Bounds on Aggregate Assets

Aggregating financial assets together to form a portfolio, commonly referred to as "asset pooling", is a standard practice in the banking and insurance industries. Determining a suitable probability distribution for this portfolio with each underlying asset is a challenging task unless several distributional assumptions are made. On the other hand, imposing assumptions on the distribution inhibits its ability to capture various idiosyncratic behaviors. It limits the model's usefulness in its ability to provide realistic risk metrics of the true portfolio distribution. In order to conquer this limitation, we propose two methods to model a pool of assets with much less assumptions on the correlation structure by way of finding analytical bounds. Our first method uses the Fréchet-Hoeffding copula bounds to calculate model-free upper and lower bounds for aggregate assets evaluation. For the copulas with specific constraints, we improve the Fréchet- Hoeffding copula bounds by providing bounds with narrower range. The improvements proposed are very robust for different types of constraints on the copula function. However, the lower copula bound does not exist for dimension three and above. Our second method tackles the open problem of finding lower bounds for higher dimensions by introducing the concept of Complete Mixability property. With such technique, we are able to find the lower bounds with specified constraints. Three theorems are proposed. The first theorem deals with the case where all marginal distributions are identical. The lower bound defined by the first theorem is sharp under some technical assumptions. The second theorem gives the lower bound in a more general setup without any restriction on the marginal distributions. However the bound achieved in this context is not sharp. The third theorem gives the sharp lower bound on Conditional VaR. Numerical results are provided for each method to demonstrate sharpness of the bounds. Finally, we point out some possible future research directions, such as looking for a general sharp lower bound for high dimensional correlation structures

VaR bounds for joint portfolios with dependence constraints

Based on a novel extension of classical Hoe ding-Fr\ue9chet bounds, we provide an upper VaR bound for joint risk portfolios with xed marginal distributions and positive dependence information. The positive dependence information can be assumed to hold in the tails, in some central part, or on a general subset of the domain of the distribution function of a risk portfolio. The newly provided VaR bound can be interpreted as a comonotonic VaR computed at a distorted con dence level and its quality is illustrated in a series of examples of practical interest

On the Size of Subclasses of Quasi-Copulas and Their Dedekind-MacNeille Completion

none4siopenDurante Fabrizio; Fernandez-Sanchez Juan; Trutschnig Wolfgang; Ubeda-Flores ManuelDurante, Fabrizio; Fernandez-Sanchez, Juan; Trutschnig, Wolfgang; Ubeda-Flores, Manue

On the Size of Subclasses of Quasi-Copulas and Their Dedekind–MacNeille Completion

We study some topological properties of the class of supermodular n-quasi-copulas and check that the topological size of the Dedekind–MacNeille completion of the set of n-copulas is small, in terms of the Baire category, in the Dedekind–MacNeille completion of the set of the supermodular n-quasi-copulas, and in turn, this set and the set of n-copulas are small in the set of n-quasi-copulas

Best-possible bounds on the set of copulas with a given value of Spearman's footrule

In this paper we find pointwise best-possible bounds on the set of copulas with a given value of the Spearman’s footrule co-efficient. We show that the lower bound is always a copula but, unlike the bounds on sets of copulas with a given value of other measures, such as Kendall’s tau, Spearman’s rho and Blonqvist’s beta, the upper bound can be a copula or a proper quasi-copula. We characterised both of these cases

A quantitative real options method for aviation technology decision-making in the presence of uncertainty

