# Compositions of Conditional Risk Measures and Solvency Capital

^{*}

## Abstract

**:**

## 1. Introduction

- to be in line with Solvency II on a one-year horizon;
- to be time consistent;
- to incorporate in the risk measurement the maturity of the product.

## 2. Framework

## 3. Conditional and Dynamic Risk Measures

**Definition**

**1.**

- (monotonicity) $X\le Y$ implies ${\rho}_{t}\left(X\right)\ge {\rho}_{t}\left(Y\right)$;
- (conditional cash invariance) for all ${m}_{t}\in {\mathrm{L}}^{1}(\mathsf{\Omega},{\mathcal{F}}_{t},\mathbb{P})$,$${\rho}_{t}(X+{m}_{t}{\iota}_{t,T})={\rho}_{t}\left(X\right)-{m}_{t};$$
- (normalization) ${\rho}_{t}\left(0\right)=0$.

**Definition**

**2.**

**Definition**

**3.**

## 4. Time Consistency and Iterated Risk Measures

**Definition**

**4.**

**Example**

**1.**

**Remark**

**2.**

**Definition**

**5.**

**Corollary**

**1.**

## 5. Compositions of Conditional Risk Measures

#### 5.1. Composition of VaRs

**Definition**

**6.**

**Definition**

**7.**

**Proposition**

**1.**

**Proof.**

**Definition**

**8.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

#### 5.2. Composition of TVaRs

**Definition**

**9.**

**Definition**

**10.**

**Proposition**

**2.**

**Proof.**

**Definition**

**11.**

**Remark**

**6.**

#### 5.3. Composition with Conditional Expectations

**Definition**

**12.**

**Definition**

**13.**

**Remark**

**7.**

**Definition**

**14.**

**Remark**

**8.**

## 6. Solvency Computation

#### 6.1. Liabilities

**Remark**

**9.**

#### 6.2. Assets

#### 6.3. Final Net Worth

#### 6.4. Solvency Capital

**Proposition**

**3.**

**Proof.**

**Remark**

**10.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 7. Numerical Illustration

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Link between Time-Consistent and Iterated Measures

**Theorem**

**A1.**

- (a)
- ρ is time consistent;
- (b)
- if:$${\rho}_{t+1}\left(X\right)={\rho}_{t+1}\left(Y\right),$$$${\rho}_{t}\left(X\right)={\rho}_{t}\left(Y\right),$$
- (c)
- ρ is iterated.

