Master Thesis∗RDF, Risk Dynamics into the future†

Bo GUAN‡

Version Final

June 21, 2010

∗This is part of the Erasmus Mundus MSc Program ”Mathmods”†This project was carried out in the R&D, AIS Group‡Email: guanbonju@gmail.com

1

Contents

1 Introduction 3

2 Background 42.1 Definitions and Theoretical Background . . . . . . . . . . . . . 4

2.1.1 Expected Loss . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Unexpected Loss . . . . . . . . . . . . . . . . . . . . . 52.1.4 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 RDF(Risk Dynamics into the Future) . . . . . . . . . . . . . . 52.2.1 About AIS Group . . . . . . . . . . . . . . . . . . . . . 52.2.2 Introduction to the RDF method . . . . . . . . . . . . 6

3 Main Problem 63.1 Macroeconomics Model . . . . . . . . . . . . . . . . . . . . . . 63.2 Loss function . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Calculating the loss distribution . . . . . . . . . . . . . . . . . 73.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4.1 Monte Carlo Simulations . . . . . . . . . . . . . . . . . 73.4.2 Delta Nascent Method . . . . . . . . . . . . . . . . . . 83.4.3 Saddlepoint Approximation . . . . . . . . . . . . . . . 10

4 Numerical tests and conclusion 164.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 An example of two segments with two variables . . . . 164.1.2 Another example of one segment with two variables . . 16

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2.1 Results from different methods for Example (4.1.1) . . 174.2.2 Results from different methods for Example (4.1.2) . . 18

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Appendices 20

A Calculating the saddlepoint using iterative method 20

2

Abstract

In this paper, I give an introduction to the RDF(Risk Dynamicsinto the Future) developed by AIS Group. The RDF method is atool to help managers to determine the risk under certain economicscenarios. And my work focused on improving the numerical methodto calculate the loss distribution. I presented three methods: MonteCarlo simulations, Nascent Delta method and a direct SaddlepointApproximation method. Comparison of the numerical results are alsoincluded for some simple examples.

1 Introduction

The financial crisis since 2007 has shown the importance of risk managementin today’s financial world. The crisis which was triggered by a liquidityshortfall in the United State Banking System, has resulted in the collapse oflarge financial institutions, the ”bail out” of banks by national governmentsand downturns in stock markets around the world. It is considered to be theworst financial crisis since the Great Depression in 1930s1.

The collapse of a global housing bubble also caused the value of securitiestied to the real estate to fall steeply. The Investor confidence was damaged,causing large losses in the stock markets. It is commonly understood thatthe credit rating agencies failed to price the risk with the mortgage-related fi-nancial product and the government did not adjust their regulatory practicesto address 21st century financial markets2.

These all shows the importance of risk management. A good understand-ing of potential risk is vital to many companies. So we seek ways to quantifyrisks. Credit risk management is risk assessment that comes in an invest-ment. Risk often comes in investing and in the allocation of capital. Therisks must be assessed so as to derive a sound investment decision. And de-cisions should be made by balancing the risks and returns. The risk of lossesthat result in the default of payment of the debtors is a kind of risk thatmust be expected. A bank to keep substantial amount of capital to protectits solvency and to maintain its economic stability. The greater the bank isexposed to risks, the greater the amount of capital must be when it comesto its reserves, so as to maintain its solvency and stability.

1Three top economists agree 2009 worst financial crisis since great depression; risksincrease if right steps are not taken. REUTERShttp://www.reuters.com/article/pressRelease/idUS193520+27-Feb-2009+BW20090227

2Declaration of G20, Whitehousehttp://georgewbush-whitehouse.archives.gov/news/releases/2008/11/20081115-1.html

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2 Background

Here, I will briefly introduce the terminologies and the AIS Group where Iconducted this work.

