Regulation, Capital Structure Decision, and …fmaconferences.org/Boston/Regulation,_Capital_Structure_Decision...Regulation, Capital Structure Decision, and Allocation ... internal

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Regulation, Capital Structure Decision, and Allocation in Banks

Woo-Young Kang, Sunil Poshakwale, and Vineet Agarwal

Abstract

The banks are now under strict financial regulation at the expense of their lower profitability.

We provide a closed-form solution for optimal internal capital allocation that the banks can

maximize their return to risk efficiency (e.g. return to risk adjusted capital) in this financial

environment. We allow no additional external capital input and reflect financial forecasting

uncertainty in our model. Our model maximizes the bank’s efficiency only with its existing

capital and minimizes the forecasting error using two additional enhancements: profitability

adjusted debt effect and Bayesian learning process. We empirically test our model using

banks from S&P500 and prove that the overall average efficiency improves using regulatory

capital (Basel regime) and equity (economic capital regime) where the improvement using

equity is higher. Then we compare the available capital and required minimum capital with

our allocation model and find that equity (economic capital regime) based model shows

better risk control. Therefore, we suggest using equity based efficiency maximization using

our model. However, we still propose that bank specific capital allocation approach is also

needed as each bank show different degree of efficiency improvements depending on

regulatory capital or equity based allocation.

Keywords: Return on Risk Adjusted Capital (RORAC), full multi-period,

Basel regime, economic capital regime,

profitability adjusted instantaneous debt effect, Bayesian learning

Cranfield School of Management, Cranfield University, Cranfield, Bedfordshire, MK43 0AL, United Kingdom

Email Addresses: [email protected] (W. Kang) +44 (0)1234 751122

[email protected] (S. Poshakwale) +44 (0)1234 754404 [email protected] (V. Agarwal) +44 (0)1234 751122

mailto:[email protected]



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1. Introduction

Since the global financial crisis in 2007-2008, the inherently high-risk characteristic of the

financial firms has received drastic attention that eventually brought the financial authority to

take place in their regulation. Consequently, the financial firms are not able to take excessive

risk anymore with the temptation for boosting their profits. Thus, sometimes they can even

miss good investment opportunities that can safely increase their returns because of the strict

financial regulation. The Bank for International Settlements (BIS) is planning to continue

raising the capital requirements with stricter regulations despite the high capital costs the

financial firms have to bear. Specifically, the capital called ‘risk capital’ (i.e. ownership

equity) is quite costly and limited as the shareholders need to bear its monitoring and agency

costs and its return is subject to corporate tax (Erel at al. 2015). Simultaneously, it is the most

decisive factor for financial firms as it backs up obligation to liability holders and reduces

debt overhang, risk shifting, and bankruptcy problems (Erel at al. 2015).

Economic capital is the minimum safety risk capital level to ensure the financial firm to

survive from insolvency due to an unexpected loss (James. 1996) and is often calculated

using Value-at-Risk. On the other hand, regulatory capital is the minimally required Tier 1

and Tier 2 capital level that must be at least a certain percentage1 of the risk-weighted-assets

at all times and is forced by the regulators. To maximize the financial firms’ returns while

sufficing the strict capital requirements of the financial authority, an efficient capital

management is the key for this success. We provide our closed-form solution for optimal

internal capital allocation to achieve this aim.

This is the first paper to derive a closed-form solution for optimal internal capital

allocation that is practical to provide empirical evidence using banks in S&P500. Our new

model is based on Buch et al (2011)’s work where we start by providing full multi-period

characteristics not enabled before. Then we provide enhancements to minimize financial

forecasting uncertainty and boost our model’s performance further: profitability adjusted

instantaneous debt effect and Bayesian learning process. We also formulate our new model to

have capital weights always summing up to one meaning that we do not allow additional

external capital input to reflect strict financial environment. The new model automatically

stops its capital allocation just before it expects its allocation to generate risk that the bank

cannot afford with its currently available capital.

1 For example, the current Basel III regulation scheme requires the financial firms to hold the total capital (Tier 1 and Tier 2 capital) at least 8% of their risk-weighted-assets.

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The empirical results show that our new model improves the banks’ overall efficiency

with no external capital input and under financial forecasting uncertainty. We show this

improvement both in regulatory capital (Basel regime) and equity (economic capital regime)

based allocations with our new model. We still see this improvement using our new model

even when we provide capital change constraints for each business unit under both Basel and

economic capital regimes. In this work, the business unit is the segment or division of a firm

such as customer banking division, investment banking division, etc. Then we prove that

equity based allocation provides better empirical result than the regulatory capital based one.

However, we also suggest using bank specific approach, either regulatory capital or equity,

for capital allocation for better efficiency improvements.

Most of the existing works on capital allocation are either in the capital budgeting or

portfolio return maximization contexts. We do not find so many research conducted in the

bank’s capital allocation context and there is no empirical works at all. The importance of

bank’s capital allocation research is rising nowadays especially when the financial regulation

continues to become stronger since the global financial crisis in 2007-2008. The existing

works on this have been mostly theoretical. We classify the existing works into three contexts

as Buch et al (2011): financial economics, insurance and mathematical finance based

contexts.

The works in the financial economic context start with the theoretical research of Merton

and Perold (1993). They provide the definition of risk capital as “the smallest amount that can

be invested to insure the value of the firm’s net assets against a loss in value relative to the

risk-free investment of those net assets”. Their work focuses on adding and subtracting the

entire business units rather than changing their sizes. Then they conclude that the firm should

not conduct risk capital allocation as it can distort the true profitability of the business units.

Erel et al (2015) contradict this point arguing that although this is true in the long-run, risk

capital allocation is needed from the short-run (e.g. on a quarterly basis) point of view. In this

aspect, Perold (2005) and Stoughton and Zechner (2007) develop risk capital allocation

models that expand or contract the business units rather than adding or subtracting them.

Perold (2005) allocates the economic capital calculated using Value-at-Risk (VAR) while

measuring the profit with Adjusted Present Value (APV). Stoughton and Zechner (2007) use

Economic Value Added (EVA) as their profit measure and define their Risk Adjusted Return

on Capital (RAROC) by dividing EVA by economic capital. Their work considers

asymmetric information, managerial opportunities, and coordination of risk-taking activities.

There work is, however, limited in its practical implementation as their model is restricted to

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use only very specific incremental VAR allocation rule with normally distributed risks (Buch

et al. 2011).

In the insurance linked context, Myers and Read (2001) provide a solvency based

allocation principle that uses business unit’s default put option per unit of liability. Sherris

(2006) considers capital allocation jointly with solvency and the fair rate of return on

insurance. Kim et al (2009) extends this model by enabling it to use real world probability

measure to consider limited liability of the shareholders in reality. However, Furman and

Zitikis (2008) use the weighted allocation principle a different approach than the solvency

principle used before. Then Dhaene et al (2010) propose a generalized risk capital allocation

model that can encompass most of the insurance linked allocation principles. In their model,

they derive the optimal capital for each business unit that minimizes the distance between its

allocated capital and loss. However, their work is limited in disregarding the profit

maximization interests of the firms. Erel et al (2015) extend Myers and Read (2001) work.

They use marginal default value, adjusted present value (APV), and credit quality measured

by the ratio of firm’s default option to default free value of its liabilities for capital allocation

decisions. Their work is, however, limited in not considering financial regulation and

empirical testing.

Then in mathematical finance context, Denault (2001) introduces the allocation principle

that incorporates coherent risk measure subject to four properties2: translation invariance,

subadditivity, positive homogeneity, and monotonicity. Kalkbrener (2005) proposes an

axiomatic capital allocation approach that entails linearity, diversification, and continuity

properties. Tasche (2004) suggests an Euler allocation principle that distributes capital to

each business unit according to its risk contribution to the firm. Buch et al (2011)’s work is

also based on Euler allocation principle but they use quadratic risk correction term to prevent

over expansion or reduction of business units. Hence they allocate more accurately estimated

capital to allocate and achieve higher efficiency than Tasche (2004)’s model. However, their

work is not a full multi-period model as it still has static conditions in expected profit, risk

measure, etc. We introduce the closed-form solution for optimal capital allocation extended

from Buch et al (2011)’s work. We overcome their incomplete multi-period setting, add

enhancements (e.g. profitability adjusted instantaneous debt effect and Bayesian learning),

2 When X and Y are portfolios and ρ is the coherent risk measure, then

Monotonicity: for all X and Y, if X ≤ Y, I have ρ(X) ≤ ρ(Y).

Subadditivity: for all and, ρ(X+Y) ≤ ρ(X) + ρ(Y)

Positive homogeneity: for all λ ≥ 0 and all X, ρ(λ X) = λ ρ(X).

Translation invariance: for all X and all real numbers α, if A is a portfolio with return α, I have ρ(X+A) = ρ(X) - α.

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and formulate into a practical model that can produce empirical results under different

financial environments.

One of the obstacles in implementation of such models is that they require business unit’s

regulatory capital or equity data that the financial statements do not report. We start by

estimating the normalized relative volatility of each business unit’s asset compared to the

overall asset’s volatility measured by realized standard deviation. Then we multiply this by

the total regulatory capital or equity to derive these values on the business unit level. Using

the selected thirteen US banks with required data from S&P500, we show our model

improves the overall return to risk profiles of these banks. The average quarterly Return on

Risk Adjusted Capital (RORAC) efficiency for these banks improve from 1.8305% to

3.2234% using regulatory capital as per regulatory capital and from 1.8498% to 3.2467%

using equity.

We take into account the financial uncertainty in our model by implementing two

enhancements: profitability adjusted instantaneous debt effect and Bayesian learning process.

The profitability adjusted instantaneous debt effect recalculates the optimal capital to allocate

by providing more capital to business units with high profitability linked with recent increase

in debt. The Bayesian learning process also recalculates the optimal capital for allocation by

considering the learning from the accumulated difference between the actual profit before

allocation and the ‘belief’ on profit after allocation over time. These enhancements minimize

the estimation errors from financial uncertainty.

This work aims to resolve the dilemma that the banks are facing between two conflicting

financial regimes for efficiency maximization: regulatory capital based allocation (Basel

regime) and equity based allocation (economic capital regime). Our empirical tests prove that

economic capital regime is relatively a better capital allocation environment compared to

Basel regime with higher efficiency improvement and better risk control. However, we also

suggest that the bank specific allocation approach is required as each bank shows different

levels of efficiency improvements under different financial regimes.

The rest of this paper proceeds as follows. In section 2 and 3, we develop our closed-form

solution for optimal capital allocation extending Buch et al (2011)’s work. We also show our

enhancements implementations, profitability adjusted instantaneous debt effect and Bayesian

learning process. Section 4 shows our empirical results using U.S. banks in S&P500 under

Basel and economic capital regimes. We then analyze the results under different financial

conditions. Section 5 concludes with implications and potential future research.