**Proof.**

## Appendix B. Proof of Proposition 3

**Lemma**

**B1.**

## References

- P. Artzner, F. Delbaen, J.M. Eber, and D. Heath. “Coherent measures of risk.” Math. Financ. 9 (1999): 203–228. [Google Scholar] [CrossRef]
- M. Frittelli, and E. Rosazza Gianin. “Putting order in risk measures.” J. Bank. Financ. 26 (2002): 1473–1486. [Google Scholar] [CrossRef]
- H. Föllmer, and A. Schied. Stochastic Finance: An Introduction in Discrete Time, 3rd ed. De Gruyter Graduate; Berlin, Germany: Walter de Gruyter, 2011. [Google Scholar]
- B. Acciaio, and I. Penner. “Dynamic Risk Measures.” In Advanced Mathematical Methods for Finance. Edited by G. Di Nunno and B. ∅ksendal. Berlin/Heidelberg, Germany: Springer, 2011, Chapter 1; pp. 1–34. [Google Scholar]
- K. Detlefsen, and G. Scandolo. “Conditional and dynamic convex risk measures.” Financ. Stoch. 9 (2005): 539–561. [Google Scholar] [CrossRef]
- G.C. Pflug, and W. Römisch. Modeling, Measuring and Managing Risk. Toh Tuck Link, Singapore: World Scientific Publishing Co. Pte. Ltd., 2007. [Google Scholar]
- P. Artzner, F. Delbaen, J.M. Eber, D. Heath, and H. Ku. “Coherent multiperiod risk adjusted values and Bellman’s principle.” Ann. Oper. Res. 152 (2007): 5–22. [Google Scholar] [CrossRef]
- P. Cheridito, and M. Stadje. “Time-inconsistency of VaR and time-consistent alternatives.” Financ. Res. Lett. 6 (2009): 40–46. [Google Scholar] [CrossRef]
- P. Cheridito, and M. Kupper. “Composition of time-consistent dynamic monetary risk measures in discrete time.” Int. J. Theor. Appl. Financ. 14 (2011): 137–162. [Google Scholar] [CrossRef]
- P. Devolder. “Revised version of: Solvency requirement for a long-term guarantee: Risk measures versus probability of ruin.” Eur. Actuar. J. 1 (2011): 199–214. [Google Scholar] [CrossRef]
- M.R. Hardy, and J.L. Wirch. “The iterated CTE: A dynamic risk measure.” N. Am. Actuar. J. 8 (2004): 62–75. [Google Scholar] [CrossRef]
- P. Devolder, and A. Lebègue. “Iterated VaR or CTE measures: A false good idea? ” Scand. Actuar. J., 2016, 1–32. [Google Scholar] [CrossRef]
- M. Kupper, and W. Schachermayer. “Representation results for law invariant time consistent functions.” Math. Financ. Econ. 2 (2009): 189–210. [Google Scholar] [CrossRef]
- P. Devolder, and A. Lebègue. “Risk measures versus ruin theory for the calculation of solvency capital for long-term life insurances.” Depend. Model., 2016, in press. [Google Scholar] [CrossRef]
- F. Black, and M.S. Scholes. “The pricing of options and corporate liabilities.” J. Political Econ. 81 (1973): 637–654. [Google Scholar] [CrossRef]
- R.C. Merton. “Theory of rational option pricing.” Bell J. Econ. Manag. Sci. 4 (1973): 141–183. [Google Scholar] [CrossRef]
- A. Pascucci. PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series; Milan, Italy: Springer, 2011, Volume 2. [Google Scholar]

**Figure 2.**Evolution of the VaR between times $t=0$ and 1 (Example 1).

**Left**: evolution for X.

**Right**: evolution for Y.

**Figure 3.**Evolution of the expectation between times $t=0$ and 1 (Example 1).

**Left**: evolution for X.

**Right**: evolution for Y.

**Figure 4.**Different choices for the confidence vector, for a maturity $T=45$ and a long-term threshold $K=8$.

**Figure 5.**Computation of the solvency capital in the proportion of the initial value of the portfolio, for maturities $T\in \{1,\dots ,45\}$ and a long-term threshold $K=8$.

**Figure 6.**Computation of the solvency capital in the proportion of the initial value of the portfolio according to the iterated VaR measure with ${\mathit{\alpha}}^{\left(\mathrm{SII}\right),2}$ and with different values for μ and σ, for maturities $T\in \{1,\dots ,45\}$ and a long-term threshold $K=8$.

**Table 1.**Parameters of the GBMobtained by the MLE method with the corresponding standard errors between brackets.

μ | σ |
---|---|

$0.05564$ ($0.03827$) | $0.18415$ ($0.00172$) |

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Devolder, P.; Lebègue, A.
Compositions of Conditional Risk Measures and Solvency Capital. *Risks* **2016**, *4*, 49.
https://doi.org/10.3390/risks4040049

**AMA Style**

Devolder P, Lebègue A.
Compositions of Conditional Risk Measures and Solvency Capital. *Risks*. 2016; 4(4):49.
https://doi.org/10.3390/risks4040049

**Chicago/Turabian Style**

Devolder, Pierre, and Adrien Lebègue.
2016. "Compositions of Conditional Risk Measures and Solvency Capital" *Risks* 4, no. 4: 49.
https://doi.org/10.3390/risks4040049