Reduced Form Capital Optimization
RReduced Form Capital Optimization
Yadong Li, Dimitri Offengenden and Jan Burgy ∗ May 16, 2019
Abstract
We formulate banks’ capital optimization problem as a classic mean variance optimization,by leveraging an accurate linear approximation to the Shapely or Constrained Aumann-Shapley(CAS) allocation of max or nested max cost functions. This reduced form formulation admitsan analytical solution, to the optimal leveraged balance sheet (LBS) and risk weighted assets(RWA) target of banks’ business units for achieving the best return on capital.
The regulatory capital requirement for banks has been on a rising trend since the 2008 financialcrisis. Given the increasing cost in capital, banks are strongly incentivized to optimize theircapital utilization among their business units to achieve better return on capital (RoC). Today,improving capital efficiency is the center piece of a bank’s business strategy.Despite being a primary concern for banks’ leadership teams, a rigorous and practical quan-titative method has not been established for capital optimization. The main challenge in capitaloptimization is its enormous complexity, as it depends on a bank’s entire business operation:including trading, lending, revenue, expenses and management strategies. Among those, theregulatory capital calculation is one of the most complex aspects.A global bank often sets up multiple subsidiary legal entities, in order to compete effectivelyin local markets. Each of the legal entities is regulated by their respective regional bankingauthorities. In addition, a bank has to report its consolidated balance sheet and capital for itstop level legal entity, which is subject to the rules and regulations of its main domestic regulator.A bank’s overall capital is therefore the greater of 1) the consolidated capital of its top legalentity, or 2) the sum of all subsidiary legal entities’ capital.The regulatory capital of a single legal entity is the greater of the required capital for riskweighted assets (RWA) and leverage balance sheet (LBS), as described in Basel Committee onBanking Supervision (2017):Legal Entity Capital ≥ max(CET1 × RWA , T1 × LBS) (1)where the CET1 is the core tier one capital ratio for RWA, and the T1 is the tier one capitalratio for LBS . Subsequently, we loosely refer to the CET1 × RWA and T1 × LBS as the RWAand LBS capital. Under Basel 3, the minimum CET1 ratio is 4.5% and the minimum T1 ratiois 3%. However, banks usually set up internal management CET1 and T1 target ratios wellabove their regulatory minima, the internal management targets factor in stress capital and ∗ Barclays, we thank Oksana Kitaychik, Art Mbanefo, Barry McQuaid, Jeff Nisen, Ariye Shater and Yishen Songfor many useful discussions and comments. The views expressed in the paper are the authors’ own, they do notnecessarily represent the views of Barclays. There is a slight difference between the capital structure corresponding to the CET1 and T1 ratios, as hinted bytheir names; the difference lies in the additional tier one (AT1) instruments, such as Coco bonds. This difference canbe compensated by adjusting the T1 ratio. a r X i v : . [ q -f i n . P R ] M a y anagement buffers. The LBS and RWA in (1) are extremely complicated nonlinear functionsdepending on many underlying factors, such as a bank’s trading position, balance sheet, marketrisk factors and even historical data.One possible approach for capital optimization is to build a structural model for a bank’sentire operation, similar to the CCAR (Federal Reserve Board (2018)). Such a structural modelwould have to be detailed enough so that bank’s revenue and capital can be related to variousunderlying driving factors. With this approach, an optimal dynamic strategy could be foundfrom the structural model by varying the driving factors. Goel et al. (2017) is a recent attemptat the structural model approach; it derives the optimal capital strategy of a stylized bank withtwo business units over three time periods. The difficulty of such structural model approach isthat it quickly becomes very complicated and intractable; and it incurs high level of model riskbecause of the large number of model assumptions and the likelihood of missing or mis-specifyingimportant factors and relationships.Instead, we take a much simpler and more practical reduced form approach, by expressing abank’s optimal capital strategy in terms of allocated capital. The allocated capital is the resultfrom a strict additive allocation of a bank’s