# Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures

## Abstract

**:**

## 1. Introduction

## 2. Coherent Risk Measures

_{p}of random variables X: Ω → ℝ. In this paper, the values of the random variable X are interpreted as the possible future values in t = 1 of a portfolio10. A special class of risk measures are the so-called monetary risk measures11:

**Definition**

**1.**

_{p}→ ℝ with ρ(0) = 0 is called a monetary risk measure if the following properties hold:

- (1)
- Monotonicity: If X
_{1}≥ X_{2}almost surely, then ρ(X_{1}) ≤ ρ(X_{2}) - (2)
- Cash invariance or translation invariance12: ρ(X + m) = ρ(X) − m for every m$\in $ ℝ

**Definition**

**2.**

- (1)
- Positive homogeneity: ρ(λX) = λ ρ(X) for all X$\in $L
_{p}and λ$\in $ℝ_{≥0}. - (2)
- Subadditivity: ρ(X
_{1}+ X_{2}) ≤ ρ(X_{1}) + ρ(X_{2}) for all X_{1}, X_{2}$\in $L_{p}

_{1}= X

_{2}. One approach to incorporate liquidity risks is to alter the distribution assumptions of the random variables by adjusting the payoffs to liquidity risks and maintaining the axioms of coherent risk measures, see (Acerbi and Scandolo 2008; Artzner et al. 1999, p. 209, Remark 2.8).

_{1}+ m

_{1}+ X

_{2}+ m

_{2}, 0) ≤ −min(X

_{1}+ m

_{1}, 0) − min(X

_{2}+ m

_{2}, 0)

## 3. Subadditivity and Bank Mergers

_{1}, D

_{1}) + min(X

_{2}, D

_{2}) ≤ min(X

_{1}+ X

_{2}, D

_{1}+ D

_{2})

_{1}− D

_{1}) + max(0, X

_{2}− D

_{2}) ≥ max(0, X

_{1}+ X

_{2}− D

_{1}− D

_{2})

_{1}and q

_{2}merge, the insolvency quota of the merged entity is denoted by q

_{m}. A bank creditor receives a lower payoff as a result of a merger if q

_{m}< q

_{i}. The following general result states that q

_{m}is always in between the two insolvency quotas q

_{1}and q

_{2}in the stand-alone case:

**Lemma**

**1.**

_{1}= min(1, X

_{1}/D

_{1}) and q

_{2}= min(1, X

_{2}/D

_{2}) denote the insolvency quotas for bank A and bank B and q

_{m}= min[1, (X

_{1}+ X

_{2})/(D

_{1}+ D

_{2})] the insolvency quota of the merged group. Assume, without loss of generality, q

_{1}≤ q

_{2}. Then for every possible realization of the asset values X:

_{1}≤ q

_{m}≤ q

_{2}

**Proof.**

## 4. Subadditivity and Deposit Insurance

**Lemma**

**2.**

_{1}and S

_{2}respectively. The default of the merged entity implies q

_{m}< 1. Assume

_{,}without loss of generality, q

_{1}< q

_{2}and that at least bank A would default in the stand-alone case, or q

_{1}< 1.

_{2}< 1, the condition that the deposit insurance scheme is worse off after a merger is given by

_{1}/D

_{1}< S

_{2}/D

_{2}

_{2}= 1, the condition is given by

_{1}/D

_{1}< S

_{2}(D

_{1}+ D

_{2}− X

_{1}− X

_{2})/(X

_{2}D

_{1}− D

_{2}X

_{1})

**Proof.**

_{2}< 1), a merger is detrimental for the deposit insurance scheme if bank A (whose creditors would receive a higher quota after a merger) has a lower proportion of insured deposits S

_{i}as part of all deposits D

_{i}as compared to bank B. It is intuitively clear that if the proportion of insured deposits is exactly the same at both banks, the deposit insurance scheme is indifferent to a merger, since it would then receive the same proportion of assets as a result of the insolvency procedures, regardless of whether a merger has taken place or not. If the proportion of insured deposits differs, a merger primarily benefits the uninsured excess claims against bank A. In the second case, where only bank A, but not bank B would default in the stand-alone case (q

_{2}= 1), a more complicated condition arises.