The developments of new technologies for commercial aviation involve significant risk for technologists as these programs are often driven by fixed assumptions regarding future airline needs, while being subject to many uncertainties at the technical and market levels. To prioritize these developments, technologists must assess their economic viability even though standard methods used for capital budgeting are not well suited to handle the overwhelming uncertainty surrounding such developments. This research proposes a framework featuring real options to overcome this challenge. It is motivated by three observations: disregarding the value of managerial flexibility undervalues long-term research and development (R&D) programs; windows of opportunities emerge and disappear and manufacturers can derive significant value by exploiting their upside potential; integrating competitive aspects early in the design ensures that development programs are robust with respect to moves by the competition. Real options analyses have been proposed to address some of these points but the adoption has been slow, hindered by constraining frameworks. A panel of academics and practitioners has identified a set of requirements, known as the Georgetown Challenge, that real options analyses must meet to get more traction amongst practitioners in the industry. In a bid to meet some of these requirements, this research proposes a novel methodology, cross-fertilizing techniques from financial engineering, actuarial sciences, and statistics to evaluate and study the timing of technology developments under uncertainty. It aims at substantiating decision making for R&D while having a wider domain of application and an improved ability to handle a complex reality compared to more traditional approaches. The method named FLexible AViation Investment Analysis (FLAVIA) uses first Monte Carlo techniques to simulate the evolution of uncertainties driving the value of technology developments. A non-parametric Esscher transform is then applied to perform a change of probability measure to express these evolutions under the equivalent martingale measure. A bootstrap technique is suggested next to construct new non-weighted evolutions of the technology development value under the new measure. A regression-based technique is finally used to analyze the technology development program and to discover trigger boundaries which help define when the technology development program should be launched. Verification of the method is performed on several canonical examples and indicates good accuracy and competitive execution time. It is applied next to the analysis of a performance improvement package (PIP) development using the Integrated Cost And Revenue Estimation method (i-CARE) developed as part of this research. The PIP can be retrofitted to currently operating turbofan engines in order to mitigate the impact of the aging process on their operating costs. The PIP is subject to market uncertainties, such as the evolution of jet-fuel prices and the possible taxation of carbon emissions. The profitability of the PIP development is investigated and the value of managerial flexibility and timing flexibility are highlighted.The developments of new technologies for commercial aviation involve significant risk for technologists as these programs are often driven by fixed assumptions regarding future airline needs, while being subject to many uncertainties at the technical and market levels. To prioritize these developments, technologists must assess their economic viability even though standard methods used for capital budgeting are not well suited to handle the overwhelming uncertainty surrounding such developments. This research proposes a framework featuring real options to overcome this challenge. It is motivated by three observations: disregarding the value of managerial flexibility undervalues long-term research and development (R&D) programs; windows of opportunities emerge and disappear and manufacturers can derive significant value by exploiting their upside potential; integrating competitive aspects early in the design ensures that development programs are robust with respect to moves by the competition. Real options analyses have been proposed to address some of these points but the adoption has been slow, hindered by constraining frameworks. A panel of academics and practitioners has identified a set of requirements, known as the Georgetown Challenge, that real options analyses must meet to get more traction amongst practitioners in the industry. In a bid to meet some of these requirements, this research proposes a novel methodology, cross-fertilizing techniques from financial engineering, actuarial sciences, and statistics to evaluate and study the timing of technology developments under uncertainty. It aims at substantiating decision making for R&D while having a wider domain of application and an improved ability to handle a complex reality compared to more traditional approaches. The method named FLexible AViation Investment Analysis (FLAVIA) uses first Monte Carlo techniques to simulate the evolution of uncertainties driving the value of technology developments. A non-parametric Esscher transform is then applied to perform a change of probability measure to express these evolutions under the equivalent martingale measure. A bootstrap technique is suggested next to construct new non-weighted evolutions of the technology development value under the new measure. A regression-based technique is finally used to analyze the technology development program and to discover trigger boundaries which help define when the technology development program should be launched. Verification of the method is performed on several canonical examples and indicates good accuracy and competitive execution time. It is applied next to the analysis of a performance improvement package (PIP) development using the Integrated Cost And Revenue Estimation method (i-CARE) developed as part of this research. The PIP can be retrofitted to currently operating turbofan engines in order to mitigate the impact of the aging process on their operating costs. The PIP is subject to market uncertainties, such as the evolution of jet-fuel prices and the possible taxation of carbon emissions. The profitability of the PIP development is investigated and the value of managerial flexibility and timing flexibility are highlighted.Ph.D

Copula-based statistical modelling of synoptic-scale climate indices for quantifying and managing agricultural risks in australia