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2. Model development

Our starting point is the model in Buch et al (2011) that derives the optimal risk capital

within the business that maximizes the financial firm’s overall return to risk measured by

Return on Risk Adjusted Capital (RORAC)3. They use Euler allocation principle to allocate

risk capital according to each business unit’s marginal RORAC. The marginal RORAC is the

partial derivative of the overall RORAC with respect to each business unit’s capital. The

decision rule is to expand or contract the business unit that has a marginal RORAC before

capital allocation greater or less than the overall RORAC, respectively, until the marginal

RORAC and overall RORAC become equal. However, we do not create newly or abandon

the business units during capital allocation.

This work is subject to two assumptions. First, the risk measure should be a homogeneous

function4 in order to be differentiable. The Euler allocation only uses homogeneous and

differentiable risk measures as it produces marginal RORAC through partial differentiation.

Therefore, it assumes the firms to have homogenous sub-businesses consisting of a

continuum of single contracts (Buch et al. 2011). Second, following Buch et al (2011)’s work,

we assume that the business unit managers have superior knowledge about the financial

change of their own business unit through expansion and reduction. Then the headquarters

centrally calculate the risk of its business unit portfolio.

This approach, as acknowledged by Buch et al (2011) themselves, is not fully multi-

period since the expected profit, risk, and capital are not time-varying. In addition to

developing a fully multi-period model, we add other restrictions such as no change in overall

capital and Basel regulation. We further improve the model by implementing profitability

adjusted instantaneous debt effect and Bayesian learning process to minimize the estimation

error arising from the future financial uncertainty.

2.1. Closed form solution for optimal allocation

2.1.1. Time-varying risk measurement

We start our work with providing the time-varying risk measure for our model. We use

the one-step-ahead expected shortfall model (Lönnbark, 2013) to estimate the time-varying

3 The RORAC is the expected profit over the allocated risk adjusted capital. Buch et al. (2011) develop their model to maximize RAROC but as they point out, it remains valid to other return to risk measures such as Risk Adjusted Return on Capital (RAROC), and Risk Adjusted

Return on Risk Adjusted Capital (RARORAC). 4 A function f such that all points (x1, ….., xn) in its domain of definition and all real t>0, the equation

f(tx1, ….., txn) = 𝑡λf(x1, ….., xn)

holds, where λ is a real number (Kudryavtsev. 2002).

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risk measure 𝜌𝑡5. Specifically, we use the absolute risk measurement form of the one-step-

ahead expected shortfall that assumes the expected profit is zero. This eliminates the

possibility of negative economic capital.

𝜌𝑡[𝑢𝑡𝑤] =

𝜙(Φ𝛼,𝑡−1)

𝛼× √𝑢𝑡

𝑤 × σ𝑡Rcov × 𝑢𝑡

𝑤 ′

⏟ Square root of weighted realized covariance

(1)

N: Number of business units

𝑢i,t : Capital of business unit i at time t

𝑢𝑖,𝑡𝑤 : Capital weight of business unit i at time 𝑡 =

𝑢𝑖,𝑡

∑ 𝑢𝑖,𝑡𝑁𝑖=1

𝜌𝑡[𝑢𝑡𝑤]: Expected shortfall with confidence interval (1 − 𝛼) at time t dependent on capital

weight of business unit i at time t, 𝑢𝑡𝑤.

𝑢𝑡𝑤: 1 x N vector of business units’ capital weights relative to the total capital at time t

𝑢𝑡𝑤 ′

: Transpose of 𝑢𝑖,𝑡𝑤

σRcov : N x N matrix of realized covariance of the business units’ returns at time t

Φ𝛼,𝑡−1 : Inverse of the cumulative distribution function (CDF) (or quartile function) of normal

distribution evaluated at confidence interval level (1 − 𝛼) at time t

We use the 97.5% confidence level which is the BIS suggested requirement for the

expected shortfall.

𝜙(•): Probability density function (PDF) of standard normal distribution

(1 − 𝛼): Confidence level

To estimate the square root of the weighted realized covariance σ𝑡Rcov, we use the return

on equity (ROE), following Barndorff-Nielsen and Neil Shephard (2002). We compute the

realized covariance matrix at time T as:

σ𝑡Rcov = ∑ (𝑅𝑂𝐸𝑖,𝑡 × 𝑅𝑂𝐸𝑗,𝑡)

𝑇𝑡=1 (2)

where segment indexes i and j range from 1 to N.

This one-step-ahead expected shortfall function satisfies all four coherent risk measure

properties for the Euler allocation used in the new model (Acerbi and Tasche. 2002). Thus

the economic capital at time t, 𝐸𝐶𝑡, is:

𝐸𝐶𝑡 = ∑ 𝑢𝑖,𝑡𝑁𝑖=1 × 𝜌𝑡[𝑢𝑡

𝑤] (3)

5 Expected shortfall provides the expected loss given that the loss exceeds the Value-at-Risk. The Basel Committee on Banking Supervision (2012) favors it over value-at-risk as a measure of risk.

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2.1.2. Time-varying expected profit process

We assume the time-varying expected profit process follows the stochastic differential

equation (SDE)6:

𝑌𝑖,𝑡[𝑢𝑡]⏟ Profit

= 𝜇𝑖,𝑡[𝑢𝑖,𝑡]⏟ Expected Profit

+ 𝜎𝑖,𝑡[𝑢𝑖,𝑡]𝑊𝑡⏟ Profit Fluctuation

(4)

The profit for business unit i at time t is its expected profit plus profit fluctuation where both

the drift 𝜇 and the diffusion 𝜎 are time varying. 𝑊𝑡 is the standard Wiener process. All the

business units in the same firm will follow this SDE process.

We then discretize this SDE process from time t-dt to t as:

𝑌𝑖,𝑡[𝑢𝑡] = 𝜇𝑖,𝑡[𝑢𝑖,𝑡] + 𝜎𝑖,𝑡[𝑢𝑖,𝑡]𝑊𝑡⇒ 𝑌𝑖,𝑡[𝑢𝑡] − 𝑌𝑖,𝑡−𝑑𝑡[𝑢𝑡−𝑑𝑡]

= (𝜇𝑖,𝑡[𝑢𝑖,𝑡] − 𝜇𝑖,𝑡−𝑑𝑡[𝑢𝑖,𝑡−𝑑𝑡]) + (𝜎𝑖,𝑡[𝑢𝑖,𝑡]𝑊𝑡 − 𝜎𝑖,𝑡−𝑑𝑡[𝑢𝑖,𝑡−𝑑𝑡]𝑊𝑡−𝑑𝑡)

(5)

Here, dt is the minimum time step of the data.

The profit 𝑌𝑡[𝑢𝑡] is a function of equity 𝑢𝑡 and return on capital at time t (𝑅𝑂𝐶𝑡), i.e.

⇔ 𝜇𝑖,𝑡[𝑢𝑘,𝑡]

= 𝜇𝑖,𝑡−𝑑𝑡[𝑢𝑘,𝑡−𝑑𝑡]

+(𝑢𝑖,𝑡 × 𝑅𝑂𝐶𝑖,𝑡 − 𝑢𝑖,𝑡−𝑑𝑡 × 𝑅𝑂𝐶𝑖,𝑡−𝑑𝑡) − (𝜎𝑖,𝑡[𝑢𝑖,𝑡]𝑊𝑡 − 𝜎𝑖,𝑡−𝑑𝑡[𝑢𝑖,𝑡−𝑑𝑡]𝑊𝑡−𝑑𝑡) (6)

Since the expected value of Weiner process is zero by definition, in expectation form

equation (5) becomes:

E [𝜇𝑖,𝑡[𝑢𝑖,𝑡]] = 𝐸[𝜇𝑖,𝑡−𝑑𝑡[𝑢𝑖,𝑡−𝑑𝑡] + (𝑢𝑖,𝑡 × 𝑅𝑂𝐶𝑖,𝑡 − 𝑢𝑖,𝑡−𝑑𝑡 × 𝑅𝑂𝐶𝑖,𝑡−𝑑𝑡)]

⇔ 𝜇𝑖,𝑡[𝑢𝑘,𝑡] = 𝜇𝑖,𝑡−𝑑𝑡[𝑢𝑘,𝑡−𝑑𝑡] + (𝑢𝑖,𝑡 × 𝑅𝑂𝐶𝑖,𝑡 − 𝑢𝑖,𝑡−𝑑𝑡 × 𝑅𝑂𝐶𝑖,𝑡−𝑑𝑡) (7)

We express this equation (6) in terms of capital weights 𝑢𝑖,𝑡𝑤

𝜇𝑖,𝑡[𝑢𝑖,𝑡𝑤 ]

= 𝜇𝑖,𝑡−𝑑𝑡[𝑢𝑖,𝑡−𝑑𝑡𝑤 ] + (∑ 𝑢𝑖,𝑡

𝑁𝑖=1 × 𝑢𝑖,𝑡

𝑤 × 𝑅𝑂𝐶𝑖,𝑡 −∑ 𝑢𝑖,𝑡−𝑑𝑡𝑁𝑖=1 × 𝑢𝑖,𝑡−𝑑𝑡

𝑤 × 𝑅𝑂𝐶𝑖,𝑡−𝑑𝑡) (8)

where 𝑢𝑖,𝑡𝑤 =

𝑢𝑖,𝑡

∑ 𝑢𝑖,𝑡𝑁𝑖=1

.

The overall return on risk-adjusted capital (RORAC) 𝑟[𝑢𝑡𝑤] is:

Overall RORAC: 𝑟[𝑢𝑡𝑤] =

𝜇𝑡[𝑢𝑡𝑤]

𝐸𝐶𝑡=

𝜇𝑡[𝑢𝑡𝑤]

∑ 𝑢𝑖,𝑡𝑁𝑖=1 ×𝜌𝑡[𝑢𝑡

𝑤] (9)

Since we are using the absolute risk measure, our overall RORAC does not subtract the

expected profit from the denominator unlike Buch et al (2011).

Similar to Buch et al (2011), we use the Euler optimization method to derive the additional

capital to allocate to business unit i such that its marginal RORAC equals the overall

RORAC. However, we work with additional capital weights rather than amount of capital.

6 This SDE is a time-varying version of the stochastic profit process used by Buch et al (2011).

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𝑟[𝜀𝑖,𝑡𝑤 |𝑢𝑖,𝑡

𝑤 ]⏟ marginal RORAC

with allocation

= 𝑟[𝑢𝑡𝑤]⏟

overall RORAC

(10)

where 𝜀𝑖,𝑡𝑤 is the additional capital weight and 𝑢𝑖,𝑡

𝑤 is the existing capital weight both for

business unit i at time t. 𝑢𝑡𝑤 is the total capital weight that is one.

