# Desirable Portfolios in Fixed Income Markets: Application to Credit Risk Premiums

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- Subadditivity: For all $X,Y\in \mathfrak{R}$, $\rho (X+Y)\le \rho (X)+\rho (Y)$.
- Positive homogeneity: For all $t\ge 0$ and $X\in \mathfrak{R}$, $\rho (tX)=t\rho (X)$.
- Translation invariance: For all $X\in \mathfrak{R}$ and all $a\in \mathbb{R}$, $\rho (X+a)=\rho (X)-a$.
- Monotonicity: For all $X,Y\in \mathfrak{R}$, if $X\le Y$ then $\rho (Y)\le \rho (X)$.
- Convexity: For all $0\le \lambda \le 1$, $\rho (\lambda X+(1-\lambda )Y)\le \lambda \rho (X)+(1-\lambda )\rho (Y).$

**Assumption**

**1.**

**Corollary**

**1.**

**Remark**

**1.**

## 3. Measurement of Desirable Portfolios

#### 3.1. First Problem: Maximizing the Income

**Definition**

**1.**

**Assumption**

**2.**

**Theorem**

**1.**

- The equivalent dual form of the primal problem is$$\begin{array}{c}\mathit{Minimize}\phantom{\rule{0.277778em}{0ex}}\theta +\alpha \beta {k}_{v},\hfill \\ \mathit{such}\phantom{\rule{4.pt}{0ex}}\mathit{that}\phantom{\rule{0.277778em}{0ex}}{p}_{j}=-{\lambda}_{j}+\alpha \beta \mathbb{E}[{\pi}_{T}^{j}{z}_{v}],\hfill \\ \phantom{\rule{2.cm}{0ex}}\overline{\lambda}\le \theta \overline{p},\hfill \\ \phantom{\rule{2.cm}{0ex}}\theta \ge 0,\phantom{\rule{1.em}{0ex}}\overline{\lambda}\ge 0,\phantom{\rule{1.em}{0ex}}\alpha \ge 0,\phantom{\rule{1.em}{0ex}}\beta \ge 0,\phantom{\rule{1.em}{0ex}}({z}_{v},{k}_{v})\in {\Delta}_{(\rho ,r)}.\hfill \end{array}$$
- $({\overline{x}}^{*},{\overline{h}}^{*})$ and $({\theta}^{*},{\overline{\lambda}}^{*},{\alpha}^{*},{\beta}^{*},{z}_{v}^{*},{k}_{v}^{*})$ solve problems in Equations (5) and (8), respectively, if and only if they satisfy the following KKT conditions$$\begin{array}{c}{p}_{j}=-{\lambda}_{j}^{*}+{\alpha}^{*}{\beta}^{*}\mathbb{E}[{\pi}_{T}^{j}{z}_{v}^{*}],\phantom{\rule{1.em}{0ex}}j=1,2,\dots ,n,\hfill \\ {\overline{\lambda}}^{*}\le {\theta}^{*}\overline{p},\hfill \\ {\alpha}^{*}{\beta}^{*}{k}_{v}^{*}+{\overline{x}}^{*}\xb7(\overline{p}+{\overline{\lambda}}^{*})=0,\hfill \\ {\overline{\lambda}}^{*}\xb7({\overline{x}}^{*}-{\overline{h}}^{*})=0,\hfill \\ {\theta}^{*}({\overline{h}}^{*}\xb7\overline{p}-1)=0,\hfill \\ {\overline{h}}^{*}\ge 0,\phantom{\rule{1.em}{0ex}}{\overline{h}}^{*}\xb7\overline{p}\le 1,\phantom{\rule{1.em}{0ex}}{\overline{x}}^{*}-{\overline{h}}^{*}\le 0,\hfill \\ {\theta}^{*}\ge 0,\phantom{\rule{1.em}{0ex}}{\overline{\lambda}}^{*}\ge 0,\phantom{\rule{1.em}{0ex}}({z}_{v}^{*},{k}_{v}^{*})\in {\Delta}_{(\rho ,r)},\phantom{\rule{1.em}{0ex}}{\alpha}^{*}\ge 0,\phantom{\rule{1.em}{0ex}}{\beta}^{*}\ge 0.\hfill \end{array}$$

**Lemma**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

#### 3.2. Second Problem: Minimizing the Risk and Maximizing the Income

**Definition**

**2.**

**Lemma**

**2.**

**Theorem**

**2.**

- The equivalent form of the dual problem is given by Equation (16);
- Furthermore, $({\overline{x}}^{*},{\overline{h}}^{*},{\xi}^{*})$ and $({\theta}^{*},{\alpha}^{*},{\beta}^{*},{\overline{\lambda}}^{*},{\overline{\mu}}^{*},{z}_{v}^{*},{k}_{v}^{*})$ solve problems in Equations (12) and (16), respectively, if and only if they satisfy the KKT conditions in Equation (17).

