# Haircut Capital Allocation as the Solution of a Quadratic Optimisation Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Risk Capital Allocation as a Quadratic Optimisation Problem

- (a)
- A function $D:\mathbb{R}\to {\mathbb{R}}^{+}$;
- (b)
- A set of positive values ${v}_{j}$, $j=1,...,n$;
- (c)
- A set of random variables ${\zeta}_{j}$ such that $\mathbb{E}\left[{\zeta}_{j}\right]>0$, $j=1,...,n$.

**Proposition**

**1.**

**Proof of Proposition**

**1.**

**Remark**

**1.**

- (b)
- A set of weights ${v}_{j}$, $j=1,...,n$;
- (c)
- A set of random variables ${\zeta}_{j}$, $j=1,...,n$, with $\frac{{v}_{j}}{\mathbb{E}\left[{\zeta}_{i}\right]}>0$.

**Remark**

**2.**

- (b)
- A set of non-negative weights ${v}_{j}$, $j=1,...,n$, such that ${\sum}_{i=1}^{n}{v}_{i}=1$;
- (c)
- A set of non-negative random variables ${\zeta}_{j}$, $j=1,...,n$, with $\mathbb{E}\left[{\zeta}_{j}\right]=1$.

## 3. Haircut Allocation Principle

**Lemma**

**1.**

- (a)
- $\mathbb{E}\left[\zeta \right]=1$;
- (b)
- $\mathbb{E}\left[\zeta X\right]=c$.

**Proof of Lemma**

**1.**

**Remark**

**3.**

**Proposition**

**2.**

- (a)
- $D\left(x\right)={x}^{2}$;
- (b)
- ${v}_{i}={\displaystyle \frac{\mathbb{E}\left[{\zeta}_{i}{X}_{i}\right]}{{\displaystyle \sum _{j=1}^{n}\mathbb{E}\left[{\zeta}_{j}{X}_{j}\right]}}}$, $i=1,\dots ,n$;
- (c)
- ${\zeta}_{i}={\displaystyle \frac{\left({Y}_{i}-\mathbb{E}\left[{Y}_{i}\right]\right){F}_{{X}_{i}}^{-1}\left(\alpha \right)+\mathbb{E}\left[{X}_{i}{Y}_{i}\right]-\mathbb{E}\left[{X}_{i}\right]{Y}_{i}}{\mathbb{E}\left[{X}_{i}{Y}_{i}\right]-\mathbb{E}\left[{X}_{i}\right]\mathbb{E}\left[{Y}_{i}\right]}},$ where ${Y}_{i}$ is a random variable such that $\mathbb{E}\left[{X}_{i}{Y}_{i}\right]\ne \mathbb{E}\left[{X}_{i}\right]\mathbb{E}\left[{Y}_{i}\right]$, for all $i=1,\dots ,n$.

**Proof of Proposition**

**2.**

**Remark**

**4.**

## 4. Examples of $\mathbf{\zeta}$ in Haircut Allocation

**Example**

**1.**

**Proof of Example**

**1.**

**Remark**

**5.**

**Remark**

**6.**

**Example**

**2.**

**Remark**

**7.**

## 5. Illustrative Application

`$\widehat{\phantom{\rule{0.277778em}{0ex}}}$GSPC`,

`$\widehat{\phantom{\rule{0.277778em}{0ex}}}$IXIC`,

`$\widehat{\phantom{\rule{0.277778em}{0ex}}}$DJI`, and

`$\widehat{\phantom{\rule{0.277778em}{0ex}}}$NYA`). We want to analyse the risk of each investment fund, on a weekly basis, if no additional information other than the index of reference is available. In addition, we are interested in ranking the investment funds based on their relative riskiness. As is shown hereinafter, these objectives may be reached with the application of the haircut capital allocation principle.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Lemma**

**A1.**

**Proof.**

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**Figure 1.**Weekly returns of the NASDAQ index for the period under study, followed by the sample values of the two weighted random variables ${\zeta}_{2}$ reported in Table 3 and Table 4, respectively. The ${\zeta}_{2}$ weighted random variable derived from Example 1 is always positive in this application.

$\mathbb{E}\left[XY\right]-\mathbb{E}\left[X\right]\mathbb{E}\left[Y\right]>0$ | $\mathbb{E}\left[XY\right]-\mathbb{E}\left[X\right]\mathbb{E}\left[Y\right]<0$ | |

$c-\mathbb{E}\left[X\right]>0$ | $Y\left(\omega \right)\le {\displaystyle \frac{c\mathbb{E}\left[Y\right]-\mathbb{E}\left[XY\right]}{c-\mathbb{E}\left[X\right]}}$ | $Y\left(\omega \right)\ge {\displaystyle \frac{c\mathbb{E}\left[Y\right]-\mathbb{E}\left[XY\right]}{c-\mathbb{E}\left[X\right]}}$ |