We define the marginal RORAC 𝑟[𝜀𝑖,𝑡𝑤 |𝑢𝑖,𝑡

𝑤 ] as:


𝑤 ] =𝜇𝑖,𝑡[𝑢𝑖,𝑡

𝑤+𝜀𝑖,𝑡𝑤 ]−𝜇𝑖,𝑡[𝑢𝑖,𝑡

𝑤 ]

∑ 𝑢𝑖,𝑡𝑁𝑖=1 ×𝜌𝑡[𝑢𝑖,𝑡

𝑤+𝜀𝑖,𝑡𝑤 ]−∑ 𝑢𝑖,𝑡

𝑁𝑖=1 ×𝜌𝑡[𝑢𝑖,𝑡

𝑤 ] (11)

We simplify the numerator of equation (11) using equation (8) and with additional capital

weight 𝜀𝑖,𝑡𝑤 as follows:

𝜇𝑖,𝑡[𝑢𝑖,𝑡𝑤 + 𝜀𝑖,𝑡

𝑤 ] − 𝜇𝑖,𝑡[𝑢𝑖,𝑡𝑤 ]

= 𝜇𝑖,𝑡−𝑑𝑡[𝑢𝑖,𝑡−𝑑𝑡] + (∑ 𝑢𝑖,𝑡𝑁𝑖=1 × (𝑢𝑖,𝑡

𝑤 + 𝜀𝑖,𝑡𝑤 ) × 𝑅𝑂𝐶𝑖,𝑡 − ∑ 𝑢𝑖,𝑡−𝑑𝑡

𝑁𝑖=1 × 𝑢𝑖,𝑡−𝑑𝑡

𝑤 ×

𝑅𝑂𝐸𝑖,𝑡−𝑑𝑡) − 𝜇𝑖,𝑡−𝑑𝑡[𝑢𝑖,𝑡−𝑑𝑡] + (∑ 𝑢𝑖,𝑡𝑁𝑖=1 × 𝑢𝑖,𝑡

𝑤 × 𝑅𝑂𝐶𝑖,𝑡 − ∑ 𝑢𝑖,𝑡−𝑑𝑡𝑁𝑖=1 × 𝑢𝑖,𝑡−𝑑𝑡

𝑤 ×

𝑅𝑂𝐸𝑖,𝑡−𝑑𝑡)

=∑ 𝑢𝑖,𝑡𝑁𝑖=1 × 𝜀𝑖,𝑡

𝑤 × 𝑅𝑂𝐶𝑖,𝑡 (12)

Then we also simplify the denominator of equation (11) by subtracting the Taylor expansion

forms for 𝜌𝑡[𝑢𝑖,𝑡𝑤 + 𝜀𝑖,𝑡

𝑤 ] and 𝜌𝑡[𝑢𝑖,𝑡𝑤 ]. We first describe 𝜌𝑡[𝑢𝑖,𝑡

𝑤 ] as a general expansion form.

𝜌𝑡[𝑢𝑖,𝑡𝑤 + 𝜀𝑖,𝑡

𝑤 ] = 𝑎0 + 𝑎1(𝑢𝑖,𝑡𝑤 + 𝜀𝑖,𝑡

𝑤 ) + 𝑎2(𝑢𝑖,𝑡𝑤 + 𝜀𝑖,𝑡

𝑤 )2+ 𝑎3(𝑢𝑖,𝑡

𝑤 + 𝜀𝑖,𝑡𝑤 )

3⋯

𝜌𝑡[𝑢𝑖,𝑡𝑤 ] = 𝑎0 + 𝑎1𝑢𝑖,𝑡

𝑤 + 𝑎2𝑢𝑖,𝑡𝑤 2+ 𝑎3𝑢𝑖,𝑡

𝑤 3 +⋯

→ 𝜌𝑡[𝑢𝑖,𝑡𝑤 + 𝜀𝑖,𝑡

𝑤 ] − 𝜌𝑡[𝑢𝑖,𝑡𝑤 ] = 𝑑𝜌𝑡[𝑢𝑖,𝑡

𝑤 ] = 𝑎1𝜀𝑖,𝑡𝑤 + 𝑎2𝜀𝑖,𝑡

𝑤 2 +⋯

𝜀𝑖,𝑡𝑤 = 𝑑𝑢𝑖,𝑡

𝑤

→ 𝜌𝑡[𝑢𝑖,𝑡𝑤 + 𝑑𝑢𝑖,𝑡

𝑤 ] − 𝜌𝑡[𝑢𝑖,𝑡𝑤 ] = 𝑑𝜌𝑡[𝑢𝑖,𝑡

𝑤 ] = 𝑎1𝑑𝑢𝑖,𝑡𝑤 + 𝑎2𝑑𝑢𝑖,𝑡

𝑤 2 +⋯

𝑑(𝜌𝑡[𝑢𝑖,𝑡𝑤 + 𝑑𝑢𝑖,𝑡

𝑤 ] − 𝜌𝑡[𝑢𝑖,𝑡𝑤 ])

𝑑𝑢𝑖,𝑡𝑤 = 𝑎1 + 2𝑎2𝑑𝑢𝑖,𝑡

𝑤 + 3𝑎3𝑑𝑢𝑖,𝑡𝑤 2 +⋯

𝑑𝑑(𝜌𝑡[𝑢𝑖,𝑡

𝑤 + 𝑑𝑢𝑖,𝑡𝑤 ] − 𝜌𝑡[𝑑𝑢𝑖,𝑡

𝑤 ])𝑑𝑢𝑖,𝑡

𝑤

𝑑𝑢𝑖,𝑡𝑤 = 2𝑎2 + 6𝑎3𝑑𝑢𝑖,𝑡

𝑤 +⋯

The first two terms of the Taylor expansion subtraction form are:

𝑑𝜌𝑡[𝑢𝑖,𝑡𝑤 ]

𝑑𝑢𝑖,𝑡𝑤 = lim

𝑑𝑢𝑖,𝑡𝑤→0

𝑑(𝜌𝑡[𝑢𝑖,𝑡𝑤 + 𝑑𝑢𝑖,𝑡

𝑤 ] − 𝜌𝑡[𝑢𝑖,𝑡𝑤 ])

𝑑𝑢𝑖,𝑡𝑤 = 𝑎1 → 𝑎1 =


𝑑𝑢𝑖,𝑡𝑤 (13)

𝑑2𝜌𝑡[𝑢𝑖,𝑡𝑤 ]

𝑑2𝑢𝑖,𝑡𝑤 = lim

𝑑𝑢𝑖,𝑡𝑤→0

𝑑𝑑(𝜌𝑡[𝑢𝑖,𝑡

𝑤 + 𝑑𝑢𝑖,𝑡𝑤 ] − 𝜌𝑡[𝑑𝑢𝑖,𝑡

𝑤 ])𝑑𝑢𝑖,𝑡

𝑤

𝑑𝑢𝑖,𝑡𝑤 = 2𝑎2 → 𝑎2 =

1

2


𝑑2𝑢𝑖,𝑡𝑤 (14)

10

We retain only the first two terms from the Taylor expansion subtraction form:

𝜌𝑡[𝑢𝑖,𝑡𝑤 + 𝜀𝑖,𝑡

𝑤 ] − 𝜌𝑡[𝑢𝑖,𝑡𝑤 ] =


𝑑𝑢𝑖,𝑡𝑤 𝜀𝑖,𝑡

𝑤 +1

2


𝑑2𝑢𝑖,𝑡𝑤 𝜀𝑖,𝑡

𝑤 2 (15)

Since we only use the first two terms of the Taylor expansion, equation (14) is subject to

approximation error given by:

f [𝜀𝑖,𝑡𝑤 ]- 𝑃𝑛 [𝜀𝑖,𝑡

𝑤 ] =𝑓(𝑛+1)(𝑐)

(𝑛+1)!(𝜀𝑖,𝑡𝑤 )𝑛+1 (16)

where f [𝜀𝑖,𝑡𝑤 ] is the actual value, 𝑃𝑛[𝜀𝑖,𝑡

𝑤 ] is the estimation from equation (15), and n is the

degree of polynomial which is two in our case. c in equation (16) is an arbitrary value that

should be between zero and 𝜀𝑖,𝑡𝑤 for a satisfactory approximation.


𝑑𝑢𝑖,𝑡𝑤 = 𝑎𝑖,𝑡 in equation (13) is the risk contribution of business unit i at time t and


𝑑2𝑢𝑖,𝑡𝑤

in equation (14) is the Hessian matrix with business units’ weights. Following Buch et al

(2011), we use the highest eigenvalue Λ𝑡 of this Hessian matrix since it provides the

maximum directional strength among the business units with the additional allocation input.

We then combine equations (12) and (15) in (11) to derive the expression for marginal

RORAC with allocation.


𝑤 ] =∑ 𝑢𝑖,𝑡𝑁𝑖=1 × 𝜀𝑖,𝑡

𝑤 × 𝑅𝑂𝐶𝑖,𝑡

∑ 𝑢𝑖,𝑡𝑁𝑖=1 × (


𝑑𝑢𝑖,𝑡𝑤 𝜀𝑖,𝑡

𝑤 +12𝑑2𝜌𝑡[𝑢𝑖,𝑡

𝑤 ]

𝑑2𝑢𝑖,𝑡𝑤 𝜀𝑖,𝑡

𝑤 2)

=𝜀𝑖,𝑡𝑤 × 𝑅𝑂𝐶𝑖,𝑡

𝑎𝑖,𝑡 𝜀𝑖,𝑡𝑤 +

12Λ𝑡𝜀𝑖,𝑡

𝑤 2 (17)

We then equate the overall RORAC (equation (8)) and the marginal RORAC with allocation

(equation (17)) and solve for the 𝜀𝑖,𝑡𝑤 .

𝜀𝑖,𝑡𝑤 × 𝑅𝑂𝐶𝑖,𝑡

𝑎𝑖,𝑡 𝜀𝑖,𝑡𝑤 +

12Λ𝑡𝜀𝑖,𝑡

𝑤 2=

𝜇𝑡[𝑢𝑡]

∑ 𝑢𝑖,𝑡𝑁𝑖=1 × 𝜌𝑡[𝑢𝑡

𝑤]

𝜀𝑖,𝑡𝑤 =

−2(𝑎𝑖,𝑡 ×𝜇𝑡[𝑢𝑡]


𝑤]− 𝑅𝑂𝐶𝑖,𝑡)

Λ𝑡 ×𝜇𝑡[𝑢𝑡]


𝑤]

(18)

We account for the Taylor approximation error by choosing 0.5𝜀𝑖,𝑡𝑤 instead of 𝜀𝑖,𝑡

𝑤 as the value

for c in equation (16). Hence, we have the expression for 𝜀𝑖,𝑡𝑤 :

𝜀𝑖,𝑡𝑤 =

−(𝑎𝑖,𝑡 ×𝜇𝑡[𝑢𝑡]


𝑤]− 𝑅𝑂𝐶𝑖,𝑡)

Λ𝑡 ×𝜇𝑡[𝑢𝑡]


𝑤]

(19)

Then we add the self-evolving existing capital weight 𝑢𝑖,𝑡𝑤 to the additional capital weight 𝜀𝑖,𝑡

𝑤

11

to produce the new capital weight 𝑢𝑛𝑒𝑤,𝑖,𝑡𝑤 .

