An Integrated Approach to Measuring Asset and Liability Risks in Financial Institutions

Transcription

1 Risk Measurement and Regulatory Issues in Business Montréal September 12, 2017 An Integrated Approach to Measuring Asset and Liability Risks in Financial Institutions Daniel Bauer & George Zanjani (both University of Alabama)

2 Page 2 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction Solvency Ratios in European Insurance Industry (2016/2017) Aegon 185% Allianz 212% [130% for group (?)] AXA 197% Generali 177% [194% for economic view (?)] Munich Re 267% Talanx 160% [186% for HDI group] Zurich 189% [based on SST] "...Solvency Ratio has other functions. Many insurance companies may use a certain level of solvency to demonstrate financial health to their customers, e.g. 150% could be a strategic goal..." (Weglarz, 2015, emphasis added) Exact level does not seem terribly important (?) Importance/relevance of Economic Capital (EC) framework for internal steering

3 Page 3 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction Primary motivations for risk-based EC models Banks: McKinsey (2011)

4 Page 3 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction Primary motivations for risk-based EC models Banks: McKinsey (2011) Insurance: Society of Actuaries (2008)

5 Page 3 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction Primary motivations for risk-based EC models Banks: McKinsey (2011) Insurance: Society of Actuaries (2008) Motivations: Pricing Performance measurement Capital allocation

6 Page 4 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction This paper (draft) Questions: How should we measure / allocate risk if motivation for capital holding is counter-parties (policyholders, depositors, trading partners,...) aversion to risk? What are the implications for solvency frameworks and risk measures?

7 Page 4 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction This paper (draft) Questions: How should we measure / allocate risk if motivation for capital holding is counter-parties (policyholders, depositors, trading partners,...) aversion to risk? Approach: What are the implications for solvency frameworks and risk measures? Solve profit maximization for financial institutions with risk-averse counter-parties (vnm-utility) Derive marginal cost of risk At the optimum, marginal premium / contribution equals marginal costs Solve for risk framework / risk measures that implement the optimum (e.g., in RAROC sense) [Extension to Bauer & Zanjani (2016) where liability risk was considered separately.]

8 Page 5 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction Conventional Framework Assets Liabilities K (capital) A = a(1 + S) = M j=1 a j (1 + S j ) I = N i=1 q N i L }{{} i = i=1 q i( L i ) =I i Available risk horizon (e.g., T = 1): K = A I ( M Risk via risk measure ρ: ρ(k ) = ρ j=1 a j (1 + S j ) + ) N i=1 q i ( L i ) No distinction between assets and liabilities in framework, allocation: ρ(k ) = M j=1 a j ρ a j + N i=1 q i ρ q i (allocations are denominator in RAROC, remember PE s comment)

9 Page 6 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction What s the problem? A simple example Two securities: Risk-free asset with interest rate of zero "Zero-beta" stock $100 in Bin. model that can go to $200 or $0, equal prob.

10 Page 6 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction What s the problem? A simple example Two securities: Risk-free asset with interest rate of zero "Zero-beta" stock $100 in Bin. model that can go to $200 or $0, equal prob. Two institutions, both have $200 of debt and $100 in equity: Institution A buys 2 shares at $200 of stocks and invests $300 $200 = $100 risk-free Institution B takes a short position of 2 shares for $200 of stocks and invests $300 + $200 = $500 risk-free

11 Page 6 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction What s the problem? A simple example Two securities: Risk-free asset with interest rate of zero "Zero-beta" stock $100 in Bin. model that can go to $200 or $0, equal prob. Two institutions, both have $200 of debt and $100 in equity: Institution A buys 2 shares at $200 of stocks and invests $300 $200 = $100 risk-free Institution B takes a short position of 2 shares for $200 of stocks and invests $300 + $200 = $500 risk-free Net Payoff (K = A I) A : = = 100 & B : = = 300

12 Page 6 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction What s the problem? A simple example Two securities: Risk-free asset with interest rate of zero "Zero-beta" stock $100 in Bin. model that can go to $200 or $0, equal prob. Two institutions, both have $200 of debt and $100 in equity: Institution A buys 2 shares at $200 of stocks and invests $300 $200 = $100 risk-free Institution B takes a short position of 2 shares for $200 of stocks and invests $300 + $200 = $500 risk-free Net Payoff (K = A I) A : = = 100 & B : = = 300 Recoveries (Equity holders debt holders trading counter-party, pari passu): 500 u: = = A: B: d: =

13 Page 6 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction What s the problem? A simple example Two securities: Risk-free asset with interest rate of zero "Zero-beta" stock $100 in Bin. model that can go to $200 or $0, equal prob. Two institutions, both have $200 of debt and $100 in equity: Institution A buys 2 shares at $200 of stocks and invests $300 $200 = $100 risk-free Institution B takes a short position of 2 shares for $200 of stocks and invests $300 + $200 = $500 risk-free Net Payoff (K = A I) A : = = 100 & B : = = 300 Recoveries (Equity holders debt holders trading counter-party, pari passu): 500 u: = = A: B: d: = Thus, if counter-parties worry about recoveries, the two positions are not equivalent as the conventional framework suggests (Collateral??)