2.1 Definitions and Theoretical Background

2.1.1 Expected Loss

In probability theory, the attribute expected always refers to an expectationor mean value, and this is also the case in risk management. The basicideas is: The bank assigns to each obligator a default probability (DP), a lossfraction called the loss given default (LGD), describing the fraction of theloan’s exposure expected to be lost in case of default, and the exposure atdefault (EAD) subject to be lost in the considered time period. The loss ofany obligor is then defined by a loss variable

L = EAD × LGD × L with L = 1D, P(D) = DP (1)

where D denotes the event that the obligor defaults in a certain period of time(most often one year), and P(D) denotes the probability of D.The model isin a probability space (Ω,F ,P), consisting of a sample space Ω, a σ-AlgebraF , and a probability measure P.

In this setting it very natural to define the expected loss(EL) of any cus-tomer as the expectation of its corresponding loss variable L, namely

EL = E[L] = EAD × LGD × P(D) = EAD × LGD ×DP (2)

To obtain representation (2) of the EL, we need some additional assump-tion on the constituents of Formula (1), for example, the assumption thatEAD and LGD are constant values. This is not necessarily the case underall circumstances. There are various random variable due to uncertaintiesin amortization, usage, and other drivers of EAD up to the chosen planninghorizon. In such cases the EL is still given by Equation (2) if one can as-sume that the exposure, the loss given default,and the default event D areindependent and EAD & LGD are the expectations of some underlying ran-dom variables. But even the independence assumption is questionable andin general very much simplifying.

2.1.2 Ratings

Basically ratings describe the creditworthiness of customers. Hereby quan-titative as well as qualitative information is used to evaluate a client. In

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practice, the rating procedure is often more based on the judgement andexperience of the rating analyst than on pure mathematical procedures withstrictly defined outcomes. It turns out that in the US and Canada, mostissuers of public debt are rated at least by two of the three main rating agen-cies Moody’s, S&P, and Fitch. Their reports on corporate bond defaults arepublicly available, either at local offices or for web access.

2.1.3 Unexpected Loss

Holding capital as a cushion against expected loss is not enough. In fact, thebank should also save money for covering unexpected losses exceeding theaverage experienced losses from past history. As a measure of the magnitudeof the deviation of losses from the EL, the standard deviation of the lossvariable L as defined in (1) is a natural choice. This quantity is called theUnexpected Loss(UL), defined by

UL =

√V[L] =

√V[EAD × SEV× L] (3)

2.1.4 Value at Risk

We seek others ways to quantify risk capital, hereby taking a target levelof statistical confidence into account. If c is the selected confidence level,VaR corresponds to the 1-c lower-tail level. For example, for a 95 percentconfidence level, the probability of loss less or equal to VaR is 95%

2.2 RDF(Risk Dynamics into the Future)

2.2.1 About AIS Group

Headquartered in Barcelona, Spain, AIS, Aplicaciones de Inteligencia Arti-ficial, S.A. is a multinational with over 20 years experience, specializing inthe development of automatic systems to help companies in decision-makingprocesses.

The activities of AIS are threefold: consultancy, software design and cre-ation of statistical and optimization models. Its solutions include: quantita-tive credit risk assessment, application of quantitative techniques in market-ing, risk control in the insurance sector, optimisation of paper sizes, monitor-ing of production, consumption forecasts and optimum distribution of energy,cash and daily publications. It also offers expert consultancy in the differentdivisions. The activities of AIS in credit risk analysis in the financial sectorhave always been its core business.

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2.2.2 Introduction to the RDF method

The RDF(Risk Dynamics into the future) method, developed by AIS, is anopen project, allowing, by way of sophisticated econometric models, the sim-ulation of unfavourable economic scenarios, the calculation of the distributionof losses in these scenarios and support for strategic planning and businessdevelopment. It provides a new method for the calculation of stress testingon the economic capital of credit risk.

3 Main Problem

One criteria that the RDF method use is that the sources of variability whichproduce the risk result essentially from the macroeconomic situation.

3.1 Macroeconomics Model

The logit of the PD is a linear combination (a linear regression) of macroe-conomic variables. The notation used is below. In the logit model, theprobability of default of an obligor in sector i is related to the sector credit-worthiness index,

Pi(Xt) =1

1 + exp(−(~βTi (B) ·Xt + γi,t))

(4)

Xt = vector of macroeconomic variables

γi,t = Error of the linear regression model of the portfolio i = 1, ..m and time t.

