# A Welfare Analysis of Capital Insurance

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

- •
- Aggregate Insurance: ${I}_{i}(X,{X}_{i})={\beta}_{i}X$, where ${\beta}_{i}\ge 0$.
- •
- Classical Insurance: ${I}_{i}(X,{X}_{i})={\beta}_{i}{X}_{i}$, where ${\beta}_{i}\ge 0$.
- •
- Aggregate-Cross Insurance: ${I}_{i}(X,{X}_{i})={\beta}_{i}{\widehat{X}}_{i}$, where ${\widehat{X}}_{i}={\sum}_{j\ne i}{X}_{j}$ is the total loss, except for the insured bank’s loss, and ${\beta}_{i}\ge 0$.

#### 2.1. Aggregate Insurance

#### 2.1.1. Optimal Load for Bank i

#### 2.1.2. Optimal Load Factor for the Regulator

**Proposition 1**

**Proof:**

#### 2.2. Classical Insurance

#### 2.2.1. Optimal Load for Bank i

#### 2.2.2. Optimal Load Factor for the Regulator

**Proposition 2**

**Proof:**

#### 2.3. Aggregate-Cross Insurance

#### 2.3.1. Optimal Load for Bank i

#### 2.3.2. Optimal Load Factor for the Regulator

**Proposition 3**

**Proposition 4**

- 1.
- If the risk-adjusted variance vector, $\left(\frac{Var({X}_{i})}{{\gamma}_{i}}\right)$, and the Sharpe ratio vector, $\left(\frac{\mathbb{E}\left[{X}_{i}\right]}{\sqrt{Var({X}_{i})}}\right)$, are co-monotonic9, then the classical insurance is preferred to the aggregate insurance.
- 2.
- If the risk-adjusted variance vector, $\left(\frac{Var({X}_{i})}{{\gamma}_{i}}\right)$, and the Sharpe ratio vector, $\left(\frac{\mathbb{E}\left[{X}_{i}\right]}{\sqrt{Var({X}_{i})}}\right)$, are counter-monotonic, and there exists one “too big to fail” bank in the sense that $\mathbb{E}{\left[X\right]}^{2}$ is close to ${\sum}_{i}\mathbb{E}{\left[{X}_{i}\right]}^{2}$, then the aggregate insurance is preferred to the classical insurance.

**Proof:**

## 3. Systemic Risk and Comparative Analysis

**Proposition 5**

**Proof:**

**Proposition 6**

**Proof:**

**Proposition 7**

**Proof:**

## 4. Discussion

#### 4.1. Disordering Loss Market and Ordering Loss Market

**Table 1.**Example 1 of a disordering loss market. This table displays a disordering loss market when each bank has the same expected loss in a one-factor model. Therefore, aggregate insurance is a better capital insurance program by Proposition 6. It can be checked that the condition in Proposition 1 is satisfied, so the equilibrium of the aggregate insurance is given in Proposition 1. We assume ${\gamma}_{i}=1$ for each $i=1,\cdots ,N$. There are $N=10$ banks.

Bank | α | σ | Risk-Adjusted Variance | Sharpe Ratio |
---|---|---|---|---|

1 | 0.1 | 0.40 | 0.170 | 0.243 |

2 | 0.1 | 0.35 | 0.133 | 0.275 |

3 | 0.1 | 0.30 | 0.100 | 0.316 |

4 | 0.1 | 0.26 | 0.078 | 0.359 |

5 | 0.1 | 0.23 | 0.063 | 0.399 |

6 | 0.1 | 0.20 | 0.050 | 0.447 |

7 | 0.1 | 0.18 | 0.042 | 0.486 |

8 | 0.1 | 0.15 | 0.033 | 0.555 |

9 | 0.1 | 0.12 | 0.024 | 0.640 |

10 | 0.1 | 0.10 | 0.020 | 0.707 |

**Table 2.**Example 2 of a disordering loss market. This table displays a disordering loss market when the percentage of specific risk in individual risk increases with respect to individual risk. Therefore, aggregate insurance is a better insurance program than the classical insurance program by Proposition 6. It can be checked that the condition in Proposition 1 is satisfied, so the equilibrium of the aggregate insurance is given in Proposition 1. We assume ${\gamma}_{i}=1$ for each $i=1,\cdots ,N$. There are $N=10$ banks.