14 Page 7 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction Asset vs. Liability Risk Asset Risk: A A A Liability Risk: A A "Asset risk is about the size of the cake, whereas liability risk affects the division of the cake"

15 Page 8 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction Consequences Asset and liability risk should be separated for the purpose of risk pricing and performance measurement: ρ L for liabilities: RAROC i = Marginal Premium i Marginal actuarial value i q i ρ L (I) τ? ρ A for liabilities: RAROC j = Marginal Excess Return j a j ρ A (A) τ? Risk measures have different form due to different influence of risk on recoveries (Bauer & Zanjani, 2016): { } ρ L has log-exponential form (I in denominator!): ρ L (I) = exp E P [log{i}], where P concentrated in default states, accounts for counter-party pref. ρ A akin to spectral version of ES (Acerbi, 2002): ρ A (A) = E P [ A A < I], where P accounts for counter-party preferences

16 Page 9 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Introduction Introduction Model: Risky Assets and Liabilities Collateral Example Conclusion

17 Page 10 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Model Setup (one period) Company with N claimants with obligation to claimant i: I i = q i L i Focus on linear contracts, extensions possible Examples: linear insurance contracts, straight debt with Li (1 + r dep ) Claims are sold at price p i = p i (q i ) collected now

18 Page 10 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Model Setup (one period) Company with N claimants with obligation to claimant i: I i = q i L i Focus on linear contracts, extensions possible Examples: linear insurance contracts, straight debt with Li (1 + r dep ) Claims are sold at price p i = p i (q i ) collected now Investment of assets a in M risky projects with return (1 + S j ) for project j Note that we assume existence of frictions / incompleteness since otherwise risk can be hedged

19 Page 10 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Model Setup (one period) Company with N claimants with obligation to claimant i: I i = q i L i Focus on linear contracts, extensions possible Examples: linear insurance contracts, straight debt with Li (1 + r dep ) Claims are sold at price p i = p i (q i ) collected now Investment of assets a in M risky projects with return (1 + S j ) for project j Note that we assume existence of frictions / incompleteness since otherwise risk can be hedged Total assets at end of period A = a (1 + S) = M j=1 a j (1 + S j ), total claims submitted I = N i=1 I i {I A} solvent states, claimants paid in full {I > A} default states, claimants paid recovery Ri = A I I i (for now, collateral...)

20 Page 10 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Model Setup (one period) Company with N claimants with obligation to claimant i: I i = q i L i Focus on linear contracts, extensions possible Examples: linear insurance contracts, straight debt with Li (1 + r dep ) Claims are sold at price p i = p i (q i ) collected now Investment of assets a in M risky projects with return (1 + S j ) for project j Note that we assume existence of frictions / incompleteness since otherwise risk can be hedged Total assets at end of period A = a (1 + S) = M j=1 a j (1 + S j ), total claims submitted I = N i=1 I i {I A} solvent states, claimants paid in full {I > A} default states, claimants paid recovery Ri = A I I i (for now, collateral...) Claimants assess coverage via utility funct.: v i = E [U i (w i p i + R i L i )]

21 Page 11 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Optimization Problem (risk-neutral company) max E {a j },{q i },{p i } i p i + j a j S j i R i τ j a j i p i s.t. v i = γ i, where: τ is the frictional cost of capital (e.g., frictional cost or capital costs in models with macro frictions, Solvency risk margin), and γ i are reservation utilities (e.g., not buy insurance (autarky) or go to a competitor) Possible to include:... additional constraints, e.g. corresponding to Solvency regulation... securities markets (Bauer & Zanjani, 2016)... additional periods (easy if capital is assets raised every period, more complex if capital is a state variable, Bauer & Zanjani, 2017; Guo, Bauer & Zanjani, 2017)... risk aversion of company E[V (...)] Key results still hold

22 Page 12 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities First Order Conditions [ [a j ] E S j ] R k τ + a j k k [ ] v i R k [q i ] λ i E + q i q i k k i [p i ] (1 + τ) λ i v i = 0, λ k v k a j = 0, λ k v k q i = 0, where λ i is the Lagrange multiplier for participation constraint i

