Portfolio Credit Risk Modelling:
A Comparative Study of Industry Frameworks
Tushar Gupta
Abstract
Portfolio Credit Risk Management has matured significantly over the past couple of
decades since the tools such as CreditMetrics and CreditRisk+ set the ball rolling back in
1997. This paper examines the framework of a new entrant - ACPM (Active Credit
Portfolio Management), the portfolio credit risk framework being implemented at Royal
Bank of Scotland vis-à-vis the widely-used industry standard frameworks – JP Morgan
Chase’s CreditMetrics, and Credit Suisse’s CreditRisk+. Whilst these competing models
follow different distributional assumptions, use different calculation methodologies, and
cover different scope; they still attempt to address the same problem with a forward
looking approach, and resulting in similar outputs.
At first glance, the ACPM being a ‘Default – No-default’ model looks more closely aligned
to CreditRisk+, which is more akin to an Insurance model that captures the event risk;
event in this case being the likelihood of default. JP Morgan’s CreditMetrics on the other
hand also captures the portfolio risk in case of credit migrations; i.e. upgrades or
downgrades in credit quality in addition to the ‘default – no-default’ scenario. A deeper
look at the implementation approaches of the frameworks, however, suggests that there
is more to it.
1
Introduction
Credit risk is the risk of loss arising from counterparty’s inability to meet its obligations.
Banks and Financial Institutions encounter credit risk with products such as outstanding
loans and leases, trading instruments, derivatives and unfunded lending commitments that
include loan commitments, letters of credit and financial guarantees.
Over the past couple of decades, credit markets have seen an unprecedented growth in
terms of both size as well as complexity of the products offered. There has been a greater
thrust towards investing in risky high-yield markets and resorting to complex credit
derivatives for risk mitigation. This relatively aggressive outlook has caused a significant
increase in the corporate defaults in the recent times. The increasing number of defaults has
also triggered further regulatory pressures being enforced on the banks to ensure that they
have enough capital to cover the risks they are exposed to. All these advancements have
encouraged leading banks and financial institutions to develop sophisticated portfolio credit
risk models that calculate the credit risk losses and the economic capital needed to be held
by the banks at aggregate portfolio level. Going a step further, these models also help the
businesses grow by providing the ability to carry out portfolio optimization, profitability
analysis in case of new additions to the portfolio, assigning portfolio credit limits, and
scenario analysis.
A portfolio credit risk model primarily captures the correlations and joint likelihood of default
between assets within a portfolio and estimate the potential credit risk losses at overall
portfolio level. Historical time series data for obligors in the portfolio is used to calculate the
asset correlations and joint default probabilities, which are in turn used to estimate the risk
contributions and profitability that a new deal/obligor brings to a portfolio. The model
essentially generates a loss distribution for the portfolio, and estimates the Portfolio VaR and
Economic Capital, which serve as the main outputs of a portfolio credit risk model.
This paper examines the implementation framework of a new entrant - ACPM (Active Credit
Portfolio Management), the portfolio credit risk framework at Royal Bank of Scotland vis-àvis the widely-used industry standard frameworks – JPMC’s CreditMetrics, and CSFP’s
CreditRisk+.
I. CreditMetrics
2
JPMC’s CreditMetrics is seen as the most widely accepted benchmark towards estimating
portfolio level credit risk. It was originally developed in 1997 by JP Morgan’s Risk
Management Research division, and subsequently spun-off to Risk Metrics Inc.
It is based on the analysis of credit migration and estimates the likelihood of an obligor to
move from one credit rating to another, including default, within a one year horizon. It uses
past credit migration data of obligors to estimate the probability of joint migrations for
obligors in a portfolio, allowing the bank to estimate the forward distribution of values of a
loan portfolio. These are in turn used as inputs to calculate portfolio risk drivers including
Marginal Risk, Value at Risk, and ascertain Economic Capital requirements.
II. CreditRisk+
Developed by Credit Suisse and launched in 1997, the CreditRisk+ methodology has
attracted much attention from practitioners, academics and the regulatory community due to
its fast analytical approach. Unlike CreditMetrics, it is not a commercial solution; instead, it is
implemented by various users in their own way. The model employs an analytical method to
derive the distribution of losses in the event of default. Both portfolio-level risk and the
individual contributions of portfolio assets to this risk are calculated.