## 5. Subadditivity and Contagion Risk

_{3}and the nominal value of the loan that bank C has given to bank B is denoted by L with L ≤ D

_{2}. First, the case where no merger has occurred is considered:

_{1}q

_{1}= min(X

_{1}, D

_{1})

_{2}− L) q

_{2}= (1 − L/D

_{2}) min(X

_{2}, D

_{2})

_{3}+ L q

_{2}, D

_{3})

_{1}− D

_{1}q

_{1}= max(X

_{1}− D

_{1}, 0)

_{2}− D

_{2}q

_{2}= max(X

_{2}− D

_{2}, 0)

_{3}+ L q

_{2}− D

_{3}, 0)

_{1}+ X

_{2}+ X

_{3}, which is the sum of all existing assets excluding interbank loans, which are netted out. The above formulas therefore show how payoffs from loans to borrowers outside the financial system are divided into payoffs to external owners and creditors of the banks.

_{1}+ D

_{2}− L) q

_{m}= [1 − L/(D

_{1}+ D

_{2})] min(X

_{1}+ X

_{2}, D

_{1}+ D

_{2})

_{3}+ L q

_{m}, D

_{3})

_{1}+ X

_{2}− (D

_{1}+ D

_{2}) q

_{m}= max(X

_{1}+ X

_{2}− D

_{1}− D

_{2}, 0)

_{3}+ L q

_{m}− D

_{3}, 0)

_{1}+ X

_{2}+ X

_{3}.

_{m}> q

_{2}, second or fourth row of Figure 1) or lower (q

_{m}< q

_{2}, third or fourth row of Figure 1) as a result of a merger of banks A and B. The payment bank C receives on the interbank loan will then be also either higher or lower in these cases. If payments to creditors of bank B increase as a result of a merger, then part of this increase will be absorbed by bank C, but external creditors of bank C do not necessarily benefit from this increase. If for example X

_{3}> D

_{3}, assets of bank C are large enough that it would never default, irrespective of the quota it receives on the interbank loan. Since it follows from lemma 1 that creditors of bank A will receive a lower or equal quota if the quota for creditors of bank B increases due to a merger, external creditors of bank B (other than bank C) are the only creditors who would benefit from a merger. Assume further that bank B has only very few external creditors and is almost exclusively funded by bank C, i.e., L ≈ D

_{2}. In this extreme case, and contrary to our result in Section 3, all external creditors combined could receive a lower payoff after a merger has taken place33.

_{m}< q

_{2}). According to the results of Section 3, creditors of bank A would then, conversely, receive a higher recovery rate. It can be concluded that all external creditors combined receive a higher payoff, since all creditors of bank A are external creditors. Note that, for the constellation given in Figure 3, our result from Section 3 that external creditors as a whole always get a higher payoff as a result of a merger can only be generalized to the case with interbank loans if q

_{m}< q

_{2}, i.e., if contagion risks are more(!) pronounced after a merger.

## 6. Summary and Discussion

- According to the subadditivity assumption of Artzner et al. (1999), a merger does not create additional risk. In order to check this assertion, a merger of two bank portfolios with given payoffs has been considered. It was assumed that the merger only alters the legal structure, but not the payoffs of the two bank portfolios.
- A merger then alters the distribution of payoffs between creditors and owners. If taken as one group, bank creditors always receive a higher payoff after a merger, and the reverse is true for bank owners. However, this does not rule out that some individual creditors or holders of certain types of debt (e.g., junior debt) can be worse off after a merger.
- If a deposit insurance scheme exists, a merger can also create a higher risk for such a scheme. It is possible that the scheme must make a higher payout than it would be the case without a merger.
- If banks are linked via interbank loans, a merger could also lead to higher contagion risks. In addition, the result that a merger always increases the payoffs for all bank creditors combined cannot be generalized to arbitrarily complex networks of interbank loans.

^{2}= 1.99%.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Lemma**

**1**

_{1}≤ q

_{2}=> q

_{1}≤ q

_{m}

_{1}/D

_{1}≤ X

_{2}/D

_{2}, then:

_{2}X

_{1}≤ D

_{1}X

_{2}

_{1}X

_{1}+ D

_{2}X

_{1}≤ D

_{1}X

_{1}+ D

_{1}X

_{2}

_{1}/D

_{1}≤ (X

_{1}+ X

_{2})/(D

_{1}+ D

_{2})

_{1}= min(1, X

_{1}/D

_{1}) ≤ q

_{m}= min[1, (X

_{1}+ X

_{2})/(D

_{1}+ D

_{2})]

_{1}/D

_{1}> X

_{2}/D

_{2}. Because of q

_{1}≤ q

_{2}this implies X

_{1}/D

_{1}> X

_{2}/D

_{2}> 1 and therefore X

_{1}+ X

_{2}> D

_{1}+ D

_{2}. It follows that q

_{1}= q

_{m}= 1.

