case where the open loop matrix polynomial is not necessarily regular.

Robustness in partial pole placement

The robustness of state feedback solutions to the problem of partial pole placement obtained by a new projection procedure is examined. The projection procedure gives a reduced-order pole assignment problem. It is shown that the sensitivities of the assigned poles in the complete closed-loop system are bounded in terms of the sensitivities of the assigned reduced-order poles, and the sensitivities of the unaltered poles are bounded in terms of the sensitivities of the corresponding open-loop poles. If the assigned poles are well-separated from the unaltered poles, these bounds are expected to be tight. The projection procedure is described in [3], and techniques for finding robust (or insensitive) solutions to the reduced-order problem are given in [1], [2].

Stability results for the time-harmonic Maxwell equations with impedance boundary conditions

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators.

Does increased hydrochemical model complexity decrease robustness?

The aim of this study was, within a sensitivity analysis framework, to determine if additional model complexity gives a better capability to model the hydrology and nitrogen dynamics of a small Mediterranean forested catchment or if the additional parameters cause over-fitting. Three nitrogen-models of varying hydrological complexity were considered. For each model, general sensitivity analysis (GSA) and Generalized Likelihood Uncertainty Estimation (GLUE) were applied, each based on 100,000 Monte Carlo simulations. The results highlighted the most complex structure as the most appropriate, providing the best representation of the non-linear patterns observed in the flow and streamwater nitrate concentrations between 1999 and 2002. Its 5% and 95% GLUE bounds, obtained considering a multi-objective approach, provide the narrowest band for streamwater nitrogen, which suggests increased model robustness, though all models exhibit periods of inconsistent good and poor fits between simulated outcomes and observed data. The results confirm the importance of the riparian zone in controlling the short-term (daily) streamwater nitrogen dynamics in this catchment but not the overall flux of nitrogen from the catchment. It was also shown that as the complexity of a hydrological model increases over-parameterisation occurs, but the converse is true for a water quality model where additional process representation leads to additional acceptable model simulations. Water quality data help constrain the hydrological representation in process-based models. Increased complexity was justifiable for modelling river-system hydrochemistry. Increased complexity was justifiable for modelling river-system hydrochemistry.

Validation of ACE-FTS satellite data in the upper troposphere/lower stratosphere (UTLS) using non-coincident measurements

CO, O3, and H2O data in the upper troposphere/lower stratosphere (UTLS) measured by the Atmospheric Chemistry Experiment Fourier Transform Spectrometer(ACE-FTS) on Canada’s SCISAT-1 satellite are validated using aircraft and ozonesonde measurements. In the UTLS, validation of chemical trace gas measurements is a challenging task due to small-scale variability in the tracer fields, strong gradients of the tracers across the tropopause, and scarcity of measurements suitable for validation purposes. Validation based on coincidences therefore suffers from geophysical noise. Two alternative methods for the validation of satellite data are introduced, which avoid the usual need for coincident measurements: tracer-tracer correlations, and vertical tracer profiles relative to tropopause height. Both are increasingly being used for model validation as they strongly suppress geophysical variability and thereby provide an “instantaneous climatology”. This allows comparison of measurements between non-coincident data sets which yields information about the precision and a statistically meaningful error-assessment of the ACE-FTS satellite data in the UTLS. By defining a trade-off factor, we show that the measurement errors can be reduced by including more measurements obtained over a wider longitude range into the comparison, despite the increased geophysical variability. Applying the methods then yields the following upper bounds to the relative differences in the mean found between the ACE-FTS and SPURT aircraft measurements in the upper troposphere (UT) and lower stratosphere (LS), respectively: for CO ±9% and ±12%, for H2O ±30% and ±18%, and for O3 ±25% and ±19%. The relative differences for O3 can be narrowed down by using a larger dataset obtained from ozonesondes, yielding a high bias in the ACEFTS measurements of 18% in the UT and relative differences of ±8% for measurements in the LS. When taking into account the smearing effect of the vertically limited spacing between measurements of the ACE-FTS instrument, the relative differences decrease by 5–15% around the tropopause, suggesting a vertical resolution of the ACE-FTS in the UTLS of around 1 km. The ACE-FTS hence offers unprecedented precision and vertical resolution for a satellite instrument, which will allow a new global perspective on UTLS tracer distributions.

Quasilimiting behavior for one-dimensional diffusions with killing

This paper extends and clarifies results of Steinsaltz and Evans [Trans. Amer. Math. Soc. 359 (2007) 1285–1234], which found conditions for convergence of a killed one-dimensional diffusion conditioned on survival, to a quasistationary distribution whose density is given by the principal eigenfunction of the generator. Under the assumption that the limit of the killing at infinity differs from the principal eigenvalue we prove that convergence to quasistationarity occurs if and only if the principal eigenfunction is integrable. When the killing at ∞ is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at ∞ is smaller, the eigenfunction is integrable only when the unkilled process is recurrent; otherwise, the process conditioned on survival converges to 0 density on any bounded interval.

