# Spectral risk measures: the risk quadrangle and optimal approximation

## Abstract

We develop a general risk quadrangle that gives rise to a large class of spectral risk measures. The statistic of this new risk quadrangle is the average value-at-risk at a specific confidence level. As such, this risk quadrangle generates a continuum of error measures that can be used for superquantile regression. For risk-averse optimization, we introduce an optimal approximation of spectral risk measures using quadrature. Lastly, we prove the consistency of this approximation and demonstrate our results through numerical examples.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1441457

- Report Number(s):
- SAND-2018-5344J

Journal ID: ISSN 0025-5610; 663229

- Grant/Contract Number:
- AC04-94AL85000; NA0003525

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Mathematical Programming

- Additional Journal Information:
- Journal Name: Mathematical Programming; Journal ID: ISSN 0025-5610

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Stochastic optimization; Risk measures; Regression; Quadrature; Average value-at-risk

### Citation Formats

```
Kouri, Drew P. Spectral risk measures: the risk quadrangle and optimal approximation. United States: N. p., 2018.
Web. doi:10.1007/s10107-018-1267-3.
```

```
Kouri, Drew P. Spectral risk measures: the risk quadrangle and optimal approximation. United States. doi:10.1007/s10107-018-1267-3.
```

```
Kouri, Drew P. Thu .
"Spectral risk measures: the risk quadrangle and optimal approximation". United States. doi:10.1007/s10107-018-1267-3. https://www.osti.gov/servlets/purl/1441457.
```

```
@article{osti_1441457,
```

title = {Spectral risk measures: the risk quadrangle and optimal approximation},

author = {Kouri, Drew P.},

abstractNote = {We develop a general risk quadrangle that gives rise to a large class of spectral risk measures. The statistic of this new risk quadrangle is the average value-at-risk at a specific confidence level. As such, this risk quadrangle generates a continuum of error measures that can be used for superquantile regression. For risk-averse optimization, we introduce an optimal approximation of spectral risk measures using quadrature. Lastly, we prove the consistency of this approximation and demonstrate our results through numerical examples.},

doi = {10.1007/s10107-018-1267-3},

journal = {Mathematical Programming},

number = ,

volume = ,

place = {United States},

year = {2018},

month = {5}

}

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