~βi(B) = vector of polynomials of lag operator

3.2 Loss function

An important fact used for computation is that, conditional on a scenario,obligor defaults are independent. In most cases, a Monte Carlo simulationcan be applied to determine portfolio conditional losses. But we want moreeffective computational tools. If a portfolio contains a very large numberof obligors, each with a small marginal contribution, then the Law of LargeNumbers can be applied to estimate the conditional portfolio losses. Asthe number of obligors approaches infinity, the conditional loss distribution

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converges to the mean losses over the scenario. Hence the total loss functionare given by the sum of the expected losses of each portfolio.

L(Xt) =∑

i=1,m

Pi(Xt)Ki =∑

i=1,m

Ki

1 + exp(−(~βTi (B) ·Xt + γi,t))

=∑

i=1,m

Ki

1 + exp(−(Si,t))

Xt = vector of macroeconomic variables

Ki = EADi × LGDi for each portfolio i = 1, ..m

γi,t = Error of the linear regression model of the portfolio i = 1, ..m and time t.

Si,t = ~βTi (B) ·Xt + γi,t

~βi(B) = vector of polynomial of lag operator

3.3 Calculating the loss distribution

In order to calculate the distribution of loss, we use the dirac delta function tocapture the variable values corresponding to a certain level of loss as follows:

P (Y = y) =

∫∫∫

X: X∈<n

Ω (X) δ (y − L (X)) dnX (5)

P ( ) Loss probability density

Y Loss

L(X) Loss function with respect to the risk indicators

Ω(X) Joint probability distribution of risk indicators

δ(s) Delta de Dirac : if δ(x) = +∞ at 0, and is 0 elsewhere. With the property+∞∫

−∞

δ(x) = 1

3.4 Methods

3.4.1 Monte Carlo Simulations

Monte Carlo Simulation is very easy but sometime it is very time-consumingfor a high level of accuracy. The method is just to generate the random

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variables according to their distributions, usually Multivariate normal distri-bution, like:

X ∼ N(M Σ) (6)

where M is the vector of mean values and Σ is the variance-covariance matrix.And for each realization of X, compute the loss

Y = L(X) (7)

Then we analysis the density of Y and calculate the relevant quantities likeVaR and Expected Loss. The number of simulations we used in the tests are10,000,000

3.4.2 Delta Nascent Method

One main difficulty in the integral is the delta function. So we try to usedifferent ways to calculate or remove the delta function. Here, the firstmethod is to approximate it. The delta function can be viewed as the limitof a sequence of functions

δ(x) = lima→0+

δa(x) (8)

where δa(x) is sometimes called a nascent delta function. This limit is meantin a weak sense:

lima→0+

+∞∫

−∞

δa(x)f(x)dx = f(0) (9)

for all smooth functions with compact support.Here we will use the Gaussian kernel as the nascent delta.

δa (x) =1√2π a

e−x2

2 a2 (10)

Then to approximate the probability density function P (·) of the loss Y, weuse

Pa (Y = y) =

∫∫∫

X: X∈<n

Ω (X) δa (y − L (X)) dnX (11)

as

Lima→0

Pa (Y = y) = P (Y = y) (12)

8

Now, we have change the problem into choosing the parameter a for nascentdelta and then calculate the approximated Pa(·).

First, we assume that we have found a good parameter a, then I will showhow to calculate the Pa(·).