III. ACPM
Active Credit Portfolio Management (ACPM) is the in-house application of Royal Bank of
Scotland, developed in collaboration with KPMG; and is the newest child on the block. The
model uses a mix of Analytical and Simulation approaches for calculation of risk drivers. The
model currently supports only corporate loan products at the bank, and is expected to be
extended to other asset classes going forward.
Working Assumptions
i. Since the current roll-out of ACPM framework supports corporate lending products only;
for the purpose of comparative analysis, this paper would restrict itself to lending
products only.
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ii. This study is based on the original versions of the models under study and refers to
their respective official publications∗.
Calibrating Asset Value Model
The AV model proposed by Robert C. Merton in 1974 has become the corner stone for
estimating portfolio credit risk. Model recognizes that the underlying value of the firm and its
volatility are unobservable variables. These unknown variables can however be inferred from
value of firm’s equity and the outstanding liabilities (both observable variables).
Merton Model characterizes company's equity as a call option on its assets. The company
defaults if the value of its assets is less than the promised debt repayment at time T. The
equity of the company is a European call option on the assets of that company with maturity
T and a strike price equal to the face value of the debt. Put-call parity is then used to price
the value of a put and this is treated as an analogous representation of the firm's credit risk.
Since according to this model, a firm defaults if its asset value falls below a critical threshold
defined by the value of liabilities; asset value correlations translate into correlations of credit
quality changes. This model is therefore used by various industry portfolio credit risk
frameworks to model credit quality correlations and in turn estimate the joint default
probabilities of assets within a portfolio.
The model takes equity spot price, equity volatility, and outstanding liabilities as inputs,
which are transformed into asset volatility. It also takes two other inputs: the default barrier,
and the volatility of the default barrier. These inputs are used to specify a diffusion process
for the asset value. The entity is deemed to have defaulted when the asset value drops
below the barrier.
Application of Merton model to fit a ‘default – no default’ scenario is straightforward as
required in case of ACPM, as the later calculates the portfolio risk and profitability measures
only in the event of default. The default barrier indicates the underlying asset value from the
distribution of asset returns at or lesser than which, the firm will default. In case of a multiasset portfolio, the default barrier is captured for a joint distribution of asset returns for the
multi-asset portfolio.
∗
These risk models may be available in different versions, tweaked to the specific ways and requirements of the
users.
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For a single asset portfolio, Probability that the firm will default can be denoted as:
Pd = P{R < Z d } = φ ( Z d / σ )
[1]
Where, φ is cumulative probability density function
Similarly, in case of a two-asset portfolio, the probability the both firms defaulting will be:
P{R < Z d , R' < Z ' d } = ∫
Z Def
−α
Where,
∑
Z ' Def
∫α
−
φ (r , r ' ; ∑ )(dr ' )dr
[2]
is covariance matrix for two asset portfolio, and can be denoted in terms of
correlation coefficient ρ as:
 σ2
∑ =  ρσσ '

ρσσ '

σ '2 
[3]
The estimates of correlation coefficient (or scaled betas) can be calculated by regressing
asset-returns of obligors against the systematic risk factors.
While ACPM is a straightforward application of Merton model in estimating the default barrier,
JP Morgan’s CreditMetrics is a more complex extension of it. Unlike the peers, CreditMetrics
also captures the portfolio credit risk in the events of credit migrations, i.e. Up/downgrades in
the credit ratings, and not just in the event of default. Hence for Credit Metrics, default is one
of the possible states along with credit migrations.
CreditMetrics assumes that there are a set of barrier levels for the asset value that will
determine a company’s credit rating at the end of the period in question. Hence there will be
multiple barriers in this case, each corresponding to a credit rating change.
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Here, the default barrier for a loan is denoted by Zd, and can be calculated in a similar
manner as illustrated Eqn [1] & [2].