overall capital to individual business units. Readersare referred to Li et al. (2017) for a detailed description of various capital allocation methodsand their properties. A business unit’s allocated capital is usually smaller than its standalonecapital, because the allocated capital takes into account hedging and diversification benefits.One of the main usages of allocated capital is to compute the RoC of individual business units,so that the hedging and diversification effects are included for fair performance comparisons.By treating allocated capital as exogenous factors, we greatly simplified the formulation ofcapital optimization because we no longer need to model any of the underlying driving factors orthe complicated business operations of a bank; as its capital is simply the sum of the allocatedcapital by construction. It is a routine practice for a bank’s senior management to set limitsor targets on individual business units’ allocated capital. Thus, an optimal capital strategyexpressed in terms of allocated capital can be easily implemented by changing the correspondingtargets on allocated capital, and relying upon the well-established management structure andprotocol to achieve them.The max function in (1) poses another challenge: it renders the nonbinding capital compo-nent (which is the smaller of the RWA and LBS capital) irrelevant for a legal entity’s overallcapital, making it very difficult for managers to understand and reason about the impact of thenonbinding capital component. As a result, it is common for banks to only actively manage thebinding capital constraint, which is easier but clearly insufficient and sub-optimal.It is straightforward to conclude that a bank can only be optimal in RoC when its con-solidated top level RWA and LBS capital are equal. Otherwise there is either “free” RWAor “free” LBS that could be used to generate additional revenue without increasing a bank’scapital. However, in practice there are various business constraints that prevent a bank fromoperating at its global optimal RoC with equal RWA and LBS capital. For example, it may notbe feasible for a bank to dramatically change the footprint of business units that caused theimbalance of its RWA and LBS capital. As these practical business constraints are difficult tomodel quantitatively, we take a local optimization approach, which is to find a local optimumof RoC near a bank’s current capital positions, instead of finding the global optimum in RoC.The local optimum is well suited for a bank’s day to day capital management as it would notbe difficult for a bank to implement small changes from its current capital position.Another challenge of capital optimization is to ensure consistency in optimizations across abank’s business unit hierarchy. In practice, capital optimization is performed at multiple levelsof a bank, where each business unit may take independent actions to optimize their own RoCs.As previously mentioned, a business unit’s RoC depends on capital allocation method for itsdenominator, therefore different capital allocation methods will result in different RoCs, leadingto different incentives for business units to optimize against. As a result, it is important fora capital allocation method to set up the correct business incentives so that when individualbusiness units attempt to optimize their own RoC it also results in an improvement in a bank’soverall RoC. The business incentives of different allocation methods were studied in Li et al. BusinessUnit RWACapital LBSCapital Revenue Allocation RoCS/A Euler Shapley Linear S/A Euler Shapley Linear
A 230 150 23 195 150 179 179 0.118 0.153 0.128 0.129B 120 250 25 212 250 218 218 0.118 0.1 0.115 0.115C 150 250 25 212 250 228 227 0.118 0.1 0.11 0.11D 250 150 25 212 150 186 185 0.118 0.167 0.134 0.135E 150 200 20 169 200 188 191 0.118 0.1 0.106 0.105