23 Page 13 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Marginal Cost of Risk (1 + τ) p i E[1 {I A} L i ] = [ φ L i a ] τ, where i q } i }{{} φl i a = a {{} marginal CoC marginal income E [ 1 {I A} S j 1 {I>A} τ ] = [ φ A j a ] τ, where j }{{}}{{} φa j a = a marginal income (return) marginal CoC τ = τ + P(I > A) E [ S ] 1{I A} }{{}}{{} cost of marginal unit of cap. benefit of capital RAROC i/j = marginal income i/j φ L/A i/j a τ and φ L/A i/j a capital allocation

24 Page 13 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Marginal Cost of Risk (1 + τ) p i E[1 {I A} L i ] = [ φ L i a ] τ, where i q } i }{{} φl i a = a {{} marginal CoC marginal income E [ 1 {I A} S j 1 {I>A} τ ] = [ φ A j a ] τ, where j }{{}}{{} φa j a = a marginal income (return) marginal CoC τ = τ + P(I > A) E [ S ] 1{I A} }{{}}{{} cost of marginal unit of cap. benefit of capital RAROC i/j = marginal income i/j φ L/A i/j a τ and φ L/A i/j a capital allocation Separate treatment of assets and liabilities

25 Page 14 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Risk Measures Liabilities (Bauer & Zanjani, 2016): φ L i = { } exp E P [log{i}], where P q i }{{} P = =ρ L (I) 1 U k I ki {I>A} k v (1 + S) k E[1 {I>A} k U k v k I ki (1 + S)] Note that qi log{i} = I i q i I Assets: =ρ A (A) "share of the cake" φ A j = E P [ A A < I], where P a j }{{} P = 1 U k I ki {I>A} k v k E[1 {I>A} k U k v k I ki ] Note that aj a j A = a j S j "size of the cake"

26 Page 14 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Model: Risky Assets and Liabilities Risk Measures Liabilities (Bauer & Zanjani, 2016): φ L i = { } exp E P [log{i}], where P q i }{{} P = =ρ L (I) 1 U k I ki {I>A} k v (1 + S) k E[1 {I>A} k U k v k I ki (1 + S)] Note that qi log{i} = I i q i I Assets: =ρ A (A) "share of the cake" φ A j = E P [ A A < I], where P a j }{{} P = 1 U k I ki {I>A} k v k E[1 {I>A} k U k v k I ki ] Note that aj a j A = a j S j "size of the cake" Separate risk measures for assets and liabilities, ρ L is not coherent (though monotone and homogeneous)

27 Page 15 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Collateral Collateralization Considerations extend to different degrees of collateralization: Uncollateralized obligations: I u i = q u i L u i, I u = i I u i Collateralized obligations: I c k = q c k L c k, I c = k Ic k Recovery Collateralized: R c k = min{i c k, A I c I c k } Recovery Uncollateralized: R u i = min{i u i, A Ic I u I u i } max{q u i },{q c k }, Have to treat collaterlized and uncollateralized obligations differently, though joint allocation. Risk measure combination.

28 Page 16 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Example Example 2 Risks Insurer: Bank: Profit = p 1 + p 2 E [ min{q 1 L 1 + q 2 L 2, a} ] τ (a p 1 p 2 ) }{{} =I [ { }] a Constraint: E exp α( p i L i + min{q i L i, q i L i } > E [exp { α L i }] q 1 L 1 + q 2 L 2 Profit = E[a 1 S 1 + a 2 S 2 ] E [ min{d(1 + r), a 1 S 1 + a 2 S 2 } ] τ (a 1 + a 2 d) }{{} =I Constraint: E [ exp { α( d + min{d(1 + r), a 1 S 1 + a 2 S 2 }}] > 1 Intuition from basic Gaussian model S 1 N(.02,.1), S 2 N(.03,.2), d = 3, r = 0.055, τ =.05, α = 0.5 a 1 = 1.209, a 2 = , a = Increasing capital costs, the company will hold less Increasing r will yield a safer asset mix (more in the first) and depresses profits

29 Page 17 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Conclusion Conclusion In this paper, we:... develop a model where capital is motivated by counter-party risk aversion... We solve for risk measure that implements optimal portfolio (in RAROC sense) Economic impact of exposure and, hence, risk measure depend on:... whether risk is on asset or liability side... collateralization Focusing on next exposure will deliver erroneous prices / portfolios / incentives

30 Page 18 Risk Measurement Workshop September 12, 2017 Bauer/Zanjani Conclusion Contact Daniel Bauer George Zanjani University of Alabama USA Thank you!