_{1}≤ q

_{2}=> q

_{m}≤ q

_{2}

_{1}/D

_{1}≤ X

_{2}/D

_{2}, then:

_{2}X

_{1}≤ D

_{1}X

_{2}

_{2}X

_{2}+ D

_{2}X

_{1}≤ D

_{2}X

_{2}+ D

_{1}X

_{2}

_{1}+ X

_{2})/(D

_{1}+ D

_{2}) ≤ X

_{2}/D

_{2}

_{m}= min[1, (X

_{1}+ X

_{2})/(D

_{1}+ D

_{2})] ≤ q

_{2}= min(1, X

_{2}/D

_{2})

_{1}/D

_{1}> X

_{2}/D

_{2}. Because of q

_{1}≤ q

_{2}this implies X

_{1}/D

_{1}> X

_{2}/D

_{2}> 1 and therefore X

_{1}+ X

_{2}> D

_{1}+ D

_{2}. It follows that q

_{1}= q

_{m}= 1. □

## Appendix B

**Proof of Lemma**

**2**

_{1}− S

_{1}q

_{1}+ S

_{2}− S

_{2}q

_{2}

_{1}+ S

_{2}− (S

_{1}+ S

_{2}) q

_{m}

_{1}+ S

_{2}− (S

_{1}+ S

_{2}) q

_{m}> S

_{1}− S

_{1}q

_{1}+ S

_{2}− S

_{2}q

_{2}

_{1}(q

_{1}− q

_{m}) > S

_{2}(q

_{m}− q

_{2})

_{1}< q

_{2}together with Lemma 1 that q

_{1}− q

_{m}< 0 and q

_{m}− q

_{2}< 0, therefore:

_{1}< S

_{2}(q

_{2}− q

_{m})/(q

_{m}− q

_{1})

_{1}< 1 and q

_{m}< 1 implies q

_{1}= X

_{1}/D

_{1}and q

_{m}= (X

_{1}+ X

_{2})/(D

_{1}+ D

_{2}). Therefore:

_{1}< S

_{2}[q

_{2}− (X

_{1}+ X

_{2})/(D

_{1}+ D

_{2})]/[(X

_{1}+ X

_{2})/(D

_{1}+ D

_{2}) − X

_{1}/D

_{1}]

_{1}< S

_{2}D

_{1}[(D

_{1}+ D

_{2})q

_{2}− (X

_{1}+ X

_{2})]/[(X

_{1}+ X

_{2}) D

_{1}− (D

_{1}+ D

_{2})X

_{1}]

_{1}< S

_{2}D

_{1}[(D

_{1}+ D

_{2})q

_{2}− (X

_{1}+ X

_{2})]/[X

_{2}D

_{1}− D

_{2}X

_{1}]

_{2}= X

_{2}/D

_{2}≤ 1, this simplifies to:

_{1}< S

_{2}D

_{1}[(D

_{1}+ D

_{2})X

_{2}/D

_{2}− (X

_{1}+ X

_{2})]/[X

_{2}D

_{1}− D

_{2}X

_{1}]

_{1}< S

_{2}D

_{1}[(D

_{1}+ D

_{2})X

_{2}− (X

_{1}+ X

_{2}) D

_{2}]/[(X

_{2}D

_{1}− D

_{2}X

_{1})D

_{2}]

_{1}/D

_{1}< S

_{2}/D

_{2}

_{2}= 1 the result is:

_{1}/D

_{1}< S

_{2}(D

_{1}+ D

_{2}− X

_{1}+ X

_{2})/(X

_{2}D

_{1}− D

_{2}X

_{1}) □

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1 | As Artzner et al. (1999, p. 209) put it, subadditivity states that “a merger does not create extra risk”. |

2 | As one example out of many, Jarrow (2017, p. 94) claims with respect to the conditions for coherent risk measures: “As axioms, the truth of these conditions should be self-evident”. |

