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Economic Capital Modeling
Closed form approximation for real-time applications
Research Co-operation between EIB / EIB Institute
and Manchester University
in the framework of the
FP7 Marie Curie ITN on Risk Management and Risk Reporting
Thomas Ribarits1 and Axel Clement1
Heikki Seppala2,∗, Hua Bai2,∗ and Ser-Huang Poon2
1 June 2014
1 European Investment Bank
2 Manchester Business School, University of Manchester
∗ Marie Curie fellow funded by the European Community’s Seventh Framework
Programme FP7-PEOPLE-ITN-2008 under grant agreement number
PITN-GA-2009-237984 (project name: RISK). The funding is gratefully acknowledged.
Abstract
Economic capital (ECap) modeling is a fundamental part of Pillar II of the
Basel framework. Indeed, ’sophisticated’ financial institutions need to have in
place internal models for the assessment of the level of the overall capital buffer
which is deemed sufficient to cover the risk of their business activities. On top,
ECap models are also frequently used for pricing purposes on an ex-ante basis:
financial institutions need to know the incremental economic capital (IECap),
i.e. the size by which the overall capital buffer needs to be increased after
addition of e.g. a single new loan to the existing portfolio. This is important
in order to be able to price such additional loan accordingly. Finally, ECap
contributions (ECapC) are also required ex-post in order to break down the
overall capital buffer to the individual obligors, products etc. within the port-
folio. Simulation of IECap and ECapC can be computationally expensive and
unstable, but it appears that closed form approximations provide accurate,
consistent and quick solutions in many cases.
The formula introduced here is based on the multi-factor approximation
from [Pykhtin,2004] applicable to a default-mode Merton type model. As such,
default correlations between obligors (stemming from a multi-factor-model) are
taken into account, but the formula also captures specific amortization sched-
ules and loss given default (LGD) values for each individual position - a feature
which is of practical relevance, but often neglected in standard default-mode
models. For the time being, credit-risk mitigants such as the existence of
guarantors on individual loans, are not captured by the formula, neither are
credit migrations or correlations between default probabilities and LGDs. Our
formula allows for approximation of all ECap contributions without extra com-
putational cost. After calculation of the ECap of the original portfolio, IECap
can be computed within few seconds and more accurately than in standard
linear approximations based on ECap contributions. JEL class: C63, G2.
Keywords: Approximation, economic capital, economic capital contributions,
incremental economic capital, risk pricing, Value at Risk, Expected Shortfall,
granularity adjustment, multi-factor adjustment, amortizing loan.
Contents
1 Executive Summary 1
2 Closed form methodology 6
2.1 Model framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Bucketing cashflows for closed form formula . . . . . . . . . . . . . . 7
2.3 Closed form approximation of VaR and ES . . . . . . . . . . . . . . 10
3 Performance of closed form approximation on homogeneous test
portfolios 17
3.1 Small portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1 Total portfolio VaR and ES . . . . . . . . . . . . . . . . . . . 20
3.1.2 Marginals and contributions . . . . . . . . . . . . . . . . . . . 23
3.2 Large portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Total portfolio VaR and ES . . . . . . . . . . . . . . . . . . . 28
3.2.2 Marginals and contributions . . . . . . . . . . . . . . . . . . . 31
4 Effect of maturities on ECap contributions 34
5 Closed form approximation on a heterogeneous test portfolio 38
5.1 Total portfolio VaR and ES . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Marginals and contributions . . . . . . . . . . . . . . . . . . . . . . . 39
6 Approximation of incremental ECap for fast calculations 43
7 Conclusion 49
A Incorporating amortization schedules 52
B Comparison of MECap, ECapC and IECap 54
C Bucketing of loans for ECap approximation 57
i
D Pykhtin’s formula 60
ii
Disclaimer
The views and opinions expressed in this report are those of the
authors and do not necessarily reflect those of the European In-
vestment Bank or the European Investment Bank Institute. All
figures shown are based on purely hypothetical test portfolios.
iii
Table 1: Abbreviations.
CF Closed Form approximation
CFES ES from CF
CFVaR VaR from CF
GA Granularity Adjustment
ECap Economic capital
ECapC ECap Contribution
EL Expected Loss
ES Expected Shortfall
ESC Expected Shortfall Contribution
Exp Exposure
IECap Incremental Economic Capital
IES Incremental ES
IVaR Incremental VaR
LGD Loss Given Default
LIECap Linear approximation of IECap
MC Monte Carlo
MA∞ Limiting Multi-Factor Adjustment
MFA Multi-Factor Adjustment
PD Probability of Default
ST Commercial Simulation Tool
STES ES from ST
STVaR VaR from ST
VaR Value at Risk
iv
1 Executive Summary
Unexpected loss risk pricing of loans has gained more attention after the financial
crisis starting in 2007: banks would like to know by how much capital buffers need to
be increased after addition of a new loan. Such capital buffers are often determined
by means of internal Economic Capital (ECap) models which measure the (credit)
Value at Risk (VaR) or Expected Shortfall (ES).
The credit VaR on a confidence level q is the q-quantile of the loss distribution
minus the expected loss (EL). The ES on confidence level q is the expected portfolio
loss conditional on losses exceeding the q-quantile, minus the EL.
VaR and ES calculations are typically performed by Monte Carlo simulations
based on the loss distribution. Simulation is a powerful method and combined with
variance reduction methods such as importance sampling, it is simple to set up and
accurate in most cases. However, for some purposes the approach is too slow or
unstable. For example the computation of the incremental ECap (IECap) of a new
loan via simulation is time consuming and potentially inconsistent due to sampling
errors.
IECap (incremental VaR or incremental ES) is defined as the difference between
the total ECap of the portfolio with and without the new loan.
An alternative for simulations is to use a closed form approximation, which
is consistent and can be designed to be fast, but less flexible and subject to an
approximation error.
The subsequent sections of the report can be summarized as:
Section 2: Closed form methodology
Section 3: Comparison of closed form approximation to Monte Carlo simula-
tion results on small and large homogeneous test portfolios
Section 4: Impact of exposure maturities on ECap calculations
1
Section 5: Comparison of closed form approximation to Monte Carlo simula-
tion results on a heterogeneous test portfolio
Section 6: Closed form approximation of incremental ECap for fast calcula-
tions
The closed form solution proposed here is a modification of the formula intro-
duced by Pykhtin (2004) (see Appendix D). The modification of Pykhtin’s formula
presented here gives approximations for VaR and ES contributions1 and most im-
portantly can be used for quick and accurate calculation of incremental VaR and
incremental ES.
ECap contributions, ECapC (VaRC or ESC), represent the change in ECap for
an infinitesimal change in the individual exposures; see formula (3) below.
The formulas for contributions, especially for VaR, are not fully justified from the
theoretical point of view and for small portfolios the closed form approximation of
VaR contributions does not perform well2. For large portfolios, however, the closed
form approximation gives reasonable results compared to the simulation results
obtained from a commercial simulation tool. The closed form formulas take the
maturities and amortization plans into account and most importantly, allow for very
fast computation of incremental ECap of a new loan. In theory, the formula could be
extended further to deal with credit migrations following the ideas developed here for
different maturities and amortizations. However, inclusion of credit migrations into
the formula this way could be computationally too heavy: the closed form formula
uses very large multi-dimensional matrices and credit migrations would increase
the number of dimensions. Closed form formulas allowing for credit migrations are
investigated in Dullmann and Puzanova (2011), Voropaev (2011) and Gordy and
1 We found out afterwards that a similar, but less general formula for contributions based on
Pykhtin (2004) was derived already in Dullmann and Puzanova (2011).2Feasibility of VaR and VaR contributions is questioned e.g. in Artzner (1999) and Kalkbrener
et al (2004).
2
Marrone (2012). Also further generalizations could be possible, such as allowing
the presence of guarantors or the correlations between default probabilities and loss
given defaults. Closed form formulas taking these into account are already available
in one factor models, see Ebert and E. Lutkebohmert (2012) and Gagliardini and
C. Gourieroux (2013).
Pykhtin’s closed form approximation of VaR is based on the second order Tay-
lor approximation of quantile function tq around a benchmark random variable L,
which is chosen such that (a) L allows for closed form solution of the quantile func-
tion and (b) the q-quantile of L is close enough to the q-quantile of portfolio loss
distribution L, see also e.g. Gourieroux et al (2000), Martin and Wilde (2002), Va-
sicek (2002) and Gordy (2003). It should be noted that Pykhtin’s formula is partly
based on an assumption that the individual losses have continuous distributions,
which is not true for constant loss given defaults. Indeed, our test portfolios have
constant loss given defaults and hence the ECap contributions from our formulas
have some deviations from the simulated ones, but the main objective is to have a
quick and accurate method for calculating incremental economic capital and for that
purpose the approximation is very good even when the loss distribution is discrete.
Pykhtin’s formula is modified here such that the total portfolio VaR is calculated
via VaR contributions, i.e. using additive decomposition of portfolio VaR. This can
be achieved using Euler’s Homogeneous Function Theorem, since EL and tq(L) are
homogeneous functions of order 1 with respect to exposures. The Theorem implies
that the credit VaR has an additive decomposition
VaRq =M∑i=1
wi∂
∂wiVaRq, (1)
where wi is the exposure attached to obligor i. This decomposition holds for ES as
well,
ESq =
M∑i=1
wi∂
∂wiESq . (2)
3
The terms in the sums above are called VaR and ES contributions (VaRC, ESC),
i.e.,
VaRCq,i = wi∂
∂wiVaRq
ESCq,i = wi∂
∂wiESq .
(3)
Pykhtin’s formula is also a homogeneous function of order one if the benchmark
random variable L is chosen appropriately. Computation of the portfolio ECap
through ECap contributions is not computationally more costly than via Pykhtin’s
original formula, which means that ECap contributions are available almost for free.
Another advantage is that the incremental ECap of a new loan can be computed
very quickly once the ECap of the original portfolio is computed. The third quantity
we discuss here is the marginal ECap (MECap):
Marginal ECap, MECap (MVaR or MES), is the difference between the ECap
of the total portfolio with and without an existing loan (or obligor).
The downside of computing the portfolio ECap through ECap contributions is
that it is necessary to make another approximation: the benchmark random variable
L suggested by Pykhtin (2004) depends on exposures, which makes the derivative of
the closed form formula with respect to exposures tedious to calculate and almost
useless in practice. Difficulties can be avoided by fixing L, i.e. assuming that
the changes in exposures have no impact on L. This is a valid assumption if the
changes in exposures are relatively small, but the approximation loses its accuracy
if the changes in exposures are large compared to the portfolio size.
According to numerical tests this simplification does not lead to big additional
approximation errors (not even for a test portfolio of only 26 loans), but theoretical
proof is not yet available. After this simplification the computation of ECap con-
tributions requires only very little extra effort or computational time compared to
the original formula, and incremental ECap of a new loan can be computed in few
seconds.
4
The performance of the closed form formula is tested by comparing it to a
commercial simulation tool (ST), which is one of the commercial simulation tools
for Economic Capital calculations. Both, the closed form formula and ST are also
tested against independent Monte Carlo simulations in the simplest cases. Basis
of all tests is a three factor default-mode Merton type model. Tests show that ST
works as well as expected for a homogeneous test portfolio, but there are some
consistency issues regarding VaR contributions of loans with very short maturities
and especially marginal VaRs if the homogeneous loan portfolio shows very low
probabilities of default (PDs). For very small portfolios3 the inconsistency of ECap
contributions using ST is even more pronounced. These issues are not a cause for
great concern, but the user should be aware of these simulation problems present
in ST. In contrast, marginal VaRs and ECap contributions in the closed form (CF)
approximation are consistent, but if the portfolio is small or factors have very low
correlations, the approximation error may be large especially when the constant
loss given defaults are used. For larger portfolios4 the performance of the closed
form formula is surprisingly good compared to ST: results from CF and ST are
very close to each other except when risk factor correlations are very low. CF
is also fast enough for pricing purposes: on a standard laptop5 the pre-run for a
heterogeneous test portfolio of 900 obligors (number of actual loans does not have
a big impact due to maturity bucketing) with six maturity buckets and three risk
factors takes about 20 minutes and after that ECap can be computed in about 3
seconds. Obviously, more powerful computers perform much faster. Computation
time grows exponentially in number of obligors, factors or maturity buckets, which
means that the formula may not be useful for very large portfolios unless the number
of maturity buckets is decreased.
3The small test portfolio consists of 26 loans.4Large test portfolio consists of 2600 loans.5Processor Intel(R) Core(TM) i5-2430M CPU @ 2.40GHz, 2401 Mhz, 2 Core(s), 4 Logical
Processor(s).
