Applications of Stochastic Control to Portfolio Selection Problems

Portfolio selection is an important problem both in academia and in practice. Due to its significance, it has received great attention and facilitated a large amount of research. This thesis is devoted to structuring optimal portfolios using different criteria. Participating contracts are popular insurance policies, in which the payoff to a policyholder is linked to the performance of a portfolio managed by the insurer. In Chapter 2, we consider the portfolio selection problem of an insurer that offers participating contracts and has an S-shaped utility function. Applying the martingale approach, closed-form solutions are obtained. The resulting optimal strategies are compared with two portfolio insurance hedging strategies, e.g. Constant Proportion Portfolio Insurance strategy and Option Based Portfolio Insurance strategy. We also study numerical solutions of the portfolio selection problem with constraints on the portfolio weights. In Chapter 3, we consider the portfolio selection problem of maximizing a performance measure in a continuous-time diffusion model. The performance measure is the ratio of the overperformance to the underperformance of a portfolio relative to a benchmark. Following a strategy from fractional programming, we analyze the problem by solving a family of related problems, where the objective functions are the numerator of the original problem minus the denominator multiplied by a penalty parameter. These auxiliary problems can be solved using the martingale method for stochastic control. The existence of a solution is discussed in a general setting and explicit solutions are derived when both the reward and the penalty functions are power functions. In Chapter 4, we consider the mean-risk portfolio selection problem of optimizing the expectile risk measure in a continuous-time diffusion model. Due to the lack of an explicit form for expectiles and the close relationship with the Omega measure, we propose an alternative optimization problem with the Omega measure as an objective and show the equivalence between the two problems. After showing the solution for the mean-expectile problem is not attainable but the value function is finite, we modify the problem with an upper bound constraint imposed on the terminal wealth and obtain the solution via the Lagrangian duality method and pointwise optimization technique. The global expectile minimizing portfolio and efficient frontier are also considered in our analysis. In Chapter 5, we consider the utility-based portfolio selection problem in a continuous-time setting. We assume the market price of risk depends on a stochastic factor that satisfies an affine-form, square-root, Markovian model. This financial market framework includes the classical geometric Brownian motion, the constant elasticity of variance (CEV) model and the Heston's model as special cases. Adopting the Backward Stochastic Differential Equation (BSDE) approach, we obtain the closed-form solutions for power, logarithm, or exponential utility functions, respectively. Concluding remarks and several potential topics for further research are presented in Chapter 6

Estimating Portfolio Risk for Tail Risk Protection Strategies

We forecast portfolio risk for managing dynamic tail risk protection strategies, based on extreme value theory, expectile regression, Copula-GARCH and dynamic GAS models. Utilizing a loss function that overcomes the lack of elicitability for Expected Shortfall, we propose a novel Expected Shortfall (and Value-at-Risk) forecast combination approach, which dominates simple and sophisticated standalone models as well as a simple average combination approach in modelling the tail of the portfolio return distribution. While the associated dynamic risk targeting or portfolio insurance strategies provide effective downside protection, the latter strategies suffer less from inferior risk forecasts given the defensive portfolio insurance mechanics

Mean-Expectile Portfolio Selection

This is a post-peer-review, pre-copyedit version of an article published in Applied Mathematics & Optimization. The final authenticated version is available online at: https://doi.org/10.1007/s00245-019-09601-1.We consider a mean-expectile portfolio selection problem in a continuous-time diffusion model. We exploit the close relationship between expectiles and the Omega performance measure to reformulate the problem as the maximization of the Omega measure, and show the equivalence between the two problems. After showing that the solution for the mean-expectile problem is not attainable but that the value function is finite, we modify the problem by introducing a bound on terminal wealth and obtain the solution by Lagrangian duality. The global expectile minimizing portfolio and efficient frontier with a terminal wealth bound are also discussed.NSERC, RGPIN-2017-04220 || NSERC, RGPIN-2016-04001

A new family of expectiles and its properties

This paper considers a risk measure called expectile. We propose a new expression defining expectile, using maximization of CVaR by changing confidence level. This expression is specified for continuous and finite discrete distribution. It is proved that the optimal value of the confidence level is equal to the CDF of expectile value. We also consider a new family of expectiles defined by two parameters. Сomparison of different new expectiles with quantile for a set of distributions shows that proposed expectiles are closer to the quantile than the standard expectile. Two variants of expectile linearization are proposed and it is shown how to use them with linear loss function. Finally, we build three fundamental risk quadrangles where expectile is a statistic and risk.Мета роботи. Як правило, експектиль порівнюється із квантилем. Наша мета – порівняти експектиль із суперквантилем (CVaR), використовуючи однаковий параметр – рівень довіри. Для цього спочатку дається нове представлення експектиля через зважену суму середнього та CVaR. Потім розглядається нове сімейство експектилей, яке задається двома параметрами. Такі експектилі порівнюються з квантилем та CVaR для різних неперервних та скінчених дискретних розподілів. Ще одна мета – побудувати регулярний ризик-квадрат, де експектиль є функцією ризику.Цель работы. Как правило, экспектиль сравнивается с квантилем. Наша задача – сравнить экспектиль с суперквантилем (CVaR), используя одинаковый параметр – уровень доверия. Для этого мы сначала даем новое представление экспектиля через взвешенную сумму среднего и CVaR. Потом рассматриваем новое семейство экспектилей, которое задается двумя параметрами. Такие экспектили сравниваются с квантилем и CVaR для разных непрерывных и конечных дискретных распределений. Еще одна цель – построить регулярный риск-квадрат, где экспектиль является функцией риска