3 | |

4 | Another example is the Swiss Solvency Test (SST) for insurance firms, in which expected shortfall is also used instead of value at risk, see FINMA (2006). By contrast, the International Association of Insurance Supervisors (IAIS) (2017, p. 64) supports the use of value at risk on practical grounds, although it is admitted that Expected Shortfall, referred to as tail-value-at-risk, is the theoretically superior risk measure. |

5 | See Basel Committee on Banking Supervision (2009, p. 21), footnote 8: “The lack of subadditivity for VaR is probably more of a concern for credit risk and operational risk than for market risk”. |

6 | For example, the IMF (2012, p. 107) concludes that Canadian and Australian banks largely avoided the financial crisis that began in 2008 because of a de facto prohibition of bank mergers. |

7 | The literature on the practical problems of empirically estimating and backtesting coherent risk measures is of no relevance for the subject of this paper. We only mention Cont et al. (2010), who have shown that there exists a conflict between robustness of a risk measurement procedure and the subadditivity of the risk measure. |

8 | |

9 | |

10 | See (Artzner et al. 1999, p. 206; Föllmer and Schied 2016, pp. 194ff.). Another interpretation defines X as the possible loss of a portfolio, see e.g., (McNeil et al. 2015, p. 72). Both interpretations are equivalent, since if the current value of a portfolio is denoted by V _{0}, then X′ = V_{0} − X. |

11 | |

12 | The cash invariance condition has been questioned in the case of uncertainty about interest rates or if no risk-free asset is available, see (Cerreia-Vioglio et al. 2011; Farkas et al. 2014). |

13 | Interest on cash deposits are neglected here for simplicity reasons. |

14 | Possible distributions of X are restricted by the fact that asset values cannot turn negative. |

15 | For a formal analysis of this see (Farkas and Smirnow 2017). |

16 | (Artzner et al. 1997, p. 69; 1999, pp. 209ff.). The dual representation theorem is a nice mathematical result that states that every coherent risk measure can be represented as the worst case expectation taken over a certain set of probability distributions, see Föllmer and Schied (2016, p. 207), Artzner et al. (1997, p. 69; 1999, pp. 209ff.) called such a set of probability distributions ‘generalized scenarios’. For example, consider all events A _{i} with prob(A_{i}) = 5% and the set of conditional probability distributions P_{i} where P_{i} is conditional on such an event A_{i}. For every of these conditional probability distributions P_{i}, the expected value can be calculated. The infimum of all these expectations is the so-called Expected Shortfall at 5% level, which averages the worst 5% results. |

17 | Let Y = −Σ _{i} min(X_{i} + m_{i}, 0) ≥ 0 denote the total sum of calls for additional funds by the exchange clearing house and Y′ ≤ Y the sum of calls for additional funds in the case in which a merger has taken place. Assume further that the exchange clearing house applies a second risk measure φ( ) and requires margin payments high enough so that φ(Y) ≤ 0. An analogous monotonicity condition for φ( ) would imply φ(Y′) ≤ φ(Y) ≤ 0 so that no additional margin payments would be required as a consequence of a merger. See Dhaene et al. (2008, pp. 371ff.) for such an approach. |

18 | |

19 | Time and costs of insolvency procedures are neglected for simplicity reasons. |

20 | This was already pointed out by Rootzén and Klüppelberg (1999, p. 553). |

21 | Junior bonds sold to retail investors played a role in the latest bank rescues in Italy and Spain. Banks that were in distress in 2017 and had sold junior debt to retail investors include Banca Monte dei Paschi di Siena, Veneto Banca, Banca Popolare di Vicenza in Italy, and Banco Popular in Spain. |

22 | Legislators and legal scholars have been aware of the fact that a merger might indeed create extra risk to creditors. Article 99 of Directive 2017/1132 of the European Union for example states that “Member States shall ensure that the creditors are authorized to apply to the appropriate administrative or judicial authority for adequate safeguards provided that they can credibly demonstrate that due to the merger the satisfaction of their claims is at stake and that no adequate safeguards have been obtained from the company.” In Germany, for example, creditors may demand a security deposit from the debtor before a merger, if the circumstances suggest the future endangerment of repayment (§ 22 Umwandlungsgesetz UmwG). However, according to German court rules, this requires a “concrete” or real danger of default. This kind of creditor protection is less developed in most English-speaking countries. |

23 | In the context of regular corporations, Banal-Estañol et al. (2013) call the results of these two possible scenarios “risk contamination” and “coinsurance”. |