Additionally, the barriers corresponding to credit downgrades and the associated
probabilities can be denoted as follows:
For a single asset portfolio,
PCCC = P{Z d < R < Z CCC } = φ ( Z CCC / σ ) − φ ( Z d / σ )
[4]
Where, ZCCC is the the barrier corresponding to a credit downgrade to CCC rating; PCCC is
the probability of downgrade to CCC rating
For a two-asset portfolio,
Pj CCC = P{Z d < R < Z CCC , Z ' d < R ' < Z ' CCC } =
∫
Z CCC
Zd
∫
Z 'CCC
Z 'd
φ ( r , r ' ; ∑ )( dr ' ) dr
[5]
Where, PjCCC is the joint probability density for both loans to downgrade to CCC rating.
In case of a multi-asset portfolio, joint probability densities can be calculated for each two
asset-pair within a portfolio in a similar fashion. These joint probabilities for two-asset
portfolio can be used to calculate the portfolio standard deviation using following relationship
(Please refer Appendix II - Mathematical Framework, for more details):
σ p2 = σ 2 (V1 + V2 ) + σ 2 (V1 + V3 ) + σ 2 (V2 + V3 ) − σ 2 (V1 ) − σ 2 (V2 ) − σ 2 (V3 )
[6]
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While the multi-state model in CreditMetrics provides a more comprehensive view of risk
associated with credit quality migrations, it brings its own share of complexities with it:
a) Storing huge volumes of data
b) Longer computation time
Treatment of Default Probability
A key difference in the methodology among these frameworks is in the modelling of the oneyear default probability distribution function. In CreditMetrics and ACPM, default/migration
probability is modelled as a discrete value based on historical data. CreditRisk+ on the other
hand assumes PD to be a variable, following a Poisson distribution around some mean
default rate. The mean default rate is modelled as a variable with a gamma distribution*.
Poisson distribution is used to calculate the probability when a given number of events take
place during a specific period of time. It is particularly useful in situations where probability of
an event occurring is low and there are a large number of debtors. Hence, it appears to be a
better approximation of default probabilities than the normal distribution for estimating the
frequency of default events.
Treatment of Recovery Rates
ACPM and CreditMetrics allows for recoveries to be variable as opposed to a set of constant
quantities in case of CreditRisk+. Recovery rate displays a variance in values making itself a
contributor of the risk. Hence to model this variation in the default values of the portfolio
assets, ACPM/CreditMetrics use Monte Carlo simulations to generate portfolio loss
distribution and simulate the values for recovery rates assuming they follow beta-distribution#.
The resulting recovery rates are used to calculate the value of the portfolio assets in the
event of a default.
By contrast, under CreditRisk+, loss severities are categorised into various buckets and
allocated into sub-portfolios, and the loss severity in any sub-portfolio is considered constant
over time. This essentially means that in case of default, the value of a loan asset in a
*
In Bayesian inference, the Gamma (α,β) distribution is the conjugate to the Poisson likelihood function, which
makes it a useful distribution to describe the uncertainty about the Poisson mean.
# Monte Carlo simulation used for generating the loss distribution, assumes that the recovery rates are distributed
according to a beta distribution
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portfolio will always be the same for a particular issuer; while in ACPM/CreditMetrics; the
values can be different due to possible variance in the recovery rates.
Correlation of Credit Events
While Credit Metrics largely rely on the Merton’s Asset Value model, it uses equity
correlations to calculate the correlations of credit events within the assets of a portfolio. This
is not completely in line with the Merton model that advocates usage of asset correlations to
model the correlations between portfolio assets. ACPM on the contrary constructs a timeseries of historical asset values from the past equity and liabilities data. Daily asset-returns
are then computed from historical asset values, which are used to calculate the portfolio
asset correlations.
While equity prices and hence equity returns are more readily available, it has been
suggested by previous studies, that in general, equity correlations are poor proxies for asset
correlations in credit risk calculations, (demonstrated by Zeng and Zhang (2001)). Hence
ACPM uses a more observable metric for assessment of default correlations between
portfolio assets.
