Skewness of returns, capital adequacy, and
mortgage lending
Paraskevi Dimou ‡
Alistair Milne †
Colin Lawrence ‡
November 2003
“The lady doth protest too much, methinks.” Hamlet, Act
3, scene 2.
Abstract
This paper presents a simulation model of credit
value-at-risk for mortgage lending, calibrated to the experience
of the UK mortgage market. Simulating this model reveals that
the IRB calculations of the new Basel accord can substantially
understate prudential capital adequacy for mortgage lending.
This is because the IRB approach does not allow for correlation
of loss given default and probability of default (LGD and PD).
To correct this understatement, pillar 2 of the accord could be
used to raise capital requirements. Any such increase has only a
minor impact on loan pricing.
†
Corresponding author, Faculty of Finance, Cass Business School, 106 Bunhill
Row, London EC1Y 8TZ, UK. email: [email protected]
‡
Cass Business School
1
Introduction
Pillar 1 of the new Basel accord has created procedures — the foundation
and advanced internal ratings based or IRB approach — for the calculation
of regulatory capital requirements for credit risk, using bank’s own
estimates of probability of default (PD) and of loss given default (LGD).
Detailed description of the IRB approach is provided in the latest Basel
committee consultation (Basel Commitee (2003)). In order to qualify for
these IRB approach banks will have to have developed and maintained
their own loan databases, covering a minimum number of years of exposure
and use these to estimate the probability of default and, for the advanced
IRB approach, the loss given default for their loan exposures. These
estimates should also reflect the intermal assessment of the quality of the
exposure (the internal rating).
The motive for the development of the IRB approach is to achieve a closer
alignment of regulatory capital requirements and ‘economic capital’, widely
used measures of the risk carried by shareholders for each loan exposure.
The data requirements for IRB qualification are similar to the data used by
many banks for calculating economic capital.
This paper examines the appropriateness of these new Basel internal
ratings based measures of capital adequacy, in a situation where returns
exhibit extreme skewness. We work with the example of mortgage lending,
identifying a problem with the proposed Basel IRB procedures. Our
analysis suggests that the IRB procedures, because they fail to allow for
correlation between loss given default and probability of default, can
seriously understate prudential capital requirements.
Although similar criticism applies to the IRB treatment of other categories
of bank lending, the point is most obvious for the case of mortgage lending.
Mortgages typically have very low levels of provisions, losses only arising in
the coincident circumstances of the borrower failing to maintain payments,
the lender choosing to ‘reposess’ the property, and the value of the property
declining so far that the principal of the loan and the accumulated interest
is not fully recovered. Nonetheless in extreme macro-economic downturns,
i.e. the very situation in which regulatory capital is intended to provide
protection, there can be a sharp rise both in probabilities of mortgage
default and and, as house prices fall, of loss given default.
To capture this correlation of probability of default (PD) and loss given
default (LGD) we develop a simple calibrated model of mortgage loan
1
losses with the following features:
1. loss given default is determined by the house price to lending ratio.
2. the distribution of the house price to lending ratios across properties
is shifted uniformly by a normally distributed unobserved
macro-economic shock
3. the probability of default for an individual loan is a logistic function of
the macro-economic shock and also of the house price to lending ratio.
4. the possibility of forced selling in extreme downturns depresses
house-prices by more than would be predicted by the aggregate shock
alone.
This model produces a highly skewed distribution of loan losses with a
pronounced downside tail, even though the underlying macro-economic
shock has a standard normal distribution. Model generated unconditional
probabilities of default (PD) and loss given default (LGD) can be input
into the Basel IRB mortgage risk-curve, to yield an IRB measure of capital
adequacy. This IRB measure can be much lower than credit value at risk or
CVaR based on the tail of the modelled loss distribution.
This failure of IRB to capture tail risk is a general problem for all
categories of exposure, not just for mortgages. The inputs to the IRB
procedures (PD and LGD) are based on a sample of the entire distribution
not on tail observations. The IRB capital measures extrapolate central
observations, based on a particular assumption about the distribution of
loan returns and ignoring correlation of PD and LGD.
Later sections of this paper address related questions. What are the
implications for loan pricing if, in order to capture tail risk, regulatory
capital requirements are increased substantially above economic capital?