5
2 Closed form methodology
2.1 Model framework
The underlying model is a multi-factor default-mode Merton type model with M
obligors. According to the model, obligor i ∈ {1, ...,M} defaults if its normalised
(log) asset returns Xi drop below a default threshold di. The default threshold
di of obligor i is determined by default probability pi. (Log-)asset returns are
assumed to be normally distributed, which means that the default threshold is
given by di = Φ−1(pi), where Φ is the cumulative distribution function of a standard
normal distribution. Asset returns depend on systematic risk factor Yf(i) and an
idiosyncratic shock εi, both following the standard normal distribution. Systematic
risk factors are correlated, but idiosyncratic shocks are independent from each other
and everything else in the model. Hence
Xi = riYf(i) +√
1− r2i εi, (4)
where factor loadings ri determine how sensitive asset returns are to the systematic
factor and
f(i) ∈ {1, 2, 3}. (5)
Obligors are divided into 3 groups, namely GroupA, GroupB and GroupC and each
group is mapped to one systemic market factor6. Furthermore, obligors within one
group have the same ri7.
Market factors are generally correlated and closed form solution requires inde-
pendent standard normally distributed factors. Independent factors Zi can be found
6The groups can represent, e.g. sectors of obligors such as ’utility companies’, ’financial institu-
tions’ or the like, depending on the level of granularity chosen in the underlying default correlation
model.7The formula works in a more general setting, i.e. number of factors can be much larger and
each obligor can be linked to more than one factor via different ri’s. More complicated structures
increase the computational cost, though.
6
by e.g. Cholesky decomposition, singular value decomposition or Gram-Schmidt or-
thogonalization. Here the singular value decomposition is used, but other methods
lead to the same final results although the decomposition may be different. Original
systematic risk factors can be recovered from independent risk factors according to
equality
Yf(i) =
3∑k=1
αf(i)kZk, where
3∑k=1
α2f(i)k = 1. (6)
In fact, independent factors themselves are not of interest, but α-coefficients are
needed.
2.2 Bucketing cashflows for closed form formula
Simulations in ST are able to take the maturities and amortization plans into ac-
count. The idea in simulations is to simulate independent systematic factors and
idiosyncratic shocks and calculate realisations of asset returns Xi. If Xi is below
the default threshold at specified time horizon τh (e.g. τh = 1 year), the next step is
to check when the default has occurred by looking at the ratio ui = Φ(Xi)pi
. Default
time is τi = uiτh. After calculation of τi, amortization plans are used to determine
the size of the loss. For closed form solution we need a different approach.
To overcome the problem of determining a time of default, the time horizon
is split into disjoint periods for the closed form approximation. As the loss of an
individual obligor i is the sum of losses on all subsequent principal payments of
the obligor in question after default time τi, cashflows are bucketed according to
payment dates, e.g., all the cashflows expected to be received during the first three
weeks are put into one bucket, cashflows that are due between three weeks and
three months are placed into the second bucket and so on. Loans of an obligor with
different maturities can also have different loss given defaults (LGD). One objective
is to calculate the derivatives of portfolio VaR and ES with respect to an individual
obligor’s total exposure. To achieve this, the idea is to consider the total exposure
7
(sum of all principal flows) related to each obligor and adjust LGD’s such that LGD
times exposure match the original LGD times exposure in each bucket. Example 1
illustrates how this is done.
Example 1
Suppose obligor 1 has three loans specified in Table 2.
Loan type Amount DtM Payment schedule LGD
Bullet 1 1M 50 1M in 50 days 0.4
Bullet 2 1.3M 90 1.3M in 90 days 0.5
Amortizing 1.5M 365 0.5M in 100 days, 0.5M in 230 days, 0.5M in 365 days 0.4
Table 2: Loan type, loan amount, days to maturity and payment schedule.
Cash inflows are placed in 3 buckets according to payment days such that the
first bucket includes cash inflows that take place between 1 and 122 days from now,
second bucket includes the cash inflows occurring in 123-244 days and the last bucket
includes the cash inflows that are anticipated to occur in 245-365 days. Buckets are
presented in Table 3.
Bucket 1-122d Bucket 123-244d Bucket 245-365d Loan
Exp E× LGD Exp E× LGD Exp E× LGD Exp E× LGD
Bullet 1 1M 0.4M 0 0 0 0 1M 0.4M
Bullet 2 1.3M 0.65M 0 0 0 0 1.3M 0.65M
Amortizing 0.5M 0.2M 0.5M 0.2M 0.5M 0.2M 1.5M 0.6M
Sum 2.8M 1.25M 0.5M 0.2M 0.5M 0.2M 3.8M 1.65M
Table 3: Exposures and exposure times LGDs related to each bucket.
Actual total cash inflows and LGDs of each bucket are shown in Table 4.
8
Bucket 1-122d Bucket 123-244d Bucket 245-365d Obligor
Exp LGD Exp LGD Exp LGD Exp LGD
Total 2.8M 0.4464 0.5M 0.4 0.5M 0.4 3.8M 0.4342105
Table 4: Bucketed exposures and LGDs of an obligor.
Eventually, LGDs are adjusted such that the exposure of each bucket is equal
to the sum of all cashflows, but exposure × LGD remain the same. This trick leads
to Table 5.
Bucket 1-122d Bucket 123-244d Bucket 245-365d Obligor
Exp LGD Exp LGD Exp LGD Exp LGD
Total 3.8M 0.3289 3.8M 0.05263 3.8M 0.05263 3.8M 0.4342105
Table 5: Adjusted LGDs of an obligor.
Rearrangement described above leads to the following representation of loss of
obligor i:
Li = wi
m∑j=1
1{Xi≤d
(j)i
}Q(j)i , (7)
where m is the number of buckets, wi is the total exposure of obligor i (3.8M in
the example above), 1{Xi≤d(j)i }
is an indicator function that indicates whether the
cashflows in bucket i are subject to default or not8, and Q(j)i is the stochastic loss
given default9 of cashflows of obligor i in bucket j. Mean and variance of Q(j)i is
denoted by µ(j)i and σ
(j)i . Total portfolio loss is then
L =
M∑i=1
Li. (8)
8In fact, d(j)i = Φ−1(p
(j)i ) where p
(j)i is the unconditional default probability of obligor i default-
ing at any time in maturity bucket j or before.9In the example above, Q
(j)i is ”stochastic” with mean µ
(j)i = 0.3289 (for bucket j = 1) and
σ(j)i = 0 (for all buckets j = 1, 2, 3).
9
2.3 Closed form approximation of VaR and ES
The starting point for the formula is the model described in Section 2.1 and 2.2.
The formula allows for a chosen number of
• obligors M (M = 26, 2600, 900 in the test portfolios of subsequent sections),
• factors N (N = 3 in our examples) and
• cashflow buckets m (m = 6 in our examples).
The formula is a function of
• factor loadings ri (see equation (4))
• α-coefficients (see equation (6))
• cashflow vectors wi holding cashflows of obligor i (wi being the same for all
maturity buckets j, see (7))
• mean LGD vectors µi = (µ(j)i )mj=1 holding mean LGD’s of obligor i placed in
time buckets j (see Table 5)
• obligor specific standard deviations of LGD’s σi
• unconditional default probabilities p(j)i corresponding to ith obligor in matu-
rity bucket j or before; note that p(1)i ≤ p
(2)i ≤ ... ≤ p
(m)i .
• and the confidence level q.
Note that the factor correlations enter into the formula through α-coefficients.
The formula gives approximations of portfolio VaR, VaR contributions, portfolio
ES and ES contributions as an output. The formula is based on [Pykhtin (2004)]
with one clear difference compared to the original formula: analytical approximation
of derivatives of Pykhtin’s formula with respect to exposures are used to calculate
10
approximations of VaR and ES contributions and total portfolio VaR and ES are
given as sums of VaR and ES contributions.
Pykhtin’s formula is a second order Taylor approximation of VaR (or ES) around
a ”suitable” benchmark random variable Y , which has a standard normal distribu-
tion. The factor Y is ”optimally” constructed from independent risk factors (Zk)Nk=1
introduced in equation (6). The Y suggested by Pykhtin has a nice feature that if
all the exposures are multiplied by the same factor, then Y remains unchanged. In
particular, Pykhtin’s formula is a homogeneous function of order one with respect
to exposures and therefore allows the decomposition (1). The ”optimal” construc-
tion of Y depends on exposures, LGD’s, default probabilities and the confidence
level. This makes the derivatives of Pykhtin’s formula with respect to exposures
complicated. However, the ”optimal” Y is not unique and the analytical formula for
Y suggested by Pykhtin may not be optimal after all. Our idea here is to assume
that Y is calibrated to the original portfolio and kept fixed after that. This as-
sumption makes the differentiation of Pykhtin’s formula with respect to exposures
very straightforward and gives an advantage in computation speed: the most time
consuming calculations involve reasonably complex functions of Y and keeping Y
fixed makes it possible to compute final outputs in two steps. The first step involves
all the most time consuming computations10. The second step is practically just to
compile all the results from the first step by using matrix operations. If the portfolio
is sufficiently large, adding new exposure would only have a small impact on Y and
hence it is possible to obtain accurate closed form results for incremental ECap by
running only the second step with updated exposures and LGD’s and subtracting
the original ECap from this. This is useful for example in loan pricing, where the
incremental ECap of a new loan is of interest11 and should be calculated on-the-fly.
10Pykhtin’s formula is actually not fast to compute if the portfolio is large.11Incremental ECap is better than ECap contribution for this purpose, because the new loan most
likely has some impact on ECap contributions of existing loans due to the correlation structure.
11
As in Pykhtin (2004) it is assumed that Y is constructed from independent
systematic risk factors, i.e.,
Y =N∑k=1
bkZk, whereN∑k=1
b2k = 1. (9)
Then the systematic part of Xi, namely Yi12, can be written as
Yi = ρiY +√
1− ρ2i ηi, (10)
where ηi ∼ N(0, 1) is independent from Y , but ηi’s can be correlated. Factor loading
ρi is the correlation between Yi and Y ,
ρi ≡ cor(Yi, Y ) =
N∑k=1
αikbk. (11)
Coefficients bk are chosen such that the weighted sum of correlations ρi,(∑M
i=1 ciρi
)over all obligors is maximised with respect to (bk)Nk=1. Optimising with Lagrange
multipliers yields
bk =
M∑i=1
(ci/λ)αik, (12)
where the Lagrange multiplier λ =
√∑Nk=1
(∑Mi=1 ciαik
)2to ensure that
∑Nk=1 b
2k =
1. Choice of weighting coefficients ci is not obvious, but one option is to use
ci = wi
m∑j=1
µ(j)i Φ
Φ−1(p(j)i ) + ri Φ−1(q)√
1− r2i
, (13)
which is the choice suggested by Pykhtin. Notice that the right-hand side of equation
(13) resembles the VaR formula in a single-factor model. This choice relies on
numerical tests and the intuition that the obligors with higher VaR figures should
have bigger impact on the choice of the single factor.
Theorem 2 Let Y be given by equation (9). Then a quantile of the portfolio loss
at the confidence level q can be calculated as
tq(L) =
M∑i=1
wi∂
∂witq(L) (14)
12As opposed to (6) we simplify notation in the sequel by replacing Yf(i) by Yi.
12
with
∂
∂witq(L) =
∂
∂witq(L) +
∂
∂wiMA∞+ max
{0,
∂
∂wiGA
}13, (15)
∂
∂wiMA∞ =
∂wiy`
2(∂y`)2
[∂yν∞ − ν∞
(∂yy`
∂y`+ y
)]− 1
2∂y`
[∂wiyν∞ −
(ν∞
(−∂wiy` ∂yy`
(∂y`)2+∂wiyy`
∂y`
)+ ∂wi
ν∞
(∂yy`
∂y`+ y
))],
(16)
and
∂
∂wiGA =
∂wiy`
2(∂y`)2
[∂yνGA − νGA
(∂yy`
∂y`+ y
)]− 1
2∂y`
[∂wiyνGA −
(νGA
(−∂wiy` ∂yy`
(∂y`)2+∂wiyy`
∂y`
)+ ∂wi
νGA
(∂yy`
∂y`+ y
))],
(17)
where Φ is the cumulative distribution function of the standard normal distribution,
y = Φ−1(1− q) (18)
`(y) =
M∑i=1
wi
m∑j=1
µ(j)i p
(j)i (y) is the expected portfolio loss on condition Y = y (19)
∂
∂witq(L) = ∂wi`(y) =
m∑j=1
µ(j)i p
(j)i (y) (20)
∂y`(y) =
M∑i=1
wi
m∑j=1
µ(j)i ∂yp
(j)i (y) (21)
∂yy`(y) =
M∑i=1
wi
m∑j=1
µ(j)i ∂yyp
(j)i (y) (22)
∂wiy`(y) =
m∑j=1
µ(j)i ∂yp
(j)i (y) (23)
∂wiyy`(y) =
m∑j=1
µ(j)i ∂yyp
(j)i (y) (24)
d(j)i (y) =
Φ−1(p(j)i )− aiy√1− a2i
(25)
p(j)i is the unconditional PD of obligor i in maturity bucket j (26)
d(j)i = Φ−1(p
(j)i ) (27)
p(j)i (y) = Φ
(d(j)i (y)
)(conditional PD on condition Y = y) (28)
13Intuitively, granularity adjustment should not be negative, but due to approximation it can
be slightly negative. Here negative granularity adjustment is not allowed since underestimation of
risk should be avoided.