Total 900 1000 118 1000 1000 1000 1000 0.118 (2017), showing that the Shapley or Constrained Aumann-Shapley (CAS) allocations give thebest business incentives among common allocation methods.To illustrate business incentives afforded by different allocation methods, we consider astylized example of a bank with five business units in Table 1. In this example, we assume thebank’s total LBS and RWA capital are simple sums of those of individual business units, i.e.we ignored the diversification benefits between different business units in either LBS or RWA.However, there remains a significant diversification benefit between the bank’s RWA and LBScapital because of the nonlinear max function . In the table, we show the results from differentallocation methods, such as Euler, Standalone (S/A in the table) and Shapley/CAS allocations,and the corresponding RoCs under a simple assumption for revenue: A business unit’s revenueis 10% of its standalone capital. The “Linear” column under “Allocation” in Table 1 is a linearapproximation to the Shapley allocation, which we will explain in detail later.The example in Table 1 shows that the Shapley/CAS allocation indeed gives a more sen-sible allocation and better business incentives than other allocation methods. For example, itrecognizes that the firm’s current binding capital constraint is in the LBS, therefore D receivesa smaller allocation than B or C (even though the three have identical standalone capital). Calso receives more allocation than B because of more RWA capital. In contrast, the Standaloneallocation offers no incentive for closing the RWA and LBS gap by giving identical allocationsto C and D. The Euler allocation goes to the opposite extreme of totally disregarding the RWAcapital, and giving identical allocation to B and C, despite C’s larger RWA capital. Overall,the Shapley allocation produces more balanced “relative costs” of the RWA and LBS capital,which not only strongly incentivizes the business units to close the gap between RWA and LBScapital, but also rewards them for reducing their non-binding RWA capital. As a result, theRoC computed from Shapley allocation in Table 1 is a much better measure of the relativeperformance of the business units than the RoCs computed from other allocation methods.In the rest of this article, we first present an important linear approximation to Shapley/CASallocation of the cost function in the format of (1). We then use this approximation to formulatethe reduced form capital optimization. Shapley allocation specifies that the allocation to a business unit is the average of its incrementalcontributions over all possible permutations of the business units. We use Ω to represent thefull set of business units in a bank, each being indexed and identified by an integer from 1 to n .Here the term “business unit” is used in a very generic sense, it could mean a business division,a trading desk, a trading book or even an individual trade. We use ˜ π to represent a randompermutation across all the business units, and ˜ π ( k ) is the position of business unit k in thepermutation ˜ π ; therefore, the S ( k ; ˜ π ) = ( i ; ∀ ˜ π ( i ) < ˜ π ( k )) is the set of business units positionedahead of the business unit k for the given permutation ˜ π . The sum of individual business units’ standalone capital is 1180, and the bank’s total capital is 1000; thus, adiversification equals 1180 − In this particular example, the CAS allocation is identical to the Shapley allocation. sing this notation, the Shapley allocation to the business unit k can be written as: α k = 1 n ! (cid:88) ˜ π ( c ( S ( k ; ˜ π ) ∪ k ) − c ( S ( k ; ˜ π ))where c ( S ) is a cost function for a given subset S ⊂ Ω of business units and the average is overall possible n ! permutations of the n business units. max( (cid:80) i ∈ Ω a i , (cid:80) i ∈ Ω b i ) We consider the following allocation problem: each business unit i in Ω has two associated costmetrics a i and b i , e.g., LBS capital and RWA capital. These two cost metrics are additive and theoverall cost of any subset of S ⊂ Ω is the greater of the two, i.e.: c ( S ) = max( (cid:80) i ∈ S a i , (cid:80) i ∈ S b i ).The total cost to be allocated is therefore c (Ω) = max( (cid:80) i ∈ Ω a i , (cid:80) i ∈ Ω b i ).In the cost function c ( S ) = max( (cid:80) i ∈ S a i , (cid:80) i ∈ S b i ), the arguments in max( · ) are linear in a i and b i . Therefore the incremental contribution of business unit k in an arbitrary permutation˜ π is its add-on to the dominant side of the running sum (cid:80) i ∈ S a i or (cid:80) i ∈ S b i , where S is the setof business units ahead of k in ˜ π ; for now we ignore the case that the dominant side may switchby adding the unit k itself to S . Therefore, if we use p ( a < b ; k ) to represent the probabilitythat the running sum up to k in a random permutation ˜ π is greater for the sum of b , then wehave a simple linear approximation of the average incremental contribution of unit k , a.k.a theShapley allocation to the unit k : α k ≈ (1 − p ( a < b ; k )) a k + p ( a < b ; k ) b k .We now consider how to compute the p ( a < b ; k ): p ( a < b ; k ) = P [ (cid:88) i ∈ S a i < (cid:88) i ∈ S b i ] = P [ (cid:88) i ∈ S ( a i − b i ) < P [ (cid:88) i i ∈ S ( a i − b i ) <