𝑢𝑛𝑒𝑤,𝑖,𝑡𝑤 = 𝜀𝑖,𝑡

𝑤 + 𝑢𝑖,𝑡𝑤 (20)

We optimize the firm’s return to risk ratio by reshuffling the existing capital while not

allowing negative capital for any business unit. We reinject any negative capital values back

into our model within the same allocation period until we derive all non-negative capital

values. To ensure no change in total capital, we require the additional capital weights 𝜀𝑖,𝑡𝑤 sum

to zero.

2.2. Profitability adjusted instantaneous debt effect

Since our capital allocation model depends on variables that evolve over time, it is

necessary to forecast their values. To minimize the forecasting error, one of the

enhancements we implement is the profitability adjusted instantaneous debt effect. We use

the instantaneous debt change 𝐷𝑖,𝑡

𝐷𝑖,𝑡−𝑑𝑡 for business unit i from time 𝑡 − d𝑡 to t where d𝑡 is a

quarter of a year. We then subtract one from 𝐷𝑖,𝑡

𝐷𝑖,𝑡−𝑑𝑡 to use only the additional debt-changing

portion. Then we add the return on capital 𝑅𝑂𝐶𝑖,𝑡 to consider the profitability effect together:

𝑢𝑛𝑒𝑤2,𝑖,𝑡𝑤

⏟ new capital weight with

profitability adjusted

instantaneous debt effect

= { 𝜀𝑖,𝑡𝑤⏟

additionalcapital weight

+ ( 𝑅𝑂𝐶𝑖,𝑡⏟ business unit′s profitability

+ (𝐷𝑖,𝑡

𝐷𝑖,𝑡−𝛥𝑡− 1

⏟ )

business unit′sinstantaneousdebt chage

) + 𝑢𝑖,𝑡𝑤

⏟existing

capital weight

} (21)

𝑢𝑛𝑒𝑤2,𝑖,𝑡𝑤 is the new capital weight with profitability adjusted instantaneous debt effect. The

instantaneous debt change enables our model to provide more capital to the business unit that

has large increase in its debt recently and vice versa. Thus, the profitability adjusted

instantaneous debt effect reduces estimation error for optimal capital weights and increases

the new model’s RORAC further.

2.3. Bayesian Learning

We further reduce the forecasting error by implementing Bayesian learning process. This

minimizes the forecasting error in expected profit that may arise from the information

asymmetry between the ‘belief’ over the expected profit after capital allocation and the

expected profit before the capital allocation. This ‘belief’ updates through accumulating

learning process as time progresses. The SDE model that we use for this is the following

(Cvitanić and Zapatero. 2004, pp 140-141):

12

𝑌𝑡 = 𝜇𝑡 + 𝜎𝑡𝑊𝑡

𝑊𝑡 = 𝑊𝑡 + ∫ (𝜇𝑢𝑡

0− 𝜇𝑢)𝑑𝑢, (𝑢 ≤ 𝑡)

(22)

𝑌𝑡 is the stochastic profit process we use in the new model that has time-varying drift 𝜇𝑡

and time-varying diffusion 𝜎𝑡. 𝜇𝑡 is the expected profit before capital allocation at time t.

𝜇𝑡 is the ‘belief’ of the expected profit after capital allocation at time t. We start by inputting

the innovation process 𝑊𝑡 into the following stochastic profit process.

𝑌𝑡 = 𝜇𝑡 + 𝜎𝑡

cov𝑊𝑡

⇔ 𝑌𝑡 = 𝜇𝑡 + 𝜎𝑡cov(𝑊𝑡 + ∫ (𝜇𝑢

𝑡

0− 𝜇𝑢)𝑑𝑢)

(23)

We use the expected shortfall risk measure 𝜌𝑡 in section 2.1.1 for the diffusion process 𝜎𝑡cov

in equation (22). We then express 𝜇𝑡 − 𝜇𝑡 using the same procedure as in equation (12).

𝜇𝑡 − 𝜇𝑡= 𝜇𝑡[𝑢𝑡] − 𝜇𝑡[𝑢𝑡 + 𝜀𝑡]= 𝜇𝑡−𝑑𝑡[𝑢𝑡−𝑑𝑡] + (𝑢𝑡 × 𝑅𝑂𝐶𝑡 − 𝑢𝑡−𝑑𝑡 × 𝑅𝑂𝐶𝑡−𝑑𝑡)−𝜇𝑡−𝑑𝑡[𝑢𝑡−𝑑𝑡] + ((𝑢𝑡 + 𝜀𝑡) × 𝑅𝑂𝐶𝑡 − 𝑢𝑡−𝑑𝑡 × 𝑅𝑂𝐶𝑡−𝑑𝑡)= (𝑢𝑡 × 𝑅𝑂𝐶𝑡) − (𝑢𝑡 + 𝜀𝑡) × 𝑅𝑂𝐶𝑡

(24)

Substituting equation (24) into equation (23):

𝑌𝑡 = 𝜇𝑡 + 𝜌𝑡(𝑊𝑡 + ∫ (𝜇𝑢

𝑡

0− 𝜇𝑢)𝑑𝑢)

= 𝜇𝑡 + 𝜌𝑡𝑊𝑡 + 𝜌𝑡 ∫ (𝑡

0(𝑢𝑡 × 𝑅𝑂𝐶𝑡) − (𝑢𝑡 + 𝜀𝑡) × 𝑅𝑂𝐶𝑡)𝑑𝑢

(25)

We take the expected value on both sides of the equation (25) to get the expression for

expected profit with Bayesian learning process:

𝐸[𝑌𝑡] = 𝐸[𝜇𝑡 + 𝜌𝑡𝑊𝑡 + 𝜌𝑡 ∫ (𝑡

0(𝑢𝑡 × 𝑅𝑂𝐶𝑡) − (𝑢𝑡 + 𝜀𝑡) × 𝑅𝑂𝐶𝑡)𝑑𝑢]

𝜇𝑡,𝐵𝑎𝑦𝑒𝑠𝑖𝑎𝑛

= 𝜇𝑡 + 𝜌𝑡 ∫ (𝑡


= 𝜇𝑡 + 𝜌𝑡 ∫ (𝑡


= 𝜇𝑡 + 𝜌𝑡 ∫ (𝑡

0(∑ 𝑢𝑖,𝑡

𝑁𝑖=1 × 𝑢𝑛𝑒𝑤,𝑡−𝑑𝑡

𝑤 × 𝑅𝑂𝐶𝑡) − (∑ 𝑢𝑖,𝑡 ×𝑁𝑖=1 𝑢𝑛𝑒𝑤,𝑡

𝑤 × 𝑅𝑂𝐶𝑡))𝑑𝑢

∴ 𝜇𝑡,𝐵𝑎𝑦𝑒𝑠𝑖𝑎𝑛 = 𝜇𝑡 + 𝜌𝑡 ∫ ∑ 𝑢𝑖,𝑡𝑁𝑖=1 (

𝑡

0(𝑢𝑛𝑒𝑤,𝑡−𝑑𝑡𝑤 × 𝑅𝑂𝐶𝑡) − (𝑢𝑛𝑒𝑤,𝑡

𝑤 × 𝑅𝑂𝐶𝑡))𝑑𝑢

(26)

The expected value of 𝑌𝑡 is 𝜇𝑡,𝐵𝑎𝑦𝑒𝑠𝑖𝑎𝑛, the expected profit value reflecting Bayesian learning

process. The expected value of 𝜇𝑡 is itself and the expected value of 𝜎𝑡cov𝑊𝑡 is zero by the

definition of Winer process. We express 𝜇𝑡,𝐵𝑎𝑦𝑒𝑠𝑖𝑎𝑛 in terms of all other variables in equation

(26). The business unit’s capital weights before and after capital allocation are 𝑢𝑛𝑒𝑤,𝑡−𝑑𝑡𝑤

and 𝑢𝑛𝑒𝑤,𝑡𝑤 , respectively. The total equity at time t is ∑ 𝑢𝑖,𝑡

𝑁𝑖=1 . The integral part of equation

(26) accumulates the difference between the actual and expected profits over time scaled by

13

our expected shortfall risk measure 𝜌𝑡. This Bayesian learning adjusted expected profit 𝜇𝑡

replaces the expected profit 𝜇𝑡[𝑢𝑡] in the new model.

3. Data and Method

For our empirical test, we select banks in S&P500 that meet the following criteria.

(1) They must have more than one business unit,

(2) The published financial statements must provide regulatory capital,

and

(3) The published financial statements must have asset and net income

data at the business unit level

Thirteen U.S. banks meet these criteria. We collect quarterly data from Bloomberg starting

with most recent quarter for each bank (September 30th 2014) and go back to the start of each

business unit for each bank. There are no merging, separating, creating, and abandoning of

business units during our internal capital allocation. We work with our existing set of

business units. Then we use the annualized expected shortfall risk measure for our

implementation by multiplying the expected shortfall in section 2.1.1 by √4. The following

Table 1 shows the descriptive statistics of the data we use.

Of course, this test is subject to two assumptions mentioned earlier: homogenous sub-

businesses using homogenous risk function and decentralized firm where business unit

managers have superior knowledge on their business units than the headquarters.

3.1. Business Unit’s Capital Data Generation

We start by generating the capital data for each business unit, as firms do not disclose this

in their financial statements. However, we do have the asset value of each business unit in the

financial statement. Thus, we derive the business unit wise capital data using the relative

asset volatility of each business unit compared to the sum of all business unit’s asset volatility

up to time t measured by realized standard deviations. Then we multiply this by the total

capital at time t, 𝑢𝑡 , to calculate business unit i’s capital, 𝑢𝑡,𝑖 as shown in equation (27).

Table 1 here

14

√∑ (𝐴𝑡,𝑖−𝐴𝑡−𝑑𝑡,𝑖)2𝑚

𝑡=1𝑡

√∑ ∑ (𝐴𝑡,i−𝐴𝑡−𝑑𝑡,𝑖)2𝑚

𝑡=1𝑁𝑖=1

𝑡

× 𝑢𝑡 = 𝑢𝑡,𝑖 (27)

At,i is the asset value for business unit i at time t, N is the total number of business unit,

and m is the number of quarters.