**Corollary**

**3.**

## 4. Model Calibration

**Remark**

**2.**

**Definition**

**3.**

- C(1)
- For all $a\in \mathbb{R}$ and $\overline{y}\in {\mathbb{R}}^{m}$, $\overline{\rho}(\overline{y}+a\mathbf{1})=\overline{\rho}(\overline{y})-a,$ where
**1**is the m-dimensional vector $(1,1,\dots ,1)$, (translation invariance). - C(2)
- For all $t\ge 0$ and $\overline{y}\in {\mathbb{R}}^{m}$, $\overline{\rho}(t\overline{y})=t\overline{\rho}(\overline{y})$, (positive homogeneity).
- C(3)
- For all vectors $\overline{y}$ and $\overline{w}$ in ${\mathbb{R}}^{m}$, if $\overline{y}\le \overline{w}$ then $\overline{\rho}(\overline{y})\ge \overline{\rho}(\overline{w})$, (monotonicity).
- C(4)
- For all vectors $\overline{y}=({y}_{1},{y}_{2},\dots ,{y}_{m})$ and $\overline{w}=({w}_{1},{w}_{2},\dots ,{w}_{m})$ in ${\mathbb{R}}^{m}$, if $({y}_{i}-{y}_{j})({w}_{i}-{w}_{j})\ge 0$ for $i\ne j$ then $\overline{\rho}(\overline{y}+\overline{w})\le \rho (\overline{y})+\rho (\overline{w})$, (comonotonic subadditivity).
- C(5)
- For any permutation $\{{i}_{1},\dots ,{i}_{m}\}$ of $\{1,2,\dots ,m\}$, we have $\overline{\rho}({y}_{1},\dots ,{y}_{m})=\overline{\rho}({y}_{{i}_{1}},\dots ,{y}_{{i}_{m}})$, (permutation invariance).

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**3.**

- If $\overline{\rho}:{\mathbb{R}}^{m}\mapsto \mathbb{R}$ is a risk statistic that satisfies subadditivity and positive homogeneity, then $\overline{\rho}(\overline{c})={sup}_{\overline{z}\in {\Delta}_{\overline{\rho}}}{\sum}_{j=1}^{m}-{z}_{j}{c}_{j},$ where ${\Delta}_{\overline{\rho}}$ is the subset of$$\{\overline{z}\in {\mathbb{R}}^{m};{z}_{i}\ge -\overline{\rho}({e}_{i}),-\overline{\rho}(\mathbf{1})\le \sum _{j=1}^{m}{z}_{j}\le \overline{\rho}(-\mathbf{1}),\mathit{and}\phantom{\rule{0.277778em}{0ex}}{z}_{j}\le {z}_{i}+\overline{\rho}({e}_{i}-{e}_{j})\phantom{\rule{0.277778em}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{0.277778em}{0ex}}j>i\}.$$
- If, in addition, the risk measure $\overline{\rho}$ satisfies monotonicity, then$${\Delta}_{\overline{\rho}}\subset \{\overline{z}\in {\mathbb{R}}^{m};\overline{z}\ge 0,-\overline{\rho}(\mathbf{1})\le \sum _{j=1}^{m}{z}_{j}\le \overline{\rho}(-\mathbf{1}),\mathit{and}\phantom{\rule{0.277778em}{0ex}}{z}_{j}\le {z}_{i}+\overline{\rho}({e}_{i}-{e}_{j})\phantom{\rule{0.277778em}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{0.277778em}{0ex}}j>i\}.$$

**Definition**

**6.**

**Remark**

**3.**

**Step 1. Calibrating an interest rate model:**for the simplicity of computation, we use the Vasicek model for the interest rate: $d{r}_{t}=(b-a{r}_{t})dt+\sigma d{B}_{t}$, where $a>0$, b, $\sigma $ are constants and ${({B}_{t})}_{t\ge 0}$ is a standard Brownian motion4. Here, we use the generalized method of moments (GMM) to estimate the parameters, see Hansen (1982).

**Step 2. Estimating $\overline{\rho}(\mathbf{1})$, $\overline{\rho}(-\mathbf{1})$, and $\overline{\rho}({e}_{i}-{e}_{j})$ for all $j>i$:**using Itô’s formula and Fubini’s theorem for stochastic processes, see Theorem 64 of Chapter 4 of Protter (2004), one can show that:

**Example**

**1.**

- The solution of the primal problem is equal to ${x}_{1}^{*}\approx 0$, ${x}_{2}^{*}\approx -0.0010$, ${x}_{3}^{*}\approx 0$, ${x}_{4}^{*}\approx 0.0099$, ${x}_{5}^{*}\approx 0$.
- The solution of the dual problem is equal to ${\theta}^{*}\approx 0.0078$, ${\lambda}_{1}^{*}\approx 0.8272$, ${\lambda}_{2}^{*}\approx 0$, ${\lambda}_{3}^{*}\approx 0.6492$, ${\lambda}_{4}^{*}\approx 0.7863$, ${\lambda}_{5}^{*}\approx 0$, ${\alpha}^{*}\approx 0.6287$, ${\overline{z}}_{1}^{*}\approx 0.0332$, ${\overline{z}}_{2}^{*}\approx 0.0327$, …

