Estimating Default Probabilities

Document Sample
Estimating Default Probabilities Powered By Docstoc
					          A Critique of Revised Basel II


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

                     Robert Jarrow         1
                  1. 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

3. Due to (1) and XYZ Theory, maintaining aggregate
   industry capital at pre-revised Basel II framework levels is

                        Robert Jarrow                             2
          2. 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).

                             Robert Jarrow                         3
                 The Bank’s Optimal Capital

The bank’s optimal capital level is defined to be that capital which
  maximizes shareholders’ wealth, independently of regulatory
                        Xt  f (t,t)

Banks may or may not know f( . , . ).
        Larger (international) banks – yes
        Smaller (regional banks) – ???

                              Robert Jarrow                            4
                   Ideal 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).

                            Zt      h(t,t)

Hypothesis 1 (Costly Externalities):

                                 Zt  Xt

                              Robert Jarrow                           5
               Ideal 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

                         Robert Jarrow                6
             Required Regulatory Capital

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

                      Yt       g(t,t)

Hypothesis 3 ( Approximate Ideal Capital from Below):

                        Yt  Zt

                           Robert Jarrow                    7
              Required Regulatory Capital

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.

                         Robert Jarrow                        8
              Required Regulatory Capital

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

     g (t,t)   0.08 j risk weighted asset values j

Will discuss later in more detail.

                          Robert Jarrow                    9
                       Theorem 1

Given hypotheses 1 and 2.
Let     g j (t,t)
for j=1,…,N represent a collection of regulatory capital
Let hypothesis 3 hold. Then,     Yt  max{ g j (t,t):all j}
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

                          Robert Jarrow                        10
               Theorem 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.

                         Robert Jarrow                      11
                        Theorem 2

Let hypotheses 1 – 3 hold.
Let g i (t,t)
for i = 1,…,m be the regulatory capital for bank i,
Then when considering a new rule

                gnew(t,t)   g i (t,t).
                 m                  m
                 i 1              i 1

                         Robert Jarrow                12
               Theorem 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.

                        Robert Jarrow                      13
         3. 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.

                         Robert Jarrow                      14
          The 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.

                       Robert Jarrow                  15
                      Credit 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…

                        Robert Jarrow                   16
                   PD, 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.

                       Robert Jarrow                 17
                   PD, 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.

                       Robert Jarrow                    18
                 Problems with VaR

Problems with the VaR measure for loss L.
              VaR(L) inf{x  0: P(L  x)  0.999}

Well-known that VaR:
 ignores distribution of losses beyond 0.999 level,
 penalizes diversification of assets (provides an
  incentive to concentrate risk).

                        Robert Jarrow                  19
       Example: 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

                  Robert Jarrow                       20
           Given 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.

                         Robert Jarrow                     21
            Given 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

                         Robert Jarrow                      22
         Given 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???

                       Robert Jarrow                    23
     Given VaR – Normal Distribution for Losses

Formula for K implies that losses (returns) are normally

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

                         Robert Jarrow                  24
          Given 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.

                        Robert Jarrow                     25
                  Significance 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.

                         Robert Jarrow                  26
                  Operational 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.

                        Robert Jarrow                 27
                     Market 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.

                        Robert Jarrow                     28
                    Liquidity Risk

Liquidity risk only included implicitly in
 credit risk (via the LGD, EAD being for an economic
 market risk (via the scale factor of 3).

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

                        Robert Jarrow                   29