0] (2)The indicator i ∈ S represents whether the unit i is ahead of k in a random permutation ˜ π .Even though the business unit k itself is not part of S when considering the allocation to k itself, we choose to include k ( a k − b k ) in the summation in (2) for a better approximationto p ( a < b ; k ). By including k ( a k − b k ), we effectively used the average of ( a k + b k ) as k ’sincremental contribution in case adding k changes the dominant side of the running sums .This is a very good approximation because the allocation to the element k is strictly between a k and b k under such circumstances. The other advantage of including the k ( a k − b k ) is thatthe p ( a < b ; k ) no longer depends on k , thus simplifying the formulation. Thus we subsequentlydenote it as p ( a < b ).The p ( a < b ) can be approximated analytically by moment matching the ˜ s = (cid:80) i i ∈ S ( a i − b i )using a normal distribution, whose mean and variance are known analytically (see appendix A): µ s = 12 (cid:88) i ( a i − b i ) σ s = 16 (cid:88) i ( a i − b i ) + 112 (cid:32)(cid:88) i ( a i − b i ) (cid:33) (3) p ( a < b ) ≈ Φ( − µ s /σ s )where Φ( · ) is the normal distribution function. Strictly speaking, the ˜ s is not normal becausethe central limit theorem does not apply as i ∈ S ( a i − b i ) are correlated with correlation of (see appendix A). Further adjustments to higher moments could be made to account for thedeviation from normality, potentially making the moment matching more accurate. However,we found that the normal moment matching is already quite accurate in practice, thus there isno need for such adjustments. This can be easily shown by enumerating all 4 possible combinations of the dominant side before and after adding k . (cid:80) i a i , (cid:80) i b i ): Correlation(MC, (4)) Figure 1 is a numerical test of the linear approximation to the Shapley allocation using (3):we varied the number of business units n from 5 to 50, for each n we drew 20 sets of a i s and b i sfrom a uniform distribution, and computed the correlation between the exact Shapley allocationresults from Monte Carlo (MC) simulation and the linear approximation from (3). Figure 1 isthe average and standard deviation of the correlations from the 20 random samples for each n , which shows that the approximation (3) is quite accurate, even for n as small as 5. Thecorrelation between the numerical solution and the linear approximation is greater than 99.5%for all n , and the accuracy improves with larger n , the correlation goes above 99.9% for n > c (Ω), therefore, we introduce a correction factor β to ensure theadditivity: α k = β ((1 − p ( a < b )) a k + p ( a < b ) b k ) (4) β = max ( (cid:80) i a i , (cid:80) i b i ) (cid:80) i a i − p ( a < b ) µ s . (5)where the µ s and p ( a < b ) are given in (3). In Table 1, we also show that the linear approximationresult from (4) is very close to the corresponding Shapley/CAS allocation. max ( f (Ω) , g (Ω)) We now consider the approximation to the Shapley allocation of the more generic cost function c ( S ) = max ( f ( S ) , g ( S )), which is exactly (1) if f ( S ) , g ( S ) are the LBS and RWA capital of asubset S ⊂ Ω of business units.We denote the Shapley allocation of f (Ω) and g (Ω) to business unit k as α fk and α gk respec-tively so that f (Ω) = (cid:80) i ∈ Ω α fi and g (Ω) = (cid:80) i ∈ Ω α gi by construction. In practice, banks docompute separately the RWA and LBS allocation to its business units. Obviously, the Shapleyallocation of max ( f (Ω) , g (Ω)) is not the max of their respective allocations because the maxfunction is nonlinear, i.e., α k (cid:54) = max( α fk , α gk ).We propose a simple approximation to the cost function in the form of c ( S ) = max ( f ( S ) , g ( S )).The key idea is to approximate f ( S ) and g ( S ) by their respective Shapley allocations, so that c ( S ) ≈ l ( S ) = max (cid:16)(cid:80) i ∈ S α fi , (cid:80) i ∈ S α gi (cid:17) . By construction, c (Ω) = l (Ω) for the entire portfolioΩ because f (Ω) = (cid:80) i ∈ Ω α fi and g (Ω) = (cid:80) i ∈ Ω α gi . The c ( S ) and l ( S ) track each other closely forany subsets S ∈ Ω, as after all the Shapley allocation α f and α g are the expected incremental c ( S ), l ( S )) for VaR contributions to f ( S ) and g ( S ). In essence, we replaced the random incremental contributionsto f ( S ) and g ( S ) by their respective expectations of α f and α g .Figure 2 is a numerical example of 99% VaR and