# Multi-Objective Stochastic Optimization Programs for a Non-Life Insurance Company under Solvency Constraints

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## Abstract

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## 1. Introduction

## 2. Portfolio Optimization Framework

#### 2.1. Preliminaries

#### 2.2. Risk Measures and Performance Indicators for an Insurance Company

**Definition 1.**Let $\left(\right)open="("\; close=")">\Omega ,\mathcal{A},\mathbb{P}$ be a probability space, ${\mathcal{M}}_{\mathcal{B}}\left(\right)open="("\; close=")">\Omega ,{\mathbb{R}}^{k}$ be the space of ${\mathbb{R}}^{k}$-valued random variables and $\mathcal{M}\left(\right)open="("\; close=")">\Omega ,\mathbb{R}\subseteq {\mathcal{M}}_{\mathcal{B}}\left(\right)open="("\; close=")">\Omega ,\mathbb{R}$ be a suitable vector subspace; a risk measure is then a map from $\mathcal{M}\left(\right)open="("\; close=")">\Omega ,\mathbb{R}$ to $\mathbb{R}$.

**Proposition 1.**Let ${L}_{\tau}(c,x)$ be the net loss defined in Equation (1), with the vector of decision variables $(c,x)\in {\mathcal{X}}_{0}$, and $\alpha \in (0,1)$ denote the confidence level. Consider the function:

- (a)
- ${F}^{\alpha}$ is finite and continuous as a function of s;
- (b)
- the conditional value-at-risk can be evaluated as:$$CVa{R}^{\alpha}\left(\right)open="("\; close=")">c,x=\underset{s\in \mathbb{R}}{inf}{F}^{\alpha}(c,x,s);$$
- (c)
- the set of solutions to the stochastic program defined in Equation (4) is the non-empty, closed and bounded interval:$${S}^{\alpha}(c,x)=\left(\right)open="["\; close="]">Va{R}^{\alpha}\left(\right)open="("\; close=")">c,x,\phantom{\rule{4pt}{0ex}}{\overline{VaR}}^{\alpha}\left(\right)open="("\; close=")">c,x$$
- (d)
- minimizing $CVa{R}^{\alpha}$ with respect to $(c,x)\in {\mathcal{X}}_{0}$ is equivalent to minimizing ${F}^{\alpha}$ with respect to $(c,x,s)\in {\mathcal{X}}_{0}\times \mathbb{R}$, i.e.,$$\underset{(c,x)\in {\mathcal{X}}_{0}}{min}CVa{R}^{\alpha}\left(\right)open="("\; close=")">c,x=\underset{(c,x,s)\in {\mathcal{X}}_{0}\times \mathbb{R}}{min}{F}^{\alpha}(c,x,s)$$
- (e)
- $CVa{R}^{\alpha}$ is convex with respect to $(c,x)$, and ${F}^{\alpha}$ is convex with respect to $(c,x,s)$.

#### 2.3. Multi-Objective Portfolio Optimization Problems with Solvency Constraint

## 3. The Normalized Normal Constraint Method

#### 3.1. Basic Concepts and Definitions

**Definition 2.**A point ${x}^{*}\in \mathcal{X}$ is Pareto optimal to Problem (18–20) if and only if there does not exist another point $x\in \mathcal{X}$ such that:

**Definition 3.**An anchor point ${f}^{i*}$ is a specific point in the objective space $\mathcal{Y}$ that corresponds to the minimum value of an i-th objective function ${f}_{i}$, $i=1,\dots ,m$, subject to Constraints (19) and (20), and is expressed as:

**Definition 4.**An hyperplane that comprises all of the anchor points is said to be the utopia hyperplane.

**Definition 5.**The point in the objective space $\mathcal{Y}$ that corresponds to all objectives simultaneously being at their best possible values is said to be utopia, or ideal:

**Definition 6.**An anti-anchor point ${f}^{i\circ}$ is a specific point in the objective space $\mathcal{Y}$ that corresponds to the maximum value of an i-th objective function ${f}_{i}$, $i=1,\dots ,m$, subject to Constraints (19) and (20), and is expressed as:

**Definition 7.**The point in the objective space $\mathcal{Y}$ that corresponds to all objectives simultaneously being at their worst possible values is said to be nadir:

**Figure 1.**Graphical representation of the reference points (on the left) and of the normal constraint (NC) method (on the right) for a bi-objective sample optimization problem.