13
∂yp(j)i (y) =
−ai√1− a2i
Φ′(d(j)i (y)
)(29)
∂yyp(j)i (y) =
a2i1− a2i
d(j)i (y) Φ′
(d(j)i (y)
)(30)
ai = riρi (31)
ρi = cor(Yi, Y ) =
N∑k=1
αikbk (32)
ν∞ =
M∑i,l=1
wiwl
m∑j=1
m∑s=1
µ(j)i µ
(s)l
(Φ2
(d(j)i (y), d
(s)l (y), ρYil
)− p(j)i (y)p
(s)l (y)
)(33)
νGA =
M∑i=1
w2i
m∑j,s=1
[(µ(j)i µ
(s)i + σ2
i
)p(j∧s)i − µ(j)
i µ(s)i Φ2
(d(j)i (y), d
(s)i (y), ρYii
)](34)
∂wiν∞ = 2
M∑l=1
wl
m∑j=1
m∑s=1
µ(j)i µ
(s)l
(Φ2
(d(j)i (y), d
(s)l (y), ρYil
)− p(j)i (y)p
(s)l (y)
)(35)
∂wiνGA = 2wi
m∑j,s=1
[(µ(j)i µ
(s)i + σ2
i
)p(j∧s)i (y)− µ(j)
i µ(s)i Φ2
(d(j)i (y), d
(s)i (y), ρYii
)](36)
∂yν∞ =
M∑i,l=1
wiwl
m∑j=1
m∑s=1
µ(j)i µ
(s)l ∂yp
(j)i (y)
Φ
d(s)l (y)− ρYil d(j)i (y)√
1−(ρYil)2
− p(s)l (y)
(37)
∂yνGA =
M∑i=1
w2i
m∑j,s=1
∂yp(j∧s)i (y)
µ(j)i µ
(s)i
1− 2 Φ
d(s)i (y)− ρYii d(j)i (y)√
1−(ρYii)2
+ σ2i
(38)
∂wiyν∞ = 2
M∑l=1
wl
m∑j,s=1
µ(j)i µ
(s)l ∂yp
(j)i (y)
Φ
d(s)l (y)− ρYil d(j)i (y)√
1−(ρYil)2
− p(s)l (y)
(39)
∂wiyνGA = 2wi
m∑j,s=1
∂yp(j∧s)i (y)
µ(j)i µ
(s)i
1− 2 Φ
d(s)i (y)− ρYii d(j)i (y)√
1−(ρYii)2
+ σ2i
(40)
ρYij =rirj
∑Nk=1 αikαjk − aiaj√
(1− a2i )(1− a2j
) (conditional asset correlation) (41)
and Φ2(·, ·, ρ) is the cumulative distribution function of the standard two-dimensional
normal distribution and j ∧ s = min{j, s}. The multi-factor adjustment MFA is
divided into two parts, limiting multi-factor adjustment MA∞ and granularity ad-
14
justment GA. In addition expected shortfall is given by
ESq =M∑i=1
wi∂
∂wiESq, (42)
with
∂
∂wiESq =
∂
∂wiESq,0 +
∂
∂wiESq MA∞+
∂
∂wiESq GA, (43)
∂
∂wiESq,0 =
1
1− q
M∑i=1
m∑j=1
µ(j)i Φ2
(d
(j)i ,Φ−1(1− q), ai
), (44)
∂
∂wiESq MA∞ = − 1
2(1− q)Φ′
(Φ−1(1− q)∂wiν∞
∂y`− ν∞∂wiy`
(∂y`)2
)(45)
and
∂
∂wiESq GA = − 1
2(1− q)Φ′
(Φ−1(1− q)∂wiνGA
∂y`− νGA∂wiy`
(∂y`)2
). (46)
Proof. In Pykhtin’s formula (see Appendix D) the loss of an individual obligor is
Li = wi1{Xi≤di}Qi. This is simply replaced by the formula given in (7),
Li = wi
m∑j=1
1{Xi≤d
(j)i
}Q(j)i ,
which allows for the amortization plans and maturities to be taken into account and
means that each wiµi in Pykhtins formula is replaced by wi∑m
j=1 µ(j)i . The second
step is to fix the correlation between Y and Yi (denoted by ρi in Appendix D) for
all i and finally differentiate the formula with respect to exposures (wi). Details of
this calculation are left for the reader.
Remark 3 (i) Expected shortfall in the subsequent tests is not exactly the same
as in the theorem above. Economic capital is usually defined as the unexpected
loss, which means that the expected loss is substracted from the ES of Theorem
2. For the same reason VaR considered in the following sections is tq minus
the expected loss.
(ii) The total conditional variance of L on condition Y = y is ν = ν∞+νGA where
ν∞ and νGA are interpreted as variances of systematic and idiosyncratic parts.
15
The most time consuming computations in the formulas above are those requir-
ing the use of loops and distribution functions. Once these are computed, the rest is
straightforward matrix computations. In the presentation above exposures wi and
LGD’s µ(j)i are visibly present only in matrix calculations, although they are used
for constructing bk’s. However, for small changes in exposures the changes in bk’s
are disregarded and the benchmark factor Y of the original portfolio is used also in
ECap computations of the updated portfolio. This allows for very fast computation
of ECap of the updated portfolio. Consequently the incremental ECap of a new
loan is very quick to compute once the ECap of the original portfolio is computed.
In short, the steps are
Step 1: Compute ci’s given in equation (13) and fix them. Then compute everything
in Theorem 2 that contains wi’s and µ(j)i ’s only through ci’s (or equivalently
ai’s), that is, equations (27), (12), (32), (31), (25), (28), (29), (30) and (41),
as well as
Φ2
(d(j)i (y), d
(s)l (y), ρYil
), Φ
d(s)l (y)− ρYil d(j)i (y)√
1−(ρYil)2
and Φ2
(d(j)i ,Φ−1(1− q), ai
)
Step 2: Perform all the computations that involve wi’s and µ(j)i ’s, e.g. compute the
sums in equations (19)-(24), (33)-(40), (16), (17), (15), (44)-(46), (43) and
finish by computing (14) and (42).
Note that the second step contains only matrix operations.
16
3 Performance of closed form approximation on homo-
geneous test portfolios
The accuracy of the closed form formula is compared to the results from a commer-
cial simulation tool (ST) using Monte Carlo simulations combined with importance
sampling and a simple Monte Carlo simulation R-code (MC). Results are compared
for two test portfolios: a small sample portfolio with 26 obligors and a larger test
portfolio with 2600 obligors.
Number of Monte Carlo simulations is 2,500,000 and number of ST simulations
is 300,000 with importance sampling. A one year time horizon and 99.9% confidence
level were used for all the tests in Section 3.
The general model structure is given in Table 6. The model consists of three
factors (N = 3) labelled GroupA, GroupB, and GroupC. Each obligor is mapped to
one factor only via ri’s of equation (4) and all obligors mapped to the same factor
carry the same ri.
GroupA GroupB GroupC R2
GroupA 1.0000 ρAB ρAC r21
GroupB ρAB 1.0000 ρBC r22
GroupC ρAC ρBC 1.0000 r23
Table 6: Borrower types, factor correlations and R2 values.
3.1 Small portfolio
The small portfolio consists of 26 obligors, each having one exposure (simple bullet
loans). The total exposure is normed to 10bne and obligors are divided into
3 groups, one group for each factor, while default probabilities vary by obligor
(depending on ratings). Results are then analysed for the following four LGD and
17
exposure settings as specified below.
Portfolio S1: Inputs from Table 7.
Portfolio S2: Equal exposures (384.6Me ), other inputs from Table 7.
Portfolio S3: Equal LGDs (10%), other inputs from Table 7.
Portfolio S4: Equal exposures (384.6Me ) and LGDs (10%), other inputs
from Table 7.
18
Exp (Me) LGD Type
GroupA1 2833 0.07 GroupA
GroupA2 881 0.07 GroupA
GroupA3 828 0.07 GroupA
GroupA4 540 0.07 GroupA
GroupA5 521 0.07 GroupA
GroupA6 514 0.07 GroupA
GroupA7 423 0.07 GroupA
GroupB1 363 0.40 GroupB
GroupB2 126 0.40 GroupB
GroupB3 252 0.40 GroupB
GroupB4 107 0.40 GroupB
GroupB5 522 0.40 GroupB
GroupB6 225 0.40 GroupB
GroupB7 85 0.40 GroupB
GroupB8 190 0.40 GroupB
GroupB9 103 0.40 GroupB
GroupC1 278 0.50 GroupC
GroupC2 173 0.50 GroupC
GroupC3 141 0.50 GroupC
GroupC4 139 0.50 GroupC
GroupC5 138 0.50 GroupC
GroupC6 136 0.50 GroupC
GroupC7 130 0.50 GroupC
GroupC8 128 0.50 GroupC
GroupC9 117 0.50 GroupC
GroupC10 108 0.50 GroupC
Table 7: Inputs: Exp = exposure, LGD = loss given default and Type = type of
the obligors (GroupA, GroupB or GroupC).
19
3.1.1 Total portfolio VaR and ES
The comparison is shown for market factor correlations as presented in Table 6
(with high correlations) as well as for an independent three factor model (zero cor-
relations) and a one factor model (perfect correlation). As shown in the result
tables, the closed form approximation is also further divided into the one factor,
multi-factor adjustment and granularity adjustment contributions.
Neither ST nor closed form (CF) approximation are designed for small portfolios
and it becomes clear when the results are compared to simple Monte Carlo simu-
lations (2.5M) which we use as a benchmark in our comparisons below. ES figures
from the closed form approximation agree quite well with Monte Carlo simulations
in most cases (see Tables 8, 9 and 11). Portfolio S3 is an exception and the figures
from CF are up to 31% less than from MC (Table 10). Reason for this is that Port-
folio S3 has very low granularity: the actual loss at the default caused by an obligor
is the exposure times LGD, and when all the LGDs are equal the largest individual
loss is almost 30% of maximal portfolio loss. VaR figures do not agree quite as
well as ES figures, but they are not completely out of range except in Portfolio S3.
Surprisingly, ST is in many cases worse than the closed form approximation.
20
Portfolio S1 total VaR and ES .
Factors ST MC CF 1-factor MA∞ GA EL
High correlationsVaR 404 249.13 241.76 149.21 4.35 90.70 2.50
ES 449 355.09 341.90 235.91 4.86 103.63 2.50
Independent (zero correlation)VaR 207 206.75 229.89 56.27 56.72 119.40 2.50
ES 284 282.14 282.82 80.93 71.66 132.73 2.50
One factor (perfect correlation)VaR 404 271.17 263.53 182.04 0.00 83.99 2.50
ES 478 391.13 389.64 295.05 0.00 97.09 2.50
Table 8: Results from simulation tool (ST), simple Monte Carlo (MC) simulation
and closed form approximation (CF) for Portfolio S1. All figures in Me.
Portfolio S2 total VaR and ES .
Factors ST MC CF 1-factor MA∞ GA EL
High correlationsVaR 381 381.07 411.03 276.57 8.99 129.02 3.55
ES 603 615.38 609.40 450.54 10.13 152.28 3.55
Independent (zero correlation)VaR 381 381.07 393.94 135.42 95.30 166.78 3.55
ES 537 512.32 524.75 205.54 132.01 190.74 3.55
One factor (perfect correlation)VaR 408 407.99 449.74 333.55 0.00 119.73 3.55
ES 696 700.92 696.79 556.69 0.00 143.65 3.55
Table 9: Results from simulation tool (ST), simple Monte Carlo (MC) simulation
and closed form approximation (CF) for Portfolio S2. Exposure attached to each
obligor is 384.6Me. All figures in Me.
21
Portfolio S3 total VaR and ES .
Factors ST MC CF 1-factor MA∞ GA EL
High correlationsVaR 334 281.45 168.49 62.59 0.66 107.05 1.81
ES 348 301.26 208.81 91.75 0.91 117.95 1.81
Independent (zero correlations)VaR 334 281.45 239.68 28.42 11.26 201.81 1.81
ES 353 292.21 283.41 37.10 14.87 233.25 1.81
One factor (perfect correlations)VaR 392 281.45 163.76 72.84 0.00 92.74 1.81
ES 435 305.74 210.85 109.87 0.00 100.99 1.81
Table 10: Results from simulation tool (ST), simple Monte Carlo (MC) simulation
and closed form approximation (CF) for Portfolio S3. LGD of each obligor is 0.1.
All figures in Me.
Portfolio S4 total VaR and ES .
Factors ST MC CF 1-factor MA∞ GA EL
High correlationsVaR 114 113.89 105.73 72.72 2.14 32.36 1.49
ES 153 151.51 152.02 113.93 2.56 37.02 1.49
Independent (zero correlations)VaR 75 75.43 95.20 25.70 26.81 44.19 1.49
ES 109 109.28 118.70 36.35 35.01 48.82 1.49
One factor (perfect correlations)VaR 114 113.89 117.61 89.11 0.00 29.98 1.49
ES 177 173.01 176.76 143.38 0.00 34.86 1.49
Table 11: Results from simulation tool (ST), simple Monte Carlo (MC) simulation
and closed form approximation (CF) for Portfolio S4. Exposure attached to each
obligor is 384.6Me and LGDs are 0.1. All figures in Me.
