Robert Jarrow 1

A Critique of Revised Basel II A Critique of Revised Basel II

Outline

1. Conclusions 2. XYZ Theory of Regulatory Capital 3. Revised Basel II Capital Rule

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1. Conclusions1. Conclusions

1. Revised Basel II framework provides a (very) rough

approximation to an ideal capital rule. 2. Due to (1) and XYZ Theory, revised Basel II should only

be used in conjunction with other rules for determining minimal capital levels:

- FDICIA leverage based rules - Parallel run and transitional floor periods are

prudent. 3. Due to (1) and XYZ Theory, maintaining aggregate

industry capital at pre-revised Basel II framework levels is prudent.

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2. XYZ Theory of Regulatory Capital2. XYZ Theory of Regulatory Capital

Randomness in the economy determined by the evolution of a set of state variables.

State variables include individual bank characteristics and business cycle characteristics (macro-variables).

t

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The Bank’s Optimal CapitalThe Bank’s Optimal Capital

The bank’s optimal capital level is defined to be that capital which

maximizes shareholders’ wealth, independently of regulatory

rules.

Banks may or may not know f( . , . ).

Larger (international) banks – yes

Smaller (regional banks) – ???

),( tt tfX

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Ideal Regulatory CapitalIdeal Regulatory Capital

Regulatory capital is needed due to costly externalities associated with bank failures.

The ideal regulatory capital is defined to be that (hypothetical) capital determined as if regulatory authorities had perfect knowledge (information).

Hypothesis 1 (Costly Externalities):    

),( tt thZ

tt XZ

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Ideal Regulatory CapitalIdeal Regulatory Capital

Hypothesis 2 (Unknown Zt):

h( . , . ) unknown to regulators

Due to:

(i) uncertainty over the exact form of the

appropriate risk measure (Value at Risk,

Coherent, …)

(ii) insufficient data to compute risk measure

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Required Regulatory CapitalRequired Regulatory Capital

Regulatory authorities specify a rule to approximate the ideal capital. This is the required regulatory capital.

Hypothesis 3 ( Approximate Ideal Capital from Below):

),( tt tgY

tt ZY

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Required Regulatory CapitalRequired Regulatory Capital

Justification:

1. Believed that many banks choose Xt > Yt for competitive reasons. Then, under hypothesis 1, Zt > Xt > Yt.

2. Rule chosen (shown later) is based on asymptotic theory where idiosyncratic risks are infinitesimal and diversified away, implies Zt > Yt.

3. Rule chosen (shown later) so that ideally, probability of failure is less than .001. Implies A credit rating or better (Moody’s). In practice, required capital does not achieve this level for many banks, so that for these banks Zt > Yt.

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Required Regulatory CapitalRequired Regulatory Capital

Example: In revised Basel II, the rule for required capital is (for illustrative purposes)

Will discuss later in more detail.

j jt valuesassetweightedrisktg 08.0),(

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Theorem 1Theorem 1

Given hypotheses 1 and 2.

Let

for j=1,…,N represent a collection of regulatory capital rules.

Let hypothesis 3 hold. Then,

is a better approximation to Zt than any single rule.

If hypothesis 3 does not hold, then no simple ordering of regulatory capital rules is possible without additional structure.

),( tj tg

}:),(max{ jalltgY tjt

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Theorem 1 - implicationsTheorem 1 - implications

1. New rules should be implemented without discarding existing rules. Implies retention of leverage based rules (FDICIA) is prudent.

2. Four year parallel run period with yearly transitional floors (95%, 90%,85%) within Basel II revised framework is prudent.

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Theorem 2Theorem 2

Let hypotheses 1 – 3 hold.

Let

for i = 1,…,m be the regulatory capital for bank i,

Then when considering a new rule

),( ti tg

.),(),(11

m

i

m

ititi

new tgtg

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Theorem 2 - implicationsTheorem 2 - implications

1. Scaling individual bank capital so that in aggregate, industry capital does not decline, is prudent. Current scale is 1.06 based on the 3rd Quantitative Impact Study. Tentative magnitude.

2. Requiring that the regulations be restudied/modified if a 10% reduction in aggregate capital results after implementation is prudent.

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3. The Revised Basel II Capital Rule3. The Revised Basel II Capital Rule

The following analysis is independent of XYZ theory.

Revised Basel II rule illustrated on a previous slide.

In revised Basel II, the risk weightings are explicitly adjusted for credit risk, operational risk, and market risk. Liquidity risk is only an implicit adjustment.