We provide a brief discussion, finding that discrepancies between regulatory
and economic capital make little difference to loan pricing. We also argue
in our conclusion that economic capital measures (unlike regulatory capital
requirements) need not be constructed so as to capture tail risk. Divergence
between regulatory and economic capital is to be expected and need not be
a big concern.
The paper is organised as follows. Section 2 describes our simulation model
of the distribution of mortgage losses. Section 3 presents the distribution of
2
losses when this model is calibrated to UK mortgage data. Section 4
compares the credit value-at-risk capital requirements, based on the results
of Section 3, with capital requirements calculated using the IRB risk-curve.
Section 5 discusses the impact on mortgage pricing, of increasing capital
requirements from IRB levels to those suggested by our model. Section 6
summarises our findings and presents a concluding discussion of the
relationship between economic and regulatory capital. A first appendix
describes the calibration of our model against UK mortgage data ánd
model solution. A second appendix provides an option based valuation of
the probability of bank failure, used for the calculations of mortgage pricing
in Section 5.
2
The simulation model
Our objective is to model the distribution of loan losses on a portfolio of
mortgage loans, allowing for probability of default (PD), for loss given
default (LGD), and for the correlation over the business cycle between
these two variables.
Loss given default depends upon both house prices and the legal and other
costs of recovery. We suppose that the underlying security of lending p
(measured net of all these recovery costs) has a log normal distribution:
ln pi ∼ ln φ + σN (0, 1)
(1)
This is a cross-sectional distribution. We use the notation pi because it can
be thought of as a distribution of individual house price to loan ratios. This
distribution is the key driver of loss given default.
φ is a measure of median recovery value, we anticipate φ > 1, i.e. the
median mortgage is more than fully collateralised even after deducting legal
and other costs of recovery. But because of the cross-sectional distribution
of house prices, not all loans are fully secured. In the event of a default a
loss arises whenever pi < 1 ; if on the other hand pi > 1 (i.e. if the security
of lending exceeds the amount loaned) there is full recovery. This can be
summarised by writing the loss given default on the individual loan as
max[0, 1 − pi ] .
The state of the housing market and wider economy is represented by an
unobserved random standardised, normally distributed variable z ∼ N (0, 1).
3
We assume that φ is a function of z the ‘state of the economy’:
ln φ = α + βz − γ max[zc − z, 0]
(2)
This function indicates how the median recovery value is affected by z. A
one standard deviation fall of z below normal reduces recovery values by β
percent. The final term is included in order to capture the possibility of a
market crisis, in which forced selling leads to a much greater decline of
realiseable security. In extreme down turns (z < zc ) there is a stronger
relationship (β + γ instead of β) between z and φ. We will run two versions
of our simulations, one with γ = 0, one with γ > 0.
The final element of our model is the probability of default for an individual
loan µi . This is assumed to be a function of recovery value pi and the state
of macro-economy z, i.e. µ(pi , z). We assume a logistic function of p and z,
of the kind commonly used for studies of bank and corporate failures:
µi =
exp (δ − εz − η ln pi )
1 + exp (δ − εz − η ln pi )
(3)
In order to apply this model we need values for eight parameters σ, α, β, γ,
zc , δ, ε, and η. These parameters are calibrated against the recent
experience of the UK housing market (see the appendix for details).
3
Simulation results
Tables 1 and 2 show the proportion of loans defaulting, average loss given
default, and total losses, in simulations of our model for selected percentiles
of z. Table 1 is the version of our model with no forced selling (γ = 0)
while Table 2 is the version with forced selling (γ = β). These results are
presented together with the unconditional values of the default probability,
loss given default, and expected losses evaluated across the entire
distribution of z.
4
Table 1: simulation results, no forced selling
Percentiles of z
%
0.1
1
z
-3.090 -2.326
PD %
15.78 9.97
LGD %
19.95 15.30
Losses % 3.15
1.53
Mean
10
-1.282
5.10
9.56
0.49
25
-0.675
3.40
6.82
0.23
50
0
2.16
4.38
0.095
R
f(z)dz
2.68
7.43
0.20
Although these simulations are not based on estimated parameters the
outcome are plausible. In a median year the probability of mortgage default
is a little over 2%, losses given default are a little over 4%, and total losses
are just under 0.1% of the mortgage portfolio. One year in ten there are
mortgages losses worth around 0.5% of the mortgage portfolio. One year in
a hundred mortgage losses rise to around 1.5% of the portfolio (a little
below the losses suffered by UK mortgage lenders in 1992).