22
3.1.2 Marginals and contributions
ECap contributions and marginal VaRs from CF and ST are compared for Portfolios
S1 and S2 in Tables 12 and 13, respectively14 considering the ”high correlations”
assumption only. Portfolios S3 and S4 are omitted in this section as they would
lead to similar observations as the ones presented below. In CF, Marginal VaRs and
VaR contributions behave quite similarly to ES contributions but are completely
different from figures given by ST. Indeed, marginal VaRs and VaR contributions
from ST do not look very good from the point of view of capital allocation as in
Portfolio S1 only two marginals and contributions are positive and in Portfolio S2
all marginal VaRs are negative. VaR contributions of Portfolio S2 from ST do not
look useful for capital allocation either. However, those are close to the theoretically
correct ones, which illustrates why VaR is not a good measure for capital allocation
purposes: as can be seen in column STVaR contributions, in Portfolio S1 small
changes only in two of the loans have an impact on the quantile of the portfolio loss
and in Portfolio S2 small changes in any GroupA or GroupB loan have zero impact
on the quantile while the exact VaR contributions for GroupC loans are positive.
The GroupA and GroupB VaR contributions should actually be negative and the
positive figures are caused by simulation error. This can be explained as follows:
VaR contributions are by definition the derivatives of total VaR with respect to
exposures multiplied by exposures. Derivatives are not directly available, so ST
calculates VaR contributions as an average loss of an obligor over all scenarios in
which the portfolio suffers a loss equal to VaR, i.e.,
VaRCi = E [Li|L = VaR] , (47)
where Li is the loss of obligor i and L is the total portfolio loss. This formula
is equivalent to contributions in formula (1), but it is very sensitive to simulation
errors.14Marginal ES is not available in ST.
23
In practice the expected value is computed by looking at an interval (length of
which is user specified) around quantile values and then computing how many times
each obligor hits the interval. Therefore VaR contributions can be very sensitive to
the choice of the length of the interval. If the interval is not chosen optimally, the
VaR contributions are not accurate and get closer to expected shortfall contributions
with different confidence level, e.g. if the original confidence level is 99.9% and the
length of the interval is 0.2% (units), leading to [99.8%, 100%], then we end up with
99.8%-confidence level ES contributions15. The chosen interval can have an impact
on VaR contributions if there are many loans such that the combinations of the
potential losses caused by them are close to VaR but not exactly the same. This
is the case in Portfolio S2, where VaR is is the sum of the losses caused by any
two GroupC obligors ((0.50+0.50)×384.62) minus the expected loss (3.55), but we
get close to, yet do not reach, this figure by summing the losses from one GroupA
obligor, one GroupB obligor and one GroupC obligor ((0.07 + 0.40 + 0.50)×384.62)
and subtracting the expected loss.
VaR contributions from CF look more sensible from the point of view of capital
allocation as they are in line with ES contributions from ST and CF, but actually
CF contributions are not accurate due to the fact that the closed form formula
is based on the assumption of continuous and smooth loss distribution with non-
negative contributions. This is a realistic assumption if LGD’s are stochastic, i.e.
they have non-zero variances.
ES contributions from ST and CF are very close to each other in Portfolio S2
(Table 13) and quite close in Portfolio S1 after the total ES is scaled to the same
level. ES contributions obtained from simulations on top of being theoretically much
better for capital allocation than VaR contributions are in general much more stable
than VaR contributions and much more reliable than marginal VaRs especially for
15In ST the length of the interval is restricted to be at most 25% of 1 − q.
24
small portfolios.
Marginal VaRs given by ST in Portfolio S2 are close to expected losses with
negative signs. This is due to the fact that the 99.9% quantile does not change if
any one of the loans is removed.16 Therefore the change in credit VaR should be the
expected loss of the portfolio without the removed loan minus the credit VaR of the
total portfolio, which results to the expected loss of the loan under consideration.
However, the marginal VaR figures in ST are not accurate and marginal VaRs are
not exactly equal to expected losses as we see in Table 13. In Portfolio S1 all but
two borrowers have slightly negative marginal VaRs. Closed form solution gives
completely different results due to the fact that the quantile in CF does not stay
the same if any of the loans is removed.
16Note that this is in contrast to the VaR contributions for GroupC loans as discussed above:
whereas the removal of a GroupC loan does not change the 99.9% quantile of the loss distribution,
the conditional expectation of loss of each obligor from GroupC, on condition that the portfolio
loss equals 99.9% VaR, is positive. See Appendix B for an illustrative comparison of MVaR, VaRC
and IVaR.
25
Portfolio S1 marginals and contributions .
Inputs Marginal Contributions
Exp LGD Exp×LGD EL CFVaR STVaR CFVaR STVaR CFES STES STES scaled
Total 10,000 1990.80 2.50 152.05 325.86 241.76 404.45 341.90 448.74 341.90
GroupA1 2832.70 0.07 198.30 0.359 17.02 148.95 28.37 197.93 32.30 61.92 47.18
GroupA2 881.30 0.07 61.69 0.007 0.27 -0.01 0.37 -0.01 0.57 0.98 0.74
GroupA3 827.80 0.07 57.95 0.047 1.31 -0.05 1.77 -0.05 2.44 3.13 2.39
GroupA4 540.00 0.07 37.80 0.046 1.00 -0.05 1.25 -0.05 1.77 2.59 1.98
GroupA5 520.70 0.07 36.45 0.004 0.13 -0.00 0.16 0.00 0.27 0.55 0.42
GroupA6 514.00 0.07 35.98 0.224 3.23 -0.22 3.96 -0.22 5.06 6.03 4.60
GroupA7 423.30 0.07 29.63 0.343 3.88 -0.34 4.55 -0.34 5.71 6.59 5.02
GroupB1 362.70 0.40 145.08 0.042 5.58 -0.04 8.75 -0.04 15.77 23.67 18.04
GroupB2 125.70 0.40 50.28 0.011 0.86 -0.01 1.08 -0.01 2.46 6.15 4.69
GroupB3 252.00 0.40 100.80 0.040 4.12 -0.04 6.16 -0.04 11.08 18.35 13.98
GroupB4 107.40 0.40 42.96 0.035 2.14 -0.04 2.80 -0.04 5.16 9.45 7.2
GroupB5 521.70 0.40 208.68 0.378 48.84 178.67 79.61 208.31 100.70 122.97 93.69
GroupB6 225.00 0.40 90.00 0.110 8.60 -0.11 13.02 -0.11 19.72 24.93 19.00
GroupB7 85.00 0.40 34.00 0.019 1.14 -0.02 1.40 -0.02 2.88 6.05 4.61
GroupB8 189.70 0.40 75.88 0.092 6.62 -0.09 9.74 -0.09 15.17 21.09 16.07
GroupB9 102.70 0.40 41.08 0.050 2.75 -0.05 3.61 -0.05 6.27 10.42 7.94
GroupC1 277.70 0.50 138.85 0.169 14.01 -0.17 25.10 -0.17 34.50 29.11 22.18
GroupC2 172.80 0.50 86.40 0.048 3.50 -0.05 5.82 -0.05 9.61 11.48 8.75
GroupC3 140.80 0.50 70.40 0.058 3.49 -0.06 5.74 -0.06 9.21 10.83 8.25
GroupC4 139.40 0.50 69.70 0.057 3.44 -0.06 5.65 -0.06 9.08 10.75 8.19
GroupC5 137.60 0.50 68.80 0.056 3.37 -0.06 5.53 -0.06 8.91 10.60 8.08
GroupC6 136.50 0.50 68.25 0.056 3.33 -0.06 5.46 -0.06 8.80 10.47 7.97
GroupC7 130.00 0.50 65.00 0.079 4.18 -0.08 6.89 -0.08 10.57 11.87 9.04
GroupC8 128.40 0.50 64.20 0.078 4.10 -0.08 6.75 -0.08 10.38 11.97 9.12
GroupC9 116.90 0.50 58.45 0.032 1.94 -0.03 3.07 -0.03 5.38 7.13 5.43
GroupC10 108.40 0.50 54.20 0.066 3.20 -0.07 5.15 -0.07 8.13 9.66 7.36
Table 12: Portfolio S1: 26 loans, exposures from Table 7.
26
Portfolio S2 marginals and contributions .
Inputs Marginal Contributions
Exp LGD Exp×LGD EL CFVaR STVaR CFVaR STVaR CFES STES
Total 10,000 3496.20 3.55 280.59 -3.55 411.03 381.05 609.40 603.54
GroupA1 384.62 0.07 26.9234 0.049 0.77 -0.16 0.86 0.74 1.24 1.77
GroupA2 384.62 0.07 26.9234 0.003 0.08 -0.19 0.09 0.21 0.15 0.25
GroupA3 384.62 0.07 26.9234 0.022 0.42 -0.11 0.46 0.75 0.70 0.90
GroupA4 384.62 0.07 26.9234 0.033 0.57 -0.19 0.63 1.00 0.93 1.20
GroupA5 384.62 0.07 26.9234 0.003 0.08 -0.04 0.09 0.22 0.15 0.25
GroupA6 384.62 0.07 26.9234 0.168 1.92 -0.06 2.15 3.26 2.87 3.24
GroupA7 384.62 0.07 26.9234 0.312 2.93 -0.16 3.28 4.76 4.22 4.55
GroupB1 384.62 0.40 153.848 0.045 4.72 -0.23 6.52 1.03 11.38 13.43
GroupB2 384.62 0.40 153.848 0.032 3.61 -0.28 4.91 0.61 9.00 11.14
GroupB3 384.62 0.40 153.848 0.061 6.14 -0.23 8.59 1.37 14.30 16.24
GroupB3 384.62 0.40 153.848 0.126 10.77 0.00 15.46 2.21 23.26 23.38
GroupB5 384.62 0.40 153.848 0.278 19.11 -0.11 27.98 6.34 37.94 35.33
GroupB6 384.62 0.40 153.848 0.187 14.45 0.00 20.97 2.30 29.91 28.56
GroupB7 384.62 0.40 153.848 0.085 7.93 -0.17 11.23 1.63 17.86 18.88
GroupB8 384.62 0.40 153.848 0.187 14.45 -0.08 20.97 2.64 29.91 29.28
GroupB9 384.62 0.40 153.848 0.187 14.45 -0.03 20.97 2.46 29.91 28.83
GroupC1 384.62 0.50 192.31 0.234 22.27 -0.19 33.38 42.91 47.70 44.58
GroupC2 384.62 0.50 192.31 0.106 11.86 -0.13 17.48 25.61 28.36 30.11
GroupC3 384.62 0.50 192.31 0.157 16.36 -0.23 24.35 31.10 37.04 36.76
GroupC4 384.62 0.50 192.31 0.157 16.36 -0.02 24.35 32.67 37.04 36.84
GroupC5 384.62 0.50 192.31 0.157 16.36 -0.31 24.35 31.14 37.04 36.82
GroupC6 384.62 0.50 192.31 0.157 16.36 -0.03 24.35 32.58 37.04 37.12
GroupC7 384.62 0.50 192.31 0.234 22.27 -0.05 33.38 42.91 47.70 44.74
GroupC8 384.62 0.50 192.31 0.234 22.27 -0.23 33.38 44.29 47.70 45.11
GroupC9 384.62 0.50 192.31 0.106 11.86 -0.16 17.48 22.41 28.36 29.47
GroupC10 384.62 0.50 192.31 0.234 22.27 -0.16 33.38 43.90 47.70 44.76
Table 13: Portfolio S2: 26 loans, equal exposures.
27
3.2 Large portfolio
The large portfolios consist of 2600 obligors and one loan for each. This portfolio is
constructed from 26 loans of the small portfolio by replicating each loan 100 times
and dividing all the exposures by 100. The correlation structure and R2 values are
unchanged (see Table 6). For these portfolios we only consider results from ST and
the closed form formula, because non-optimized Monte Carlo simulations for large
portfolios are time consuming. Therefore we only compare the results from ST and
CF to each other and consider ST as a benchmark.
Similar to the analysis for the small portfolio, results are analysed for the fol-
lowing four LGD and exposure settings as specified below where large portfolios L1,
L2, L3 and L4 are constructed from Portfolios S1, S2, S3 and S4 respectively.
Portfolio L1: Inputs from Table 7.
Portfolio L2: Equal exposures (384.6Me ), other inputs from Table 7.
Portfolio L3: Equal LGDs (10%), other inputs from Table 7.
Portfolio L4: Equal exposures (384.6Me ) and LGDs (10%), other inputs
from Table 7.
3.2.1 Total portfolio VaR and ES
Again, the comparison is done for high market factor correlations as well as for an
independent three factor model (zero correlations) and a one factor model (perfect
correlations). As shown in the result tables, the closed form approximation is also
further divided into the one factor, limiting multi-factor adjustment and granularity
adjustment contributions.
Tables 14, 15, 16 and 17 show that VaR and ES figures from ST and CF match
almost perfectly for all four portfolios, except when the factor correlations are very
low. When the factors are independent, CF underestimates VaR and ES of test
28
portfolios by 15-42% due to the inaccuracy in multi-factor adjustment. This is far
too much and thus the closed closed form formula should be used very carefully if
the risk factor correlations are low.
Portfolio L1 total VaR and ES .
