Optimal Capital Requirements in Financial Networks with Fire Sales

I explore and analyze a problem of finding the socially optimal capital requirements for financial institutions considering two distinct channels of contagion: direct exposures among the institutions, as represented by a network and fire sales externalities, which reflect the negative price impact of massive liquidation of assets.These two channels amplify shocks from individual financial institutions to the financial system as a whole and thus increase the risk of joint defaults amongst the interconnected financial institutions; this is often referred to as systemic risk. In the model, there is a trade-off between reducing systemic risk and raising the capital requirements of the financial institutions. The policymaker considers this trade-off and determines the optimal capital requirements for individual financial institutions. I provide a method for finding and analyzing the optimal capital requirements that can be applied to arbitrary network structures and arbitrary distributions of investment returns.

In particular, I first consider a network model consisting only of direct exposures and show that the optimal capital requirements can be found by solving a stochastic linear programming problem. I then extend the analysis to financial networks with default costs and show the optimal capital requirements can be found by solving a stochastic mixed integer programming problem. The computational complexity of this problem poses a challenge, and I develop an iterative algorithm that can be efficiently executed. I show that the iterative algorithm leads to solutions that are nearly optimal by comparing it with lower bounds based on a dual approach. I also show that the iterative algorithm converges to the optimal solution.

Finally, I incorporate fire sales externalities into the model. In particular, I am able to extend the analysis of systemic risk and the optimal capital requirements with a single illiquid asset to a model with multiple illiquid assets. The model with multiple illiquid assets incorporates liquidation rules used by the banks. I provide an optimization formulation whose solution provides the equilibrium payments for a given liquidation rule.

I further show that the socially optimal capital problem using the ``socially optimal liquidation" and prioritized liquidation rules can be formulated as a convex and convex mixed integer problem, respectively. Finally, I illustrate the results of the methodology on numerical examples and

discuss some implications for capital regulation policy and stress testing.

A soft robust model for optimization under ambiguity

In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decision maker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant. Our primary objective is to develop this framework and relate it to the standard notion of robustness, which deals with only a single guarantee across one uncertainty set. First, we show that our approach connects closely to the theory of convex risk measures. We show that the complexity of this approach is equivalent to that of solving a small number of standard robust problems. We then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach. Finally, we illustrate theme thodology on an asset allocation example consisting of historical market data over a 25-year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable risk-return trade-off: each case results in a higher out-of-sample expected return for a relatively minor degradation of out-of-sample downside performance. © 2010 INFORMS.

Lognormal and gamma mixed negative binomial regression

In regression analysis of counts, a lack of simple and efficient algorithms for posterior computation has made Bayesian approaches appear unattractive and thus underdeveloped. We propose a lognormal and gamma mixed negative binomial (NB) regression model for counts, and present efficient closed-form Bayesian inference; unlike conventional Poisson models, the proposed approach has two free parameters to include two different kinds of random effects, and allows the incorporation of prior information, such as sparsity in the regression coefficients. By placing a gamma distribution prior on the NB dispersion parameter r, and connecting a log-normal distribution prior with the logit of the NB probability parameter p, efficient Gibbs sampling and variational Bayes inference are both developed. The closed-form updates are obtained by exploiting conditional conjugacy via both a compound Poisson representation and a Polya-Gamma distribution based data augmentation approach. The proposed Bayesian inference can be implemented routinely, while being easily generalizable to more complex settings involving multivariate dependence structures. The algorithms are illustrated using real examples. Copyright 2012 by the author(s)/owner(s).

Explanatory Model for the Use of Traditional Medicines in Kilimanjaro, Tanzania

Introduction: Traditional medicines are one of the most important means of achieving total health care coverage globally, and their importance in Tanzania extends beyond the impoverished rural areas. Their use remains high even in urban settings among the educated middle and upper classes. They are a critical component healthcare in Tanzania, but they also can have harmful side effects. Therefore we sought to understand the decision-making and reasoning processes by building an explanatory model for the use of traditional medicines in Tanzania.

Methods: We conducted a mixed-methods study between December 2013 and June 2014 in the Kilimanjaro Region of Tanzania. Using purposive sampling methods, we conducted focus group discussions (FGDs) and in-depth interviews of key informants, and the qualitative data were analyzed using an inductive Framework Method. A structured survey was created, piloted, and then administered it to a random sample of adults. We reported upon the reliability and validity of the structured survey, and we used triangulation from multiple sources to synthesize the qualitative and quantitative data.

Results: A total of five FGDs composed of 59 participants and 27 in-depth interviews were conducted in total. 16 of the in-depth interviews were with self-described traditional practitioners or herbal vendors. We identified five major thematic categories that relate to the decision to use traditional medicines in Kilimanjaro: healthcare delivery, disease understanding, credibility of the traditional practices, health status, and strong cultural beliefs.

A total of 473 participants (24.1% male) completed the structured survey. The most common reasons for taking traditional medicines were that they are more affordable (14%, 12.0-16.0), failure of hospital medicines (13%, 11.1-15.0), they work better (12%, 10.7-14.4), they are easier

to obtain (11%, 9.48-13.1), they are found naturally or free (8%, 6.56-9.68), hospital medicines have too many chemical (8%, 6.33-9.40), and they have fewer side effects (8%, 6.25-9.30). The most common uses of traditional medicines were for symptomatic conditions (42%), chronic diseases (14%), reproductive problems (11%), and malaria and febrile illnesses (10%). Participants currently taking hospital medicines for chronic conditions were nearly twice as likely to report traditional medicines usage in the past year (RR 1.97, p=0.05).

Conclusions: We built broad explanatory model for the use of traditional medicines in Kilimanjaro. The use of traditional medicines is not limited to rural or low socioeconomic populations and concurrent use of traditional medicines and biomedicine is high with frequent ethnomedical doctor shopping. Our model provides a working framework for understanding the complex interactions between biomedicine and traditional medicine. Future disease management and treatment programs will benefit from this understanding, and it can lead to synergistic policies with more effective implementation.