Bank | α | σ | Risk-Adjusted Variance | Sharpe Ratio |
---|---|---|---|---|

1 | 0.10 | 0.200 | 0.050 | 0.447 |

2 | 0.15 | 0.315 | 0.122 | 0.430 |

3 | 0.20 | 0.440 | 0.234 | 0.414 |

4 | 0.25 | 0.575 | 0.393 | 0.399 |

5 | 0.30 | 0.720 | 0.608 | 0.385 |

6 | 0.35 | 0.875 | 0.888 | 0.371 |

7 | 0.40 | 1.040 | 1.242 | 0.359 |

8 | 0.45 | 1.215 | 1.679 | 0.347 |

9 | 0.50 | 1.400 | 2.210 | 0.336 |

10 | 0.55 | 1.595 | 2.847 | 0.326 |

**Table 3.**An Example of an ordering loss market. This table displays an ordering loss market when the percentage of specific risk in individual risk decreases with respect to individual risk. Therefore, classical insurance is a better insurance program than the aggregate insurance program by Proposition 6. We assume ${\gamma}_{i}=1$ for each $i=1,\cdots ,N$. There are $N=10$ banks.

Bank | α | σ | Risk-adjusted Variance | Sharpe ratio |
---|---|---|---|---|

1 | 0.10 | 0.400 | 0.170 | 0.243 |

2 | 0.15 | 0.350 | 0.145 | 0.394 |

3 | 0.20 | 0.300 | 0.130 | 0.555 |

4 | 0.25 | 0.260 | 0.130 | 0.693 |

5 | 0.30 | 0.230 | 0.143 | 0.794 |

6 | 0.35 | 0.200 | 0.163 | 0.868 |

7 | 0.40 | 0.180 | 0.192 | 0.912 |

8 | 0.45 | 0.150 | 0.225 | 0.949 |

9 | 0.50 | 0.120 | 0.264 | 0.972 |

10 | 0.55 | 0.100 | 0.313 | 0.984 |

#### 4.2. Low Correlation Market and High Correlation Market

#### 4.3. Systemic Risk

#### 4.4. Identification and Implementation of “Too Big to Fail”

- How to implement the aggregate insurance program, i.e., how to characterize the equilibrium in a general situation.
- How to distinguish the “too big to fail” banks that are forced to purchase aggregate insurance from the other banks. Alternatively, how to identify those “too big to fail” banks.

**Table 4.**Example 3 of a disordering loss market. This table displays a disordering loss market when each bank has the same expected loss in a one-factor model. Therefore, aggregate insurance is a better program by Proposition 6. However, the condition in Proposition 1 is not satisfied, as shown for $i=11,12,\cdots ,15$. There are $N=15$ banks, and each ${\gamma}_{i}=1$.

Bank | α | σ | Risk-Adjusted Variance | Sharpe Ratio | $\frac{Cov({X}_{i},X)}{Var(X)}$ |
---|---|---|---|---|---|