Pa (y) =

∫∫∫

X

e−(y−L(X))2

2a2

√2πa

Ω (X) dX

=

∫∫∫

X

e−(y−L(X))2

2a2

√2πa

e−12(X−M)Tr.Σ−1.(X−M)

(2π)n2 |Σ|

12

dX

=1

a(2π)n+1

2 |Σ|12

∫∫∫

X

e−(y−L(X))2

2a2 − 12(X−M)TrΣ−1(X−M)dX

Now we consider the exponent part

Elip (X, y) = −(y − L (X))2

2a2− 1

2(X −M)TrΣ−1 (X −M) (13)

We use the second order Taylor expansion around a fixed point X0, then

Elip (X, y) ≈ Elip (X0, y) + Elip′ (X0, y) (X −X0)

+1

2(X −X0)

TrElip′′ (X0, y) (X −X0) (14)

And we want to eliminate the first order term. Notice that the Elip functionis differentiable, so for fixed y, the local maxima has the property that makesthe first order derivative zero. i.e.

∂Elip (X0,y, y)

∂X=

y − L (X0,y)

a2L′ (X0,y)− (X0,y − µ)TrΣ−1 = 0 (15)

The calculation of X0,y has to be done on each value of y(amout of losses),we could use Newton’s Method or other Quasi-Newton Methods. In theimplementation, we used BFGS method, one of Quasi-Newton Methods.

Then, equation (14) becomes

Elip (X, y) ≈ Elip (X0,y, y) +1

2(X −X0,y)

Tr.Elip′′ (X0,y, y) . (X −X0,y)

= −1

2

(y − L (X0,y))2

a2− 1

2(X0,y −M)TrΣ−1 (X0,y −M) +

1

2(X −X0,y)

Tr.[y − L (X0,y)

a2L′′ (X0,y)− 1

a2L′ (X0,y) .L′(X0,y)

Tr − Σ−1

]. (X −X0,y)

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And

Pa (y) =1

(2π)12a

1

(2π)n2 |Σ|

12

e−(y−L(X0,y))

2

2a2 − 12(X0,y−M)TrΣ−1(X0,y−M)

∫∫∫

X

e− 1

2(X−X0,y)Tr

[Σ−1−

y−L(X0,y)a2 L′′(X0,y)+ 1

a2 L′(X0,y).L′(X0,y)Tr

](X−X0,y)

dX

Now we can calculate the integral analytically using the quadratic formula,like

∫∫∫

X

exp

(−1

2(x−M)TrΛ (x−M)

)dX =

(2π)n/2

|Λ|1/2(16)

And in our case

Λ = Σ−1 − y − L (X0,t)

a2L′′ (X0,y) +

1

a2L′ (X0,y) .L′(X0,y)

Tr (17)

Then, we have

Pa (y) =e−

(y−L(X0,y))2

2a2 − 12(X0,y−M)TrΣ−1(X0,y−M)

(2π)12a

∣∣∣I − y−L(X0,y)

a2 Σ.L′′ (X0,y) + 1a2 Σ.L′ (X0,y) .L′(X0,y)

Tr∣∣∣12

(18)

3.4.3 Saddlepoint Approximation

I was mainly focusing on the following method, saddlepoint approximationwith lagrangian multiplier

∫

X:<n

δ (y − f [X])e−

12(X−M)TrΣ−1(X−M)

|Σ|12 (2π)

n2

dXn (19)

First, we introduce Xy is such a point that, y = f [Xy] being of and havingthe maximum value of probability density , a Lagrangian will be formed tocalculate it.

LagElip[X,λ]Xy :Max Elip[X]s.t.:y=f [Xy ]

= −12(X −M)Tr · Σ−1 · (X −M)− λ (y − f [X])

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With first derivatives, with respect to X and λ

∂LagElip∂X

= −(X −M)Tr · Σ−1 + λfX,Tr [X]

∂LagElip∂λ

= y − f [X]

We are looking for the critical points such that make both equation equalzero, i.e.

−(Xy −M)Tr · Σ−1 + λ fX,Tr [Xy] = 0

y − f [Xy] = 0(20)

We introduced an efficient numerical method to compute the saddlepoint inAppendix A, which is an iterative method.

