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

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TV

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ean

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SS

TE

Sm

ean

min

max

Tota

l10

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148.

7614

5.65

64.1

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8.63

151.9

715

3.48

141.

2116

6.73

239.3

024

1.28

235.5

324

7.8

3

Gro

up

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2832

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60

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up

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50.

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up

A3

827.

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up

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up

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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

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4462

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451

.24

472.

69

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3.80

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1.16

1.41

1.32

1.2

51.

38

Gro

up

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120.

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20

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22

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up

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384.

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580.

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up

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384.

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up

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up

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2.78

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Gro

up

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384.

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Gro

up

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028

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Gro

up

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384.

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14.3

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5122

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415

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up

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384.

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8.54

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06

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015

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Gro

up

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384.

6215

.15

14.6

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5122

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415

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816

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Gro

up

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384.

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5122

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415

.13

14.1

615

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322.

82

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923

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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

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Gro

up

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384.

6211

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10.7

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5919

.12

11.7

811

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10.8

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20.9

222.

48

21.9

423

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Gro

up

C3

384.

6215

.66

14.8

27.

5322

.92

15.9

916

.05

14.7

017

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26.8

327.

67

26.9

128

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Gro

up

C4

384.

6215

.66

14.5

07.

5326

.77

15.9

915

.96

14.3

617

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26.8

327.

59

26.8

628

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Gro

up

C5

384.

6215

.66

14.3

87.

5322

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15.9

916

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14.6

217

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67

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428

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Gro

up

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384.

6215

.66

15.0

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5322

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15.9

916

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14.7

817

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64

27.0

528

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Gro

up

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384.

6220

.78

20.0

311

.30

30.4

121

.41

21.4

519

.98

22.9

634

.03

33.

63

32.7

334

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Gro

up

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384.

6220

.78

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911

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.41

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Gro

up

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384.

6211

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1619

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11.7

811

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Gro

up

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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-

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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