**Remark**

**4.**

## 5. Application to Market Integration

**Corollary**

**4.**

**Example**

**2.**

## 6. Application to Credit Premium Measurement

**Proposition**

**1.**

**Remark**

**5.**

**Example**

**3.**

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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1 | VaR is a risk measure that is neither coherent nor convex; see the properties listed before and the comment on VaR after Equation (1). |

2 | Note that the maturity T could be replaced by a date even shorter than the earliest maturity and no essential modifications of the paper would be required. However, the numerical examples of the paper are under the assumption that T is the longest maturity within the bonds of the portfolio. |

3 | This refers to the set ${\Delta}_{(\rho ,r)}$ in Assumption 1. |

4 | This dynamic is under the physical measure. |

5 | The data are taken from the US Department of the Treasury available at: http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield. |

6 |

UNITED STATES TREAS NTSAs of 15 October 2014 | UNITED STATES TREAS NTSAs of 15 October 2014 | ||

Price: | 106.22 | Price: | 101.00 |

Coupon(%): | 3.250 | Coupon(%): | 0.875 |

Maturity Date: | 31 December 2016 | Maturity Date: | 31 December 2016 |

Fitch Rating: | AAA | Fitch Rating: | AAA |

Coupon Payment Frequency: | Semi-Annual | Coupon Payment Frequency: | Semi-Annual |

First Coupon Date: | 30 June 2011 | First Coupon Date: | 15 May 2011 |

Type: | US Government Note | Type: | US Government Note |

Callable: | No | Callable: | No |

Debt Type: | Debt Type : | ||

Bond 1 | Bond 2 | ||

UNITED STATES TREAS NTSAs of 15 October 2014 | UNITED STATES TREAS NTSAs of 15 October 2014 | ||

Price: | 100.72 | Price: | 107.09 |

Coupon(%): | 0.750 | Coupon(%): | 2.375 |

Maturity Date: | 15 January 2017 | Maturity Date: | 15 January 2017 |

Fitch Rating: | AAA | Fitch Rating: | AAA |

Coupon Payment Frequency: | Semi-Annual | oupon Payment Frequency: | Semi-Annual |

First Coupon Date: | 15 July 2014 | First Coupon Date: | 15 July 2007 |

Type: | US Government Note | Type: | Inflation Indexed Security |

Callable : | No | Callable : | No |

Debt Type: | Debt Type : | ||

Bond 3 | Bond 4 | ||

US TREAS INFLATION INDEXED NTSAs of 15 October 2014 | CITIGROUP INCAs of 15 October 2014 | ||

Price: | 100.97 | Price: | 100.25 |

Coupon(%): | 0.875 | Coupon(%): | 1.250 |

Maturity Date: | 31 January 2017 | Maturity Date: | 15 January 2016 |

Fitch Rating: | AAA | Fitch Rating: | A |

Coupon Payment Frequency: | Semi-Annual | Coupon Payment Frequency: | Semi-Annual |

First Coupon Date: | 31 July 2012 | First Coupon Date: | 15 July 2013 |

Type: | US Government Note | Type: | US Corporate Debentures |

Callable: | No | Callable: | No |

Debt Type: | Debt Type: | Senior Note | |

Bond 5 | Bond 6 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|

1 | −0.0008 | −0.0016 | −0.0092 | −0.0099 | −0.0107 | −0.0177 | −0.0184 | −0.0191 | |

2 | 0.0008 | −0.0008 | −0.0084 | −0.0091 | −0.0099 | −0.0169 | −0.0176 | −0.0183 | |

3 | 0.0016 | 0.0008 | −0.0076 | −0.0083 | −0.0091 | −0.0161 | −0.0168 | −0.0175 | |

4 | 0.0095 | 0.0087 | 0.0079 | −0.0007 | −0.0014 | −0.0084 | −0.0091 | −0.0098 | |

5 | 0.0103 | 0.0095 | 0.0087 | 0.0008 | −0.0007 | −0.0077 | −0.0083 | −0.0090 | |

6 | 0.0111 | 0.0103 | 0.0095 | 0.0016 | 0.0008 | −0.0069 | −0.0076 | −0.0083 | |

7 | 0.0185 | 0.0177 | 0.0168 | 0.0091 | 0.0083 | 0.0075 | −0.0006 | −0.0012 |

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## Share and Cite

**MDPI and ACS Style**

Garrido, J.; Okhrati, R.
Desirable Portfolios in Fixed Income Markets: Application to Credit Risk Premiums. *Risks* **2018**, *6*, 23.
https://doi.org/10.3390/risks6010023

**AMA Style**

Garrido J, Okhrati R.
Desirable Portfolios in Fixed Income Markets: Application to Credit Risk Premiums. *Risks*. 2018; 6(1):23.
https://doi.org/10.3390/risks6010023

**Chicago/Turabian Style**

Garrido, José, and Ramin Okhrati.
2018. "Desirable Portfolios in Fixed Income Markets: Application to Credit Risk Premiums" *Risks* 6, no. 1: 23.
https://doi.org/10.3390/risks6010023