3.2. Regulatory Capital and Economic Capital

In our new model, we apply the current financial regulation environment, Basel. We use

all the regulatory capital data of the most current business unit (i.e. segment) set back until

the past where this business unit set started to exist. We derive the regulatory capital for each

business unit using the total regulatory capital and relative risk fluctuation method as shown

in equation (27). The empirical result derived with this regulatory capital is our Basel regime.

In the similar manner, we derive the equity value for each business unit using the total equity

and relative risk fluctuation method in equation (27) back until the most current business unit

set initially existed. We derive our empirical result using this business unit wise equity values

and call this our economic capital7 regime. The return to risk efficiency calculations in both

regimes are expected net income over regulatory capital.

3.3. Stopping time for capital allocation

In order to restrict the risk from capital allocation not to exceed the risk that the financial

firm can afford, we provide an automatic stopping procedure in our new model. With this,

our new model ceases its capital allocation when it expects the generated risk to exceed the

firm’s risk tolerance level. In specific, our new model allocates capital unless the expected

shortfall risk measure in section 2.1.1 multiplied by the annualized factor √4 becomes greater

than one. Otherwise, the financial firm goes into bankruptcy, as it requires more total capital

than its currently available one to remain solvent.

4. Results

4.1. Initial Empirical Result

We start our empirical test using one sample firm, Huntington Bancshares Inc., to show

how our new model improves the RORAC efficiency over time under Basel and economic

capital regimes. In Table 2, we compare the overall RORAC and expected net income with

7 Economic capital is the minimum safety capital level to ensure the financial firm to survive from insolvency due to an unexpected loss (James. 1996).

15

the available capital (i.e. regulatory capital or equity) under Basel and economic capital

regime against the existing historical data with no capital allocation, our benchmark. The

benchmark overall RORAC is expected net income over available regulatory capital.

The available regulatory capital and equity are the same existing data for these three

scenarios as we strive to maximize the efficiency with no additional total capital input.

The capital allocation starts from the third quarter, March 31st 2011, until June 30th 2014,

as we need at least two initial quarters to measure the risk from our expected shortfall. The

overall RORAC under Basel regime and the benchmark are the expected net income over

available regulatory capital while under economic capital regime it is the expected net

income over available total equity. We then see that the overall RORAC under Basel and

economic capital regimes are on average greater than the benchmark that proves our new

model’s effectiveness as shown in Table 2. Our new model’s capital allocation automatically

stops just before the risk from capital allocation exceeds the firm’s affordable risk level.

However, we do not encounter such situation in case of Huntington Bancshares Inc.

We see similar evolution patterns of overall RORAC, expected net income, additional

business unit capital, and new business unit capital with the new model between Basel and

economic capital regimes in Figure 1. On the business unit level, we find that both regimes

show the highest capital injection to the Automobile Finance and Commercial Real Estate

Unit at the expense of Treasury/Other unit and Regional and Commercial Banking unit

(Figure 1.1.3 and 1.2.3). In addition, Retail and Business Banking unit shows the most stable

capital change over time in both regimes (Figure 1.1.3, 1.1.4, 1.2.3, and 1.2.4). However, we

find dissimilarity between Basel and economic capital regimes in their capital variation level

that is somewhat higher under Basel regime than under economic capital regime (Figure 1.1.4

and 1.2.4).

Figure 1 here

4.2. Full Empirical Result

We then extend the new model’s application to our full sample of thirteen U.S. banks

listed in S&P500. We show our full empirical result under Basel and economic capital

regimes with and without our efficiency enhancement features (i.e. profitability adjusted

instantaneous debt effect and Bayesian learning process). If our new model is applied under

Table 2 here

16

Basel (or economic capital) regime with our efficiency enhancement features, we call it

Basel+ (or economic capital+) regime while we call it only Basel (or economic capital) regime

if no efficiency enhancement features are applied at all. Thus, we show four full empirical

results that have under two different regimes with two different versions: Basel (Table 3.1),

Basel+ (Table 3.2), economic capital (Table 3.3), and economic capital+ (Table 3.4).

Table 3 here

Among our four full empirical results, we find that the one under economic capital+ shows

the highest average RORAC efficiency improvement from 1.8498% to 3.2467% per quarter.

This proves that our new model is the most effective under economic capital+ regime.

Besides, the total quarters of estimation period are slightly higher under economic capital

regimes compared to Basel regimes that proves a better risk control during capital allocation

under economic capital regimes compared to Basel regimes. Although our new model is not

perfectly improving the efficiencies for all financial firms (i.e. 2 out of 13 firms under Basel+

and economic capital+ regimes exhibit negative efficiencies), the overall average efficiency

improvement is positive for all four regimes. Moreover, our enhancements (i.e. profitability

adjusted instantaneous debt effect and Bayesian learning) indeed improve the efficiencies

further in both Basel and economic capital regimes. However, our enhancement features are

more effective under economic capital regime, from 2.7533% to 3.2467%, than under Basel

regime, from 2.7918% to 3.2234%.

We then find that there are different preferences of the financial firms towards capital

allocation: (1) the ones preferring Basel regime (e.g. Bank of America Corp., Citigroup Inc.,

Huntington Bancshares Inc., PNC Financial Services Group Inc., and US Bancorp/MN), (2)

economic capital regime (e.g. BB&T Corp., KeyCorp., People’s United Financial Inc.,

Regions Financial Corp., and SunTrust Banks Inc.), and (3) no capital allocation (e.g. Fifth

Third Bancorp and JPMorgan Chase & Co). Thus we suggest different capital allocation

approach, either Basel+ or economic capital+ regime, for different financial firm to maximize

their efficiencies. However, we still conclude that the overall financial firms prefer capital

allocation under economic capital+ regime the most where we see the highest average

RORAC efficiency with longer capital allocation periods with more effective risk control.

Besides, we find that our new model almost do not allocate capital if we allow external

capital input that our total equity can increase. This is because the risk level increases

drastically at once and exceeds the available capital cushion that our new model

17

automatically stops allocating immediately as explained in section 3.3. Therefore, we need

reasonable capital constraints to test this that we show in section 4.3.

Although our new model improves the overall RORAC under both regimes, we suggest a

more firm-specific capital allocation approach. In other words, while knowing that economic

capital+ regime shows the highest efficiency improvement, it is crucial to recognize that each

bank improves efficiency differently in Basel and economic capital regimes. In addition, even

though our Bayesian learning process captures the information asymmetry problem in our

capital allocation, it is only the part of the agency problem. Therefore, other agency issues

should be regarded as well: power and connection of the CEOs and divisional managers

(Graham, Harvey, and Puri. 2010, Duchin et al. 2013, Glaser et al. 2013), rent seeking

activity (Duchin et al. 2013), managers’ empire building tendencies (Xuan et al. 2009,

Hoechle et al. 2012), overconfidence (Hoechle et al. 2012), voluntary risk-adversity

(Stoughton et al. 2007), different risk appetites and incentives (Roper et al. 2012) that may

result in inefficient investment with over or under allocations (Hund et al. 2010, Lamont.

1997, Shin and Stulz. 1998, Rajan el al. 2000, and Ozbas and Scharfstein. 2010), and more

others (Villalonga. 2004). Besides, as no financial forecasting can be perfect, our new model

is still subject to the forecasting error as well. For instance, in case of financial firms such as

JPMorgan Chase & Co., its capital volatility is generally quite high which makes our new

model difficult to predict the best capital allocation procedure over time.

In Table 4, we classify the high/medium/low levels of new RORAC with capital allocation

under high/medium/low levels of firm size (i.e. total asset) (Table 4.1), total equity (Table

4.2), regulatory capital (i.e. risk based capital) (Table 4.3), and volatility (Table 4.4) in Basel+

and Economic Capital+ regimes. Table 5 shows the summary comparison result of Table 4.

The firm size (i.e. total asset) shows negative relationship with the new average RORAC

level with capital allocation (i.e. the larger the firm size is, the lower the average RORAC

with capital allocation). In both Table 4 and 5, we do not include Wells Fargo & Co (i.e.

WFC US Equity) as its risk level rises above its existing available capital if it starts internally

allocate its capital in the first allocation quarter in Basel+ and Economic Capital+ regimes.

Therefore, its capital allocation process automatically stops even before it starts allocating

capital and we do not include this result in our Table 4 and 5.

However, the relationship of average RORAC for total equity, regulatory capital, and

volatility are not apparent. Therefore, our test shows that the firm size is the best criteria for

predicting the capital allocation effect than total equity, regulatory capital, and volatility.

However, Table 4 and 5 are showing the average trends where RORAC of each bank within

18

the same high/medium/low class level can vary much in some cases. Therefore, it is still quite

important to have bank specific RORAC analysis with respect to firm size, total equity,

regulatory capital, and volatility.

Table 4

Table 5

4.3. Full Empirical Result with Constraints

However, the capital change in each business unit may be restricted. The financial firm

may prefer more granular business unit wise capital change for lower transaction costs, more

stabilized business strategies, etc. Therefore, we also test our new model when there are

capital change constraints in each business unit. Specifically, we impose minimum and

maximum outside constraints that each business unit cannot exceed. Then we impose

additional minimum and maximum inside constraints that each business unit must exceed to

allocate capital. The latter constraints are within the former ones. We use four different

constraints in our test: ±2%/±20%, ±2%/±30%, ±5%/±20%, and ±5%/±30%. For instance, the

inside constraints ±2% from ±2%/±20% means that the our model allocates capital to each

business unit only when its capital change with allocation is more than 2% or less than -2%

of its capital before allocation. The outside constraints ±20% from ±2%/±20% indicates the

capital change of a business unit cannot exceed ±20% of its capital before allocation. We test

the new model with these constraints under our four regimes: Basel, Basel+, economic

capital, and economic capital+. This test result is shown in Table 6.

Table 6 here

In this test with constraints, we find longer capital allocation periods with less allocation

stopping incidents from better risk control compared to the test without constraints.

However, this is at the expense of profitability that the efficiency improvements with

constraints are lower than the ones without constraints. Then we see that our enhancements

applied in this test with constraints strictly improve the overall efficiency further in all

19

different constraints and regimes. Thus, our enhancements improve the overall efficiency in

all tests with and without constraints. We find the best efficiency improvement with

constraints ±2%/±30% under Basel+ regime while the worst one has the constraints

±2%/±20% under economic capital regime. Then we compare the efficiency improvements

with different constraints controlling the regimes to analyze the constraints effect on

efficiency improvement in Table 6. We find that it is the outside constraints, ±20 and ±30,

that determine the efficiency improvements where ±30 constraints show higher efficiency

than ±20 constraints. Therefore, we suggest allowing larger outside constraints to increase the

efficiency maximally.

