Capital Allocation

Abstract

Banks and other financial institutions should allocate capital in proportion to the marginal default value of each line of business, which is the derivative of the value of the bank's default put with respect to a change in the scale of the business. Marginal default values give a unique allocation that adds up exactly. Cross subsidies are avoided if capital allocations are set so that capital-adjusted marginal default values are the same for all lines. We include a series of examples showing how our procedures work and why the allocations are different from allocations based on VaR and the allocations implicit in risk-weighted bank capital requirements. We explain how capital allocations should be "priced" and charged to each line of business.

Full text

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Electronic copy available at: http://ssrn.com/abstract=1411190
Charles A. Dice Center for
Research in Financial Economics

Capital Allocation



Isil Erel,
The Ohio State University, Department of Finance

Stewart C. Myers,
MIT Sloan School of Management

James A. Read, Jr.
The Brattle Group, Inc.
Dice Center WP 2009-10
Fisher College of Business WP 2009-03-010

May 2009
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Electronic copy available at: http://ssrn.com/abstract=1411190


CAPITAL ALLOCATION

Isil Erel, Stewart C. Myers and James A. Read, Jr.*

May 26, 2009


Abstract
Banks and other financial institutions should allocate capital in proportion to the marginal
default value of each line of business, which is the derivative of the value of the bank’s
default put with respect to a change in the scale of the business. Marginal default values
give a unique allocation that adds up exactly. Cross subsidies are avoided if capital
allocations are set so that capital-adjusted marginal default values are the same for all
lines. We include a series of examples showing how our procedures work and why the
allocations are different from allocations based on VaR and the allocations implicit in
risk-weighted bank capital requirements. We explain how capital allocations should be
“priced” and charged to each line of business.



*Ohio State University, MIT Sloan School of Management and The Brattle Group, Inc.
We have received helpful comments from Richard Derrig, Ken Froot, Hamid Mehran,
Rene Stulz and seminar participants at the University of Michigan, the NBER, the New
York Federal Reserve Bank, Ohio State University and University College Dublin.

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Electronic copy available at: http://ssrn.com/abstract=1411190

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CAPITAL ALLOCATION


1. Introduction
This paper investigates procedures for allocating capital in financial institutions. We
focus on banks, but the procedures apply generally. We show that the capital allocation
for a line of business should be set at marginal default value, which is the derivative of
the value of the bank’s default put with respect to a change in the scale of the business.
Consider a bank that diversifies across different activities and asset classes, which
may include lending; trading and market making; investment banking; asset management;
and retail services, such as credit-card operations. Financing comes from deposits and
debt and from risk capital, which is primarily common equity. The capital has to be
sufficient to satisfy regulators and lenders. That is, the bank has to carry enough capital
to keep the probability of default or financial distress to an acceptably tiny level. The
amount of capital required depends on the risk of the bank's various lines of business.
Some businesses are riskier than others, and risky businesses require more capital
backing than safe ones.
If the bank can identify capital requirements by lines of business, then it can
allocate capital back to the businesses. Allocation is important if capital is costly, say
because of tax, information problems or agency costs, or if capital is limited. Limited
capital also has a shadow price if the firm is forced to pass up positive-NPV investments.
If capital is costly, capital allocation (explicit or implicit) is required to assess the
profitability of each line of business and to set incentives and compensation correctly.
Allocation is relevant for pricing products and services. The more capital a product or
service requires, the higher the break-even price. Allocation is also necessary to calculate

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the net benefits of hedging or securitization. For example, suppose that a credit-default
swap transaction can offset one half of the risk of a loan portfolio, freeing up half of the
capital that the portfolio would otherwise require. The bank must then compare the costs
of hedging to the value of the capital released. To do that, the bank has to know how
much capital was properly allocated to the loan portfolio in the first place. We show how
to calculate allocations. We also show how capital allocations should be “priced” and
changed back to lines of business.
If the bank had only one line of business, its capital requirement could be based
on stand-alone risk -- on its value-at-risk (VaR), for example. Capital allocations are
implicit in RAROC (risk-adjusted return on capital) calculations, where lines of business
are assessed capital charges proportional to their VaRs. But VaRs for several lines of
business don't add up. Thanks to diversification, the VaR for the bank as a whole is less
than the weighted sum of the stand-alone VaRs of the individual businesses. The proper
procedures for allocating this diversification benefit are not obvious, and it appears that
various procedures are used in practice.1 Moreover, there is influential opinion that the
bank should not even attempt a complete allocation of capital:2

[T]he risk capital of a multi-business firm is less than the aggregate risk capital
of the businesses on a stand-alone basis. Full allocation of risk capital across the
individual businesses of the firm is generally not feasible. Attempts at such a
full allocation can significantly distort the true profitability of individual
businesses.


1 See Helbekkmo (2006), for example.
2 Merton and Perold (1993), p. 30.

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It's true that the reduction in risk from diversification across lines of business
cannot be uniquely allocated to the lines. But we show that the lines' marginal capital
requirements are unique and do add up exactly. We argue that capital allocations should
be based on these marginal requirements, both in principle and in practice.
Our conclusion that marginal capital requirements add up is not wholly original.
One special case has been noted already: Contribution VaRs, which depend on the
covariances or betas of line-by-line returns vs. returns for the bank as a whole, do add up.
See Saita (1999) and Stulz (2003),3 for example. This result falls out from mean-variance
portfolio theory. For example, the variance of a bank's overall return -- the return on its
portfolio of businesses -- can be expressed as a weighted sum of covariances of each
business’s return with the overall portfolio return. But it appears that contribution VaRs
are rarely used for capital allocation. Also, contribution VaRs only work in a mean-
variance setting. We derive a general adding-up theorem, which works for any joint
probability distribution of returns. Our only assumption is complete markets, complete
enough that individual lines of business would have well-defined market values if they
could be traded separately.
This paper extends Myers and Read (2001), who analyze capital (surplus)
allocation for insurance companies.4 Principles are similar here, although this application
is different and our proofs are more general. For example, insurance risks come from the

3 Contribution VaRs appear in Froot and Stein (1998, pp. 67-68), Stoughton and Zechner (2007), Saita
(1999), Stulz (2003, pp. 99-103) and no doubt in other places. The label varies: synonyms for
“contribution” include “marginal,” for example in Saita and Stulz. Others refer to “incremental VaR,”
which is not the same thing. Incremental VaR is the discrete change in VaR from adding or subtracting an
asset or business from the bank’s overall portfolio. Merton and Perold (1993), Perold (2005) and Turnbull
(2000) focus on incremental VaR.
4 Follow-on articles in the insurance literature include Cummins, Lin, and Phillips (2006) and Grundl and
Schmeiser (2007).