Now multiplying the first equation in (20) by (X −Xy), we have

λ fX′Tr [Xy] · (X −Xy) = (Xy −M)Tr · Σ−1 · (X −Xy) (21)

Then if we do

elip[X] = −12(X −M)Tr · Σ−1 · (X −M)

= −12(Xy −M)Tr · Σ−1 · (Xy −M)− (Xy −M)Tr · Σ−1 · (X −Xy)

−12(X −Xy)

Tr · Σ−1 · (X −Xy)

Substitute using (21)

elip[X] = −12(X −M)Tr · Σ−1 · (X −M)

= −12(Xy −M)Tr · Σ−1 · (Xy −M)

−λ fX′Tr [X] · (X −Xy) − 1

2(X −Xy)

Tr · Σ−1 · (X −Xy)

Also, we develop Taylor expansion of the loss function at Xy

y − f [X] ∼= y − f [Xy]− f ′X [Xy].(X −Xy)

Now come back to the original integral (19), we have

P ∗(y) =

∫

X:<n

δ(−f ′X

Tr[Xy](X −Xy)

)(22)

e−12(Xy−M)Tr·Σ−1·(Xy−M)−λ f ′X

Tr[Xy ] ·(X−Xy)−12(X−Xy)Tr·Σ−1·(X−Xy)

|Σ|12 (2π)

n2

dXn

Now, we will perform a change of variable in order to be able to integratethe dirac delta function. First, we write

X =

(v

V

)Xy =

(vy

Vy

)

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f ′X (Xy) =

(f ′vf ′V

)Σ =

(svv svV

sV v sV V

)Σ−1 =

(rvv rvV

rV v rV V

)

And make the change of variable

(f ′v, f

′V

Tr)·(

v − vy

V − Vy

)→ u

and keep the rest (n-1) variables the same. Then

v − vy =u

f ′v− 1

f ′vf ′Tr

V · (V − Vy)

And the Jacobian Matrix is:

J = ∇(

u

V

)=

f ′v1

11

If we write C = Xy −M , the integral (22) becomes

∫

V

∫

u

δ (u)e

−12CTr·Σ−1·C−λ (f ′v ,f ′V

Tr)·

uf ′v− 1

f ′vf ′V

Tr (V − Vy)

V − Vy

|Σ|12 (2π)

n2 |J |

e

−12

(uf ′v− 1

f ′v(V−Vy)f ′V ,(V−Vy)Tr

)Σ−1·

uf ′v− 1

f ′vf ′V

Tr (V − Vy)

V − Vy

du dV n−1

∫

V

∫

u

δ (u)e

−12

CTrΣ−1C−λ u−12

(u

fv′− 1

fv′ (V−Vy)f ′V ,(V−Vy)Tr

)Σ−1·

ufv′ − 1

fv′f ′V

Tr · (V − Vy)

V − Vy

|Σ|12 (2π)

n2 |J |

du dV n−1

(23)

Now we can integrate the dirac delta function inside the integral first.

∫

V ∈<n−1

δ (u)e

−12

CTrΣ−1C −12

(− 1

f ′v(V−Vy)f ′V ,(V−Vy)Tr

)Σ−1

− 1

f ′vf ′V

Tr (V − Vy)

V − Vy

|Σ|12 (2π)

n2 |J |

dV n−1

12

(24)

Rearrange, we have

∫

V :<n−1

e

−12

CTrΣ−1C−12(V−Vy)Tr·

(− 1

f ′vf ′V I

)· rvv rvV

rV v rV V

·

− 1

f ′vf ′V

Tr

I

·(V−Vy)

|Σ|12 (2π)

n2 |J |

dV n−1

(25)

Then

∫

V :<n−1

e−1

2CTrΣ−1C−1

2(V−Vy)Tr·

(1

f ′v2 rvvf ′V ·f ′V Tr− 1f ′v (rV v ·f ′V Tr+f ′V ·rvV )+rV V

)·(V−Vy)

|Σ|12 (2π)

n2 |J |

dV n−1

(26)

Now the integral takes a quadratic form, we can calculate it and obtainthe following result.

e−12

CTr·Σ−1·C

(2π)12 |Σ|

12

∣∣∣ 1f ′v

2 rvvf ′V · f ′V Tr − 1f ′v

(rV v · f ′V Tr + f ′V · rvV

)+ rV V

∣∣∣12 |J |

(27)

|J | = |f ′v| the absolute value of the determinant of the Jacobian Matrix

C = Xy −M

Xy = Vector on <n defines the Saddlepoint.