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The Revised Basel II Capital RuleThe Revised Basel II Capital Rule

Two approaches:

1. Standard (based on tables and rules given in revised Basel II framework).

2. Internal ratings/ Advanced approach (based on internal models).

For my analysis, concentrate on internal ratings/advanced approach.

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Credit RiskCredit Risk

Risk weights determined based on bank’s internal estimates of PD, LGD and EAD.

These estimates input into a formula for capital (K) held for each asset. Capital K based on:

1. Value at Risk (VaR) measure over a 1-year horizon with a 0.999 confidence level.

2. Asymptotic single-factor model, with constant correlation assumption.

3. An adjustment for an asset’s maturity.

Discuss each in turn…

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PD, LGD, EADPD, LGD, EAD

PD is 1-year long term average default probability

– not state dependent.

LGD is computed based on an economic downturn

– quasi-state dependent.

EAD is computed based on an economic downturn

– quasi-state dependent.

These do not change with business cycle.

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PD, LGD, EADPD, LGD, EAD

Ideal regulatory capital should be state dependent.

Pro: Makes bank failures counter-cyclic.

Con: Makes bank capital pro-cyclic. Could adversely effect interest rates (investment). But, monetary authorities have market based tools to reduce this negative impact.

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Problems with VaRProblems with VaR

Problems with the VaR measure for loss L.

Well-known that VaR: ignores distribution of losses beyond 0.999 level, and penalizes diversification of assets (provides an

incentive to concentrate risk).

}999.0)(:0inf{)( xLPxLVaR

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Example: Concentrating RiskExample: Concentrating Risk

Loss P(LA) P(LB) P(L(A+B)/2)

$0 0.9991 0.9991 0.9982

$.50 0 0 0.0018

$1 0.0009 0.0009 0.0000

VaR(LA) = 0 and VaR(L(A+B)/2) = $.50

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Given VaR – Portfolio InvarianceGiven VaR – Portfolio Invariance

Capital K formulated to have portfolio invariance, i.e. the required capital for a portfolio is the sum of the required capital for component assets.

Done for simplicity of implementation.

But, it ignores benefits of diversification, provides an incentive toward concentrating risk.

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Given VaR – Single Risk FactorGiven VaR – Single Risk Factor

The asymptotic model (to get portfolio invariance) has a single risk factor.

The single risk factor drives the state variables vector.

Inconsistent with evidence, e.g.

Duffee [1999] needed 3 factors to fit corporate bond prices.

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Given VaR – Common CorrelationGiven VaR – Common Correlation

When implementing the ASRF model, revised Basel II assumes that all assets are correlated by a simple function of PD, correlation bounded between 0.12 and 0.24.

No evidence to support this simplifying assumption???

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Given VaR – Normal Distribution for LossesGiven VaR – Normal Distribution for Losses

Formula for K implies that losses (returns) are normally distributed.

Inconsistent with evidence: Ignores limited liability (should be lognormal) Ignores fat tails (stochastic volatility and jumps)

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Given VaR – Maturity AdjustmentGiven VaR – Maturity Adjustment

Capital determination based on book values of assets.

This ignores capital gains/losses on assets over the 1-year horizon.

Gordy [2003] argues that a maturity adjustment is necessary to capture downgrades of credit rating in long-dated assets.

Do not understand. Asset pricing theory has downgrade independent of maturity. Maturity (duration) adjustment only (roughly) captures interest rate risk.

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Significance of ErrorSignificance of Error

P. Kupiec constructs a model – Black/Scholes/Merton economy, correlated geometric B.M.’s for assets. Considers a portfolio of zero-coupon bonds.

Computes ideal capital, compares to revised Basel II framework capital.

Finds significant differences.

Conclusion: revised Basel II capital rule is a (very) rough approximation to the ideal rule.

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Operational RiskOperational Risk

Basic indicator and standard approach: capital is proportional to income flow.

Advanced measurement approach: internal models approach based on VaR, 1-year horizon, 0.999 confidence level.

Jarrow [2005] argues operational risk is of two types: system or agency based.

Income flow captures system type risk. Agency risk is not captured by income flow. More

important of the two types. Only possibly captured in advanced measurement approach.

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Market RiskMarket Risk

Standardized and internal models approach.

Concentrate on internal models approach.

Internal models approach is VaR based with 10-day holding period and 0.99 confidence level with a scale factor of 3.

Why the difference from credit risk?

Could lead to regulatory arbitrage if an asset could be classified as either.

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Liquidity RiskLiquidity Risk

Liquidity risk only included implicitly in credit risk (via the LGD, EAD being for an economic

downturn) market risk (via the scale factor of 3).

Better and more direct ways of doing this are available, see Jarrow and Protter [2005].