1 | 0.05 | 0.40 | 0.1625 | 0.124 | 0.1170 |

2 | 0.05 | 0.38 | 0.1469 | 0.130 | 0.1070 |

3 | 0.05 | 0.36 | 0.1321 | 0.138 | 0.0990 |

4 | 0.05 | 0.34 | 0.1181 | 0.145 | 0.0907 |

5 | 0.05 | 0.32 | 0.1049 | 0.154 | 0.0829 |

6 | 0.05 | 0.30 | 0.0925 | 0.164 | 0.0755 |

7 | 0.05 | 0.28 | 0.0809 | 0.176 | 0.0686 |

8 | 0.05 | 0.26 | 0.0701 | 0.189 | 0.0622 |

9 | 0.05 | 0.24 | 0.0601 | 0.204 | 0.0563 |

10 | 0.05 | 0.22 | 0.0509 | 0.222 | 0.0509 |

11 | 0.05 | 0.20 | 0.0425 | 0.243 | 0.0459 |

12 | 0.05 | 0.18 | 0.0349 | 0.268 | 0.0414 |

13 | 0.05 | 0.16 | 0.0281 | 0.298 | 0.0374 |

14 | 0.05 | 0.14 | 0.0221 | 0.336 | 0.0338 |

15 | 0.05 | 0.12 | 0.0169 | 0.385 | 0.0307 |

**Table 5.**Implementation of Example 3. This table displays the equilibrium of Example 3. We note that when i starts from 14, $\frac{Cov({X}_{i},X)}{{\gamma}_{i}}$ is strictly greater than $\frac{{\sum}_{j=1}^{i}Cov({X}_{j},X)}{2{\sum}_{j=1}^{i}{\gamma}_{j}}$. Then, the last two banks are not “too big to fail”. The optimal load factor is ${\rho}^{*}=8.1\%$.

Bank | $\frac{Cov({X}_{i},X)}{{\gamma}_{i}}$ | $\frac{Cov({X}_{i},X)}{Var(X)}$ | $\frac{{\sum}_{j=1}^{i}Cov({X}_{j},X)}{2{\sum}_{j=1}^{i}{\gamma}_{j}}$ | ${\beta}^{i,a}$ |
---|---|---|---|---|

1 | 0.1975 | 0.1170 | 0.09875 | 8.10 % |

2 | 0.1819 | 0.1070 | 0.09485 | 7.18 % |

3 | 0.1671 | 0.0990 | 0.09108 | 6.30 % |

4 | 0.1531 | 0.0907 | 0.08745 | 5.47 % |

5 | 0.1399 | 0.0829 | 0.08395 | 4.69 % |

6 | 0.1275 | 0.0755 | 0.08058 | 3.95 % |

7 | 0.1159 | 0.0686 | 0.07735 | 3.27 % |

8 | 0.1051 | 0.0622 | 0.07425 | 2.63 % |

9 | 0.0951 | 0.0563 | 0.07128 | 2.03 % |

10 | 0.0859 | 0.0509 | 0.06845 | 1.49 % |

11 | 0.0775 | 0.0459 | 0.06575 | 0.99 % |

12 | 0.0699 | 0.0414 | 0.06318 | 0.54 % |

13 | 0.0631 | 0.0374 | 0.06075 | 0.14 % |

14 | 0.0571 | 0.0338 | 0.05845 | 0 |

15 | 0.0519 | 0.0307 | 0.05628 | 0 |

## 5. Conclusion

## Acknowledgements

## Conflicts of Interest

## References

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## Appendix A. Proofs

**Lemma 1**

- 1.
- If the vector $\kappa =({\kappa}_{i})$ is co-monotonic to the vector $\frac{b}{c}=(\frac{{b}_{i}}{{c}_{i}})$, then:$$\begin{array}{c}\hfill \frac{{\sum}_{i=1}^{n}{b}_{i}{\kappa}_{i}}{{\sum}_{i=1}^{n}{b}_{i}}>\frac{{\sum}_{i=1}^{n}{c}_{i}{\kappa}_{i}}{{\sum}_{i=1}^{n}{c}_{i}}.\end{array}$$
- 2.
- If the vector $\kappa =({\kappa}_{i})$ is counter-monotonic to the vector $\frac{b}{c}=(\frac{{b}_{i}}{{c}_{i}})$, then:$$\begin{array}{c}\hfill \frac{{\sum}_{i=1}^{n}{b}_{i}{\kappa}_{i}}{{\sum}_{i=1}^{n}{b}_{i}}<\frac{{\sum}_{i=1}^{n}{c}_{i}{\kappa}_{i}}{{\sum}_{i=1}^{n}{c}_{i}}.\end{array}$$