σ = Covariance Matrix

|Z| = Determinant of Z if Z is a square matrix

Absolute Value of r if r is a scalar

(v, V ) = X, parts of X(rvv rvV

rV v rV V

)= Σ−1Blocks of inverse of covariance matrix

Remark We need to point out that the matrix

1

f ′v2 rvvf

′V · f ′V Tr − 1

fv′(rV v · f ′V Tr

+ f ′V · rvV

)+ rV V (28)

need to be positive definite in order to have rational square root. But it isnot trivial to see this in such a complex form.

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So this innovate us to find a method with more compact form. Beforethe change of variables, first, we perform a transform Z = U(X - Xy) i.e.UTrZ = (X - Xy) such that U is a unitary matrix and Σ = UTrDU whereD is a diagonal matrix of the eigenvalues of Matrix , this could be done bySingular Value Decomposition or Eigendecomposition.

Notice that f ′X [Xy] and Xy − M are just constants and we denote C =Xy −M . So from equation (22), we have

∫

Z:<n

δ(−f ′X

Tr[Xy].U

TrZ) e−

12CTrΣ−1C−λ f ′X

Tr[Xy ] UTrZ−12ZTrD−1Z

|Σ|12 (2π)

n2

abs(∣∣UTr

∣∣)dZ

(29)

Because U is an unitary matrix , then∣∣UTr

∣∣ = ±1, so abs(∣∣UTr

∣∣) is 1.Now, we rewrite the variables

Z →(

zZZ

)U · f ′X [Xy] →

(t

T

)

D →

d1 · · · 0...

. . ....

0 · · · dn

D−1 →

d−11 · · · 0...

. . ....

0 · · · d−1n

Then the equation (29) becomes

∫∫

ZZ :<n−1

z:<1

δ

(− (

t, T Tr) (

zZZ

) )e−1

2CTrΣ−1C−λ (t,T Tr)

z

ZZ

−1

2

z

ZZ

TrD−1·

z

ZZ

|Σ|12 (2π)

n2

dzdZZ

Change the expression inside the dirac,

(t, T Tr

) ·(

zZZ

)→ u z =

u

t− 1

tT Tr ZZ Jacobian =

t T Tr

0 1 · · · 0...

.... . .

...0 0 · · · 1

|Jacobian| = t

Then we have

∫

ZZ∈<n−1

∫

u∈<1

δ (u)e−1

2CTrΣ−1C−λ u−1

2

(ZZ

Tr,1)

−T

tI

ut

0

D−1

−T Tr

tut

I 0

ZZ

1

|Σ|12 (2π)

n2 |t|

du dZZ

14

Now, we can integrate out the dirac delta,

∫

ZZ∈<n−1

e

−12

CTr·Σ−1·C−12

(ZZ

Tr·( −T

tI

))·

d−11 · · · 0...

. . ....

0 · · · d−1n

·

−T Tr

t

I

·ZZ

|Σ|12 (2π)

n2 |t|

dZZ

=

∫

ZZ∈<n−1

e

−12

CTr·Σ−1·C−12

ZZTr

d−11t2·T ·T Tr+

d−12 · · · 0...

. . ....

0 · · · d−1n

ZZ

|Σ|12 (2π)

n2 |t|

dZZ

Integrating as a quadratic form and simplifying

e−12

CTr·Σ−1·C

(2π)12 |Σ|

12 |t|

∣∣∣∣∣∣∣d−11

t2· T · T Tr +

d−12 · · · 0...

. . ....

0 · · · d−1n

∣∣∣∣∣∣∣

12

Xy = Vector on <n defines the Saddlepoint.