Modelling of Concentration Risk
CreditMetrics models the concentration risk by sector allocation, and derives the borrower
correlation from sector correlations. Asset returns are simulated and mapped to credit
migrations. CreditRisk+ also allocates each position to the sectors that represent countries
and industries; but assumes that sectors are conditionally assumed to be independent, and
only one individual sector can be used for modelling the risk of positions. Unfortunately,
there can be realistic scenarios where correlations between sectors do exist. This may be
marked as a key downside of CreditRisk+, since it fails to capture sector correlations.
ACPM models concentration risk considering the industries and regions as nodes of a tree
structure, and individual obligors at the bottom nodes of the tree. The correlations between
any two nodes can be captured by this approach, providing more accurate and realistic
estimations of correlations across obligors, sectors and regions. ACPM uses ‘beta’
sensitivities (along with R-squared value) for estimation of correlations between asset
returns by regressing the asset value time-series of an obligor against the systematic risk
factor. These betas are then scaled to calculate the correlation coefficients between the
obligors. In terms of correlation coefficient it can be denoted as:
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β =ρ
σ1
σ2
[7]
Where, σ 1 and σ 2 are the standard deviations of asset returns for an obligor and the
systematic risk factor, against which the obligor is being regressed, respectively.
The comparison of approaches to capture concentration risk by competing frameworks
suggest that the tree-based regression model used by ACPM is a simpler and yet intuitive
and powerful approach when compared to its peers.
An interesting case worth examining here would be the situation where a company belongs
to more than one sector and country. For example – an Obligor with a 40% presence in U.S.,
and 60% presence in U.K., and diversified with 70% market capitalisation in Oil & Gas, and
30% in Retail (a typical example could be BP). Thanks to the intuitive approach used by
ACPM, the latter need not implement a specialized treatment for such cases, since the
impact of each sector-geography on the obligor is automatically captured via the userdefined tree-based factor model and regression schemes, resulting in required beta
sensitivities more realistically and inherently. CreditMetrics, on the other hand, needs to
address such instances by assigning industry-region weights to the obligor. For example, for
the same illustration above, where an Obligor has 40% presence in U.S., and 60% in U.K.,
and diversified with 70% market cap in O&G, and 30% in Retail, it would translate in 28%
participation in U.S. Oil & Gas industry, 12% in U.S. Retail, 42% in U.K. O&G, and 18% in
U.K. Retail industry.
It is this tree-based factor model that provides ACPM with the added flexibility to carry out
scenario analysis in any state of the world and in any possible situation - be it assessing the
impact of an obligor on industry sector, obligor on region, sector on region, sector on world
factor, or region on world factor. With all possible permutations of betas pre-calculated from
the tree model, it provides the ability to analyze various real-life scenarios with lower lead
times.
Analytical vs. Simulation Approach
CreditRisk+ employs an analytical approach towards risk estimation as opposed to
simulation-based approach unlike its counterparts. Based on the convenient distributional
assumptions (Poisson distribution for individual loans and the fixed recovery rate assumption
for each sub-portfolio) the model uses analytical solution for estimating the probability
density function of losses. It is this closed-form calculation approach that makes it a lot faster,
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with calculations taking seconds, and not minutes or hours as may be case in simulation
based approaches.
ACPM and CreditMetrics on the other hand, apply Monte Carlo simulations to generate an
approximate aggregate distribution of loan portfolio values, and thus a VaR. Monte Carlo
simulation methodology has longer run-times when compared to analytic solution of
CreditRisk+; however, where it scores considerably is the accuracy of risk estimates in case
of large portfolios with multiple obligors. Monte Carlo is able to model and capture the losses
more effectively in such cases where there are multiple dependencies between large
numbers of obligors in a portfolio.
Evaluation of Marginal Risk
Addition of a new deal in an existing portfolio brings along the change in the risk shape of
the portfolio due to change in concentration or diversification. The concept of marginal risk is
based on applying the marginal change in the reference portfolio’s loss volatility due to the
addition of the new transaction to the portfolio. This marginal risk (or the increase in portfolio
loss volatility) can be equated to an increase in risk capital, which is then assigned to the
new transaction for profitability calculations. This makes it an important parameter from
regulatory perspective as well.
It is therefore obvious that all three frameworks acknowledge the significance of marginal
risk, and calculate the risk contributions for any new addition/deduction to the portfolio.