For assessing capital adquacy we focus on the ‘three standard deviation’
event, of z = −3.090, corresponding to a credit Value-at-Risk evaluated at
the 0.1% threshold i.e. a one-in-a-thousand year outcome. As noted in the
following section, this is exactly the same probability threshold adopted by
the Basel committee for the IRB approach of the new accord. If the model
is reliable, then this would represent a considerable margin of safety for
lenders, but a wide prudential margin is appropriate in order to allow for
model uncertainties. In this case, with no correction for the model to
capture forced selling, nearly 16% of mortgages default and, with loss given
default of 20%, loan losses are a little over 3% of the mortgage portfolio.
Table 2 differs from Table 1, by allowing for a more pronounced decline of
house prices, and hence of defaults, for values of the macro-disturbance z
below -1 (i.e. it presents results for the case γ = 1). We label this as ‘forced
selling’, since in extreme downturns we would expect distressed house sales
and consequently more severe declines of prices. The magnitude of this
effect is arbitrary, but it would be mistaken to ignore it altogether.
Table 2: simulation results, forced selling
%
z
PD %
LGD %
Losses %
Percentiles of z
0.1
1
-3.090 -2.326
21.98 12.57
33.70 23.97
7.41
3.01
Mean
10
-1.282
5.38
11.05
0.59
5
25
-0.675
3.40
6.82
0.23
50
0
2.16
4.38
0.095
R
f(z)dz
2.81
9.51
0.27
The effect of this forced selling becomes relatively large for extreme
negative values of z. Now, one year in a hundred mortgage losses rise to
just over 3% of the portfolio. In the ‘three standard deviation’ event, of
z = −3.090, nearly 22% of mortgages default and, with loss given default of
33%, loan losses are nearly 7 12 % of the mortgage portfolio.
4
4.1
IRB versus credit VaR calculations
The Basel mortgage risk-curve
The IRB calculations of capital for mortgage exposures are described on
pages 59-60 of the third Basel committee consultation on the new Basel
accord (Basel Committee 2003). We quote the relevant paragraph in full:
“(i) Residential Mortgage Exposures.
298.
For exposures secured or partly secured by residential mortgages as
defined in paragraph 199, risk weights will be assigned based on the
following formula:
Correlation (R)= 0.15
Capital requirement (K) = LGD × N[(1-R)^-0.5 × G(PD) +
(R/(1-R))^0.5×G(0.999)]
Risk-weighted assets = K ×12.50 × EAD ”
In this quotation EAD refers to exposure at default, G() is the inverse of
the standard normal distribution, N[] is the standard normal distribution,
and LGD is the fixed and imposed loss given default, fixed either by the
regulator (foundation IRB) or estimated by the bank itself (advanced IRB).
PD is always estimated by the bank itself as the PD associated with the
internal borrower grade to which the mortgage has been assigned, subject
(Basel Committee (2003) paragraph 302) to being at least 0.03%.
This ‘risk-curve’, like the other risk-curves used for computing the IRB
capital requirements, assumes that loan default is triggered when the value
of an underlying asset fall below some default boundary. The risk-curve
describes the 0.1% tail of aggregate losses, for the case of a loan portfolio in
which the underlying assets are all jointly normally distributed with a
6
correlation coefficient of R.(1) Loss given default is a constant across all
exposures. The 0.1% tail is reflected in the use of the coefficient 0.999 in
the risk curve.
The rationale for this risk-curve is best understood by considering some
special cases. In the special case where loan defaults are uncorrelated
(R = 0) aggregate losses are completely predictable (there is no variability
of loss from year to year) and K = LGD × P D. In the special case where
loan defaults are perfectly positively correlated (R = 1) then all loans
default with a probability of P D. In this case K = LGD (assuming
P D > 0.1%) i.e. enough capital must be held to cover the possibility of all
loans defaulting at once. For 0 < R < 1 the risk curve yields a value of K
satisfying P D × LGD < K < LGD.