its linear approximation. We generated 20sets of random PnL vectors for each of the n business units with randomized PnL volatility andcorrelations, then compared the mean and standard deviation of the 20 resulting correlationsbetween f ( S ) = VaR( S ) and its linear approximation (cid:80) i ∈ S α fi under many random permuta-tions. It shows that the linear approximation to the cost function is very accurate; the averagecorrelation between the actual and linearized cost function is greater than 99.5% when n > n as small as 5.We do expect this high level of accuracy from the linearized cost function l ( S ) in practicebecause most risk and capital measures, such as VaR, do exhibit linear-ish behaviors at the toplevel business units within legal entities. The reason is that these top level business units tendto have complicated and diverse portfolios, thus their risk and capital metrics are unlikely toexhibit strong non-linear behaviors.The Shapley allocation of c ( S ) can then be well approximated by that of l ( S ), which is givenby (4). Therefore: α k ≈ β (cid:0) − p (cid:0) α f < α g (cid:1)(cid:1) α fk + βp (cid:0) α f < α g (cid:1) α gk (6)where p (cid:0) α f < α g (cid:1) is given in (3) and β is defined in (5).Similar to Shapley allocation, the CAS allocation is linear with respect to its cost function,therefore the same linear approximation (6) also holds true under CAS for all the businessunits in the same legal entity. The simple linear relationship between the RWA allocation, LBSallocation and capital allocation of a business unit in (6) offers a simple, intuitive and consistentway to understand and reason about both the LBS and RWA consumptions; it overcomes thedifficulty of the max function in (1) and allows a bank to effectively and consistently manageand optimize both the binding and non-binding capital components.It is a significant and useful result that the Shapley/CAS allocation of a complicated non-linear cost function c ( S ) = max ( f ( S ) , g ( S )) can be well approximated by a simple weightedaverage of α f , α g in (6). This property is critical for the formulation of the reduced form capitaloptimization in the next section. It is a common practice for global banks to set up multi-level legal entity structures, in order tomeet the regulatory and business requirements of regional markets. The linear approximationcan be extended to more complicated legal entity hierarchies. However, the CAS allocation has o be used in such cases to remain consistent across the organizational structure, as explainedin Li et al. (2017).Let’s consider a realistic example of a bank with two non-overlapping subsidiary legal entities, X and Y ; each of them are regulated by a regional regulator with different minimal capital ratios,and the bank’s consolidated legal entity with the whole portfolio of Ω = X ∪ Y regulated by itsdomestic regulator. For a subset of business unit S ⊂ Ω, we can write its cost function as: c ( S ) = max (cid:0) max (cid:0) f θ ( S ) , g θ ( S ) (cid:1) , max ( f x ( S ∩ X ) , g x ( S ∩ X )) + max ( f y ( S ∩ Y ) , g y ( S ∩ Y )) (cid:1) (7)i.e, the bank’s capital is the greater of 1) the consolidated entity’s capital and 2) the sum of twosubsidiaries’ capital. Within each legal entity, the capital is the greater of LBS and RWA capital,which are represented by f ( · ) and g ( · ); the superscripts θ, x and y represent the consolidatedand subsidiary legal entities. X and Y are non-overlapping: each of the top level business unitof the bank belongs to either X or Y, but not both, i.e., X ∩ Y = ∅ .Following the same approach described in the previous section, we approximate the f ( S ) , g ( S )by their CAS allocations: l ( S ) = max( θ ( S ) , x ( S ) + y ( S )) (8)= max (cid:32) max (cid:32)(cid:88) i ∈ S α f θ i , (cid:88) i ∈ S α g θ i (cid:33) , max (cid:32) (cid:88) i ∈ S ∩ X α f x i , (cid:88) i ∈ S ∩ X α g x i (cid:33) + max (cid:32) (cid:88) i ∈ S ∩ Y α f y i , (cid:88) i ∈ S ∩ Y α g y i (cid:33)(cid:33) where the α fi s and α gi s are the CAS allocation of respective capital component and legal entity. θ ( S ), x ( S ), y ( S ) are defined to be the corresponding terms to ease the notation. The c (Ω) = l (Ω)still holds for the bank’s entire portfolio by construction.Following the same logic and notation as used in section 2.1, the Shapley allocation ofbusiness unit k to the cost function l ( S ) can be approximated as: α k ≈ p ( θ > x + y, α f θ > α g θ ) α f θ k + p ( θ > x + y, α f θ < α g θ ) α g θ k (9)+ p ( θ < x + y, α f x > α