Lagrangean in this point is zero

C = Xy −M

Σ = Covariances Matrix

= UTrDU,where D is diagonal matrix and U is unitary matrix

i.e. Singular Value Decomposition of Σ.

|Z| = Determinant of Z if Z is a square matrix

Absolute Value of r if r is a scalar(t

T

)= Uf ′X

Tr[Xy]

t = U(1, ·) · f ′XTr[Xy]

T = U(2 : n, ·) · f ′XTr[Xy]

C = Xy −M

Remark : d−11 TT Tr + t2

d−12 · · · 0...

. . ....

0 · · · d−1n

is positive definite.

15

4 Numerical tests and conclusion

4.1 Examples

Here we will use two simple examples to illustrate the advantages and disad-vantages of different methods.

4.1.1 An example of two segments with two variables

We consider an example of two segments with two variables. In this case, weare not able to obtain the loss distribution analytically, so we will comparethe results from Monte Carlo Simulations, Nascent Delta Method and DirectMethod

y = g (x) =10000000

1 + e−x1+

8000000

1 + e−x2

(x1

x2

)∼ N

(( −0.1−0.2

),

(0.2 0.10.1 1.3

))

4.1.2 Another example of one segment with two variables

Here we consider an example of one segment with two variables.

y = g (x) =10000000

1 + e−x1−x2

(x1

x2

)∼ N

(( −0.1−0.2

),

(0.2 0.10.1 0.3

))

In this case, the good news is we could obtain the analytical solution as:

• First, let z = x1 + x2 , find the distribution of z.

mean(z) = mean(x1) + mean(x2) = 0.5 + 0.6 = 1.1

V ar(z) = V ar(x1) + V ar(x2) + 2Cov(x1, x2) = 0.9 + 1.2 + 2*0.8 = 3.7

• Then the loss function becomes y = g (z) = 100000001+e−z , we could compute

the distribution of y analytically.

pdfY (y) =pdfX(x)

|gx(x)| =1√2πσ

e−(x−µ)2

2σ2 /(106e−x/(1 + e−x)2)

=10−6

√2πσ

e−(x−µ)2

2σ2 ex(1 + e−x)2

where y = g(x) i.e. x = log( y106−y

)

16

Figure 1: Two segments with two variables. Ex (4.1.1)

4.2 Results

Here we present the results for different methods. We fix the number of simu-lation 10,000,000 for Monte Carlo method and we use the same discretizationfor the Nascent Delta method, the direct saddlepoint method and the ana-lytical solution. We discretize the loss interval into 2,000 pieces. The VaR iscalculated at 99.9% quantile. The parameter a for the Nascent Delta is setto the same as the step length for discretizing the

4.2.1 Results from different methods for Example (4.1.1)

Here, since we do not have analytical solution in this case, we use the MonteCarlo simulations as benchmark.

17

First, for Expected Loss

Expected Loss ”Error” RelativeError(percentage)MonteCarlo 8388314Saddlepoint 8378436 −9878 0.118%

NascentDelta 8387252 −1062 0.013%

And for Value at Risk

V alue at Risk ”Error” RelativeErrorMonteCarlo 13552067Saddlepoint 13607536 55469.3 0.41%

NascentDelta 13553536 1469.3 0.011%

The time consumed:

Time(seconds)MonteCarlo 27.28Saddlepoint 11.06

NascentDelta 45.47

4.2.2 Results from different methods for Example (4.1.2)

In this example, we are able to obtain the analytical solution. So we will useit as the benchmark.First, for Expected Loss

Expected Loss Error RelativeError(percentage)AnalyticalSol 435295.1MonteCarlo 435310.7 15.6 0.0036%Saddlepoint 435295.1 5.5× 10−08 1.2× 10−11%

NascentDelta 435294.9 −0.16 3.8× 10−05%

And for Value at Risk

V alue at Risk Error RelativeError(percentage)AnalyticalSol 907757.5MonteCarlo 907725 −32.5 0.0036%Saddlepoint 907757.5 0 0%

NascentDelta 907757.5 0 0%

Please note that here the error ”0” does not mean that the method is accuratewithout any error. Here the error is 0 because, we are using the same gridfor discretizing the loss. And computing the VaR depends on the this grid.Since the methods have very high levels of accuracy, they found the samepoint to be the VaR (99.9% quantile).