24 | If both banks are of equal size in terms of assets and both have an equity ratio of 8%, a medium sized shock would be a shock that triggers an asset write-down in the range of 8% to 16% of the assets of the respective bank. |

25 | q = 1 could also indicate that a default has taken place but creditors are nevertheless repaid in full. |

26 | Since market forces are suspended, the Modigliani–Miller theorem on the irrelevance of the capital structure is no longer valid in a world with deposit insurance. |

27 | For example, Sweden in the 1990s and Ireland in 2008 increased deposit insurance to an unlimited amount. The German government also stated in 2008 that it would guarantee all private savings accounts. However, such guarantees do not always cover all liabilities of a bank. |

28 | It is neglected in the following that a merger might reduce the amount of insured deposits if a depositor has accounts at both banks involved in the merger and the guaranteed amount is calculated per depositor and bank. |

29 | If instead claims filed by the insurance scheme would rank before excess claims, then a merger would always be beneficial for the deposit insurance scheme, because it increases the liability mass. Payments for excess claims would only be made after the insurance scheme has already been fully refunded. |

30 | In the EU, Article 9 of Directive 2014/49/EU states that “the DGS (Deposit Guarantee Scheme) shall have a claim against the relevant credit institution for an amount equal to its payments. That claim shall rank at the same level as covered deposits under national law governing normal insolvency proceedings”. |

31 | |

32 | The capitalization of the banks also plays a role. If banks are better capitalized, it is more likely that losses can be absorbed by the combined capital of both banks, and vice versa. |

33 | This is at least true if creditors of bank A receive a strictly lower quota in case of a merger (q _{1} < q_{m}) and L = D_{2}. For continuity reasons, this result would then also be valid for some L < D_{2}. On the other hand, if L = 0, there is no interbank loan, and the result of Section 3 again applies. |

34 | (Föllmer and Schied 2002; 2016, p. 197; Frittelli and Rosazza Gianin 2002). If cash invariance holds (see Definition 1(2) above), then quasi-convexity and convexity are equivalent, for a proof see e.g., Farkas and Smirnow (2017). A positive homogeneous risk measure (see Definition 2(1) above) is convex if and only if it is subadditive. Therefore, there is only a difference between convexity and subadditivity if the assumption of positive homogeneity is rejected, possible reasons for this are discussed in Section 2. The most prominent example for a risk measure that is quasi-convex and convex but, unlike expected shortfall, not subadditive is the entropic risk measure or exponential premium principle, see for example (Föllmer and Schied 2016, p. 203). |

35 | See Merton (1974), who applies an option-theoretical model to explain the market value of equity and debt. The resulting agency conflicts have been studied by Jensen and Meckling (1976) and the following literature on asset substitution. |

36 | This does not necessarily also apply to junior debt. Junior debtholders are long a call with strike price equal to the nominal value of senior debt only and short a call with strike price equal to the nominal value of all debt. It is not clear how this combination of long and short positions is affected by risk reduction, see e.g., Kolb (1995, p. 309). |

37 | There is also a literature showing that the probability of many simultaneous bank failures could increase if banks diversify their risks in similar ways, see (Battiston et al. 2012; Beale et al. 2011; Stiglitz 2010; Wagner 2010). The banking system is then more vulnerable to a common shock. |

38 | See (Basel Committee on Banking Supervision 2005, p. 3). In the internal-ratings based (IRB) approach of the Basel framework, α is set at 0.1% with a one-year time horizon, i.e., 99.9% of the tail of the distribution is covered by value at risk. Of course, much lower confidence levels are used if value at risk is applied to market risk, which is problematic for assets that most of the time deliver steady returns but very rarely suffer large losses. |

39 | In this context see also (Ibragimov and Prokhorov 2016) who show that diversification does not reduce value at risk for heavy tailed distributions with a tail index below one. |

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**MDPI and ACS Style**

Rau-Bredow, H.
Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures. *Risks* **2019**, *7*, 91.
https://doi.org/10.3390/risks7030091

**AMA Style**

Rau-Bredow H.
Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures. *Risks*. 2019; 7(3):91.
https://doi.org/10.3390/risks7030091

**Chicago/Turabian Style**

Rau-Bredow, Hans.
2019. "Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures" *Risks* 7, no. 3: 91.
https://doi.org/10.3390/risks7030091