4.2
The comparison with CVaR
In this sub-section we compare the Basel mortgage risk curve measure of
prudential capital with the calculations of credit value-at-risk (CVaR)
emerging from our simulation model. In order to obtain the appropriate PD
and LGD inputs for the IRB mortgage risk curve, and thus ensure
comparability with our own model, we use unconditional expected PD and
expected LGD as calculated from our own simulation model. These are
input into the Basel risk curve, to yield a model-consistent ‘through the
cycle’ measure of IRB capital adequacy. Both the IRB calculation and our
measure of CVaR are calculated for the 0.1% threshold of the loss
distribution over a one year holding period.
We compute a credit value-at-risk (CVaR) from the 0.1% thresholds of the
loss distribution reported in Tables 1 and 2. Our CVaRs are 1% less than
level of losses recorded in Tables 1 and 2. This deduction reflects our
assumption that expected net income of 1% of mortgage lending is available
as a first protection against default. In order for a mortgage bank to avoid
failure, it needs to hold equity capital that is 1% less than the amount of
credit losses covered.
(1)
See Gordy (2000) for formal discussion. This assumption would apply if all loan
defaults were determined according to the Merton (1974) model of corporate default. It
also applies in a number of other models of default.
7
Table 3: IRB versus credit VaR capital requirements
Simulation
No forced selling
No forced selling
Forced selling
Forced selling
Basel 1988
PD
LGD
RWA K
IRB
2.68 7.43 19.8
CVaR
19.95 39.4
IRB
2.81 9.51 26.2
CVaR
33.70 92.6
50.0
1.58
2.15
2.09
6.41
4.00
As indicated in Table 3, the IRB calulations can result in very much lower
capital charges than a comparable calculation of credit value-at-risk. We
compute the IRB capital requirements using the unconditional values of PD
and LGD from the two simulations (Table 1, no forced selling, and Table 2,
forced selling). These IRB calculations match relatively well to those
emerging from the Basel committee’s quantitative impact studies, working
out around half or less of those required under the 1988 Basel accord.
With no forced selling, our CVaR calculation is around one-third greater
than the corresponding IRB calculation. Allowing for the possibility of
forced selling, substantially increases CVaR, so while the model consistent
IRB is increased (it is now computed using the unconditional PD and LGD
of our model with forced selling), CVaR is increased much more and is now
some three times as great as the IRB and around half as high again as the
capital requirement for mortgage lending in the 1988 Basel accord.
Both models, our simulation model and the model underlying the IRB risk
curve are highly schematic. We do not go so far as to claim that our model
captures the “correct” level of prudential capital for a portfolio of UK
mortgages. But our calculations do show that is perfectly conceivable that
the IRB levels of capital may prove insufficient to protect bank solvency in
the event of a extreme negative shock to mortgage markets. The question of
how much prudential regulatory capital is required for solvency protection
in extreme circumstances is a difficult professional judgement, one that
cannot be reduced to a simple formula, however good the quality of bank
data. It is not possible to entirely rule out the possibility of a decline in
house prices or a rise in defaults wholly out line with previous experience.
Our analysis does suggest that supervisors would be well to advised to
apply pillar 2 of the new accord, increasing prudential capital requirements
for mortgage lending well above those emerging from the IRB approach.
8
5
Capital adequacy and the pricing of bank
exposures
The application of the IRB approach to calculating regulatory capital
requirements for mortgage lending has been welcomed by practitioners,
because it results in regulatory capital requirements that are mujch closer
to their own calculations of economic capital than the 50% risk weighting of
the 1988 accord. Raising capital requirements for mortgage lending above
the IRB levels will meet with the objection that this will increase the cost
of conducting mortgage business and lead to substantial increases mortgage
interest rates. But how large an impact would there be on mortgage
pricing? The key issue here, one that has received scant previous attention
from either academics or practitioners, is the appropriate pricing of
regulatory capital requirements.
This section explores this issue. Applying standard tools of corporate
finance, it turns out the pricing impact of regulatory capital requirements
are relatively small. This in turn suggests that practitioner objections to
discrepancies between measures of regulatory and economic capital are
overstated, exagerrating the impact on supply and pricing for all categories
of lending.