Spectral analysis of diffusions with jump boundary

In this paper we consider one-dimensional diffusions with constant coefficients in a finite interval with jump boundary and a certain deterministic jump distribution. We use coupling methods in order to identify the spectral gap in the case of a large drift and prove that there is a threshold drift above which the bottom of the spectrum no longer depends on the drift. As a corollary to our result we are able to answer two questions concerning elliptic eigenvalue problems with non-local boundary conditions formulated previously by Iddo Ben-Ari and Ross Pinsky.

How difficult is it to recover from dangerous levels of global warming?

Climate models provide compelling evidence that if greenhouse gas emissions continue at present rates, then key global temperature thresholds (such as the European Union limit of two degrees of warming since pre-industrial times) are very likely to be crossed in the next few decades. However, there is relatively little attention paid to whether, should a dangerous temperature level be exceeded, it is feasible for the global temperature to then return to safer levels in a usefully short time. We focus on the timescales needed to reduce atmospheric greenhouse gases and associated temperatures back below potentially dangerous thresholds, using a state-of-the-art general circulation model. This analysis is extended with a simple climate model to provide uncertainty bounds. We find that even for very large reductions in emissions, temperature reduction is likely to occur at a low rate. Policy-makers need to consider such very long recovery timescales implicit in the Earth system when formulating future emission pathways that have the potential to 'overshoot' particular atmospheric concentrations of greenhouse gases and, more importantly, related temperature levels that might be considered dangerous.

Predictability and uncertainty in a regional climate model

The evaluation of the quality and usefulness of climate modeling systems is dependent upon an assessment of both the limited predictability of the climate system and the uncertainties stemming from model formulation. In this study a methodology is presented that is suited to assess the performance of a regional climate model (RCM), based on its ability to represent the natural interannual variability on monthly and seasonal timescales. The methodology involves carrying out multiyear ensemble simulations (to assess the predictability bounds within which the model can be evaluated against observations) and multiyear sensitivity experiments using different model formulations (to assess the model uncertainty). As an example application, experiments driven by assimilated lateral boundary conditions and sea surface temperatures from the ECMWF Reanalysis Project (ERA-15, 1979–1993) were conducted. While the ensemble experiment demonstrates that the predictability of the regional climate varies strongly between different seasons and regions, being weakest during the summer and over continental regions, important sensitivities of the modeling system to parameterization choices are uncovered. In particular, compensating mechanisms related to the long-term representation of the water cycle are revealed, in which summer dry and hot conditions at the surface, resulting from insufficient evaporation, can persist despite insufficient net solar radiation (a result of unrealistic cloud-radiative feedbacks).

The unsteady flow of a weakly compressible fluid in a thin porous layer II: three-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional layer, composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Numerical solution of this three-dimensional evolution problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l, a situation which occurs frequently in the application to oil and gas reservoir recovery and which leads to significant stiffness in the numerical problem. Under the assumption that $\epsilon\propto h/l\ll 1$, we show that, to leading order in $\epsilon$, the pressure field varies only in the horizontal directions away from the wells (the outer region). We construct asymptotic expansions in $\epsilon$ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive expressions for all significant process quantities. The only computations required are for the solution of non-stiff linear, elliptic, two-dimensional boundary-value, and eigenvalue problems. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the layer, $\epsilon$, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighbourhood of wells and away from wells.

The unsteady flow of a weakly compressible fluid in a thin porous layer III: three-dimensional computations

We describe a novel method for determining the pressure and velocity fields for a weakly compressible fluid flowing in a thin three-dimensional layer composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Our approach uses the method of matched asymptotic expansions to derive expressions for all significant process quantities, the computation of which requires only the solution of linear, elliptic, two-dimensional boundary value and eigenvalue problems. In this article, we provide full implementation details and present numerical results demonstrating the efficiency and accuracy of our scheme.

Reviewing the sustainability/stationarity of current account imbalances with tests for bounded integration

We investigate for 26 OECD economies whether their current account imbalances to GDP are driven by stochastic trends. Regarding bounded stationarity as the more natural counterpart of sustainability, results from Phillips–Perron tests for unit root and bounded unit root processes are contrasted. While the former hint at stationarity of current account imbalances for 12 economies, the latter indicate bounded stationarity for only six economies. Through panel-based test statistics, current account imbalances are diagnosed as bounded non-stationary. Thus, (spurious) rejections of the unit root hypothesis might be due to the existence of bounds reflecting hidden policy controls or financial crises.