There is however slight observable difference in the implementation approaches.
CreditMetrics and ACPM define the marginal risk contribution of an asset to the portfolio as
the change in the portfolio’s standard deviation due to addition of a new asset in the portfolio.
CreditRisk+ follows a slightly different approach by defining the marginal risk contribution as
the change in the portfolio standard deviation due to change in the weight of the asset in a
portfolio.
Conclusion
At first glance, the ACPM being a ‘Default – No-default’ model looks more closely aligned to
CreditRisk+, which is more akin to an Insurance model that captures the event risk; event in
this case being the likelihood of default. JP Morgan’s CreditMetrics on the other hand also
captures the portfolio risk in case of credit migrations; i.e. upgrades or downgrades in credit
quality in addition to the ‘default – no-default’ scenario. Hence the foremost conclusion that
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would be of interest to practitioners is that while all three achieve the objective of providing
fair estimates of capital requirements; it is CreditMetrics that becomes the model of choice
when the focus is equally on internal control processes like estimation of reserves and
provisions, where rating migrations play an important role. A deeper look at the
implementation approaches of the frameworks, however, suggests that there is more to it.
ACPM and CreditRisk+ both derive default correlations from asset return correlations;
CreditMetrics however relies on a similar approach, but employ equity return correlations as
a substitute for asset returns; which are a poor proxy for asset correlations in credit risk
calculations.
Credit Risk+ enjoys the advantage of being an attractive choice from a computational point
of view. As discussed in the paper, this is primarily attributable to two reasons – i) an
analytical approach leads to closed-form expressions for the probability of losses making it
computationally a lot more effective as compared to simulation approaches; ii) It focuses on
‘default’ only, and therefore requires relatively fewer estimates, lesser inputs, and lower
storage capacity. For each instrument, only the probability of default and the loss given
default statistics are required. The key disadvantage however is that it ignores Marginal Risk;
i.e. the exposure for each obligor is not sensitive to potential future changes in the credit
quality of the issuer. Furthermore, it may not be as effective a tool when compared to its
peers in calculating the risk for large portfolios with multiple obligors, as simulation
approaches capture the multiple dependencies between large numbers of obligors in a more
accurate manner.
It can be concluded that although the three frameworks under study follow different
distributional assumptions, different calculation methodologies, different scope, and different
approaches (analytical/simulations based); they still end-up addressing the same problem in
a forward looking manner resulting in similar outputs. While CreditMetrics appears to be a
more comprehensive and wholesome solution that assesses the risk for any significant
change in the market variables; CreditRisk+ scores on grounds of computational efficiency
by implementing simpler and much faster analytical calculations for modelling credit risk.
ACPM at the same time seems to emerge as a more effective tool on various other grounds.
It follows a more empirical approach towards using ‘asset’ correlations leading to more
accurate correlation between portfolio assets; a more comprehensive approach towards
capturing industry-geography impact on obligors from the tree structure; and a more intuitive
approach towards calculating scaled regression coefficients for estimating correlations;
making it a simpler, configurable, yet powerful and effective portfolio credit risk framework.