The arguments here are based on Lawrence and Milne (2003), who review
the concept of economic capital in the context of standard corporate
finance theory. Under the special circumstances assumed by Modigliani and
Miller (1958) (absence of taxes and no capital market frictions) an increase
in regulatory capital requirements has no impact on loan pricing or loan
supply. In practice regulatory capital requirements will have impacts on
loan pricing and loan supply. These can be quantified by taking specific
account of all potential departures from the Modigliani-Miller assumptions.
Three such departures are relevant, all of which might affect loan pricing:
1. Taxation. Interest payments on debt, including subordinated debt
which is counted as a component of “tier 2” regulatory capital, are
deducted from bank income before tax. Dividends are paid after tax.
Substitution of shareholder equity for any of these forms of debt thus
increases taxable income and tax paid, imposing a cost equal to the
corporate tax rate times the interest paid on the debt, and hence
reducing the value of the bank to shareholders. This tax impact will
increase the price of mortgage lending.
9
2. ‘Agency costs’ of equity. Increasing the share of equity on the balance
sheet increases the amount of cash flow under management control. A
demonstrated in standard agency cost models, such as that of and
Myers (1973), this gives management greater freedom to pursue
objectives conflicting with those of shareholders. Three points can be
made here. Firstly such a conflict between the objectives of managers
and shareholders will be reflected in a higher cost of equity capital
relative to debt. Second, management objectives may include the
expansion of the volume of loans, beyond the level that would be
desired by shareholders, in which case a rise in the share of equity
finance has an ambiguous impact on the supply and cost of bank
lending (increased willingness to lend offsetting the higher cost of
funding; to our knowledge there is no formal model of this mechanism
in the literature). Third, as Brealey (2001) argues, regulatory capital
is, from an agency perspective, more like debt than equity, since it
also imposes a discipline on management behaviour. Brealey argues
that there will be no agency costs of regulatory capital at all.
3. Deadweight costs of financial distress or liquidation. There is a loss of
value in the event that a bank or company is liquidated, or comes so
close to liquidation that its credit standing is seriously damaged. An
increase in regulatory capital requirements will reduce the probability
of liquidation or financial distress and increase the value of the bank,
and be reflected in a lower loan price ie it operates in the opposite
direction to the tax impact. In order to quantify the impact on loan
pricing of higher regulatory capital requirements, we price these
deadweight costs as a ‘digital’ option, allowing for: (i) changes in the
probability of financial distress or liquidation; (ii) the magnitude of
deadweight costs; (iii) the valuation of returns in outcomes where
default is triggered.
Our specific assumptions for calculating the price impact of regulatory
capital are as follows. To capture the impact of taxation we assume that
half of the additional capital requirement is met through an increase in
equity capital, a corporate tax rate applied to bank earnings of τ = 0.333
and that there is a market rate of interest on debt finance of 6%. The cost
of financial distress amounts to a loss of 20% of the present value of all
future expected bank revenues. Agency costs are ignored altogether (since
the direction of their impact is ambiguous and the applicability of agency
theory to regulatory capital is doubtful.) For a first comparison we also
ignore the impact of reduced probability of financial distress or liquidation.
10
Consider the impact on mortgage pricing of increasing the regulatory
capital requirement for the financing mortgage lending from the IRB
requirement of 2.1% of total exposure to our own Credit VaR calculation of
6.4% (the comparison in the the final column of Table 3 for the case with
forced selling). We assume that half the additional capital requirement is
tier 1 (equity capital) and half is tier 2 (subordinated debt.) The following
tabulation shows the tax impact on the price of mortgage lending:
Table 4: tax impact of increased capital requirement
Mortgage assets
Capital requirement
Capital requirement
∆Capital requirement
∆Debt
∆Pre-tax income
∆Tax
IRB
CVaR
(3) − (2)
−(4)/2
(5)×6%
(6)×0.333
(1)
(2)
(3)
(4)
(5)
(6)
(7)
100
2.09
6.41
4.32
−2.16
0.1296
0.0432
As indicated in this table, the tax impact is worth 6%
×0.333 × (7.41 − 2.09%)/2 or 4.3 basis points of total lending. This
increase in the cost of mortgage lending is not large. The extent to which
this increase is passed on to mortgage borrowers will depend also upon the
aggregate long-run interest elasticity of the demand for mortgage loans.