g x ) α f x k + p ( θ < x + y, α f x < α g x ) α g x k + p ( θ < x + y, α f y > α g y ) α f y k + p ( θ < x + y, α f y < α g y ) α g y k where all the p ( · , · )s are joint probabilities of the relative ordering of respective running sumsup to k in a random permutation. Note that the α fk , α gk for X or Y is zero if the business unit k is not part of the corresponding regional legal entity.The joint probabilities in (9) can be computed very efficiently by running a small scale MonteCarlo simulation using the CAS allocation of the few top level business units. We can also definea scaling factor β similar to the one in (5) to ensure the additivity of c (Ω) = β (cid:80) i ∈ Ω α i . Then the w j = βp j ( · , · ), where p j ( · , · )s are the individual joint probabilities in (9), can be interpreted asthe “exchange rates” from the corresponding α fk , α gk to the business unit’s CAS capital allocation α k . The same set of “exchange rates” apply to all the business units in the bank, making itconvenient to actively manage and optimize all the capital components from all legal entities. As discussed in the beginning, the Shapley/CAS allocations do set up the correct incentives forindividual business units to take actions that improve a bank’s overall RoC. A key benefit ofthe linear approximation is that it preserves and accentuates the correct business incentive fromthe Shapley/CAS allocations.Figure 3 is a scenario analysis using the same data as in Table 1, but after scaling the RWAcapital of individual business units so that the RWA capital of the bank varies from 500 to 1500,while holding LBS capital constant at 1000. The color bars in the figure are the corresponding“exchange rates” of the RWA and LBS capital for a given scenario. Figure 3 shows that when the bank is operating in a suboptimal state with unequal RWA and LBS capital, the “exchangerate” on the dominant constraint is greater, thus incentivizing the utilization of the non-bindingconstraint. The “exchange rates” are more skewed against the dominant constraint when thereis a wider gap between RWA and LBS capital, providing a stronger incentive to close the gap.When the bank is at its optimal state with equal RWA and LBS capital, the “exchange rates”become equal at , giving no incentive to move away from the optimal state.In addition, since the linear “exchange rates” established a single and consistent conversionbetween the allocated LBS, allocated RWA and the allocated capital across all business unitsand all legal entities, it allows us to derive a convenient analytical solution to local capitaloptimization if we further assume a linear revenue model.We use the vector (cid:126)h to represent the allocated LBS and RWA capital of banks’ n businessunits. For the example we considered in section 2.3, the (cid:126)h would be a vector of length 6 n including all six α f , α g elements on the RHS of (9); (cid:126)h should not be confused with (cid:126)α , which is avector of length n including only the LHS of (9). There are two main reasons to formulate thecapital optimization using the more granular (cid:126)h rather than (cid:126)α . The first is to be able to solve theoptimal LBS and RWA allocation of every individual business unit, not just its overall capitalallocation; the second is that the legal entity’s capital ratio constraints are specified on its LBSand RWA, which can be expressed in (cid:126)h but not in (cid:126)α .We then use (cid:126)r to represent the RoC vector for the corresponding allocated LBS and RWAcapital, so that the dot product (cid:126)r · (cid:126)h is the total revenue of the bank. Similarly, we use (cid:126)w torepresent the corresponding “exchange rates” from the linear approximation presented in theprevious section, so that the dot product (cid:126)w · (cid:126)h is the sum of the allocated capital to all businessunits, which is exactly the bank’s total capital. The (cid:126)w is a function of (cid:126)h , and the RoC vector (cid:126)r is assumed to be constant.By linearizing both the revenue and capital, we can write down a reduced form local capitaloptimization similar to the classic mean variance portfolio optimization: (cid:126)δ ∗ = argmin (cid:126)δ (cid:16) (cid:126)w T ( (cid:126)h + (cid:126)δ ) + (cid:15) (cid:126)δ T V − (cid:126)δ (cid:17) (10)subject to: (cid:126)r T (cid:126)δ = z, i.e., subject to a constant change z in the bank’s revenue, find a change (cid:126)δ from the currentallocated capital (cid:126)h so that the resulting capital position (cid:126)h + (cid:126)δ minimizes the firm’s overallcapital. The V is a constant covariance matrix of (cid:126)h ’s daily changes, which can be convenientlyestimated from historical time series of (cid:126)h following the spirit of the reduced form modeling. The BusinessUnit RWACapital LBSCapital Revenue Return onRWA/C Return onLBS/C RoC ThresholdRWA/C LBS/C