#### 3.2. A New Variant of the Normal Constraint Algorithm

**Step 1**- Generation of reference points:Use (24) to identify the m anchor points ${f}^{1*},\dots ,{f}^{m*}$ and calculate the utopia point according to (25).
**Step 2**- Objective normalization:To avoid scaling deficiencies, objective functions are transformed according to the following normalization equation:$${\overline{f}}_{i}(x)=\frac{{f}_{i}(x)-{f}_{i}^{U}(x)}{{f}_{i}^{N}(x)-{f}_{i}^{U}(x)},\phantom{\rule{1.em}{0ex}}i=1,\dots ,m$$Note that ${\overline{f}}_{i}(x)$ has values between zero and one, for all $x\in \mathcal{X}$. The optimization will now take place in the normalized objective space.
**Step 3**- Utopia line vector:Let ${\overline{N}}_{k}$ indicate the difference between the k-th normalized anchor point and the normalized anchor point corresponding to dimension m:$${\overline{N}}_{k}\stackrel{\u25b5}{=}{\overline{f}}^{k*}-{\overline{f}}^{m*},\phantom{\rule{1.em}{0ex}}k=1,\dots ,m-1.$$In this way, $m-1$ utopia line vectors are defined, all of which point to ${\overline{f}}^{m*}$.
**Step 4**- Normalized utopia line increments:Based on the prescribed number of utopia line points ${m}_{k}$, $k=1,\dots ,m-1$, that the user fixes along each normalized utopia line direction ${\overline{N}}_{k}$, a normalized increment ${\delta}_{k}$ is given using the equation:$${\delta}_{k}=\frac{\parallel {\overline{N}}_{k}\parallel}{{m}_{k}-1},\phantom{\rule{1.em}{0ex}}k=1,\dots ,m-1$$
**Step 5**- Hyperplane point generation:Evaluate a set of evenly-distributed points on the normalized utopia hyperplane as:$${Y}_{pj}=\sum _{k=1}^{m}{\alpha}_{kj}{\overline{f}}^{k*}\phantom{\rule{1.em}{0ex}}j=1,\dots ,J$$
**Step 6**- Pareto point generation:Using the set of points ${Y}_{pj}$, $j=1,\dots ,J$, on the normalized utopia hyperplane, generate a corresponding set of Pareto optimal points on the PF by solving the following series of single-objective problems:$$\underset{x\in \mathcal{X}}{min}{\overline{f}}_{m}(x)$$$${\overline{N}}_{j}{\left(\right)}^{\overline{f}}T\le 0$$The plane Constraints (29) enforce the optimization to operate on the portion of the objective space in which the vectors pointing to ${Y}_{pj}$ are in opposition to the normalized utopia hyperplane. When Constraints (29) reduces to being equal to zero, it represents the equation of a hyperplane orthogonal to the normalized utopia hyperplane.
**Step 7**- Pareto design metric values:Evaluate the non-normalized metrics by using the equation:$${f}_{i}(x)={\overline{f}}_{i}(x)\left(\right)open="["\; close="]">{f}_{i}^{N}(x)-{f}_{i}^{U}(x)+{f}_{i}^{U}(x),\phantom{\rule{1.em}{0ex}}i=1,\dots ,m.$$

