This paper studies a probabilistic characteristic called buffered probability of exceedance (bPOE). It is a function of a random variable and a real-valued threshold. By definition, bPOE is the pro...

Buffered Probability of Exceedance: Mathematical Properties and Optimization

Conditional value-at-risk (CVaR) and value-at-risk, also called the superquantile and quantile, are frequently used to characterize the tails of probability distributions and are popular measures of risk in applications where the distribution represents the magnitude of a potential loss. buffered probability of exceedance (bPOE) is a recently introduced characterization of the tail which is the inverse of CVaR, much like the CDF is the inverse of the quantile. These quantities can prove very useful as the basis for a variety of risk-averse parametric engineering approaches. Their use, however, is often made difficult by the lack of well-known closed-form equations for calculating these quantities for commonly used probability distributions. In this paper, we derive formulas for the superquantile and bPOE for a variety of common univariate probability distributions. Besides providing a useful collection within a single reference, we use these formulas to incorporate the superquantile and bPOE into parametric procedures. In particular, we consider two: portfolio optimization and density estimation. First, when portfolio returns are assumed to follow particular distribution families, we show that finding the optimal portfolio via minimization of bPOE has advantages over superquantile minimization. We show that, given a fixed threshold, a single portfolio is the minimal bPOE portfolio for an entire class of distributions simultaneously. Second, we apply our formulas to parametric density estimation and propose the method of superquantiles (MOS), a simple variation of the method of moments where moments are replaced by superquantiles at different confidence levels. With the freedom to select various combinations of confidence levels, MOS allows the user to focus the fitting procedure on different portions of the distribution, such as the tail when fitting heavy-tailed asymmetric data.

Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation

We develop recursive, data-driven, stochastic subgradient methods for optimizing a new, versatile, and application-driven class of convex risk measures, termed here as mean-semideviations, strictly generalizing the well-known and popular mean-upper-semideviation. We introduce the MESSAGEp algorithm, which is an efficient compositional subgradient procedure for iteratively solving convex mean-semideviation risk-averse problems to optimality. We analyze the asymptotic behavior of the MESSAGEp algorithm under a flexible and structure-exploiting set of problem assumptions. In particular: 1) Under appropriate stepsize rules, we establish pathwise convergence of the MESSAGEp algorithm in a strong technical sense, confirming its asymptotic consistency. 2) Assuming a strongly convex cost, we show that, for fixed semideviation order $p>1$ and for $\epsilon\in\left[0,1\right)$, the MESSAGEp algorithm achieves a squared-${\cal L}_{2}$ solution suboptimality rate of the order of ${\cal O}(n^{-\left(1-\epsilon\right)/2})$ iterations, where, for $\epsilon>0$, pathwise convergence is simultaneously guaranteed. This result establishes a rate of order arbitrarily close to ${\cal O}(n^{-1/2})$, while ensuring strongly stable pathwise operation. For $p\equiv1$, the rate order improves to ${\cal O}(n^{-2/3})$, which also suffices for pathwise convergence, and matches previous results. 3) Likewise, in the general case of a convex cost, we show that, for any $\epsilon\in\left[0,1\right)$, the MESSAGEp algorithm with iterate smoothing achieves an ${\cal L}_{1}$ objective suboptimality rate of the order of ${\cal O}(n^{-\left(1-\epsilon\right)/\left(4\bf{1}_{\left\{ p>1\right\} }+4\right)})$ iterations. This result provides maximal rates of ${\cal O}(n^{-1/4})$, if $p\equiv1$, and ${\cal O}(n^{-1/8})$, if $p>1$, matching the state of the art, as well.

Recursive Optimization of Convex Risk Measures: Mean-Semideviation Models

We propose a new risk-constrained reformulation of the standard Linear Quadratic Regulator (LQR) problem. Our framework is motivated by the fact that the classical (risk-neutral) LQR controller, although optimal in expectation, might be ineffective under relatively infrequent, yet statistically significant (risky) events. To effectively trade between average and extreme event performance, we introduce a new risk constraint, which explicitly restricts the total expected predictive variance of the state penalty by a user-prescribed level. We show that, under rather minimal conditions on the process noise (i.e., finite fourth-order moments), the optimal risk-aware controller can be evaluated explicitly and in closed form. In fact, it is affine relative to the state, and is always internally stable regardless of parameter tuning. Our new risk-aware controller: i) pushes the state away from directions where the noise exhibits heavy tails, by exploiting the third-order moment (skewness) of the noise; ii) inflates the state penalty in riskier directions, where both the noise covariance and the state penalty are simultaneously large. The properties of the proposed risk-aware LQR framework are also illustrated via indicative numerical examples.

Risk-Constrained Linear-Quadratic Regulators

Bond immunization is an important topic in portfolio management. This paper presents a scenario based optimization framework for solving a cash flow matching problem. In this problem, the time horizon of the cash flow generated by the liability is longer than the maturities of the available bonds, and the interest rates are uncertain. Bond purchase decisions are made each period to generate cash flows to cover the obligations due in the future. We use buffered probability of exceedance (bPOE) and conditional value-at-risk (CVaR) to control for the risk of shortfalls. The initial cost of the hedging portfolio of bonds is minimized and optimal positions in bonds are calculated at all time periods. We also study the methodology when solving the optimization problem to minimize bPOE instead of CVaR, which has important practical relevance. The methodology we present in this paper is quite general and can be extended to other financial optimization problems. We use portfolio safeguard optimization package to solve the optimization problems.

Cash flow matching with risks controlled by buffered probability of exceedance and conditional value-at-risk

Bond immunization is an important topic in portfolio management. This project demonstrates a scenario based optimization framework for solving a cash ow

/pdf/cash-flow-matching-problem-with-cvar-constraints-a-case-5g6dmzl5d1.pdf

CASH FLOW MATCHING PROBLEM WITH CVaR CONSTRAINTS: A CASE STUDY WITH PORTFOLIO SAFEGUARD

Bond immunization is an important topic in portfolio management. This paper demonstrates a scenario based optimization framework for solving a cash

/pdf/cash-flow-matching-with-risks-controlled-by-bpoe-and-cvar-2c9ek579ec.pdf

CASH FLOW MATCHING WITH RISKS CONTROLLED BY bPOE AND CVaR

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Danjue Shang

Author Tools

Chat about Author

Papers

Cash flow matching with risks controlled by buffered probability of exceedance and conditional value-at-risk

CASH FLOW MATCHING PROBLEM WITH CVaR CONSTRAINTS: A CASE STUDY WITH PORTFOLIO SAFEGUARD

CASH FLOW MATCHING WITH RISKS CONTROLLED BY bPOE AND CVaR