A 230 150 23 0.0605 0.0605 0.0365 0.0879B 120 250 25 0.0676 0.0676 0.0365 0.0879C 150 250 25 0.0625 0.0625 0.0365 0.0879D 250 150 25 0.0625 0.0625 0.0365 0.0879E 150 200 20 0.0571 0.0571 0.0365 0.0879
Total
900 1000 118covariance matrix V captures the “modes” of the co-movements among capital components, e.g.the LBS and RWA capital changes of the same business unit are often highly correlated. The (cid:112) δ T V − (cid:126)δ is a Mahalanobis distance, which is the magnitude of (cid:126)δ measured in the unit of thestandard deviation of (cid:126)α ’s changes. Mahalanobis distance can also be interpreted as a measureof plausibility (Mouy et al. (2017)): the greater the (cid:112) δ T V − (cid:126)δ , the less likely (cid:126)δ can occur underthe given covariance matrix V . The (cid:15) > (cid:126)δ solutions,which also controls the trade-off between capital efficiency and plausibility.Interestingly, there is no need to add explicit minimum ratio constraints of various legalentities to (10), as the optimization is expressed in the allocated capital (cid:126)h and the effects ofcapital ratios are implicitly captured by the RoC vector (cid:126)r . Other business constraints can beeasily added as linear constraints to (10), such as the maximum RWA of a legal entity is $10billion; or the ratio of RWA/LBS of a business unit has to be between 3 and 5. In the mostgeneral case, (10) can be solved numerically using quadratic programming. However, if we ignorethe additional business constraints, an analytical solution to (10) can be easily obtained usingLagrange multiplier: (cid:126)δ ∗ = ( J + (cid:15)V − ) − ( λ(cid:126)r − (cid:126)w − J(cid:126)h ) (11)where J = ∂ (cid:126)w∂(cid:126)h is the Jacobian matrix, λ is the Lagrange multipler which can be determinedusing the constraint (cid:126)r · (cid:126)δ = z : λ = z + (cid:126)r T ( J + (cid:15)V − ) − (cid:126)w + (cid:126)r T ( J + (cid:15)V − ) − J(cid:126)h(cid:126)r T ( J + (cid:15)V − ) − (cid:126)r The Jacobian matrix J can be easily computed numerically or analytically.If we ignore the changes in the capital “exchange rates” (cid:126)w under small changes in (cid:126)h bysetting J = 0, (11) reduces to a simpler “crude” approximation: (cid:126)δ ∗ = λV (cid:18) (cid:126)r − (cid:126)wλ (cid:19) (12)where (cid:126)wλ is a RoC threshold.The crude solution (12) has a simple interpretation when V is diagonal: it dictates thegrowth of the business units with high RoCs at the expense of those with low RoCs. It mimicsthe rule-of-thumb approach that is often used in the absence of a rigorous quantitative solutionfor capital optimization. The (12) shows that the rule-of-thumb approach can only be validunder the following two conditions: 1) V is diagonal, i.e., there is zero correlation betweenthe changes in different capital components 2) the RoCs are computed using Shapley or CASallocation, as it is a pre-requisite to set up the local optimization problem.Table 3 shows the results of (11) and (12) for the data in Table 2, with different z, (cid:15) and V , and a simple revenue model assuming that the return on LBS capital and RWA capital areidentical for the same business unit. The column “RWA/C” and “LBS/C” are the optimalchanges in the allocated RWA and LBS capital, i.e, (cid:126)δ ∗ ; the Total column is the change in Local Optimum (cid:126)δ ∗ with z = 0 , (cid:15) = 0 . , V = I BusinessUnit Local Optimal (11) Crude Solution (12)RWA/C LBS/C Total RWA/C LBS/C Total
A 1.45 -1.73 1.09 1.49 -1.70 1.18B 2.03 -1.15 -4.23 1.93 -1.26 -4.41C 1.61 -1.57 -3.85 1.61 -1.58 -3.91D 1.61 -1.57 1.80 1.61 -1.58 1.85E 1.17 -2.01 -2.84 1.28 -1.91 -2.75
Total (cid:126)δ ∗ with z = 0, Corr(dLBS, dRWA)=0.95 BusinessUnit Local Optimal (11) Crude Solution (12)RWA/C LBS/C Total RWA/C LBS/C Total
A -2.70 -5.15 -3.10 -2.05 -4.62 -2.42B 13.49 11.04 9.05 11.72 9.14 6.94C 1.84 -0.61 -2.31 1.81 -0.76 -2.58D 1.84 -0.61 1.98 1.81 -0.76 1.98E -10.48 -12.93 -13.87 -8.66 -11.24 -12.16
Total (cid:126)δ ∗ with z = 2, Corr(dLBS, dRWA)=0.95 BusinessUnit Local Optimal (11) Crude Solution (12)RWA/C LBS/C Total RWA/C LBS/C Total
A 0.44 -2.01 0.05 0.89 -2.86 0.35B 16.98 14.53 12.54 21.39 17.64 14.43C 5.08 2.63 0.91 6.64 2.88 0.22D 5.08 2.63 5.22 6.64 2.88 6.89E -7.51 -9.95 -10.89 -8.96 -12.71 -14.06