## 4. Estimation Methods

#### 4.1. Liability Distribution and Asset Return Scenario Generation

#### 4.2. Quantile-Based Risk Measures and Objective Estimation

## 5. Empirical Results and Discussion

#### 5.1. Data Description and Model Estimation

**Table 1.**Descriptive statistics for the daily asset log-returns from 3 January 2005–9 July 2011 for a total of 1714 observations: EMUTracker Government Bond Index with 1–3 year maturities (TEMGVG1); Markit iBoxx Euro Corporate Bond Index with 1–3 year maturities (IBCRP13); Markit iBoxx Benchmark Collateralized Index (IBCOLAL); Markit iBoxx Euro High Yield Fixed Rate Bonds Index (IBEHYFR); MSCI EMU Index (MSEMUIL); MSCI BRIC Index (MSBRICL).

TEMGVG1 | IBCRP13 | IBCOLAL | IBEHYFR | MSEMUIL | MSBRICL | |
---|---|---|---|---|---|---|

descriptive statistics | ||||||

Min | −0.0044 | −0.0047 | −0.0073 | −0.0354 | −0.0818 | −0.1015 |

Max | 0.0041 | 0.0034 | 0.0072 | 0.0347 | 0.0993 | 0.1223 |

Mean | 0.0001 | 0.0001 | 0.0001 | 0.0002 | 0.0001 | 0.0005 |

SD | 0.0007 | 0.0008 | 0.0017 | 0.0035 | 0.0137 | 0.0156 |

Skewness | −0.0050 | −0.6480 | −0.0914 | −1.5211 | −0.0106 | −0.1709 |

Kurtosis | 8.3629 | 6.8593 | 3.8708 | 28.3298 | 10.5776 | 12.8170 |

**Table 2.**Statistical tests for normality, autocorrelation and conditional heteroskedasticity relative to the daily asset log-returns from 3 January 2005–29 July 2011 for a total of 1714 observations. The p-values corresponding to the test statistics are reported in parentheses. ARCH(2) is Engle’s LM test for the ARCH effect in the residuals up to the second order.

TEMGVG1 | IBCRP13 | IBCOLAL | IBEHYFR | MSEMUIL | MSBRICL | |
---|---|---|---|---|---|---|

normality | ||||||

Jarque–Bera | 2054.01 ** | 1183.66 ** | 56.54 ** | 46,481.79 ** | 4100.75 ** | 6891.07 ** |

(0.001) | (0.001) | (0.001) | (0.001) | (0.001) | (0.001) | |

Lilliefors | 0.07 ** | 0.08 ** | 0.02 * | 0.17 ** | 0.09 ** | 0.10 ** |

(0.001) | (0.001) | (0.022) | (0.001) | (0.001) | (0.001) | |

Shapiro–Wilk | 0.93 ** | 0.95 ** | 0.99 ** | 0.73 ** | 0.91 ** | 0.88 ** |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | |

autocorrelation | ||||||

Ljung-Box | 56.25 ** | 166.22 ** | 20.05 | 1546.96 ** | 42.68 ** | 107.62 ** |

(0.000) | (0.000) | (0.455) | (0.000) | (0.002) | (0.000) | |

conditional heteroskedasticity | ||||||

ARCH(2) | 39.75 ** | 190.53 ** | 51.90 ** | 331.18 ** | 160.05 ** | 430.06 ** |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |

**Table 3.**Parameter estimates of the AR(1)-GJR-GARCH(1,1) model for each log-return series for the period from 3 January 2005–29 July 2011. The t-ratios are given in parentheses.