Factors ST CF 1-factor MA∞ GA EL
High correlationsVaR 153.56 151.97 149.21 4.35 0.91 2.50
ES 240.86 239.3 235.91 4.86 1.04 2.50
Independent (zero correlations)VaR 140.62 111.68 56.27 56.72 1.19 2.50
ES 189.67 151.42 80.93 71.66 1.33 2.50
One factor (perfect correlation)VaR 181.70 180.38 182.04 0.00 0.84 2.50
ES 294.33 293.52 295.05 0.00 0.97 2.50
Table 14: Results from simulation tool (ST) and closed form approximation (CF)
for Portfolio L1. All figures in Me.
Portfolio L2 total VaR and ES .
Factors ST CF 1-factor MA∞ GA EL
High correlationsVaR 285.07 283.29 276.57 8.99 1.29 3.55
ES 464.84 458.636 450.54 10.13 1.52 3.55
Independent (zero correlations)VaR 277.80 228.83 135.42 95.30 1.67 3.55
ES 393.04 335.92 205.54 132.01 1.91 3.55
One factor (perfect correlations)VaR 333.08 331.2 333.55 0.00 1.20 3.55
ES 556.10 554.58 556.69 0.00 1.44 3.55
Table 15: Results from simulation tool (ST) and closed form approximation (CF)
for Portfolio L2. Exposure attached to each obligor is 384.6Me . All figures in Me.
29
Portfolio L3 total VaR and ES .
Factors ST CF 1-factor MA∞ GA EL
High correlationsVaR 63.13 62.52 62.59 0.66 1.07 1.81
ES 92.76 92.03 91.75 0.91 1.18 1.81
Independent (zero correlations)VaR 47.35 41.69 28.42 11.26 2.02 1.81
ES 90.10 52.49 37.10 14.87 2.33 1.81
One factor (perfect correlations)VaR 73.05 71.95 72.84 0.00 0.93 1.81
ES 109.68 109.07 109.87 0.00 1.01 1.81
Table 16: Results from simulation tool (ST) and closed form approximation (CF)
for Portfolio L3. LGD of each obligor is 0.1 and other inputs are given in Table 7.
All figures in Me.
Portfolio L4 total VaR and ES .
Factors ST CF 1-factor MA∞ GA EL
High correlationsVaR 75.04 73.69 72.72 2.14 0.32 1.49
ES 117.00 115.37 113.93 2.56 0.37 1.49
Independent (zero correlations)VaR 67.74 51.45 25.70 26.81 0.44 1.49
ES 86.42 70.36 36.35 35.01 0.49 1.49
One factor (perfect correlations)VaR 88.51 87.92 89.11 0.00 0.30 1.49
ES 142.67 142.24 143.38 0.00 0.35 1.49
Table 17: Results from simulation tool (ST) and closed form approximation (CF)
for Portfolio L4. Exposure attached to each obligor is 384.6Me and LGD of each
obligor is 0.1. All figures in Me.
30
3.2.2 Marginals and contributions
Again, ECap contributions and marginal VaRs from CF and ST are compared
for Portfolios L1 and L2 in Tables 18 and 19 considering the ”high correlations”
assumption only. Portfolios L3 and L4 are omitted in this section as they would lead
to similar observations as Portfolios L1 and L2. All 100 loans replicated from original
loans should have identical ECap contributions and marginal VaRs. However, this
is not exactly what ST gives as can be seen in Tables 18 and 19. In fact marginal
VaRs from ST are extremely unstable. VaR contributions are more stable, the
highest figures are about 20% higher than the lowest figures for identical loans.
ES contributions are much more stable and the highest figures are only about 5%
higher than the lowest figures. The closed form approximation is by construction
stable and all identical loans have exactly the same marginals and contributions.
The overall level from CF for marginal VaRs and ECap contributions are fairly close
to the average figures over 100 identical loans in ST. It appears that the closed form
solution performs as well as ST even in the case of VaR contributions if the portfolio
is large. It is likely that neither of them gives exactly the right VaR contributions,
but ES contributions are more credible.
31
Inp
uts
Mar
gin
alC
ontr
ibu
tion
s
Exp
CF
VaR
ST
VaR
mea
nm
inm
axC
FV
aRS
TV
aR(m
ean
)m
inm
axC
FE
SS
TE
Sm
ean
min
max
Tota
l10
,000
148.
7614
5.65
64.1
925
8.63
151.9
715
3.48
141.
2116
6.73
239.3
024
1.28
235.5
324
7.8
3
Gro
up
A1
2832
.70
8.66
10.3
66.
3013
.88
8.82
10.3
09.
3711
.66
12.2
213
.00
12.5
513.
60
Gro
up
A2
881.
300.
310.
06-0
.01
0.36
0.31
0.37
0.27
0.49
0.50
0.6
50.
590.
70
Gro
up
A3
827.
801.
410.
79-0
.03
2.84
1.40
1.68
1.44
1.98
2.04
2.3
02.
152.
41
Gro
up
A4
540.
001.
240.
50-0
.05
2.00
1.23
1.46
1.25
1.64
1.75
1.9
31.
852.
00
Gro
up
A5
520.
700.
180.
040.
000.
300.
180.
230.
140.
290.
290.3
80.
330.
42
Gro
up
A6
514.
003.
693.
210.
198.
133.
694.
263.
964.
594.
814.7
44.
634.
87
Gro
up
A7
423.
304.
504.
670.
158.
554.
505.
124.
875.
415.
695.3
85.
295.
51
Gro
up
B1
362.
705.
406.
841.
498.
865.
336.
265.
507.
0210
.41
13.6
013
.22
14.
06
Gro
up
B2
125.
701.
430.
840.
012.
501.
401.
691.
491.
912.
873.9
43.
754.
05
Gro
up
B3
252.
004.
846.
141.
558.
854.
815.
715.
016.
338.
9511
.23
10.9
611.
60
Gro
up
B4
107.
403.
572.
940.
347.
093.
574.
113.
814.
456.
006.9
36.
797.
11
Gro
up
B5
521.
7029
.53
31.3
423
.36
41.2
031
.07
34.8
433
.03
36.9
746
.89
48.8
548
.00
49.
67
Gro
up
B6
225.
009.
9011
.07
8.30
14.1
810
.04
11.4
510
.76
12.6
516
.00
17.5
117
.20
17.
95
Gro
up
B7
85.
002.
091.
560.
006.
292.
082.
422.
222.
713.
704.4
74.
374.
56
Gro
up
B8
189.
708.
359.
467.
0213
.01
8.46
9.70
9.11
10.2
013
.48
14.7
714
.53
15.
13
Gro
up
B9
102.
704.
535.
121.
558.
844.
565.
214.
815.
437.
287.9
97.
868.
15
Gro
up
C1
277.
7013
.24
13.4
28.
7118
.67
13.8
311
.16
10.1
312
.05
21.3
618
.16
17.7
618.
71
Gro
up
C2
172.
804.
723.
930.
3510
.42
4.76
3.75
3.37
4.08
8.13
7.2
97.
057.
58
Gro
up
C3
140.
805.
144.
110.
9411
.30
5.23
4.21
3.89
4.67
8.49
7.4
47.
187.
69
Gro
up
C4
139.
405.
094.
350.
3510
.26
5.18
4.14
3.72
4.58
8.40
7.3
47.
197.
55
Gro
up
C5
137.
605.
033.
840.
519.
155.
114.
103.
644.
388.
297.2
67.
057.
41
Gro
up
C6
136.
504.
994.
100.
2912
.15
5.07
4.05
3.65
4.44
8.23
7.2
07.
067.
36
Gro
up
C7
130.
006.
275.
281.
8711
.49
6.43
5.20
4.79
5.50
9.94
8.5
08.
268.
71
Gro
up
C8
128.
406.
195.
430.
6211
.44
6.35
5.16
4.69
5.59
9.82
8.3
98.
198.
65
Gro
up
C9
116.
903.
201.
980.
008.
263.
212.
562.
262.
935.
494.9
54.
795.
08
Gro
up
C10
108.
405.
244.
250.
388.
615.
354.
364.
034.
788.
287.0
86.
937.
30
Tab
le18:
Por
tfol
ioL
1:26
00(1
00id
enti
cal
inea
chgr
oup
)lo
ans,
orig
inal
exp
osu
res.
32
Inp
uts
Mar
gin
alC
ontr
ibu
tion
s
Exp
CF
VaR
ST
VaR
mea
nm
inm
axC
FV
aRS
TV
aR(m
ean
)m
inm
axC
FE
SS
TE
Sm
ean
min
max
Tot
al10
000.
0027
9.27
266.
6013
6.46
436.
60283.2
928
3.37
260.
4030
7.71
458.6
4462
.02
451
.24
472.
69
Gro
up
A1
384.
621.
041.
02-0
.05
3.80
1.02
1.06
0.92
1.16
1.41
1.32
1.2
51.
38
Gro
up
A2
384.
620.
120.
010.
000.
900.
120.
120.
090.
170.
180.
20
0.1
80.
22
Gro
up
A3
384.
620.
580.
62-0
.02
3.82
0.57
0.59
0.50
0.73
0.81
0.79
0.7
40.
84
Gro
up
A4
384.
620.
780.
86-0
.03
3.81
0.76
0.79
0.69
0.92
1.07
1.02
0.9
71.
08
Gro
up
A5
384.
620.
120.
010.
000.
920.
120.
120.
080.
170.
180.
20
0.1
70.
22
Gro
up
A6
384.
622.
473.
28-0
.17
3.68
2.43
2.52
2.30
2.71
3.15
2.78
2.7
12.
85
Gro
up
A7
384.
623.
693.
660.
487.
383.
623.
763.
504.
064.
563.
91
3.8
34.
01
Gro
up
B1
384.
625.
294.
95-0
.04
11.4
95.
155.
074.
425.
749.
4910.
50
10.2
210
.80
Gro
up
B2
384.
624.
103.
96-0
.03
7.66
3.95
3.88
3.28
4.62
7.61
8.71
8.4
09.
05
Gro
up
B3
384.
626.
796.
293.
7811
.48
6.66
6.59
5.85
7.84
11.7
512.
60
12.2
512
.98
Gro
up
B4
384.
6211
.51
11.2
87.
4419
.10
11.4
811
.39
10.3
912
.36
18.5
518.
68
18.2
719
.11
Gro
up
B5
384.
6219
.66
18.6
511
.26
26.6
419
.96
19.8
718
.52
21.3
729
.51
27.
70
27.1
028
.19
Gro
up
B6
384.
6215
.15
14.3
87.
5122
.89
15.2
415
.11
14.1
216
.47
23.5
322.
80
22.2
823
.31
Gro
up
B7
384.
628.
647.
973.
7619
.15
8.54
8.43
7.78
9.77
14.4
715.
06
14.7
015
.43
Gro
up
B8
384.
6215
.15
14.6
77.
5122
.89
15.2
415
.15
14.2
816
.14
23.5
322.
84
22.2
723
.31
Gro
up
B9
384.
6215
.15
14.3
67.
5122
.89
15.2
415
.13
14.1
615
.93
23.5
322.
82
22.3
923
.35
Gro
up
C1
384.
6220
.78
20.2
810
.78
26.6
921
.41
21.4
619
.84
22.7
234
.03
33.
61
32.9
734
.43
Gro
up
C2
384.
6211
.64
10.7
17.
5919
.12
11.7
811
.83
10.8
813
.11
20.9
222.
48
21.9
423
.10
Gro
up
C3
384.
6215
.66
14.8
27.
5322
.92
15.9
916
.05
14.7
017
.64
26.8
327.
67
26.9
128
.16
Gro
up
C4
384.
6215
.66
14.5
07.
5326
.77
15.9
915
.96
14.3
617
.32
26.8
327.
59
26.8
628
.21
Gro
up
C5
384.
6215
.66
14.3
87.
5322
.92
15.9
916
.08
14.6
217
.60
26.8
327.
67
27.1
428
.16
Gro
up
C6
384.
6215
.66
15.0
67.
5322
.92
15.9
916
.06
14.7
817
.38
26.8
327.
64
27.0
528
.46
Gro
up
C7
384.
6220
.78
20.0
311
.30
30.4
121
.41
21.4
519
.98
22.9
634
.03
33.
63
32.7
334
.25
Gro
up
C8
384.
6220
.78
19.1
911
.30
26.6
921
.41
21.4
719
.84
22.9
034
.03
33.
64
32.9
334
.43
Gro
up
C9
384.
6211
.64
11.2
35.
1619
.12
11.7
811
.91
10.8
612
.92
20.9
222.
51
21.9
723
.05
Gro
up
C10
384.
6220
.78
20.4
311
.30
30.5
421
.41
21.5
119
.66
23.0
034
.03
33.
65
33.0
134
.31
Tab
le19:
Por
tfol
ioL
2:26
00(1
00id
enti
cal
inea
chgr
oup
)lo
an
s,eq
ual
exp
osu
res.