18

Figure 2: One segment with two variables. Ex (4.1.2)

4.3 Conclusion

As we could see from the examples, the Nascent Delta Method has very highlevel of accuracy, but it is very time-consuming and one may need extraeffort and computation to determine a good parameter a for the method.The Monte Carlo Simulations method is the standard way to calculate theloss distribution, but in order to achieve high level of accuracy, one needsa very large number of simulations, which results in long-time computingand low efficiency. The direct Saddlepoint method has the advantage thatit is much faster than the other two methods and the error is tolerable. Inthe fast-paced financial world, one second ahead of the counterparty maysave the company a lot of money. So the direct saddlepoint method shouldbe considered as a very good complementary method for the Monte CarloSimulations.

19

Appendices

A Calculating the saddlepoint using iterative

method

To compute the saddlepoint which satisfies these conditions in (20), i.e.:−(Xy −M)Tr · Σ−1 + λ f ′X

Tr[Xy] = 0

y − f [Xy] = 0(30)

Rearrange the first equation and make Taylor expansion at Xt we have−Σ f ′X [Xy] λ = M −Xy

y − f [Xt]− f ′XTr

[Xt] (Xy −Xt) = 0(31)

Make the iteration scheme as:

(Xt+1 −Xt)− Σ f ′X [Xt] λ = M −Xt

f ′XTr

[Xt] (Xt+1 −Xt) = y − f [Xt](32)

Make the matrix form(

I −Σ · f ′X [Xt]

f ′XTr [Xt] 0

)·(

Xt+1 −Xt

λ

)=

(M −Xt

y − f [Xt]

)(33)

Take the inverse matrix,

(I −Σ f ′X [Xt]

f ′XTr [Xt] 0

)−1

=1

∆t

(∆tId − Σ f ′X [Xt] f

′X

Tr [Xt] Σ f ′X [Xt]

−f ′XTr [Xt] 1

)

(34)

where ∆t = f ′XTr [Xt] · Σ · f ′X [Xt]

So the explicit solution is:

(Xt+1 −Xt

λ

)=

1

∆t

(∆tId − Σf ′X [Xt] f

′X

Tr [Xt] Σf ′X [Xt]

−f ′XTr [Xt] 1

)((M −Xt)y − f [Xt]

)

(35)

The iteration we will use in numerical computation is:

Xt+1 = M +(− (Σ f ′X [Xt]) f ′X

Tr[Xt] (M −Xt) + (Σ f ′X [Xt])× (y − f [Xt])

) 1

∆t

(36)

where ∆t = f ′XTr [Xt] · Σ · f ′X [Xt]

20

Acknowledgement

I would to thank Ferran Carrascosa, Guillermo Nebot for their help, adviseand discussion during my work at AIS. Also I would to thank Ramon Trias,Lluisa Pares and professor Aureli Alabert who together made this work possi-ble. Finally, I want to thank my academic advisor, professor Joan del Castillofor very helpful discussion and comments on my work.

References

[1] C. Bluhm, C. Wagner, L. Overbec An Introduction to Credit Risk Mod-eling. Chapman & Hall/CRC, Boca Raton, 2002.

[2] Nisso Bucay and Dan Rosen Applying Portfolio Credit Risk Models toRetail Portfolios Enterprise Credit Risk Using Mark-to-Future, 263-292Algorithmics 2001.

[3] Yasuhiro Yamai and Toshinao Yoshiba Comparative Analyses of ExpectedShortfall and Value-at-Risk: Their Estimation Error, Decomposition, andOptimization Monetary and Economic Studies, January 2002, 87-122

[4] Ramon Trias Capella, Ferran Carrascosa Mallafre, David Fernadez, LluisaPares and Guillermo Nebot. The RDF Method(Risk Dynamics into thefuture) AIS Group April 2009

21