Total llocated capital (cid:126)α for the business unit after applying the optimal change (cid:126)δ ∗ . The optimizationresults in Table 3 are quite sensible and intuitive: the bank’s total LBS and RWA capital aremore balanced; when the revenue is kept constant with z = 0, the optimization reduces thebank’s overall capital; when a revenue increase of z = 2 is required, the bank has to increase itscapital. Table 3 also shows that the crude solution produces similar results to the full solution(11).Table 3 also verifies the scope where the rule-of-thumb may apply. When V = I , the rule-of-thumb approach correctly identifies the general optimal direction of change in capital, whichis to increase capital allocation in A and D, the business units with the greatest overall RoCunder Shapley allocation (see Table 1). However, when V implies a 95% correlation between thechanges in RWA and LBS of the same business unit, the rule-of-thumb approach fails to identifythe optimal direction of change, which is to increase the capital allocation in B at the expenseof E. It is worth noting that in this case, the large LBS and RWA changes of the same businessunit are all to the same direction in the optimal solution, due to the preference for plausiblesolutions. This result shows that the rule-of-thumb approach is inadequate in practice.The solution (11) may break down for large (cid:126)δ ∗ because its underlying assumptions, such asconstant RoC (cid:126)r and Jacobian matrix J , may fail with large deviation from the current capitalposition. Nonetheless, the analytical local optimal solution of (11) offers a valuable quantitativeframework for a bank’s day to day capital management, where small changes in capital positionare much more common and relevant. The accurate linear approximation to the Shapley/CAS allocation of cost function of (nested)max function is an important and useful result. It establishes a single set of “exchange rates”between the capital components of all legal entities, making it easy for a bank’s management toreason about and actively manage their RWA and LBS capital.By leveraging this linear approximation, we formulated banks’ capital optimization problemas a simple mean/variance optimization and obtained an analytical local optimal solution thatmaximizes the bank’s overall RoC; it is a tractable and practical quantitative framework forbank’s day to day capital management.Many well-established classic portfolio optimization techniques and concepts, such as efficientfrontier, can be applied to capital optimization following this reduced form formulation, greatlyexpanding the available toolkit for capital management.
References
Basel Committee on Banking Supervision (2017). Basel iii: Finalising post-crisis reforms.
Bankfor International Settlements .Federal Reserve Board (2018). Comprehensive capital analysis and review 2018: Assessmentframework and results.
Federal Reserve Board Reports and Publications .Goel, T., Lewrick, U., and Tarashev, N. (2017). Bank capital allocation under multiple con-straints.
BIS Working Papers, No. 666 .Li, Y., Naldi, M., Nisen, J., and Shi, Y. (2017). Organizing the allocation.
Risk .Mouy, P., Archer, Q., and Selmi, M. (2017). Extremely (un)likely: a plausibility approach forstress testing.
Risk . Derivation of p ( a < b ; k ) We consider the following two random variables for a random permutation ˜ π of n units:˜ a = n (cid:88) i =1 i a i (13)˜ b = n (cid:88) i =1 i b j . (14)The i is the indicators of whether the unit i is ahead of a given unit k in a random permutation.These indicators are not independent. One can easily prove that the cov( i , j ) = for any k ,by noticing: E [ i ] = E [ j ] = 12 (15) E [ i j ] = 1 n + 1 n (cid:88) m =0 m ( m − n ( n −
1) = 13 (16)where m ( m − n ( n − is the expectation of i j conditioned on there are m units ahead of k , and n +1 is because there is equal probability for there are 0, 1, ..., n units ahead of k . Thereforecov( i , j ) = E [ i j ] − E [ i ] E [ j ] = , by noting E [ i ] = . Therefore the correlation between i and j is by noting the variance of i is .Therefore we have: E [˜ a ] = (cid:80) i a i , E [˜ b ] = (cid:80) i b i and the covariance:cov(˜ a, ˜ b ) = (cid:88) i,j cov( i a i , j b j ) = (cid:88) i,j a i b j cov( i , j )= 14 (cid:88) i a i b i + 112 (cid:88) i (cid:54) = j a i b j = 16 (cid:88) i a i b i + 112 ( (cid:88) i a i )( (cid:88) j b j ) (17)The variance of ˜ a is therefore cov(˜ a, ˜ a ).).