Overall we conclude that if capital requirements for UK mortgage lending
were based on our credit-VaR calculations in the case of forced selling,
resulting in capital requirements about three times as large as those
required when applying the IRB mortgage risk-curve, then the mortgage
interest rate spreads would increase by no more than 4 basis points.
The calculations presented so far make no allowance for the reduced
probability of liquidation or financial distress resulting from operating with
a higher ratio of prudential capital relative to debt. This reduction will
offset the tax impact and reduce the impact on mortgage spreads to below
4 basis points.
In order to calculate this second impact we assume we are dealing with a
specialised mortgage bank that holds only mortgages and no other assets,
i.e. we ignore the possibility of diversification against other bank assets. We
continue to assume net interest revenue, net of all costs other than loan loss
provisions, of 1%. As shown in Table 2, in the case of forced selling
11
expected loans losses are 0.27%, so expected return on assets are 0.73%.
We will assume that the deadweight costs of financial liquidation amount to
20% of the net present value of future bank returns and that the bank
always operates with the regulatory minimum level of capital.
We can then complete the calculation in one of two ways. The first is
assuming investor risk-neutrality. Suppose there is a nominal growth of
bank assets of 4% per annum. Then, applying the assumed safe rate of
interest of 6%,the present value of current and future bank returns for
risk-neutral investors amounts to 0.73%/ (6% − 4%) = 36.5% of bank
assets. This in turn implies that the deadweight cost of financial distress is
then 20% × 36.5% = 7.3% of bank assets.
The valuation impact of the increase in regulatory capital is then simply
this deadweight loss times the reduction in the probability of failure
resulting from the higher capital requirement. As indicated by Table 2, this
probability falls from 2% to 0.1%, indicating an increase in bank value of
1.9% of 7.3% or 14 basis points. This is equivalent to a narrowing of
mortgage spreads of 14/50 or about one third of a basis point of total
lending.
The assumption of investor risk-neutrality understates the impact of
reduced probability of liquidation or financial distress, since the event of
default, and hence the deadweight loss, occurs in periods when aggregate
returns are low. A better approach is to to assume risk-averse investors and
apply risk-neutral probabilities for the distribution of the stochastic
variable z. The second appendix to this paper describes such a calculation.
This suggests that the impact of reduced probability of liquidation or
financial distress should be increased to between [] and [] basis points, still
relatively small in comparison to the tax impact. Overall we conclude that
the impact of our recommended tripling of the capital requirement for
mortgage lending will be an increase in mortgage lending rates of between 2
and 4 basis points.
6
Summary and conclusions
This paper has examined the appropriateness of the Basel IRB calculations
for computing prudential capital adequacy, in a situation where loan returns
exhibit extreme skewness. Our results, based on a calibrated model for UK
mortgage lending, suggest that ignoring the correlation of default and loss
12
given default leads to a substantial downward bias in IRB calculations of
prudential capital adequacy. We suggest that in order to correct this bias
supervisors should utilise pillar 2 of the accord to raise capital requirements
for mortgage lending above those emerging from the IRB calculations.
Such a response would reverse one of the biggest impact of the new accord
on regulatory bank capital. Reduced capital requirements for mortgage
lending are quantitatively much the largest impact of pillar 1 of the new
accord. The big ‘winners’ (in terms of reduced regulatory capital
requirements) from the new accord are banks with substantial mortgage
business. If our suggestion is followed, then action taken under pillar 2 of
the new accord will offset much of the impact of piller 1.
Would this matter? The Basel committee has encouraged banks to collect
databases of loan peformance history, by allowing banks to use these data
bases to compute regulatory capital requirements and hence to achieve a
closer alignment between regulatory and economic capital. Greater use of
data to manage and price bank exposures is welcome, but the use of these
databases to compute regulatory capital is not so obviously desirable as is
often supposed. Economic and regulatory capital measure different things.
Economic capital is calculated as a measure of the variability of returns
over the past observed distribution of loan performance. This is a sensible
thing to do. Economic capital can be used to support pricing and
performance measurement across a variety of bank products and business
lines. Requiring a minimum required return upon economic capital imposes
an appropriate discipline, encouraging decisions that create value for bank
shareholders. But economic capital calculations do not capture tail risks,
including increases in loss-given-default such as we model in this paper, and
also the evaporation of liquidity in extreme market situations.