TEMGVG1 | IBCRP13 | IBCOLAL | IBEHYFR | MSEMUIL | MSBRICL | |
---|---|---|---|---|---|---|

AR(1) model | ||||||

c | 8.58E-05 ** | 1.23E-04 ** | 1.16E-04 ** | 2.00E-04 ** | 5.57E-04 ** | 8.88E-04 ** |

(6.280) | (7.480) | (2.979) | (6.808) | (2.614) | (3.702) | |

θ | 0.1176 ** | 0.1402 ** | 0.0648 * | 0.4350 ** | −0.0139 | 0.1824 ** |

(4.486) | (5.153) | (2.508) | (20.591) | (−0.516) | (7.259) | |

GJR-GARCH(1,1) model | ||||||

κ | 2E-07 ${}^{+}$ | 2E-07 ${}^{+}$ | 2E-07 ${}^{+}$ | 2E-07 | 2.2784E-06 ** | 5.2582E-06 ** |

(1.660) | (1.670) | (1.656) | (1.242) | (2.865) | (3.671) | |

α | 0.4544 ** | 0.4564 ** | 0.8466 ** | 0.6913 ** | 0.8837 ** | 0.8464 * |

(8.190) | (10.075) | (55.928) | (30.266) | (55.716) | (44.279) | |

ϕ | 0.4800 ** | 0.3647 ** | 0.0912 ** | 0.2436 ** | – | 0.0480 ** |

(4.395) | (4.444) | (4.061) | (5.559) | (2.472) | ||

ψ | −0.2037 | −0.0633 | −0.0186 | 0.1302 * | 0.2010 ** | 0.1526 ** |

(−1.610) | (−0.689) | (−0.658) | (2.353) | (7.107) | (4.703) | |

t distribution | ||||||

$DoF$ | 3.2411 ** | 4.2279 ** | 24.2761 ** | 3.5620 ** | 9.8494 ** | 6.8035** |

(11.943) | (8.383) | (6.06E+06) | (14.036) | (4.413) | (5.533) |

**Table 4.**Generalized Pareto distribution parameter estimates for the standardized residuals from the AR(1)-GJR-GARCH(1,1) model for each log-return series for the period from 3 January 2005–29 July 2011.

TEMGVG1 | IBCRP13 | IBCOLAL | IBEHYFR | MSEMUIL | MSBRICL | |
---|---|---|---|---|---|---|

l | −0.9577 | −1.0920 | −1.2775 | −1.1539 | −1.3427 | −1.2551 |

${\xi}_{l}$ | 0.2713 | 0.0753 | −0.1543 | 0.3202 | −0.0585 | −0.1005 |

${\beta}_{l}$ | 0.4083 | 0.6323 | 0.6093 | 0.6305 | 0.6699 | 0.7073 |

u | 1.0050 | 1.1145 | 1.2844 | 1.1038 | 1.2142 | 1.1771 |

${\xi}_{u}$ | 0.0981 | 0.0281 | −0.0866 | 0.0996 | −0.1323 | −0.0287 |

${\beta}_{u}$ | 0.5041 | 0.5177 | 0.5176 | 0.6019 | 0.4834 | 0.5830 |

**Table 5.**Parameter estimates of the multivariate Student t-Copula involving the standardized residuals from the AR(1)-GJR-GARCH(1,1) model associated with each log-return series for the period from 3 January 2005–29 July 2011. The t-ratios are given in parentheses.