33
4 Effect of maturities on ECap contributions
Assuming the default probabilities (p) for different maturities are scaled down lin-
early from original 1-year default probabilities, it is obvious that the expected loss
scales down linearly as well,
EL = p× Exp× LGD, (48)
where Exp is exposure and LGD is the loss given default, but ECap contributions do
not scale down linearly. This section analyses the behaviour of ECap contributions
depending on the obligor default probability. This is of special importance since
the use of a closed form formula requires bucketing of payments by maturities and
it is not obvious which probabilities should be associated to each bucket. The
natural option is to take the mid-point of each bucket and to choose the probability
accordingly, e.g. for 0-3 week bucket the 1.5 week default probability would be
used. However, as the behaviour is non-linear the mid-point doesn’t guarantee
an average ECap estimation. The first impression is that the VaR contribution
increases with maturity and a conservative, i.e. over-estimating, choice would be
the right end-point of the bucket. However, this is not always the case because the
EL is subtracted from the quantile and when the considered time period is long
enough the default probability is close to 1 and EL is close to the maximal loss. On
the other hand, the VaR contribution is close to 0 if the default probability is close
to 1. Therefore, the question is to determine when VaR (and ES) contributions
start to decrease.
This question is investigated by using portfolios of 1001 identical bullet loans
(same default probability, nominal (=100me ) and loss given default (=100%)).
A one factor model is used, asset correlations between all obligors are constant at
0.5, the time horizon is chosen to be 3 years and the quantile is 99.97%. All loans
except one have a maturity of 3 years (1095 days) and for the remaining loan the
34
maturity is set to 3, 10, 30, 100, 300, 400, 500, 600, 700, 800 and 1095 days in
different simulation runs. Default probabilities are scaled down linearly from the 3
years probability (p3Y ) because this is how it is done in the simulation tool. Scaling
up from 1 year to 3 years is done by using a constant hazard rate.
In Figure 1 it is shown how the ECap contributions behave when the maturity
of the exposure changes. For high default probabilities ECap contributions increase
rapidly at first, but then start to decrease. If the default probability is lower, the
increase is not as fast, but it lasts longer. Figures from the closed form approxima-
tion and the simulation tool indicate similar behaviour, but the simulation tool is
less stable and has issues with handling the combination of low default probability
and short maturity (Figure 2).
In summary, for low default portfolios, choosing the right end-point of the ma-
turity bucket yields conservative ECap contributions.
35
(a) CF
0 200 400 600 800 1000
020
4060
80
Days to maturity
EC
apC
(b) ST
0 200 400 600 800 1000
020
4060
80
Days to maturity
EC
apC
0 200 400 600 800 1000
010
2030
4050
6070
Days to maturity
EC
apC
0 200 400 600 800 1000
010
2030
4050
6070
Days to maturity
EC
apC
0 200 400 600 800 1000
05
1015
2025
Days to maturity
EC
apC
0 200 400 600 800 1000
05
1015
2025
Days to maturity
EC
apC
Figure 1: Effect of changes in maturity to VaR contributions (black solid) and ES
contributions (red dashed) for obligors with p3Y = 25% (at the top), p3Y = 5% (in
the middle) and p3Y = 1% (at the bottom), calculated using the closed form formula
CF, (left) and simulation tool ST, (right). Straight lines illustrate the non-linearity
of VaR and ES contributions with respect to maturity.
36
(a) CF
0 200 400 600 800 1000
010
020
030
040
0
Days to maturity
EC
apC
/EL
(b) ST
0 200 400 600 800 1000
010
020
030
040
0
Days to maturity
EC
apC
0 200 400 600 800 1000
020
040
060
080
010
00
Days to maturity
EC
apC
/EL
0 200 400 600 800 1000
020
040
060
080
010
00
Days to maturity
EC
apC
/EL
0 200 400 600 800 1000
050
015
0025
0035
00
Days to maturity
EC
apC
/EL
0 200 400 600 800 1000
050
015
0025
0035
00
Days to maturity
EC
apC
Figure 2: Effect of changes in maturity to VaRC/EL ratios (black solid) and
ESC/EL ratios (red dashed) for obligors with p3Y = 25% (at the top), p3Y = 5% (in
the middle) and p3Y = 1% (at the bottom), calculated using closed form formula
(left) and ST (right). 37
5 Closed form approximation on a heterogeneous test
portfolio
The heterogeneous test portfolio consists of 900 obligors with a total of 3750 bullet
and amortizing loans. In this case, the obligors are not simply ’copies’ of each other
(like in the case of the large homogeneous portfolio in Section 3.2), but are chosen
to be heterogeneous. Loans for the same obligor can have different exposures and
LGD’s. Credit risk is calculated on a 3-year horizon and the confidence level is set
to 99.97%. The ”high correlation” assumption for the 3-factor model is used.
Most of the obligors have low default probabilities and some of them have very
large exposures. The loans are bucketed into 6 buckets depending on their maturities
as follows
Bucket 0-3 weeks 3 weeks - 3 months 3-8 months 8-15 months 15-27 months 27+ months
PD 2 weeks 2 months 6 months 1 year 2 year 3 year
Table 20: Bucketed cashflows according to the time to payment. Probabilities are
scaled starting from 1-year default probabilities using a constant hazard rate model.
The choice of buckets and default probabilities takes the non-linear behaviour of
the ECap contributions into account (as discussed in Section 4). This is explained
in more detail in Appendix C. The choice is further justified by the results shown in
Table 21 as it leads to a very good approximation. It is noted that the homogeneous
portfolios considered in Section 3 only contained bullet loans with different maturi-
ties, because at the time when tests were performed, the simulation tool contained
a bug when dealing with amortization plans. Technically testing with bullet loans
with different maturities is sufficient since the bucketing in CF treats the cash flows
from bullet loans and amortization plans identically. In the meanwhile the problems
in ST were fixed and quick tests show that the closed form approximation agrees
38
closely with ST also when there are amortizing loans in the portfolio; see Appendix
A.
5.1 Total portfolio VaR and ES
The analysis of the test portfolio shows that total ECap figures from the closed
form approximation (CF) and the simulation tool (ST) with 30M simulations17 are
practically the same. The choice of buckets leads to a slight over estimation of
about 3%, see Table 21.
Method Amortization VaR ES EL
CF Yes 7.23% 8.80% 0.427%
ST Yes 7.00% 8.53% 0.423%
CF No 7.46% 9.05% 0.438%
ST No 7.43% 9.01% 0.437%
Table 21: Portfolio ECap and EL figures from the closed form approximation (CF)
and the simulation tool (ST) with 30M simulations. Figures to be read as percent-
ages of total portfolio size.
5.2 Marginals and contributions
In the following, ECap contributions from CF and ST are compared for the het-
erogeneous test portfolio. We choose the portfolio without amortization schedules,
since it is subject to smaller approximation error caused by bucketing. The choice
of buckets and default probabilites have a big impact on accuracy of ECap contribu-
tions. Left panels of Figure 3 show some significant differences between CF and ST
ECap contributions in particular for small exposures. More detailed investigation
17While the maximum number of simulations in ST is typically 300,000, we were able to once
perform a batch run with 30 million simulations upon specific request.
39
revealed that CF gives much higher ECap figures (leading to negative differences
in Figure 3) in cases where the obligor had only one loan with a maturity of just
few days. This is expected as all loans with maturities of less than 3 weeks are
assigned a two-week probability, which leads to over-estimation of ECap contribu-
tions of loans with extremely short maturities. On the other hand, cases where ST
gives 100% higher ECap figures than CF (leading to positive differences in Figure
3) are the ones where the obligor has defaulted already. Indeed, a further analysis
shows that this can be attributed to a problem in scaling of default probabilities
of defaulted loans (p1Y = 1) in the ST if the horizon is more than one year, rather
than an approximation error of the closed form formula. Indeed, default proba-
bilities are scaled down linearly in ST, which leads to an under-estimation of the
expected loss of a defaulted loan if the loan contains payments before time horizon
τh. On the right panels of Figure 3 these outliers are removed and it seems that
there is no clear structural difference, although CF tends to slightly over-estimate
ECap contributions of obligors with large exposures.
Figure 4 shows ECap/EL ratios vs. default probabilities from CF and ST.
Again, results look quite similar except for the fact that ST exhibits few outliers
which would be outside the scale provided on the right hand side of Figure 4, and
thus do not appear.
40
0.0 0.5 1.0 1.5 2.0 2.5 3.0
−30
0−
100
010
020
030
0
Exposure (%)
VaR
C d
iffer
ence
(%
)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
−40
−20
020
40
Exposure (%)
VaR
C d
iffer
ence
(%
)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
−30
0−
100
010
020
030
0
Exposure (%)
ES
C d
iffer
ence
(%
)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
−40
−20
020
40
Exposure (%)
ES
C d
iffer
ence
(%
)
Figure 3: The relative differences of ECap contributions (ST−CFST ) from the closed
form approximation (CF) and simulation tool (ST) (3 million simulations), plotted
against exposure size. Top panels show the differences in VaRC figures and bottom
panels show differences in ESC figures. Black circles are GroupA obligors, red
triangles are GroupB obligors and blue +-signs are GroupC obligors. The right
hand side shows the same information but excludes outliers.
41
(a) CF
0 1 2 3 4 5
010
020
030
040
050
0
Default probability (%)
VaR
C/E
L ra
tio
(b) ST
0 1 2 3 4 5
010
020
030
040
050
0
Default probability (%)
VaR
C/E
L ra
tio
0 1 2 3 4 5
050
010
0015
00
Default probability (%)
ES
C/E
L ra
tio
0 1 2 3 4 5
050
010
0015
00
Default probability (%)
ES
C/E
L ra
tio
Figure 4: ECap/EL ratios from the closed form approximation (CF, left) and simula-
tion tool (ST, right). Upper panels show VaRC/EL ratios and lower panels ESC/EL
ratios plotted against default probabilities. Black circles are GroupA obligors, red
triangles are GroupB obligors and blue +-signs are GroupC obligors.
42
6 Approximation of incremental ECap for fast calcula-
tions
So far, results were presented for total ECap figures as well as ECap contributions
and marginal ECap. We will now deal with the calculation of incremental ECap
figures. As previously introduced, the full calculation of the closed form is time con-
suming and therefore the idea is to split the calculation procedure into two steps: a
pre-calculation and subsequently a final calculation where Y is fixed for calculating
the derivative of Pykhtin’s formula with respect to exposures. In that way, the first
step involves all the most time consuming computations. The second step is prac-
tically just to compile all the results from the first step by using matrix operations.
If the portfolio is sufficiently large, addition of a new mid-sized exposure would only
have a small impact on Y and hence it is possible to obtain accurate approxima-
tions for incremental ECap by running only the second step with updated exposures
and LGD’s and then subtracting the original ECap from this. In the following, the
performance of the closed form approximation with fixed Y in incremental ECap
computations is tested against non-fixed Y for the heterogeneous test portfolio of
section 5 with total exposure 100bne using actual amortization schedules.
Method Amortization VaR ES EL
CF Yes 7.23% 8.80% 0.427%
ST Yes 7.00% 8.53% 0.423%
Table 22: Portfolio VaR, ES and expected loss of the heterogeneous test portfo-
lio from the closed form approximation (CF) and simulation tool (ST) with 30M
simulations. (Computation time: pre-run 1360.89 s, compilation 2.81 s)
Table 22 shows the total ECap and EL figures of the portfolio computed using
the closed form approximation. In Tables 23, 24 and 25 we show the comparisons
43
of incremental ECap with fixed and non-fixed Y of the 10 highest contributing
obligors from each group (GroupA, GroupB and GroupC) assuming that the new
loan to be added to the portfolio has a nominal of 34Me(representing 0.34% of the
total portfolio size), an LGD of 0.4 and a maturity of three years. These figures
are also compared to the incremental ECap figures of similar new obligors without
existing exposure. Differences in incremental ECap figures are relatively small,
but the computation time drops from little over 20 minutes to less than 3 seconds
when choosing a fixed Y . Linear approximation of a contribution is widely used in
practice and available also in the simulation tool. In the linear approximation the
incremental ECap is given by the nominal (wi,new) times LGD (µi,new) of the new
loan times the directional derivative of ECap, i.e.,
LIECapi = wi,newµi,new
µi
∂
∂wiECap . (49)
This can also be written with the help of contributions:
LIECapi =wi,newµi,new
wiµiECapCi . (50)
If the loans have amortization schedules, formulas (49) and (49) are just a bit more
complicated. When the closed form approximation is used this linear approximation
(LinCF) works fairly well and in some cases is even closer to the full calculation
than the approximation with fixed Y . However, the overall performance is not
as good and it is likely that it yields worse results: because all the loan sizes are
different, it is quite obvious that many of the VaR contributions are negative18 (see
Section 3.1.2), which would lead to negative incremental VaRs even if the added
exposure is very large. In the following we compare also the linear approximations
using contributions from CF and ST. As the closed form formula over estimates the
ECap figures due to bucketing, we have scaled the contributions from ST in Tables
18It is recalled that Tables 23-25 only show the 10 most important obligors in terms of VaR
contributions. Hence, no negative incremental VaRs appear for these obligors.
44
23-25 to the same level by multiplying by the ratio of closed form VaR and ST VaR
(= 7.23/7).
Assigning the loan to a completely new obligor with identical characteristics (see
column ”New ob./Exact CF” in the tables below) leads to lower incremental ECap
figures in most cases, as one would normally expect.