4.4. Available capital comparison with risk tolerance

We then analyze the available capital and risk tolerance aspect of our new model’s capital

allocation based on the tests without constraints under Basel+ and economic capital+ regimes

in Figure 2. For each bank, we show the available regulatory capital and equity evolution

against the required minimum regulatory capital and equity calculated from our expected

shortfall risk measure. The required minimum regulatory capital or equity stops evolving

before they exceed the available regulatory capital or equity, respectively. This ensures that

the available regulatory capital or equity are always greater than the required minimum ones

for the firms to remain solvent.

We find three types of capital evolution in Figure 2: (1) the required minimum regulatory

capital intermediately stops evolving before the required minimum equity (Figure 2.1 and

2.3), (2) both the required minimum regulatory capital and equity intermediately stop

evolving simultaneously (Figure 2.9, 2.12, and 2.13), and (3) both of these do not stop

evolving until the final quarters (Figure 2.2, 2.4, 2.5, 2.6, 2.7, 2.8, 2.10, and 2.11). However,

there is no case where the required minimum equity stops evolving before the required

minimum regulatory capital ceases its evolution. This implies that equity based allocation

achieves better risk control with longer allocation periods and higher efficiency (section 4.2)

compared to regulatory capital based allocation with our model. Therefore, we suggest using

economic capital regimes in our new capital allocation model.

5. Conclusion

Figure 2 here

20

This paper presents a closed-form solution for a dynamic optimal capital allocation

extended upon Buch et al (2011)’s model, a hypothetical incomplete multi-period model with

no empirical application. We provide a complete multi period characteristic to this model

using time varying risk using expected shortfall and expected return measures decomposed

by discretization. Of course, all other financial variables are time varying as well. Under this

dynamic financial condition, we derive a closed-form solution for the optimal capital weight

that each business unit should possess to maximize the overall firm’s return to risk efficiency,

Return to Risk Adjusted Capital (RORAC). Then we reflect highly limited financial

condition where we do not allow any additional external capital input. We achieve this by

making the optimal capital weights always sum up to one by reinjecting any negative capital

weights back into our new model until all capital weights are equal to or greater than zero.

We strive to improve the firm’s efficiency further under financial uncertainty using two

additional enhancements: profitability adjusted instantaneous debt effect and Bayesian

learning process. The profitability adjusted instantaneous debt effect assigns more capital to

the business units with high both in profitability and instantaneous debt change. The Bayesian

learning process recalculates the optimal capital weights with accumulated learning from the

differences between the actual profit after allocation and the “believed” profit before

allocation. These two enhancements minimize the financial forecasting errors and boost the

financial firm’s efficiency.

We provide empirical evidence on financial firm’s return to risk efficiency improvement

with our capital allocation model using thirteen banks from S&P500. We start by generating

regulatory capital and equity on the business unit level that is generally not available in the

financial statements. We achieve this by multiplying the relative risk fluctuation of each

business unit by the total regulatory capital or equity. Then we maximize the financial firms’

efficiencies using regulatory capital (Basel regime) and equity (economic capital regime),

respectively. We further improve the efficiencies with enhancements with regulatory capital

(Basel+ regime) and equity (economic capital+ regime) as well. Our new model automatically

ceases its capital allocation process just before its required minimum capital exceeds the

available capital to ensure solvency.

We improve the average quarterly RORAC efficiency from 2.7918% to 3.2234% under

Basel+ regime and from 2.7533% to 3.2467% under economic capital+ regime using our new

model with enhancements. We find each financial firm shows different efficiency

improvements using our new model in different regimes. However, on average, our new

model improves the financial firms’ efficiency better under economic capital regimes where

21

the economic capital+ regime shows the highest improvement. Then we extend our test under

four different capital constraints (inside/outside constraints8: ±2%/±20%, ±2%/±30%,

±5%/±20%, and ±5%/±30%) under four different regimes (Basel, Basel+, economic capital,

and economic capital+). In this case, we find that Basel+ under ±2%/±30% constraint

performs the best. We also find that it is the outside constraint that determines the efficiency

improvement that becomes higher with larger outside constraint range. We then compare the

capital allocation length under Basel+ and economic capital+ regimes. The economic capital+

regime shows longer capital allocation periods with less intermittent stopping incidents. It is

because the required minimum capital stays longer below the available capital level

compared to the Basel+ regime. This indicates that the risk control is better under economic

capital+ regime than under Basel+ regime. Therefore, we conclude that the economic capital+

regime is the best capital allocation environment for our new model. However, we also

suggest considering firm specific capital allocation approach as each firm shows different

efficiency improvements under different regimes and constraints.

This work provides implications both for the financial authorities and banks. We prove

that our internal capital allocation process for the banks indeed maximize their efficiency

even with their existing capital and no external capital input. This becomes quite important

especially these days when the banks are very conservative that they are not able to take high

risk to raise their profits than before as the financial regulation is becoming stricter. We also

suggest the financial authority to rethink whether they are regulating the banks too

monotonically and bluntly without regarding each bank’s characteristics enough as shown in

our internal capital allocation result.

The agency problem application with empirical test can be an interesting extension to this

work. The challenge will come from collecting good quality data related with agency issues

and transforming them into practical implementations to differentiate with other existing

works that mostly provide ad-hoc empirical test results.

8 As mentioned earlier, the inside and outside constraints are the limits that each business unit of a firm should and should not exceed for

capital allocation, respectively.

22

References

Acerbi, C. (2002). Spectral measures of risk: a coherent representation of subjective risk

aversion. Journal of Banking and Finance. 26(7). pp 1505-1518.

Acerbi, C. and Tasche, D (2002). On the coherence of expected shortfall. Journal of Banking

and Finance. 26(7). pp 1487-1503.

Artzner, P., Delbaen, F., Eber, J.M., Heath, D. (1999). Coherent measures of risk.

Mathematical Finance. 9(3). pp 203-228.

Barndorff-Nielsen, Ole E. and Shephard, Neil (2002). Econometric analysis of realised

volatility and its use in estimating stochastic volatility models. Journal of the Royal

Statistical Society, Series B 64(2). pp 253-280.

Buch, A., Dorfleitner, G. and Wimmer, M. (2011). Risk capital allocation for RORAC

optimization. Journal of Banking and Finance. 35(11). pp 3001-3009.

Buch, A., Dorfleitner, G. (2008). Coherent risk measures, coherent capital allocations

and the gradient allocation principle. Insurance: Mathematics and Economics. 42(1). pp 235-

242.

Cai, J. (2005). Conditional Tail Expectations for Multivariate Phase Type Distributions.

Journal of Applied Probability. 42(3). pp 810-825.

Cvitanić, Jakša and Zapatero, Fernando. (2004). Introduction to the Economics and

Mathematics of Financial Markets. USA, MIT Press, pp 140-141.

Denault, M. (2001). Coherent allocation of risk capital. Journal of Risk 4. pp 1–34.

Dhaene, J., Tsanakas, A., Valdez, E. a. and Vanduffel, S. (2012). Optimal Capital Allocation

Principles. Journal of Risk and Insurance. 79 (1). pp 1-28.

Duchin, R. & Sosyura, D. (2013). Divisional managers and internal capital markets. Journal

of Finance. 68(2). pp 387-429.

Erel, I., Myers, S.C., and Read Jr., J.A. (2015). A theory of risk capital. Journal of Financial

Economics. 118(3). pp 620-635

https://en.wikipedia.org/wiki/Ole_Barndorff-Nielsen

https://en.wikipedia.org/wiki/Neil_Shephard

http://www.jstor.org/action/showPublication?journalCode=japplprob

http://www.jstor.org/action/showPublication?journalCode=japplprob

23

Furman, E., Zitikis, R. (2008). Weighted risk capital allocations. Insurance: Mathematics and

Economics. 43(2). pp 263–270.

Gatzert, N., Schmeiser, H. (2008). Combining fair pricing and capital requirements for non-

life insurance companies. Journal of Banking and Finance. 32(12). pp 2589-2596.

Glaser, M., Lopez-de-Silanes, F. and Sautner, Z. (2013). Opening the Black Box: Internal

Capital Markets and Managerial Power. The Journal of Finance. 68(4). pp 1577-1631.

Glasserman, Paul. (2003). Monte Carlo Methods in Financial Engineering. New York, USA,

pp 134-142.

Graham, R., Harvey, C.R., and Puri, M. (2015). Capital allocation and delegation of decision-

making authority within firms. Journal of Financial Economics. 115(3). pp 449-470.

Grundl, H., Schmeiser, H. (2007). Capital allocation for insurance companies- what good is

it? Journal of Risk and Insurance. 74(2). pp 301-317.

Hoechle, D., Schmid, M., Walter, I. and Yermack, D. (2012). How much of the

diversification discount can be explained by poor corporate governance? Journal of Financial

Economics. 103 (1). pp 41-60.

Hund, J., Monk, D., and Tice, S. (2010). Uncertainty about average profitability and the

diversification discount. Journal of Financial Economics. 96(3). pp 463-484.

James, C. (1996). RAROC based capital budgeting and performance evaluation: A case study

of bank capital allocation. Working paper. Wharton Financial Institutions Center

Kalkbrener, M. (2005). An axiomatic approach to capital allocation. Mathematical Finance

15(3). pp 425–437.

Kim, J., and Hardy, M. (2009). A capital allocation based on a solvency exchange option.

Insurance: Mathematics and Economics. 44.(3). pp 357-366.

Kudryavtsev, L.D. (2002). Homogeneous function. Encyclopedia of mathematics. Springer:

Berlin.

Lamont, O. (1997). Cash Flow and Investment: Evidence from Internal Capital Markets. The

Journal of Finance. 52(1). pp 83-109.

Lönnbark, C. (2013). On the role of the estimation error in prediction of expected shortfall.

Journal of Banking and Finance. 37(3). pp 847-853.

http://www.sciencedirect.com/science/journal/01676687/44/3


24

Merton, R.C., Perold, A.F. (1993). Theory of risk capital in financial firms. Journal of

Applied Corporate Finance. 6(3). pp 16-32.

Myers, S.C. and Read, J.A. (2001). Capital allocation for insurance companies. Journal of

Risk and Insurance. pp 545–580.

Ozbas, O. and Scharfstein, D.S. (2010). Evidence on the dark side of internal capital markets.

Review of Financial Studies. 23 (2). pp 581-599.

Perold, A.F. (2005). Capital allocation in financial firms. Journal of Applied Corporate

Finance. 17(3). pp 110-118.

Phillips, R.D., Cummins, J.D., and Allen, F. (1998). Financial pricing of insurance in the

multiple-line insurance company. Journal of Risk and Insurance. 65(4). pp 597-636.