Regulatory capital requirements, in contrast, are specifically designed to
absorb the extreme downside tail of loan returns. Because tail events such
as a deep recession are rare, regulatory capital can never be estimated with
any accuracy or reliability. Regulators should presumably play safe, and set
requirements that are clearly high relative to the outcome of statistical
modelling. Their judgment can be supplemented by simulations, such as
the ones reported in this paper, designed to take account of major
macro-economic fluctuations. But the setting of regulatory capital, unlike
economic capital, will ultimately always rest on subjective judgement.
It may seem that Basel committee could deal with our specific criticism of
IRB calculations, by introducing an allowance for correlation between LGD
13
and PD. This is unlikely to work because the correlation we explore is
evident only in the tail of the distribution of outcomes. It would take
several hundred years worth of back data to obtain statistically precise
estimates of this correlation. Even then it would be unclear how relevant
these estimates would be to describing future risk exposures.
For these reasons appropriate regulatory capital requirements can often be
larger, possibly much larger than economic capital. Our simulations suggest
that this discrepancy will be more pronounced, where correlation between
the probability of default and loss given default results in a return
distribution with a highly pronounced downward tail.
Does this discrepancy matter? Banks will object that raising regulatory
capital above economic capital requirements reduces the supply of loans
and increases the cost of borrowing. For practitioners reduced levels of
regulatory capital, for certain safer banking exposures, are the principal
benefit of the new accord. They will not want to see this taken away.
On this point we echo Hamlet, believing that “The lady doth protest too
much.” The calculations of Section 5 reveal that even for the tripling of
regulatory capital suggested by our simulation model, the consequence is an
increase in bank mortgage lending rates of only between 2 and 4 basis
points. These calculations are based on standard arguments of corporate
fianance arguments. These indicates that the primary economic impact of
altering regulatory capital is the tax impact, and even this small impact is
offset by reduced probability of financial distress. Provided increases in
capital requirements are introduced gradually and with good notice,
regulators can safely set their capital requirements in excess of economic
capital, for mortgages or for other exposures, without imposing material
costs on the banking industry or its customers.
Regulators could even neutralise the tax implications of a discrepancy
between economic and regulatory capital, by allowing a larger proportion of
total regulatory capital requirements to be satisfied through the issue of
subordinated debt (at present subordinated debt qualifies only as ‘tier 2’
regulatory capital and the qualifying amount of subordinated debt is
limited to only 50% of ‘tier 1’ capital).
Our analysis has emphasised the difficulties of estimating tail risks and the
low compliance costs of imposing regulatory capital requirements that
exceed economic capital. This does not mean that we oppose the greater
use of bank loan data bases.These are welcome, not for calculating
14
regulatory capital, but because collection and analysis of loan data allows
lending decisions to be assessed against the past performance of similar
exposures. The widespread acceptance of this discipline, especially within a
culture of promoting shareholder value, will improve bank governance,
safety, and soundness. With the benefit of hindsight, this, rather than any
of the complicated recommendations for calculating capital adequacy, may
be understood as the principal achievement of the new accord.
7
7.1
Appendix
Parameter calibration
Altogether there are eight parameters in this model, to be determined
before this model can be simulated. These parameters are α, β, γ, δ, ε,
η,zc , σ.
The following table shows house prices fluctuations, mortgage defaults, and
mortgage loans losses in the UK, from 1986-2002.
In a ‘median’ year, ie when z = 0, the distribution of losses is determined
by a sub-set of these parameters: α, δ, η, σ. These are chosen to match
average probabilities of default, and the distribution of loss given default
for the UK.
The expected security, in a median year, is φ|z=0 = α. We impose α = 1.15.
The median default rate ... (choose δ̇ so this is 2%). η should capture
observed correlation of default and loss given default. Choose σ to capture
the observed distribution of loss given default.
Two parameters β and ε capture the business cycle variability of defaults
and of loss given default. These are calibrated against the experience of the
UK housing market in the early 1990s.