45
Incr
emen
tal
VaR
(Me
)In
crem
enta
lE
S(Me
)
Obli
gor
Exp
VaR
C(C
F)
ESC
(CF
)E
LE
xis
tin
gobli
gor
New
ob.
Exis
tin
gob
ligor
New
ob.
Exac
tCF
Fix
edY
CF
Lin
CF
Lin
ST
Exac
tCF
Exact
CF
Fix
edY
CF
Lin
CF
Lin
ST
Exact
CF
Gro
up
A1
239.
3338
.20
43.6
93.
954.
65
4.5
14.
374.
334.
24
5.3
75.
19
4.99
4.8
94.9
2
Gro
up
A2
676.
2521
.06
24.1
42.
214.
22
4.3
74.
263.
534.
24
4.8
95.
05
4.88
3.9
84.9
2
Gro
up
A3
632.
7820
.19
23.1
22.
134.
21
4.3
74.
303.
434.
24
4.8
75.
04
4.92
3.8
44.9
2
Gro
up
A4
407.
0418
.10
19.2
46.
487.
13
7.5
57.
637.
277.
54
7.6
28.
05
8.11
7.0
98.0
6
Gro
up
A5
122.
5617
.21
17.8
611
.50
6.81
7.2
27.
297.
327.
23
7.1
27.
52
7.56
6.9
17.5
3
Gro
up
A6
1040
.91
16.9
520
.50
1.00
2.26
2.3
12.
232.
032.
18
2.7
62.
82
2.69
2.5
32.6
8
Gro
up
A7
698.
2515
.30
17.2
12.
095.
05
5.2
55.
245.
065.
18
5.7
25.
94
5.90
5.4
55.8
9
Gro
up
A8
168.
309.
3310
.12
2.25
6.18
6.9
07.
036.
936.
92
6.7
87.
55
7.63
6.9
97.5
7
Gro
up
A9
340.
828.
519.
212.
106.
45
6.9
67.
095.
756.
98
7.0
67.
60
7.68
5.9
57.6
2
Gro
upA
1049
2.82
7.91
9.58
0.48
2.12
2.2
32.
191.
992.
18
2.6
12.
73
2.66
2.5
82.6
8
Tab
le23:
Ten
Gro
up
Aob
ligor
sw
ith
the
hig
hes
tV
aRco
ntr
ibu
tion
s.T
he
tab
lesh
ows
exp
osu
res,
VaR
and
ES
contr
ibu
tion
sfr
omth
e
close
dfo
rmfo
rmu
la,
exp
ecte
dlo
sses
,in
crem
enta
lV
aRan
dE
Sof
add
ed34
Me
loan
from
run
nin
gcl
osed
form
form
ula
twic
e(E
xac
tCF
),
incr
emen
tal
VaR
an
dE
Sfr
omfa
stap
pro
xim
atio
n(F
ixed
YCF
),li
nea
rap
pro
xim
atio
ns
bas
edon
der
ivat
ives
ofV
aRan
dE
Sfr
omth
ecl
osed
form
form
ula
(Lin
CF
),E
Cap
contr
ibu
tion
sfr
omsi
mu
lati
onto
ol(L
inST
),an
dth
e”e
xac
t”in
crem
enta
lV
aRan
dE
Sof
an
ewob
ligo
rw
ith
iden
tica
lch
ara
cter
isti
csas
the
exis
tin
gob
ligo
rin
the
sam
ero
w(N
ewob
.E
xac
tCF
).A
llfi
gure
sin
Me
.
46
Incr
emen
tal
VaR
(Me
)In
crem
enta
lE
S(Me
)
Ob
ligo
rE
xp
VaR
C(C
F)
ES
C(C
F)
EL
Exis
tin
gob
ligo
rN
ewob
.E
xis
tin
gob
ligor
New
ob
.
Exac
tCF
Fix
edY
CF
Lin
CF
Lin
ST
Exact
CF
Exact
CF
Fix
edY
CF
Lin
CF
Lin
ST
Exact
CF
Gro
up
B1
2328
.28
245.
4830
1.30
3.02
6.08
6.1
05.4
34.8
54.8
07.4
87.4
96.6
76.3
76.0
9
Gro
up
B2
3013
.45
227.
1129
9.89
1.79
4.42
4.4
33.8
13.1
03.2
05.8
45.8
45.0
34.7
44.3
5
Gro
up
B3
2011
.46
176.
1222
4.61
1.79
4.90
4.9
24.4
03.7
93.9
56.2
86.2
95.6
05.2
85.1
8
Gro
up
B4
2391
.97
109.
2915
5.32
0.68
2.71
2.7
22.4
02.0
02.1
13.8
83.8
93.4
23.4
43.0
6
Gro
up
B5
1834
.55
87.5
212
2.12
0.63
3.03
3.0
52.4
82.1
22.5
64.2
24.2
43.4
63.5
03.6
0
Gro
up
B6
1584
.45
83.6
112
1.33
0.51
2.57
2.5
72.0
31.7
22.1
13.7
03.7
02.9
42.9
23.0
6
Gro
up
B7
685.
3070
.95
87.6
11.
005.
095.1
65.0
24.5
94.8
06.4
06.4
66.2
06.0
36.0
9
Gro
up
B8
264.
5960
.32
64.8
93.
629.
9810
.29
9.9
49.8
410.2
710
.69
10.9
910.7
09.9
611.0
3
Gro
up
B9
258.
7358
.30
64.1
92.
649.
399.4
98.9
28.7
59.3
910
.25
10.3
39.8
29.2
310.3
6
Gro
up
B10
239.
7856
.99
60.9
33.
709.
9810
.27
10.
24
10.
41
10.2
710
.71
10.9
910.9
410.2
911.0
3
Tab
le24:
Ten
Gro
upB
ob
ligor
sw
ith
the
hig
hes
tV
aRco
ntr
ibu
tion
s.T
he
tab
lesh
ows
exp
osu
res,
VaR
and
ES
contr
ibu
tion
sfr
omth
e
close
dfo
rmfo
rmu
la,
exp
ecte
dlo
sses
,in
crem
enta
lV
aRan
dE
Sof
add
ed34
Me
loan
from
run
nin
gcl
osed
form
form
ula
twic
e(E
xac
tCF
),
incr
emen
tal
VaR
an
dE
Sfr
om
fast
app
roxim
atio
n(F
ixed
YCF
),li
nea
rap
pro
xim
atio
ns
bas
edon
der
ivat
ives
ofV
aRan
dE
Sfr
omth
ecl
osed
form
form
ula
(Lin
CF
),E
Cap
contr
ibu
tion
sfr
omsi
mu
lati
onto
ol(L
inST
),an
dth
e”e
xac
t”in
crem
enta
lV
aRan
dE
Sof
an
ewob
ligo
rw
ith
iden
tica
lch
ara
cter
isti
csas
the
exis
tin
gob
ligo
rin
the
sam
ero
w(N
ewob
.E
xac
tCF
).A
llfi
gure
sin
Me
.
47
Incr
emen
tal
VaR
(Me
)In
crem
enta
lE
S(Me
)
Ob
ligo
rE
xp
VaR
C(C
F)
ES
C(C
F)
EL
Exis
tin
gob
ligo
rN
ewob
.E
xis
tin
gob
ligo
rN
ewob
.
Exac
tCF
Fix
edY
CF
Lin
CF
Lin
ST
Exac
tCF
Exac
tCF
Fix
edY
CF
Lin
CF
Lin
ST
Exact
CF
Gro
up
C1
331.
3212
0.66
130.
065.
3110
.66
10.4
610
.16
10.6
110
.16
11.3
611.
1610
.96
10.4
810.9
4
Gro
up
C2
495.
7512
0.10
142.
741.
987.8
17.6
56.
636.6
67.
09
8.98
8.8
07.8
77.7
38.2
8
Gro
up
C3
316.
9511
9.42
128.
435.
3210
.69
10.4
610
.30
10.9
810
.16
11.3
911.
1611
.08
10.7
210.9
4
Gro
up
C4
599.
2498
.55
125.
500.
985.1
75.1
34.
765.0
04.
56
6.44
6.4
06.0
66.3
75.7
7
Gro
up
C5
359.
9994
.58
110.
651.
737.6
77.5
27.
197.7
37.
09
8.83
8.6
78.4
18.5
08.2
8
Gro
up
C6
326.
2089
.24
89.3
173
.71
7.6
27.4
87.
487.3
97.
48
7.61
7.4
97.4
96.5
77.4
9
Gro
up
C7
478.
6581
.58
104.
310.
815.1
15.0
24.
664.7
84.
56
6.38
6.2
85.9
66.2
05.7
7
Gro
up
C8
670.
1481
.49
110.
130.
634.2
64.1
93.
333.6
53.
73
5.52
5.4
44.5
05.3
04.8
8
Gro
up
C9
512.
8475
.09
98.8
80.
634.2
24.1
64.
004.1
53.
73
5.46
5.3
95.2
75.6
04.8
8
Gro
up
C10
425.
8771
.08
90.9
40.
725.0
44.9
54.
565.0
04.
56
6.30
6.2
05.8
46.2
85.7
7
Tab
le25:
Ten
Gro
up
Cob
ligor
sw
ith
the
hig
hes
tV
aRco
ntr
ibu
tion
s.T
he
tab
lesh
ows
exp
osu
res,
VaR
and
ES
contr
ibu
tion
sfr
omth
e
close
dfo
rmfo
rmu
la,
exp
ecte
dlo
sses
,in
crem
enta
lV
aRan
dE
Sof
add
ed34
Me
loan
from
run
nin
gcl
osed
form
form
ula
twic
e(E
xac
tCF
),
incr
emen
tal
VaR
an
dE
Sfr
omfa
stap
pro
xim
atio
n(F
ixed
YCF
),li
nea
rap
pro
xim
atio
ns
bas
edon
der
ivat
ives
ofV
aRan
dE
Sfr
omth
ecl
osed
form
form
ula
(Lin
CF
),E
Cap
contr
ibu
tion
sfr
omsi
mu
lati
onto
ol(L
inST
),an
dth
e”e
xac
t”in
crem
enta
lV
aRan
dE
Sof
an
ewob
ligo
rw
ith
iden
tica
lch
ara
cter
isti
csas
the
exis
tin
gob
ligo
rin
the
sam
ero
w(N
ewob
.E
xac
tCF
).A
llfi
gure
sin
Me
.
48
7 Conclusion
A closed form formula for calculating economic capital contributions and incremen-
tal economic capital in the multi-factor default-mode model based on Pykhtin (2004)
was introduced. The performance of the formula was tested against a commonly
used commercial simulation tool and independent Monte Carlo simulations. The
results show that the closed form approximation is very good for approximating the
total economic capital of a portfolio (except if risk factor correlations are very low)
and also the incremental economic capital of a newly added loan. For ECap con-
tributions the formula can sometimes be inaccurate. Specifically, problems occur in
the computation of VaR contributions for small portfolios, partly because the loss
distribution in the tested model is piecewice constant, which violates the continuity
assumption used for the derivation of the approximate formula. The problem could
be possibly solved by having stochastic loss given defaults, which would also be more
realistic from the practical point of view. Despite these problems VaR contributions
for large portfolios in the closed form approximation are in most cases reasonable
and within the range of simulation error of the commercial simulation tool used for
comparison. For expected shortfall contributions the results are even better.
Another observation is that VaR contributions in general are not good for linear
approximation of incremental ECap, because even very large new loans could have
negative incremental ECap figures if VaR contributions are negative whereas the
actual difference between updated and original ECap figures could be a large positive
number. Linear approximation of incremental ECap using ES contributions is safer,
as ES takes the whole tail of the loss distribution into account and does not yield
completely unacceptable results, although the approximation error can be well over
10%.
The performance of the closed form approximation (CF) and the commercial
simulation tool (ST) are summarized in Tables 26 and 27.
49
Factor correlation
High correlation Zero correlation Correlation one
Portfolio size VaR ES VaR ES VaR ES
CF ST CF ST CF ST CF ST CF ST CF ST
SMALL 4(1) ∼ 4(1) ∼ ∼ 4(1) 4(1) ∼ ∼ ∼ 4(1) ∼
LARGE homog. 4 4 4 4 6 4 6 4 4 4 4 4
LARGE heterog. 4 4 4 4 N/A N/A N/A N/A N/A N/A N/A N/A
Table 26: Performances of total portfolio ECap for CF and ST. 4= good, ∼= fair,
6= poor. (1) Except for very low granularity (Portfolio S3).
ECap contribution Incremental ECap
Portfolio size VaRC ESC IVaR IES
CF ST CF ST Fixed Y CF LinST Fixed Y CF LinST
SMALL 6(1) 6(2) ∼ 6(3) N/A N/A N/A N/A
LARGE homog. ∼ ∼(4) 4 4(4) N/A N/A N/A N/A
LARGE heterog. ∼ ∼(5) 4 4(5) 4 ∼(6) 4 ∼(6)
Table 27: Performance of ECap contributions and incremental ECap for CF and
ST for highly correlated risk factors. 4= good, ∼= fair, 6= poor. (1) More suitable
for capital allocation purposes, but not accurate due to non-stochastic LGD’s. (2) Not useful for
capital allocation, but for Portfolio S2 actually theoretically correct. (3) Very close to CF for
Portfolio S2, but much worse on Portfolio S1 because total ES on Portfolio S1 is inaccurate for ST.