Australia is an agricultural nation characterised by one of the most naturally diverse climates in the world, which translates into significant sources of risk for agricultural production and subsequent farm revenues. Extreme climatic events have been significantly affecting large parts of Australia in recent decades, contributing to an increase in the vulnerability of crops, and leading to subsequent higher risk to a large number of agricultural producers. However, attempts at better managing climate related risks in the agricultural sector have confronted many challenges. First, crop insurance products, including classical claim-based and index-based insurance, are among the financial implements that allow exposed individuals to pool resources to spread their risk. The classical claim-based insurance indemnifies according to a claim of crop loss from the insured customer, and so can easily manage idiosyncratic risk, which is the case where the loss occurs independently.Nevertheless, the existence of systemic weather risk (covariate risk), which is the spread of extreme events over locations and times (e.g., droughts and floods), has been identified as the main reason for the failure of private insurance markets, such as the classical multi-peril crop insurance, for agricultural crops. The index-based insurance is appropriate to handle systemic but not idiosyncratic risk. The indemnity payments of the index-based insurance are triggered by a predefined threshold of an index (e.g., rainfall), which is related to such losses. Since the covariate nature of a climatic event, it sanctions the insurers to predict losses and ascertain indemnifications for a huge number of insured customers across a wide geographical area. However, basis risk, which is related to the strength of the relationship between the predefined indices used to estimate the average loss by the insured community and the actual loss of insured assets by an individual, is a major barrier that hinders uptake of the index-based insurance. Clearly, the high basis risk, which is a weak relationship between the index and loss, destroys the willingness of potential customers to purchase this insurance product. Second, the impact of multiple synoptic-scale climate mode indices (e.g., Southern Oscillation Index (SOI) and Indian Ocean Index (IOD)) on precipitation and crop yield is not identical in different spatial locations and at different times or seasons across the Australian continent since the influence of large-scale climate heterogeneous over the different regions. The occurrence, role, and amplitude of synoptic-scale climate modes contributing to the variability of seasonal crop production have shifted in recent decades. These variables generally complicate the climate and crop yield relationship that cannot be captured by traditional modelling and analysis approaches commonly found in published agronomic literature such as linear regression. In addition, the traditional linear analysis is not able to model the nonlinear and asymmetric interdependence between extreme insurance losses, which may occur in the case of systemic risk. Relying on the linear method may lead to the problem that different behaviour may be observed from joint distributions, particularly in the upper and lower regions, with the same correlation coefficient. As a result, the likelihood of extreme insurance losses can be underestimated or overestimated that lead to inaccuracies in the pricing of insurance policies. Another alternative is the use of the multivariate normal distribution, where the joint distribution is uniquely defined using the marginal distributions of variables and their correlation matrix. However, phenomena are not always normally distributed in practice. It is therefore important to develop new, scientifically verified, strategic measures to solve the challenges as mentioned above in order to support mitigating the influences of the climate-related risk in the agricultural sector. Copulas provide an advanced statistical approach to model the joint distribution of multivariate random variables. This technique allows estimating the marginal distributions of individual variables independently with their dependence structures. It is clear that the copula method is superior to the conventional linear regression since it does not require variables have to be normally distributed and their correlation can be either linear or non-linear. This doctoral thesis therefore adopts the advanced copula technique within a statistical modelling framework that aims to model: (1) The compound influence of synoptic-scale climate indices (i.e., SOI and IOD) and climate variables (i.e., precipitation) to develop a probabilistic precipitation forecasting system where the integrated role of different factors that govern precipitation dynamics are considered; (2) The compound influence of synoptic-scale climate indices on wheat yield; (3) The scholastic interdependencies of systemic weather risks where potential adaptation strategies are evaluated accordingly; and (4) The risk-reduction efficiencies of geographical diversifications in wheat farming portfolio optimisation. The study areas are Australia’s agro-ecological (i.e., wheat belt) zones where major seasonal wheat and other cereal crops are grown. The results from the first and second objectives can be used for not only forecasting purposes but also understanding the basis risk in the case of pricing climate index-based insurance products. The third and fourth objectives assess the interactions of drought events across different locations and in different seasons and feasible adaptation tools. The findings of these studies can provide useful information for decision-makers in the agricultural sector. The first study found the significant relationship between SOI, IOD, and precipitation. The results suggest that spring precipitation in Australia, except for the western part, can be probabilistically forecasted three months ahead. It is more interesting that the combination of SOI and IOD as the predictors will improve the performance of the forecast model. Similarly, the second study indicated that the largescale climate indices could provide knowledge of wheat crops up to six months in advance. However, it is noted that the influence of different climate indices varies over locations and times. Furthermore, the findings derived from the third study demonstrated the spatio-temporally stochastic dependence of the drought events. The results also prove that time diversification is potentially more effective in reducing the systemic weather risk compared to spatially diversifying strategy. Finally, the fourth objective revealed that wheat-farming portfolio could be effectively optimised through the geographical diversification. The outcomes of this study will lead to the new application of advanced statistical tools that provide a better understanding of the compound influence of synoptic-scale climatic conditions on seasonal precipitation, and therefore on wheat crops in key regions over the Australian continent. Furthermore, a comprehensive analysis of systemic weather risks performed through advanced copula-statistical models can help improve and develop novel agricultural adaptation strategies in not only the selected study region but also globally, where climate extreme events pose a serious threat to the sustainability and survival of the agricultural industry. Finally, the evaluation of the effectiveness of diversification strategies implemented in this study reveals new evidence on whether the risk pooling methods could potentially mitigate climate risks for the agricultural sector and subsequently, help farmers in prior preparation for uncertain climatic events

Stochastics of Environmental and Financial Economics

Systems Theory, Contro