This leaves two parameters, introduced to capture the impact of a potential
market crisis. For our baseline we will assume, no crisis, γ = 0, so zc is not
relevant. For the alternative simulation with forced selling we set γ = β,
zc = −1.
15
7.2
Solving the model
Solution is straightforward. Matlab coding is available from the authors on
request. We first obtain from (1) and (2):
ln pi ∼ ln φ(z) + σw
where w has ths standard normal distribution f(w) and from (3):
µi = µ(z, pi )
We then obtain PD from the integration:
Z +∞
exp −w 2
dw
PD(z) =
µ(z, p(w)) √
2π
−∞
and expected losses from the integration:
EL(z) =
Z
+∞
−∞
exp −w2
dw
µ(z, p(w)) max(0, 1 − p(w)) √
2π
Average loss given default (LGD(z)) is recovered from EL=PD×LGD
Finally, in order to obtain the unconditional values for the probability of
default and expected losses we integrate across the entire distribution of z,
using:
Z
+∞
PD(z)f(z)dz
PD =
−∞
and
EL =
8
Z
+∞
EL(z)f (z)dz
−∞
Appendix: valuing the deadweight costs
of bank failure with risk-averse investors
This appendix corrects the valuation of the deadweight costs of bank failure
based on the assumption of investor risk-neutrality given in Section 5 of the
main text. It instead assumes investor risk-aversion, and develops an
16
alternative calculation using risk-neutral probabilities to value the impact
of capital requirements on the probability of bank failure. All other
assumptions are exactly as in the main text.
We have no comparable traded assets on which to base this calculation. We
therefore appeal to the ‘consumption CAPM’, assuming that aggregate
per-capital consumption is also driven by the macroeconomic disturbance z:
ln c = ln c̄ + κz
and that investor preferences, and hence investment valuations, can be
represented as the choices of an expected utility maximising representative
consumer with an iso-elastic expected utility function:
1 1−λ
c
1−λ
where λ is the representative consumer’s coefficient of relative risk aversion
and ν is the per period discount rate. This implies that the future value of
returns for given z (the marginal utility of consumption evaluated at the
time consumption) is given by:
c(z)−λ = c̄−λ exp (κz)−λ
In order to value the impact of a change in capital on we need to choose
values for the parameters κ, λ. First note that the standard deviation of
aggregate real consumption in the UK, around trend levels, is around 1%,
so we can assume κ = 0.1. We will consider alternative values of λ of 5, 10,
and 20.
Uncertainty is To determine ν we note that in steady state growth with
trend consumption c̄ growing at 4% we will require that c̄−λ
= (1 + r)νc̄−λ
t
t+1
implying that ν = (c̄t+1 /c̄t+1 )λ /(1 + r) = (1.04)λ /(1.06) ≈ 1.
There are then two steps to the valuation. First we must evaluate the
present value of future mortgage returns. The future value of a single
period of mortgage returns is given by:
Z +∞
π=
[1% − EL(z)] c̄−λ exp (κz)−λ f(z)dz
−∞
On a perpetual basis these have a present value of π/ (6% − 4%)
(uncertainty or returns are independent from period to period, so we can
17
apply the same formula as for risk-neutrality). Deadweight losses, in the
event of a bank default, are then 20% of this this present value i.e.
0.2 × π/ (6% − 4%)
In order to value the increased capital requirement, we must comparte the
expected per period deadweight losses, under the IRB and Credit VaR
levels of regulatoryu capital. Refer to the corresponding values of z as zIRB
and zCVaR respectively. For z < zCVaR the bank will default for both levels
of regulatory capital. For z > zIRB the bank will not default, for both levels
of regulatory capital. The gain to the bank, of the higher capital
requjirement, in terms of reduced probabilty of default, is then given by:
Z
zIR B
[0.2 × π/ (6% − 4%)] c̄−λ exp (κz)−λ f(z)dz
zC VaR
Z zIR B
−λ
= [0.2 × π/ (6% − 4%)] c̄
exp (κz)−λ f(z)dz
ω =
zC Va R
Using Matlab to carry out the necessary integrations, we find that π = []
and ω = [].
References
Basel Committee (2003)
Brealey (2001)
Gordy (2000)
Lawrence, Colin and Alistair Milne (2003)
Merton (1974)
18