(4) Slightly worse for capital allocation than CF because identical obligors have non identical VaRC
and ESC, respectively. (5) ST produces incorrectly high ECap contributions for already defaulted
loans, while CF can produce (too) high ECap contributions for loans with very short maturity due
to conservative bucketing approach. (6) Very close to CF for the obligors investigated more closely,
but overall performance worse due to the general problem of linear approximation, particularly for
obligors with negative (or zero) VaR contributions, including new obligors.
50
References
Artzner, P., Delbaen F., Eber J.-M. and Heath, D. (1999), Coherent Measures of
Risk, Mathematical Finance 9, 1999, pages 203–228.
K. Dullmann and N. Puzanova (2011), Systemic risk contributions: a credit port-
folio approach, Discussion Paper Series 2: Banking and Financial Studies 08.
Deutsche Bundesbank
S. Ebert and E. Lutkebohmert (2012), Treatment of double default effects within
the granularity adjustment for basel II, Journal of Credit Risk 7, 2011, pages 3–33.
P. Gagliardini and C. Gourieroux, Granularity adjustment for risk measures: Sys-
tematic vs unsystematic risks, International Journal of Approximate Reasoning
54, 2013, pages 717– 747.
Gordy, M. (2003), A risk-factor model foundation for ratings-based bank capital
rules, Journal of Financial Intermediation 12(3), 2003, pages 199–232.
M. Gordy and J. Marrone (2013), Granularity adjustment for mark-to-market
credit risk models, Journal of Banking and Finance 36, 2012, pages 1896–1910.
Gourieroux C., Laurent J-P., Scaillet, O. (2000), Sensitivity analysis of values at
risk, Journal of Empirical Finance 7, 2000, pages 225–245.
Kalkbrener, M. (2005), An axiomatic approach to capital allocation, Mathematical
Finance 15(3), 2005, pages 425–437.
Martin, R., Wilde, T. (2002), Unsystematic credit risk, Risk, November 2002,
pages 123–128.
Pykhtin, M. (2004), Multi-factor adjustment, Risk, March 2004, pages 17–31.
Voropaev, M. (2011), Analytical framework for credit portfolio risk measures, Risk,
May 2011, pages 72–78.
51
A Incorporating amortization schedules
The comparison of the ST and CF calculations following the recent update of the
simulation tool now confirms that both approaches lead to approximately the same
results irrespective of amortization schedules. ST and CF are compared based on the
test portfolios ”Portfolio S2” and ”Portfolio L1”. All the inputs for these portfolios
are kept the same as described in the main report except for the time horizon which
is now 3 years, the quantile which is set at 99.7%, and (if Amortization = Yes in
the two tables below) all loans are assumed to be repaid in three equal parts; first
payment in 1 year, second payment in 2 years and the last payment in 4 years.
Tables 28 and 29 show the detailed results.
Method Amortization Exposure VaR ES EL
CF Yes 10000 347.55 555.07 7.09
ST Yes 10000 351.88 556.47 7.09
CF No 10000 479.83 730.43 10.64
ST No 10000 489.36 734.29 10.64
Table 28: ”Portfolio S2”: 99.7% 3-year portfolio VaR, ES and EL with and without
yearly amortization, using the closed form approximation (CF) and the simulation
tool (ST). All figures in Me.
52
Method Amortization Exposure VaR ES EL
CF Yes 10000 144.26 237.85 4.99
ST Yes 10000 143.30 235.85 4.99
CF No 10000 199.09 313.62 7.49
ST No 10000 197.95 311.87 7.49
Table 29: ”Portfolio L1”: 99.7% 3-year portfolio VaR, ES and EL with and without
yearly amortization, using the closed form approximation (CF) and the simulation
tool (ST). All figures in Me.
53
B Comparison of MECap, ECapC and IECap
Suppose we have a portfolio of three (uncorrelated) obligors each having one loan:
loan to obligor A has notional wA = 1Me and default probability pA = 5%, loan
to obligor B has notional wB = 3Me and default probability pB = 5%, and loan to
obligor C has notional wC = 3.5Me and default probability pC = 2.5%. We assume
that the loss given default of each loan is one. Then the expected losses of A, B
and C are ELA = 50ke, ELB = 150ke and ELC = 87.5ke. The loss distribution of
the portfolio is shown in Figure 30.
Defaults Probability Loss (Me ) Cumulative Probability
None 87.99% - 87.99%
A 4.63% 1.00 92.63%
B 4.63% 3.00 97.26%
C 2.26% 3.50 99.51%
AB 0.24% 4.00 99.76%
AC 0.12% 4.50 99.88%
BC 0.12% 6.50 99.99%
ABC 0.01% 7.50 100.00%
Defaults Probability Loss (Me ) Cumulative Probability
None 87.99% - 87.99%
A 4.63% 1.00 92.63%
C 2.26% 3.50 94.88%
AC 0.12% 4.50 95.00%
B 4.63% 5.00 99.63%
AB 0.24% 6.00 99.88%
BC 0.12% 8.50 99.99%
ABC 0.01% 9.50 100.00%
80 85 90 95 100
02
46
8
Probabililty (%)
Loss
(M
€)
80 85 90 95 100
02
46
8
Probabililty (%)
Loss
(M
€)
Table 30: Loss distribution before (left) and after (right) the added loan of 2Me for
obligor B. Before addition: EL=0.2875Me and VaR (99%)=3.2125Me . After ad-
dition: EL=0.3875Me and VaR (99%)=4.6125Me . The graphs show the loss dis-
tribution, reduced by the expected portfolio loss. The red circle represents the 99%
VaR; it corresponds to a loss state in which only C defaults (left) and in which only
B defaults (right).
54
It is obvious that 99% VaR is 3.2125Me = 3.5Me-0.2875me and that only C
is contributing to the VaR. Derivatives of 99% quantile of the loss distribution with
respect to exposures are zero to directions of loans to obligors A and B, and hence
for i ∈ {A,B}
∂
∂wiVaR99,i = − ∂
∂wiELi = − ∂
∂wipiwi = −pi = −0.05.
Therefore, the VaR contribution for obligor B is negative, namely 3Me ×
(-0.05) = - 150ke .
Suppose now that B needs a new loan of 2Me. VaR of the updated portfolio
is 4.6125Me= 5Me-0.3875Me (see Figure 30) and the actual incremental VaR
is 1.4Me.
If we remove the loan B, the 99% VaR increases by 150ke, because the 99%
quantile remains the same, but EL reduces by 150ke: the marginal VaR for
obligor bf B is also negative.
If we use the linear approximation using either VaRC or MVaR for determining
the incremental ECap using formula (3) and (50), the result is
2Me
3Me×VaRC = −100ke
In other words, linear approximation suggests that we can reduce the VaR by 100ke,
whereas the actual credit VaR increases by 1.4Me. For expected shortfall, linear
approximation leads at least to a positive incremental ES estimate for obligor B,
although it still underestimates the true intremental ES in our example; see Table
31 for details and additional comparisons for the other obligors.
This also illustrates why credit VaR is not the best measure for economic
capital: the first row of Table 31 shows that the credit VaR of the portfolio actually
decreases when a new loan of 2Me is given to obligor A, due to the fact that the
VaR does not change although the EL increases. This anomaly is not present when
using expected shortfall: addition of a 2Me loan to any of the obligors does lead to
a positive incremental expected shortfall.
55
Obligor Exp PD EL MVaR VaRC LI.VaRM LI.VaRC IVaR ESC LI.ESC IES
A 1Me 5% 50 -50 -50 -100 -100 -100 84.4 168.8 168.8
B 3Me 5% 150 -150 -150 -100 -100 1,400 253.2 168.8 1,310.8
C 3.5Me 2.5% 87.5 412.5 3,412.5 235.7 1,950 1,950 3,101.6 1,772.3 1,923.3
Total 7.5Me - 287.5 - 3,212.5 - - - 3,439.2 - -
Table 31: All figures are to be understood in ke unless otherwise stated. Compar-
ison of linear approximations and exact incremental VaR and incremental ES (in
grey columns) when a loan of 2Me is added for obligor A, B and C, respectively.
Exp = exposure, PD = default probability, LI.VaRM = linear approxiation of IVaR
using MVaR (marginal VaR), LI.VaRC/LI.ESC = linear approxiation of IVaR/IES
using VaRC (VaR contribution) and ESC (ES contribution), and IVaR/IES = exact
incremental VaR/ES.
56
C Bucketing of loans for ECap approximation
The closed form formula for ECap requires bucketing of cashflows according to
payment dates. For 3-year ECap calculations cashflows are bucketed into 6 buckets:
Bucket 0-3 weeks 3 weeks - 3 months 3-8 months 8-15 months 15-27 months 27+ months
PD 2 weeks 2 months 6 months 1 year 2 year 3 year
Table 32: Buckets and attached default probabilities.
The reason for bucketing is that the computation time in the closed form solution
increases significantly if the number of cashflows is large. This is due to the fact that
the closed form solution requires a large number of matrix operations and the sizes
of the matrices depend on the number of factors, loans and cashflow dates. The
dependence is quadratic i.e., doubling the number of factors, loans or maturities,
quadruples the size of the largest matrices.
There is no obvious way to choose the buckets or default probabilities associated
to each bucket. The current choice of buckets and default probabilities of Table 32
is based on the following:
1. The number of loans in each bucket is of the same magnitude (except the last
one, which is much larger).
2. For the first bucket, 2 weeks default probability is used because using longer
PD’s could lead to a significant over-estimation of ECap contributions of loans
with very short maturities.
3. The buckets and PD’s are chosen such that the ratio between actual maturity
and used ”PD-rating” is not much less than 12 or much more than 2 (except
for the first and last bucket for which this is impossible to control).
57
4. The default probabilities are not chosen to be the mid-points of buckets, be-
cause in most cases a longer PD implies higher VaRC and we want to be
rather conservative. On the other hand it is not desirable to attach e.g. the
3 months PD (right end point of the bucket) to a 22 days loan. Also, taking
longer default probability does not always mean that ECap contributions are
overestimated; according to Figure 1 in the main part of this paper the impact
can be the opposite especially for obligors with high default probabilities.
58
06
PD
2w
3w
3m
6
PD
2m
8m
6
PD
6M
15m
6
PD
1Y
27m
6
PD
2Y
-
6
PD
3Y
Figure 5: Buckets and attached default probabilities.
59
D Pykhtin’s formula
The approximate formula for q-quantile of portfolio loss distribution tq(L) given by
[Pykhtin, 2004] is
tq(L) = tq(L)− 1
2`′(y)
[ν ′(y)− ν(y)
(`′′(y)
`′(y)+ y
)] ∣∣∣∣y=Φ{−1}(1−q)
(51)
where Φ is the cdf of standard normal distribution, µi and σi are the expectation
and variance of loss given default of obligor i,
tq(L) = `(y) :=M∑i=1
wiµipi(y) (q-quantile of one factor approximation),
`′(y) =M∑i=1
wiµip′i(y)
`′′(y) =
M∑i=1
wiµip′′i (y)
pi(y) = Φ
Φ−1(pi)− aiy√1− a2
i
(conditional default probability on condition Y = y)
p′i(y) = − ai√1− a2
i
Φ′
Φ−1(pi)− aiy√1− a2
i
p′′i (y) =
a2i
1− a2i
Φ−1(pi)− aiy√1− a2
i
Φ′
Φ−1(pi)− aiy√1− a2
i
ai = riρi
ρi = cor(Yi, Y )
ν(y) = ν∞(y) + νGA(y) (conditional variance of tq)
ν∞(y) =M∑i=1
M∑j=1
wiwjµiµj[Φ2(Φ−1(pi(y)),Φ−1(pj(y)), ρYij)− pi(y)pj(y)
]νGA(y) =
M∑i=1
w2i
(µ2i
[pi(y)− Φ2(Φ−1(pi(y)),Φ−1(pi(y)), ρYii )
]+ σ2
i pi(y))
60
ν ′∞(y) = 2
M∑i=1
M∑j=1
wiwjµiµj p′i(y)
Φ
Φ−1(pj(y))− ρYij Φ−1(pi(y))√1− (ρYij)
2
− pj(y)
ν ′GA(y) =
M∑i=1
w2i p′i(y)
µ2i
1− 2 Φ
Φ−1(pi(y))− ρYii Φ−1(pi(y))√1− (ρYii )
2
+ σ2i
and
ρYij =rirj
∑Nk=1 αikαjk − aiaj√(
1− a2i
) (1− a2
j
) (conditional asset correlation).
Expected shortfall is given by
ESq(L) = ESq
(L)
+ ∆ ESq(L),
where
ESq
(L)
=1
1− q
M∑i=1
wiµi Φ2
(Φ−1(pi),Φ
−1(1− q), ai) ∣∣∣∣
y=Φ{−1}(1−q)
∆ ESq(L) =1
2(1− q)ϕ (y)
ν(y)
`′ (y)
∣∣∣∣y=Φ{−1}(1−q)
and ϕ is the density function of the standard normal distribution.
61