Rajan, R., Servaes, H. and Zingales, L., (2000). The Cost of Diversity: The Diversification

Discount and Inefficient Investment. The Journal of Finance. 55(1). pp 35-80.

Rockafellar, R.T., and Uryasev, S. (2002). Conditional value-at-risk for general loss

distributions. Journal of Banking and Finance. 26(7). pp 1443-1471.

Roper, A.H., Ruckes, M.E. (2012). Intertemporal capital budgeting. Journal of Banking and

Finance. 36(9). pp 2543-2551.

Sherris, M. (2006). Solvency, capital allocation and fair rate of return in insurance. Journal of

Risk and Insurance. 73(1). pp 71-96.

Shin, H.H., Stulz, R.M. (1998). Are internal capital markets efficient? The Quarterly Journal

of Economics. pp 531-552.

Stoughton, N.M. and Zechner, J. (2007). Optimal capital allocation using RAROCTM and

EVA®. Journal of Financial Intermediation. 16 (3). pp 312-342.

Tasche, D. (2004). Allocating Portfolio Economic Capital to Sub-Portfolios. Economic

Capital: A Practitionerʼs Guide, Risk Books. pp 1-20.

Tasche, D. (2002). Expected shortfall and beyond. Journal of Banking and Finance. 26(7).

pp 1519-1533.

Tsanakas, A. (2008). To split or not to split: capital allocation with convex risk measures.

Insurance: Mathematics and Economics.44(2). pp 268-277.



25

Urban, M., Dittrich, J., Klüppelberg, C., and Stölting, R. (2004). Allocation of risk capital to

insurance portfolios. Blätter der DGVFM. 26(3). pp 389-406.

Venter, G.G. (2004). Capital allocation survey with commentary. North American Actuarial

Journal. 8(2). pp 96-107.

Villalonga, B. (2004). Diversification Discount or Premium? New Evidence from the

Business Information Tracking Series. Journal of Finance. 59(2). pp 479-506.

Xuan, Y. (2009). Empire-Building or Bridge-Building? Evidence from New CEOs’ Internal

Capital Allocation Decisions. Review of Financial Studies. 22(12). pp 4919-4948.

Zaik, E., Walter,J., Kelling,G. and James,C. (1996). RAROC at Bank of America: from

theory to practice. Journal of Applied Corporate Finance. 9(2). pp 83-93.

26

Table 1. Descriptive Statistics

The following table shows the descriptive statistics of the data we use. Table 1.1 shows the

average, minimum, maximum, standard deviation (i.e. Std.), and total count (i.e. N) of the

total asset, total equity, regulatory capital (i.e. risk based capital), and total net income. The

units for these are million US dollars rounded up to third decimal points. The dates are

quarterly and we show the minimum, maximum, and total count for these. Table 1.2 shows

each bank’s names, total quarters of estimation, and total number of business units that we

use for our empirical test.

Table 1.1. Financial Data

Date Total Asset Total Equity Regulatory Capital Total Net Income

Average 674022.791 72706.705 70943.435 1410.019

Min 3/31/2006 30188.700 4568.400 3105.400 -8826.000

Max 12/31/2014 2573126.000 238681.000 229094.000 6529.000

Std. 772325.187 78740.697 72756.889 1882.792

N 220 220 220 220 220

Table 1.2. Banks’ Characteristics

Banks Total Quarters of Estimation Total Number of Business Units

Bank of America Corp 15 5

BB&T Corp 11 7

Citigroup Inc 21 3

Fifth Third Bancorp 15 4

Huntington Bancshares Inc/OH 16 4

JPMorgan Chase & Co 8 5

KeyCorp 22 3

People's United Financial Inc 6 3

PNC Financial Services Group Inc 23 5

Regions Financial Corp 6 3

SunTrust Banks Inc 11 4

US Bancorp/MN 31 5

Wells Fargo & Co 35 3

27

Table 2. New model’s result comparison using Huntington Bancshares Inc.

This table shows the new model’s result using Huntington Bancshares Inc. for its overall

RORAC and expected net income with no allocation (i.e. benchmark), with allocation under

Basel regime, and with allocation under economic capital regime. The last two columns show

the total available regulatory capital and equity amount that do not change during allocation

while we use only the available regulatory capital in our overall RORAC formula for

comparison purpose. The capital allocation starts from March 31st 2011. The units other than

the overall RORAC are all million US dollars.

Benchmark

(No Allocation)

New Model (Allocation

with Basel)

New Model (Allocation

with Equity)

Available Regulatory

Capital

Available Total

Equity

Overall

RORAC (No

Allocation)

Expected

Overall

Net Income

(No

Allocation)

Overall

RORAC (With

Allocation)

Expected

Overall

Net Income

(With

Allocation)

Overall

RORAC (With

Allocation)

Expected

Overall

Net Income

(With

Allocation)

2010-09-30 6285.00 4980.54

2010-12-31 6389.60 5038.60

2011-03-31 2.101% 137.93 2.194% 137.93 2.174% 137.93 6565.00 5252.64

2011-06-30 2.130% 142.77 2.082% 144.00 2.081% 142.71 6704.00 5400.48

2011-09-30 1.756% 119.04 2.424% 139.60 2.371% 139.51 6778.00 5418.10

2011-12-31 1.949% 134.41 2.187% 164.32 2.182% 160.74 6895.00 5549.83

2012-03-31 1.892% 130.62 1.797% 150.78 1.787% 150.46 6904.00 5649.23

2012-06-30 2.168% 149.83 2.093% 124.05 2.034% 123.37 6912.00 5807.60

2012-09-30 1.914% 132.63 2.110% 144.70 2.087% 140.61 6928.00 5790.21

2012-12-31 2.079% 144.95 2.452% 146.20 2.363% 144.59 6973.00 5867.14

2013-03-31 2.154% 150.88 1.697% 170.95 1.701% 164.74 7005.00 5783.52

2013-06-30 2.577% 184.07 2.606% 118.90 2.601% 119.16 7144.00 5961.58

2013-09-30 2.193% 158.72 2.108% 186.19 2.098% 185.85 7239.00 6090.15

2013-12-31 2.188% 158.06 2.502% 152.60 2.533% 151.89 7225.00 6176.23

2014-03-31 2.309% 167.40 2.545% 180.76 2.546% 183.03 7250.00 6240.79

2014-06-30 2.140% 156.23 2.320% 184.50 2.337% 184.58 7302.00 6284.21

Overall RORAC = Expected Overall Net Income

Available Regulatory Capital

Units except overall RORAC: million US $

28

Figure 1. Empirical result using Huntington Bancshares Inc.

These figures show the overall RORAC, expected net income, and business units’ new

capital (i.e. regulatory capital or equity) after allocation. We compare these with the

benchmark with no capital allocation against Basel (Figure 1.1) and economic capital (Figure

1.2) regimes. There are four business units and the allocation starts at time 3 (3rd quarter or

3/31/2011) until time 16 (16th quarter or 6/30/2014). The units other than the overall RORAC

are million US dollars.

Figure 1.1 Full result under Basel versus the Benchmark

29


30

Figure 1.2 Full result under Economic Capital versus the Benchmark

31


32

Table 3. Full empirical result

These tables show the empirical result of the new model using thirteen banks from S&P500.

We compare the benchmark RORAC without using the new model and the optimized

RORAC using the new model for each bank. Then we show the average difference per

quarter, aggregate difference, minimum difference, and maximum difference for these

RORACs. We show these under four different regimes: Basel (Table 3.1), Basel+ (Table 3.2),

Economic Capital (Table 3.3), and Economic Capital+ (Table 3.4) regimes. Each bank has

different total quarters of estimation since we use the most current set of business units (i.e.

segments) that existed from different points of time in the past.

Table 3.1. Full empirical result under Basel regime

Difference = New Model's RORAC - Benchmark RORAC

Benchmark: Overall RORAC without risk capital allocation with Basel Total quarters of estimation: Number of quarters prior to 01/01/2015