$c-\mathbb{E}\left[X\right]<0$ | $Y\left(\omega \right)\ge {\displaystyle \frac{c\mathbb{E}\left[Y\right]-\mathbb{E}\left[XY\right]}{c-\mathbb{E}\left[X\right]}}$ | $Y\left(\omega \right)\le {\displaystyle \frac{c\mathbb{E}\left[Y\right]-\mathbb{E}\left[XY\right]}{c-\mathbb{E}\left[X\right]}}$ |

**Table 2.**Estimation of the haircut allocation principle for the four investment funds of the illustration. The expected return of the fund (expectation of $-{X}_{i}$) is approximated by the sample mean with a negative sign, while the risk of the fund is approximated by the historical VaR of the sample.

i | Index of Reference | $\mathbb{E}[-{\mathit{X}}_{\mathit{i}}]$ | ${\mathbf{VaR}}_{51/52}\left({\mathit{X}}_{\mathit{i}}\right)$ | ${\mathit{K}}_{\mathit{i}}$ |
---|---|---|---|---|

1 | S&P500 | 0.242% | 5.338% | 229.08 |

2 | NASDAQ | 0.326% | 6.218% | 266.84 |

3 | DJI | 0.198% | 5.975% | 256.42 |

4 | NYA | 0.099% | 5.771% | 247.66 |

**Table 3.**Expressions of ${\zeta}_{i}$ defined in Example 1 to accommodate the haircut allocation principle into the framework (1). Note that $\alpha $ is approximated by $0.98084$, ${F}_{{X}_{i}}^{-1}\left(\alpha \right)$ is approximated by the historical VaR, and the (conditional) expectations are approximated by the sample means. Therefore, the $\zeta $s are also approximations.

i | Index | ${\mathit{\zeta}}_{\mathit{i}}$ |
---|---|---|

1 | S&P500 | $\frac{\left(\mathbb{1}\left[{X}_{1}\le 5.338\%\right]-0.98084\right)5.338\%-0.378\%+0.242\%\mathbb{1}\left[{X}_{1}\le 5.338\%\right]}{-0.378\%+0.242\%0.98084}$ |

2 | NASDAQ | $\frac{\left(\mathbb{1}\left[{X}_{2}\le 6.218\%\right]-0.98084\right)6.218\%-0.462\%+0.326\%\mathbb{1}\left[{X}_{2}\le 6.218\%\right]}{-0.462\%+0.326\%0.98084}$ |

3 | DJI | $\frac{\left(\mathbb{1}\left[{X}_{3}\le 5.975\%\right]-0.98084\right)5.975\%-0.345\%+0.198\%\mathbb{1}\left[{X}_{3}\le 5.975\%\right]}{-0.345\%+0.198\%0.98084}$ |

4 | NYA | $\frac{\left(\mathbb{1}\left[{X}_{4}\le 5.771\%\right]-0.98084\right)5.771\%-0.276\%+0.099\%\mathbb{1}\left[{X}_{4}\le 5.771\%\right]}{-0.276\%+0.099\%0.98084}$ |

**Table 4.**Expressions of ${\zeta}_{i}$ defined in Example 2 to accommodate the haircut allocation principle into the framework (1). Note that ${F}_{{X}_{i}}^{-1}\left(\alpha \right)$ is approximated by the historical VaR, and the expectations and variances are approximated by the sample means and sample variances, respectively. Therefore, the $\zeta $s are also approximations.

i | Index | ${\mathit{\zeta}}_{\mathit{i}}$ |
---|---|---|

1 | S&P500 | $\frac{\left({X}_{1}+0.242\%\right)5.338\%+0.062\%+0.242\%{X}_{1}}{0.061\%}$ |

2 | NASDAQ | $\frac{\left({X}_{2}+0.326\%\right)6.218\%+0.084\%+0.326\%{X}_{2}}{0.083\%}$ |

3 | DJI | $\frac{\left({X}_{3}+0.198\%\right)5.975\%+0.064\%+0.198\%{X}_{3}}{0.063\%}$ |

4 | NYA | $\frac{\left({X}_{4}+0.099\%\right)5.771\%+0.070\%+0.099\%{X}_{4}}{0.070\%}$ |

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**MDPI and ACS Style**

Belles-Sampera, J.; Guillen, M.; Santolino, M.
Haircut Capital Allocation as the Solution of a Quadratic Optimisation Problem. *Mathematics* **2023**, *11*, 3846.
https://doi.org/10.3390/math11183846

**AMA Style**

Belles-Sampera J, Guillen M, Santolino M.
Haircut Capital Allocation as the Solution of a Quadratic Optimisation Problem. *Mathematics*. 2023; 11(18):3846.
https://doi.org/10.3390/math11183846

**Chicago/Turabian Style**

Belles-Sampera, Jaume, Montserrat Guillen, and Miguel Santolino.
2023. "Haircut Capital Allocation as the Solution of a Quadratic Optimisation Problem" *Mathematics* 11, no. 18: 3846.
https://doi.org/10.3390/math11183846