ν = 9.4947 ** | IBCRP13 | IBCOLAL | IBEHYFR | MSEMUIL | MSBRICL |
---|---|---|---|---|---|

(13.156) | |||||

TEMGVG1 | 0.8145 ** | 0.7828 ** | −0.0272 | −0.3179 ** | −0.1812 ** |

(114.414) | (95.986) | (−1.075) | (−14.081) | (−7.368) | |

IBCRP13 | 0.8067 ** | 0.1426 ** | −0.2498 ** | −0.1168 ** | |

(109.054) | (5.721) | (−10.603) | (−4.609) | ||

IBCOLAL | −0.0003 | −0.3339 ** | −0.1674 ** | ||

(−0.012) | (−14.986) | (−6.783) | |||

IBEHYFR | 0.2905 ** | 0.3029 ** | |||

(12.617) | (13.209) | ||||

MSEMUIL | 0.5929 ** | ||||

(39.938) |

#### 5.2. Pareto Frontiers and Portfolio Analysis

- (i)
- Deb’s diversity metric Δ [18] measures the spread of solutions in the given optimal set S. It is defined as:$$\Delta (S)=\sum _{i=1}^{\left|S\right|-1}\frac{|{d}_{i}-\overline{d}|}{\left|S\right|-1}$$
- (ii)
- Hypervolume $HV$ [55] is defined as the volume in the objective space that is dominated by the optimal solution set S. It measures both the convergence to the true Pareto front and the diversity information. In fact, the larger is the value of $HV$, the closer are the solutions of S to the true Pareto front. At the same time, a higher $HV$ could indicate that the solutions of S are scattered more evenly in the objective space.

**Table 6.**Performance metric results of the proposed version of the NNCmethod and of the ϵ-constraint method for Problem (14).

Proposed NNC | ϵ-Constraint | |
---|---|---|

Δ | 0.0570 | 0.0987 |

$HV$ | 8.2265E-04 | 8.1539E-04 |

**Figure 2.**Graphical representations of the efficient frontier for the problem given by (14) drawn by the NNC method (on the left) and by the ϵ-constraint method (on the right).

**Figure 3.**Graphical representations of the efficient frontier for the problem given by (15) drawn by the proposed version of the NNC method.

**Figure 4.**The projection of the efficient frontier for the problem given by (15) onto the $\sigma -CVaR$ plane is reported in (a), the projection onto the $\sigma -RORAC$ is displayed in (b) and (c) provides the projection onto the $CVaR-RORAC$ plane.

**Figure 5.**Graphical representations of the efficient frontier for the problem given by (17) drawn by the proposed version of the NNC method.

**Figure 6.**The projection of the efficient frontier for the problem given by (17) onto the $SR-ROC$ plane is reported in (a), the projection onto the $SR-RORAC$ is displayed in (b) and (c) provides the projection onto the $RORAC-ROC$ plane.

**Figure 7.**Scatter plot displaying displaying the link between $CVaR$ and invested capital (on the left) and between $CVaR$ and portfolio volatility (on the right) for the 275 portfolios detected on the Pareto front for the problem given by (17).

ROC | RORAC | |
---|---|---|

$SR$ | 0.7741 | −0.6307 |

$ROC$ | −0.4322 |

**Table 8.**Correlation of the conditional value-at-risk with shareholders’ invested capital and with portfolio volatility, respectively.

Invested Capital | σ | |
---|---|---|

$CVaR$ | −0.7626 | 0.8148 |

## 6. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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^{1}In this article, asset gross returns at time t are defined as $r}_{i,t}=\frac{{P}_{i,t}}{{P}_{i,t-1}$, $i=1,\dots ,n$, where ${P}_{i,t}$ is the price of asset i at time t, while the corresponding gross returns over the period τ become $R}_{i,\tau}=\prod _{t=1}^{\tau}{r}_{i,t$. As a result, the vector ${R}_{\tau}$ is nonnegative.^{2}Since the argmin is, in general, not uniquely defined, we will refer to the solution identified by the numerical method used in the optimization process as the anchor point.

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**MDPI and ACS Style**

Kaucic, M.; Daris, R.
Multi-Objective Stochastic Optimization Programs for a Non-Life Insurance Company under Solvency Constraints. *Risks* **2015**, *3*, 390-419.
https://doi.org/10.3390/risks3030390

**AMA Style**

Kaucic M, Daris R.
Multi-Objective Stochastic Optimization Programs for a Non-Life Insurance Company under Solvency Constraints. *Risks*. 2015; 3(3):390-419.
https://doi.org/10.3390/risks3030390

**Chicago/Turabian Style**

Kaucic, Massimiliano, and Roberto Daris.
2015. "Multi-Objective Stochastic Optimization Programs for a Non-Life Insurance Company under Solvency Constraints" *Risks* 3, no. 3: 390-419.
https://doi.org/10.3390/risks3030390