Benchmark Quarterly

Average

RORAC

New

Model's

Quarterly Average

RORAC

Average

Difference Per Quarter

Aggregate

Difference

Minimum


Maximum


Total

Quarters of Estimation

Bank of America

Corp 0.6142% 4.1557% 3.5414% 21.2487% 0.4078% 7.0634% 6

BB&T Corp 2.4849% 4.3912% 1.9063% 15.2504% -1.0972% 5.8825% 8

Citigroup Inc 1.6347% 1.0669% -0.5678% -6.8132% -3.7994% 3.3023% 12

Fifth Third Bancorp 1.7438% 1.4043% -0.3395% -4.4132% -1.2019% 0.2991% 13

Huntington Bancshares Inc/OH

2.1106% 2.2374% 0.1268% 1.7755% -0.5397% 1.1087% 14

JPMorgan Chase &

Co 2.1041% 1.7890% -0.3151% -1.5756% -4.8352% 1.9078% 5

KeyCorp 1.6367% 1.7851% 0.1485% 2.8211% -1.0199% 2.8972% 19

People's United

Financial Inc 1.8314% 2.1805% 0.4064% 1.6255% 0.2699% 0.5380% 4

PNC Financial

Services Group Inc 1.5754% 2.4724% 0.8969% 9.8663% -0.3825% 2.5211% 11

Regions Financial Corp

1.9477% 7.0324% 5.0847% 15.2540% 0.3299% 10.6763% 3

SunTrust Banks Inc 2.1510% 3.4106% 1.2595% 10.0762% -1.6486% 5.8267% 8

US Bancorp/MN 3.9616% 4.3675% 0.4060% 1.2179% 0.2621% 0.5278% 3

Wells Fargo & Co 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0

Average 1.8305% 2.7918% 0.9657%

33

Table 3.2. Full empirical result under Basel+ regime



Benchmark

Quarterly

Average RORAC

New

Model's Quarterly

Average

RORAC

Average Difference

Per Quarter

Aggregate

Difference

Minimum Difference

Per Quarter

Maximum Difference

Per Quarter

Total Quarters of

Estimation

Bank of America

Corp 0.6142% 4.7928% 4.1786% 25.0715% 0.5330% 7.2332% 6

BB&T Corp 2.4849% 4.8357% 2.3508% 18.8062% -0.1779% 6.7975% 8

Citigroup Inc 1.6347% 2.1395% 0.5048% 6.0576% -2.3991% 1.8176% 12



2.1106% 2.2227% 0.1121% 1.5696% -0.4565% 0.6681% 14

JPMorgan Chase & Co

2.1041% 0.5943% -1.5098% -7.5491% -4.8352% 2.9722% 5

KeyCorp 1.6367% 1.7079% 0.0712% 1.3532% -1.0199% 3.2825% 19

People's United

Financial Inc 1.8314% 2.0367% 0.2572% 1.0288% -0.3268% 0.5380% 4

PNC Financial


Regions Financial

Corp 1.9477% 8.7799% 6.8322% 20.4966% 0.5873% 10.6763% 3


US Bancorp/MN 3.9616% 6.0760% 2.1145% 6.3435% 0.3140% 3.5549% 3

Wells Fargo & Co 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0

Average 1.8305% 3.2234% 1.3969%

34

Table 3.3. Full empirical result under economic capital regime



Benchmark

Quarterly

Average RORAC

New

Model's Quarterly

Average

RORAC

Average Difference

Per Quarter

Aggregate

Difference

Minimum Difference

Per Quarter

Maximum Difference

Per Quarter

Total Quarters of

Estimation

Bank of America

Corp 0.9670% 3.8845% 2.9175% 26.2576% -1.1385% 7.0634% 9

BB&T Corp 2.3508% 4.4563% 2.1055% 16.8443% -1.0915% 6.7975% 8

Citigroup Inc 1.6676% 1.5130% -0.1546% -1.7006% -3.7994% 2.9604% 11



2.1106% 2.2387% 0.1281% 1.7939% -0.5398% 1.1484% 14

JPMorgan Chase & Co

2.1041% 1.9141% -0.1900% -0.9501% -4.8352% 2.3460% 5

KeyCorp 1.6367% 1.7769% 0.1403% 2.6651% -1.0199% 3.1126% 19

People's United

Financial Inc 1.8314% 2.1776% 0.4041% 1.6165% 0.2609% 0.5380% 4

PNC Financial


Regions Financial

Corp 1.9477% 6.0500% 4.1023% 16.4092% 0.3292% 10.6763% 4


US Bancorp/MN 3.9616% 4.4450% 0.4834% 1.4503% 0.3498% 0.5773% 3

Wells Fargo & Co 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0

Average 1.8498% 2.7533% 0.9080%

35

Table 3.4. Full empirical result under economic capital+ regime



Benchmark

Quarterly

Average RORAC

New

Model's Quarterly

Average

RORAC

Average Difference

Per Quarter

Aggregate

Difference

Minimum Difference

Per Quarter

Maximum Difference

Per Quarter

Total Quarters of

Estimation

Bank of America

Corp 0.9670% 4.8002% 3.8333% 34.4994% 0.3615% 6.6756% 9

BB&T Corp 2.3508% 4.7798% 2.4290% 18.8062% -0.1779% 6.7975% 8

Citigroup Inc 1.6676% 2.0217% 0.3541% 3.8946% -2.3365% 2.2987% 11



2.1106% 2.2069% 0.0963% 1.3486% -0.4527% 0.6153% 14

JPMorgan Chase & Co

2.1041% 0.7039% -1.4002% -7.0010% -4.8352% 2.3995% 5

KeyCorp 1.6367% 1.7243% 0.0876% 1.6650% -1.0199% 3.5060% 19

People's United

Financial Inc 1.8314% 2.1556% 0.3795% 1.5178% 0.1623% 0.5380% 4

PNC Financial


Regions Financial

Corp 1.9477% 8.8496% 6.9018% 27.6073% 0.5897% 10.6763% 4

SunTrust Banks Inc 2.1510% 5.2899% 3.1388% 25.1105% 0.2819% 5.9597% 8

US Bancorp/MN 3.9616% 6.0040% 2.0425% 6.1274% 0.2908% 3.5549% 3

Wells Fargo & Co 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0

Average 1.8498% 3.2467% 1.4011%

36

Table 4. Full empirical result with different firm size, total equity, regulatory capital,

and volatility levels.

The following tables show the average RORAC levels with capital allocation under Basel+

and Economic Capital+ regimes. We classify the average RORAC with capital allocation into

high/medium/low categories under high/medium/low firm size (i.e. total asset) (Table 4.1),

total equity (Table 4.2), regulatory capital (Table 4.3), and volatility levels (Table 4.4). The

units for firm size, total equity, and regulatory capital are million US dollars. The units for

RORAC and volatility are percentages rounded up to third decimal points.

Table 4.1. Full empirical result under different firm size levels

Firm Size

Level Banks Volatility

RORAC

(Basel+)

RORAC

(Economic Capital+)

High

JPMorgan Chase & Co 2473696 0.594% 0.704%

Bank of America Corp 2188006 4.793% 4.800%

Citigroup Inc 1655995 2.053% 2.022%

US Bancorp/MN 229608.8 6.097% 6.044%

Medium

PNC Financial Services Group Inc 184118.4 2.213% 2.204%

BB&T Corp 183173.9 4.836% 4.914%

SunTrust Banks Inc 173906.6 5.157% 5.290%

Fifth Third Bancorp 130147.1 1.348% 1.467%

Low

KeyCorp 84392.05 1.708% 1.724%

Regions Financial Corp 84205.6 8.780% 8.910%

Huntington Bancshares Inc/OH 47163.27 2.223% 2.207%

People's United Financial Inc 31617.45 2.089% 2.211%

Firm Size Level

Average

RORAC

(Basel+)

Average

RORAC

(Economic Capital+)

Average

RORAC

Level

High 3.384% 3.393% Low

Medium 3.389% 3.469% Medium

Low 3.700% 3.763% High

37

Table 4.2. Full empirical result under different total equity levels

Total

Equity

Level

Banks Volatility RORAC

(Basel+)

RORAC

(Economic Capital+)

High

Bank of America Corp 232714.3 4.793% 4.800%

JPMorgan Chase & Co 217555.5 0.594% 0.704%

Citigroup Inc 172192.3 2.053% 2.022%


Medium

BB&T Corp 22064.4 4.836% 4.914%

SunTrust Banks Inc 21286.2 5.157% 5.290%

US Bancorp/MN 21092.4 6.097% 6.044%


Low


KeyCorp 10460.48 1.708% 1.724%



Total Equity Level

Average

RORAC

(Basel+)

Average

RORAC

(Economic Capital+)

Average

RORAC

Level

High 2.413% 2.432% Medium

Medium 6.217% 6.290% High

Low 1.842% 1.902% Low

38

Table 4.3. Full empirical result under different regulatory capital levels

Regulatory

Capital

Level

Banks Volatility RORAC

(Basel+)

RORAC

(Economic Capital+)

High

Bank of America Corp 206290.5 4.793% 4.800%

JPMorgan Chase & Co 203422.5 0.594% 0.704%

Citigroup Inc 165888.9 2.053% 2.022%


Medium

US Bancorp/MN 26432.6 6.097% 6.044%

BB&T Corp 19024.1 4.836% 4.914%

SunTrust Banks Inc 18626 5.157% 5.290%


Low


KeyCorp 12989.81 1.708% 1.724%



Regulatory Capital Level

Average

RORAC

(Basel+)

Average

RORAC

(Economic Capital+)

Average

RORAC

Level

High 2.413% 2.432% Low

Medium 4.359% 4.429% High

Low 3.700% 3.763% Medium

39

Table 4.4. Full empirical result under different volatility levels

Volatility

Level Banks Volatility

RORAC

(Basel+)

RORAC

(Economic Capital+)

High

Bank of America Corp 2.293% 4.793% 4.800%

KeyCorp 2.038% 1.708% 1.724%

SunTrust Banks Inc 1.424% 5.157% 5.290%

JPMorgan Chase & Co 1.212% 0.594% 0.704%

Medium

Citigroup Inc 1.072% 2.053% 2.022%

Huntington Bancshares Inc/OH 0.709% 2.223% 2.207%

PNC Financial Services Group Inc 0.502% 2.213% 2.204%

Fifth Third Bancorp 0.487% 1.348% 1.467%

Low

BB&T Corp 0.478% 4.836% 4.914%

US Bancorp/MN 0.379% 6.097% 6.044%

Regions Financial Corp 0.249% 8.780% 8.910%

People's United Financial Inc 0.091% 2.089% 2.211%

Volatility Level

Average

RORAC

(Basel+)

Average

RORAC

(Economic Capital+)

Average

RORAC

Level

High 3.063% 3.130% Medium

Medium 1.959% 1.975% Low

Low 5.450% 5.520% High

40

Table 5. Summary comparison of average RORAC under different firm size, total

equity, regulatory capital, and volatility.

The following table shows the summary comparison result of Table 4. It shows different

average RORAC levels with capital allocation under different firm size (i.e. total asset), total

equity, regulatory capital, and volatility levels. We show these with high/medium/low classes.

The classes of average RORAC with capital allocation are identical in Basel+ and Economic

Capital+ regimes that we do not show these results separately.

Firm

Size Average

RORAC Total

Equity Average

RORAC Regulatory

Capital Average

RORAC Volatility Average

RORAC High Low High Medium High Low High Medium

Medium Medium Medium High Medium High Medium Low

Low High Low Low Low Medium Low High

41

Table 6. Full empirical result with constraints

These tables show the empirical result of the new model using thirteen banks from S&P500

with four different constraints: ±2%/±20%, ±2%/±30%, ±5%/±20%, and ±5%/±30. We

compare the benchmark RORAC without using the new model and the optimized RORAC

using the new model for each bank. Then we show the average difference per quarter,

aggregate difference, minimum difference, and maximum difference for these RORACs. We

show these under four different regimes for each constraint: Basel, Basel+, Economic Capital,

and Economic Capital+ regimes. Each bank has different total quarters of estimation since we

use the most current set of business units (i.e. segments) that existed from different points of

time in the past. The figures are rounded up to the third decimal points.

Benchmark Quarterly

Average RORAC New Model's Quarterly

Average RORAC Average Difference

Per Quarter

±2%/±20%

under Basel regime 2.017% 2.143% 0.126%

±2%/±20% under Basel+ regime

2.017% 2.305% 0.288%

±2%/±20%

under economic capital regime 1.999% 2.121% 0.121%

±2%/±20% under economic capital+ regime 1.999% 2.209% 0.210%

±2%/±30%


±2%/±30%

under Basel+ regime 2.005% 2.375% 0.370%

±2%/±30%


±2%/±30%

under economic capital+ regime 2.066% 2.350% 0.284%

±5%/±20%


±5%/±20%


±5%/±20%


±5%/±20%


±5%/±30%


±5%/±30%


±5%/±30%


±5%/±30%


42

Figure 2. Available capital comparison with risk tolerance

The following thirteen graphs compares the available regulatory capital, available equity,

required minimum regulatory capital, and required minimum equity during the capital

allocation process for thirteen financial firms in S&P500. The red and blue lines are required

minimum equity and required minimum regulatory capital, respectively. The black and grey

lines are the available regulatory capital and available equity, respectively. The unit of the x-

axis is the quarter of a year where the last period for all graphs are June 30th 2014 where the

total number of estimating quarters vary depending on each financial firm’s available data set.

The unit of the y-axis is million US dollars.

43

44

45

X-axis unit: quarter of a year

Y-axis unit: million US $

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