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APPENDIX I: Comparative Summary
Definition of Risk
Credit Events
Model Inputs
Numerical
Approach
Distributional
assumptions (PD)
Correlation of
Credit Events
Recovery Rates
ACPM
Default - No-default
(Actuarial / Eventbased)
Default
Equity Prices,
Liabilities data
(Balance Sheet),
Default Rates (PD),
Credit Exposures
(EAD),
Recovery Rates
(LGD),
Risk-free rates and
Forex rates
Simulation / Analytical
Gaussian probability
distribution∗
Asset-return
correlations
Random (beta
distribution)#
Product Coverage
Model Outputs
∗
Lending Products
Portfolio Loss
Distribution,
Economic Capital,
Portfolio VaR,
Exceedance
Probability,
Expected Loss,
Expected Shortfall,
Marginal Risk
Contributions,
Profitability Metrics
CreditMetrics
CreditRisk+
Default - No-default
(Actuarial / Eventbased)
Credit Migration
(Marked-to-Market )
Credit
Downgrade/Default
Equity Prices,
Obligor Credit
Ratings (Transition
Matrix),
Default Rates (PD),
Credit Exposures
(EAD),
Recovery Rates
(LGD),
Risk-free rates and
Forex rates
Simulation /
Analytical
Analytical
Standard Normal
Equity-return
correlations
Random (beta
distribution)#
Corporate/Retail
Loans, Derivatives,
Traded Bonds,
Letters of Credit &
Commitments
Poisson
Asset-return
correlations
Deterministic; fixed for
each band
Corporate/Retail
Loans, Derivatives,
Traded Bonds, Letters
of Credit &
Commitments
Portfolio Value
Distribution,
Economic Capital,
Portfolio VaR,
Marginal Risk
Contributions,
Migration Risk,
Credit Limits,
Profitability Metrics
Portfolio Loss
Distribution,
Economic Capital,
Portfolio VaR,
Expected Loss,
Expected Shortfall,
Unexpected Loss,
Credit Limits,
Profitability Metrics
Default
Default rates (PD),
Credit Exposures
(EAD),
Recovery Rates
(LGD),
Default Rate Volatility
Risk-free rates and
Forex rates
Merton’s Asset Value model assumes the default probability to follow Gaussian distribution.
#
Monte Carlo simulation (used for simulating loss distribution) assumes that the recovery rates are distributed
according to a beta distribution
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Appendix II: Mathematical Framework
i. Portfolio Standard Deviation and Joint probability densities
i) Standard deviation of a two-asset portfolio can be denoted as:
s
σ =
∑pv
i i
− µ2
..[1]
i =1
Where,
pi = joint probability of a particular combination of states for two-assets in a portfolio
(possible state could be a default or a rating change)
vi = value of two-asset portfolio within each state
s = number of possible states (e.g. for 8 possible ratings in a two-asset portfolio, s = 8*8
= 64.
s
And, Portfolio Mean, µ =
∑pv
i i
i =1
ii) Standard Deviation for a multi-asset portfolio can be denoted as:
σ p2 = σ 2 (V1 ) + σ 2 (V2 ) + σ 2 (V3 ) + 2 * COV (V1 , V2 ) + 2 * COV (V1 , V3 ) + 2 * COV (V2 , V3 )
σ 2 (V1 + V2 ) = σ 2 (V1 ) + σ 2 (V2 ) + 2 * COV (V1 , V2 )
σ p2 = σ 2 (V1 + V2 ) + σ 2 (V1 + V3 ) + σ 2 (V2 + V3 ) − σ 2 (V1 ) − σ 2 (V2 ) − σ 2 (V3 )
..[2]
With Eqn. [1], portfolio standard deviation can be expressed in terms of the standard
deviations of single assets and the standard deviations of two-asset sub-portfolios. Thus, the
portfolio standard deviation can be calculated based on Eqn. [1] and [2] illustrated above.
ii. Relationship between β , σ and ρ
Standard formula for Correlation ( ρ ) in terms of Covariance can be denoted as:
ρ12 =
σ 12
σ 1σ 2
..[3]
Where, ρ12 denotes the correlation between asset and market factor; σ 12 denotes
covariance between asset and market factor; σ 1 & σ 2 are std. dev of asset and market
factor respectively.
13
Standard formula for Beta ( β ) of an asset in a portfolio in terms of covariance can be
denoted as:
β of =
σ 12
(σ 2 ) 2
..[4]
Where, (σ 2 ) 2 denotes the market index (factor) variance
Combining Eqn. [3] & [4], we get the relationship between β and ρ :
β12 = ρ12
σ1
σ2
..[5]
iii. Covariance matrix for a two-asset portfolio
 σ 11
Using matrix notation, variance between any two variables can be denoted as: 
 σ 21
Correlation coefficient is denoted as ρ =
σ 12 

σ 22 
σ 12
.
σ 1σ 2
Substituting the values of σ 12 in above equation, we obtain the covariance matrix in terms
of ρ :
 σ 12
=
∑  ρσ σ
 1 2
ρσ 1σ 2 

σ 22 
..[6]
14
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