Scenario aggregation method for portfolio expectile optimization

The statistical functional expectile has recently attracted the attention of researchers in the area of risk management, because it is the only risk measure that is both coherent and elicitable. In this article, we consider the portfolio optimization problem with an expectile objective. Portfolio optimization problems corresponding to other risk measures are often solved by formulating a linear program (LP) that is based on a sample of asset returns. We derive three different LP formulations for the portfolio expectile optimization problem, which can be considered as counterparts to the LP formulations for the Conditional Value-at-Risk (CVaR) objective in the works of Rockafellar and Uryasev [43], Ogryczak and Śliwiński [41] and Espinoza and Moreno [21]. When the LPs are based on a simulated sample of the true (assumed continuous) asset returns distribution, the portfolios obtained from the LPs are only approximately optimal. We conduct a numerical case study estimating the suboptimality of the approximate portfolios depending on the sample size, number of assets, and tail-heaviness of the asset returns distribution. Further, the computation times using the three LP formulations are analyzed, showing that the formulation that is based on a scenario aggregation approach is considerably faster than the two alternatives

Elicitability and backtesting: Perspectives for banking regulation

Conditional forecasts of risk measures play an important role in internal risk management of financial institutions as well as in regulatory capital calculations. In order to assess forecasting performance of a risk measurement procedure, risk measure forecasts are compared to the realized financial losses over a period of time and a statistical test of correctness of the procedure is conducted. This process is known as backtesting. Such traditional backtests are concerned with assessing some optimality property of a set of risk measure estimates. However, they are not suited to compare different risk estimation procedures. We investigate the proposal of comparative backtests, which are better suited for method comparisons on the basis of forecasting accuracy, but necessitate an elicitable risk measure. We argue that supplementing traditional backtests with comparative backtests will enhance the existing trading book regulatory framework for banks by providing the correct incentive for accuracy of risk measure forecasts. In addition, the comparative backtesting framework could be used by banks internally as well as by researchers to guide selection of forecasting methods. The discussion focuses on three risk measures, Value-at-Risk, expected shortfall and expectiles, and is supported by a simulation study and data analysis

Risk-aware linear bandits with convex loss

In decision-making problems such as the multi-armed bandit, an agent learns sequentially by optimizing a certain feedback. While the mean reward criterion has been extensively studied, other measures that reflect an aversion to adverse outcomes, such as mean-variance or conditional value-at-risk (CVaR), can be of interest for critical applications (healthcare, agriculture). Algorithms have been proposed for such risk-aware measures under bandit feedback without contextual information. In this work, we study contextual bandits where such risk measures can be elicited as linear functions of the contexts through the minimization of a convex loss. A typical example that fits within this framework is the expectile measure, which is obtained as the solution of an asymmetric least-square problem. Using the method of mixtures for supermartingales, we derive confidence sequences for the estimation of such risk measures. We then propose an optimistic UCB algorithm to learn optimal risk-aware actions, with regret guarantees similar to those of generalized linear bandits. This approach requires solving a convex problem at each round of the algorithm, which we can relax by allowing only approximated solution obtained by online gradient descent, at the cost of slightly higher regret. We conclude by evaluating the resulting algorithms on numerical experiments

Advancing Systematic and Factor Investing Strategies using Alternative Data and Machine Learning

This thesis advances systematic and factor investing strategies using alternative data and machine learning techniques. The first chapter studies the relevance of high-frequency news data for low-frequency factor investing strategies. We build various news-based equity factors for an investable global equity universe to investigate the factors’ ability to extend the information inherent in standard factor models. Specifically, we document that incorporating news-based equity factors benefits multi-factor equity investments, employing diversified multi-factor equity allocations but also more dynamic factor timing strategies. The second chapter examines dynamic asset allocation strategies that focus on explicit downside risk management. We investigate suitable risk models that best inform tail risk protection strategies. In addition to forecasting portfolio risk based on standalone models such as extreme value theory or copula-GARCH, we propose a novel expected shortfall (ES) and value-at-risk (VaR) forecast combination approach that utilizes a loss function that overcomes the lack of elicitability for ES. This forecast combination method dominates simple and sophisticated standalone models as well as a simple average combination approach in terms of statistical accuracy. While the associated dynamic risk targeting or portfolio insurance strategies provide effective downside protection, the latter strategies suffer less from inferior risk forecasts, given the defensive portfolio insurance mechanics. The third chapter extends the above ES and VaR forecast combination approach using machine learning techniques. Building on a rich predictor set of VaR and ES forecasts from an array of econometric models (including GARCH, CAViaR-EVT, dynamic GAS and realized range models), we leverage shrinkage and neural network models to form combination forecasts. Such machine-learned VaR and ES forecasts outperform a set of competing forecast combination approaches in terms of statistical accuracy as well as economical relevance in dynamic tail risk protection strategies

Multivariate Growth Models

This thesis deals with models for (human) growth. We argue that growth is a multivariate phenomenon that should also be analysed as such. This is why we need multivariate growth models and multivariate reference curves. An example of a multivariate growth model is the Multivariate Superimposition by Translation And Rotation model (MSITAR), an extension of the SITAR model by Cole. In the SITAR model, there is a general, or template function for the overall growth pattern which is modelled by a spline. From this template the individual growth curves are then derived by shifting it horizontally and vertically and stretching it in the horizontal direction. Multivariate reference curves can be created by projecting the data in a large number of directions and then calculating the reference for that direction in the usual (univariate) manner. Quantile can be used to this end. Here we show that expectiles (are a generalisation of the mean in the same way that quantiles generalise the median) are an interesting alternative