Foster-Hart Optimal Portfolios

Abstract

We reinvestigate the classic portfolio optimization problem where the notion of portfolio risk is captured by the "Foster-Hart risk"-a new, bankruptcy-proof, reserve based measure of risk, extremely sensitive to left tail events (Foster and Hart, 2009). To include financial market frictions induced by market microstructure, we employ a general, ex-ante transaction cost function with fixed, linear and quadratic penalty terms in the objective function. We represent the US equity market by the Dow Jones Industrial Average (DJIA) index and study the performance of the Foster-Hart optimal DJIA portfolio. In order to capture the skewed and leptokurtotic nature of real life stock returns, we model the returns of the DJIA constituents as an ARMA-GARCH process with multivariate "normal tempered stable" innovations. We demonstrate that the Foster-Hart optimal portfolio's performance is superior to those obtained under several techniques currently in use in academia and industry.

Systemic Risk Modeling with Lévy Copulas

“…To price options by no-arbitrage arguments, a price process must be a semi-martingale (Shiryaev 1999), for example an NTS process, which can be characterized as a solution to a backward PDE. VaR and CVaR under the ARMA-GARCH with Tempered Stable Innovation have been studied in (Kim et al 2011), and the multidimensional ARMA-GARCH-NTS was investigated in (Anand et al 2016). Based on the previous research, we propose a CoVaR model with an NTS copula and innovations to produce semi-martingale price processes.…”

“…For the marginal models, we consider that the market log returns (X j (t), t ≥ 0) for institution j follow an ARMA-GARCH process. Using this model, first the marginal residuals are extracted and then they are analyzed further (Anand et al 2016). Specifically, we adopt the common assumption of ARMA(1,1)-GARCH(1,1) stochastic process:…”

Spatial Graphic Interpretation of the Foster-Hart Formula

“…In practice, stock exchange investors may be interested in it even on the grounds that it disregards the preferences, pointing to instruments that would be too risky for them. It should be mentioned that according to a recent study, it has been shown that the optimal portfolios built with its use have high performance [Anand et al 2016]. This means that the graphic interpretation, unprecedented in literature, may be regarded as interesting, especially if it is used in the process of evaluating various investment assets, including financial instruments, as well as for the construction of portfolios.…”

Cryptocurrency portfolio optimization with multivariate normal tempered stable processes and Foster-Hart risk

“…Our time series model is the multivariate normal tempered stable (MNTS) distributed GARCH model. The MNTS distribution (Kim et al, 2012) has demonstrated excellent fit to joint dynamics of physical asset returns in a number of empirical studies (Anand et al, 2016;Anand et al, 2017;Bianchi and Tassinari, 2020;Kim et al, 2015;Kurosaki and Kim, 2013a;Kurosaki and Kim, 2013b;Kurosaki and Kim, 2019;and Shao et al, 2015). Our portfolio optimization strategy is based on Foster-Hart risk, which is very sensitive to risky left tail events.…”