**Proof:**

**Lemma 2**

- •
- If those numbers, ${a}_{1},\cdots ,{a}_{n}$, are close enough in the sense that $VAR(a)\le \mathbb{E}{\left[a\right]}^{2}VAR(b)/\mathbb{E}{\left[b\right]}^{2}$, then $\frac{\sum {a}_{i}^{2}}{\sum {b}_{i}^{2}}\le \frac{{(\sum {a}_{i})}^{2}}{{(\sum {b}_{i})}^{2}}$.
- •
- If those numbers, ${b}_{1},\cdots ,{b}_{n}$, are close enough in the sense that $VAR(b)\le \mathbb{E}{\left[b\right]}^{2}VAR(a)/\mathbb{E}{\left[a\right]}^{2}$, then $\frac{\sum {a}_{i}^{2}}{\sum {b}_{i}^{2}}\ge \frac{{(\sum {a}_{i})}^{2}}{{(\sum {b}_{i})}^{2}}$.

**Proof:**

**Proof of Proposition 4.**

**Proof of Proposition 5.**The welfare of each insurance contract in the one-factor model is computed as follows.

**Proof of Proposition 6.**

**Proof of Proposition 7.**As the risk-adjusted variance is co-monotonic to the Sharpe ratio across each bank, Lemma 1 yields that:

## Appendix B. A General Solution of the Equilibrium of Aggregate Insurance

**Lemma 3**

^{1}Under the standards set forth in section 113 of the Dodd-Frank Act, a bank holding company or “non-bank financial company” poses a potential systemic risk if “material financial distress at the company, or the nature, scope, size, scale, concentration, interconnectedness, or mix of the activities of the company, could pose a threat to the financial stability of the United States.” Therefore, we focus only on these companies with systemic risks (too big to fail).^{2}Precisely, when a risk-adjusted covariance of the loss portfolio is co-monotonic to the Sharpe ratio of the loss portfolio, we say it is an ordering loss market. If both of these sequences are counter-monotonic to each other, we say that the market is a disordering loss market. See Propositions 4, 6 and 7 below.^{3}By [4], the optimal sharing rules must increase with respect to the aggregate endowment. Our setting is different from Borch’s equilibrium setting in the presence of the central bank.^{4}We follow the same mean-variance setting as in [12], in which the aggregate uncertainty insurance is considered, as we focus on the aggregate or systematic risk.^{5}It is easy to see that $Var({W}^{i})=Var({X}_{i})+{\beta}_{i}^{2}Var(X)-2{\beta}_{i}Cov({X}_{i},X)$. Then, ${\beta}^{i,a}(\rho )$ follows from the first-order condition in Equation (3).^{6}It is the beta in the capital asset pricing model when the loss variable is replaced by the return variable.^{7}Since $X={\sum}_{i}{X}_{i}$, ${\sum}_{i}Cov({X}_{i},X)=Var(X)$.^{8}It is different from a traditional insurance contract on individual loss exposure. The load factor for a traditional insurance contract is either given exogenously or depends on the specific loss vector in equilibrium. Classical insurance in our setting, however, is characterized in a rational expectation equilibrium with banks and a regulator^{9}Given two vectors $a=({a}_{1},\cdots ,{a}_{n}),b=({b}_{1},\cdots ,{b}_{n})$, a and b are counter-monotonic if $({a}_{i}-{a}_{j})({b}_{i}-{b}_{j})\le 0,\forall i,j$, and at least one inequality is strict; a and b are co-monotonic if $({a}_{i}-{a}_{j})({b}_{i}-{b}_{j})\ge 0,\forall i,j$, and at least one inequality is strict.^{10}We write $x\gg y$ to denote $y/x\to 0$.^{11}By two functions, $f\sim g$, we mean that ${lim}_{Var(Y)\to \infty}\frac{f}{g}=1$.

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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Panttser, E.; Tian, W.
A Welfare Analysis of Capital Insurance. *Risks* **2013**, *1*, 57-80.
https://doi.org/10.3390/risks1020057

**AMA Style**

Panttser E, Tian W.
A Welfare Analysis of Capital Insurance. *Risks*. 2013; 1(2):57-80.
https://doi.org/10.3390/risks1020057

**Chicago/Turabian Style**

Panttser, Ekaterina, and Weidong Tian.
2013. "A Welfare Analysis of Capital Insurance" *Risks* 1, no. 2: 57-80.
https://doi.org/10.3390/risks1020057