## The basics of quantitative portfolio selection

The seminal work of Harry Markowitz from the 1950s is the first scientific approach towards portfolio selection based on the idea of diversification, constituting a quantitative setup whose core ideas are still prominent in today’s financial industry. The content of the present article consists of three parts. First, the Markowitz theory is summarized, with an emphasis on its relation with the concept of the Sharpe ratio. Second, its limitations and potential generalizations are discussed. Third, it is demonstrated in the particular case of our fund XAIA Credit Curve Carry how the related concept of Sharpe ratio maximization can assist with managing daily portfolio adjustments.

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## Portfolio optimization for credit-risky assets under extreme dependence

We consider power utility maximization in a multivariate Black-Scholes model that is enhanced by credit risk via the Marshall-Olkin exponential distribution. On the practical side, the model is analytically tractable, easy to interpret, and thus simple to implement. On the theoretical side, the model constitutes a well-justified and intuitive mathematical wrapping to study the effect of extreme and higher-order dependence on optimal portfolios. In particular, we show that it is rich enough to model both, situations in which diversification is beneficial and situations in which this is not the case.

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## Geostatistical modeling of financial data

We explore how the joint modeling of financial assets can utilize methodologies from geostatistical modeling. The considered approach is essentially based on modeling data as realizations of a (Gaussian) random field. This allows for a parsimonious representation of the dependence structure by means of a covariance function taken to be a function of the distance between observations. A key beneft of this ansatz is the possibility to include new data points, i.e. to consider new companies in financial applications. Consequently, geostatistical modeling has appealing benefits in the contexts of covariance matrix estimation and missing data imputation. We thoroughly discuss the necessary adjustments when applying geostatistical methods to the high-dimensional framework that entails the modeling of financial data, instead of the 2D/3D coordinate space encountered in original applications of the method. We illustrate the two use cases of covariance matrix estimation and missing data imputation on a data set of CDS spreads of constituents of the iTraxx universe.

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## Simulating realistic correlation matrices for financial applications

The major part of observed correlation matrices in financial applications exhibits the Perron-Frobenius property, namely a dominant eigenvector with only positive entries. We present a simulation algorithm for random correlation matrices satisfying this property, which can be augmented to take into account a realistic eigenvalue structure. From the construction principle applied in our algorithm, and the fact that it is able to generate all such correlation matrices, we are further able to compute explicitly the proportion of Perron-Frobenius correlation matrices in the set of all correlation matrices in a fixed dimension.

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## Consistent iterated simulation of multivariate defaults

A current market-practice to incorporate multivariate defaults in global risk-factor simulations is the iteration of (multiplicative) i.i.d. survival indicator increments along a given time-grid, where the indicator distribution is based on a copula ansatz. The underlying assumption is that the behavior of the resulting iterated default distribution is similar to the one-shot distribution. It is shown that in most cases this assumption is not fulfilled and furthermore numerical analysis is presented that shows sizable differences in probabilities assigned to both “survival-of-all” and “mixed default/survival” events. We furthermore present a survey of those copula families that make the aforementioned methodology work.

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