nonparametric estimation of conditional expected shortfall … · 2019. 10. 24. · 1050 e. torsen,...

Applied Mathematical Sciences, Vol. 13, 2019, no. 22, 1049 - 1060HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2019.99135

Nonparametric Estimation of Conditional Expected

Shortfall Admitting a Location-Scale Model

Emmanuel Torsen

Department of Mathematics, Pan African UniversityInstitute of Basic Sciences, Technology, and Innovation, Kenya

&Department of Statistics and Operations Research

Modibbo Adama University of Technology Yola, Nigeria

Peter N. Mwita

Department of Mathematics, Machakos University, Kenya

Joseph K. Mung’atu

Department of Statistics and Actuarial SciencesJomo Kenyatta University of Agriculture and Technology, Kenya

This article is distributed under the Creative Commons by-nc-nd Attribution License.

Copyright c© 2019 Hikari Ltd.

Abstract

The Expected Shortfall as the most preferred measure of risk takesinto account all the possible losses that exceed the severity level cor-responding to the Value-at-Risk. In this paper at hand, we have pro-posed a nonparametric method of estimation for conditional expectedshortfall assuming that the returns on an asset or portfolio admit alocation-scale model. We have verified the properties of the estimatorthrough simulation using the bootstrap method, the variance, bias, andasymptotic mean square error are relatively small, meaning that theestimator performs relatively good. Financial market data for TOTALcompany quoted on the Nigerian Stock Exchange was used to illustratethe applicability of the estimator.

Keywords: Expected Shortfall, Location-Scale Model, Nonparametric Es-timation, Three-Step, Conditional Value-at-Risk

1050 E. Torsen, P. N. Mwita and J. K. Mung’atu

1 Introduction

A major concern for financial institutions owners and regulators is the riskanalysis, Value-at-Risk (VaR) is one of the most used and common measuresof risk used in finance [6]. It measures the down-side risk and is determined fora given probability level τ . Typically, in measuring losses, VaR is the lowestvalue which exceeds this level (the quantile of the loss distributions). TheExpected Shortfall (ES) due to [1] is the average of the 100(1 − τ)% worstlosses. The ES takes into account all the possible losses that exceeds theseverity level corresponding to the VaR. Since the first Basel Accord [15], theVaR and recently, the ES [3] forms the essential basis for the determination ofmarket risk capital [12]. [2] proposed ES to solve the problems inherent in VaR.ES is the conditional expectation of loss given that the loss is beyond the VaRlevel, as defined in section (1.2). The ES shows the average loss when the lossexceeds the VaR level. In other words, ES is a numerical figure indicating theaverage of the possible losses that could be incurred beyond a selected tolerancelevel, a vast literature refer to this as the Tail Conditional Expectation (TCE),Conditional Loss (CL) or Tail Loss (TL). Taking the ES implies a selection ofa benchmark value (VaR). The values beyond this benchmark value (VaR) arethen averaged to come up with the ES [16].

ES and CES have been known and popular for more than two decades nowamong actuary sciences and industries, see [13, 4]. There exists a large studiesin the literature on nonparametric estimation of unconditional VaR and ES,for example., [14], [5], [4], [10] and [9].

2 Estimation of Conditional Expected Short-

fall (CES)

Here, the estimation of CES(X)τ for processes Yt that admit a location-scalerepresentation given as

Yt = m(Xt) +√h(Xt)�t (1)

where m and h > 0 are nonparametric functions defined on the range of Xt, �t is independent of Xt, and �t is an independent and identically distributed(iid) innovation process with E(�t) = 0, Var(�t) = 1 and the unknown distri-bution function F�.

From equation (1) we have;

CV aR(X)τ := QY |X(τ |x) = m(Xt) +√h(Xt)q(τ) (2)

where QY |X(τ |x) is the conditional τ−quantile associated with F (y|x) andq(τ) is the τ−quantile associated with the error innovation F�. The estimator

Nonparametric estimation of conditional expected ... 1051

of (2), its properties, and the prediction intervals has been studied in [18] and[20] respectively.

Hence,

CES(X)τ ≡ E(Yt/Yt > QY |X(τ |x), Xt = x) = m(Xt) +√h(Xt)E(�t|�t > q(τ))

(3)Where QY |X(τ |x) is the conditional τ−quantile associated with FY |X(y|x)

and q(τ) is the τ−quantile associated with F�.The problem of estimating m(X) and h(X) in equation (3) was studied by

[7] and [8]. Now, with the estimators of the mean and variance functions, wehave the sequence of squared residuals {ri = {Yi − m̂(x)}2}ni=1.

These estimators the mean and variance functions are then used to get asequence of Standardized Nonparametric Residuals (SNR) {�̂i}ni=1, where;

�̂i =

Yi − m̂(X)√

ĥ(X), if ĥ(X) > 0

0, if ĥ(X) ≤ 0(4)

We use these SNR to obtain the cumulative conditional density estimatorof F�, see our previous paper for details [19].

With the estimators of the mean function m(X), the variance functionh(X) and the unknown error innovation, our estimator for Conditional Value-at-Risk (CVaR), discussed in [18] is given as;

ĈV aR(X)τ := Q̂Y |X(τ |x) = m̂(x) + ĥ1/2(x)q̂(τ) (5)

and hence, our estimator for Conditional Expected Shortfall is

ĈES(X)τ := E(Yt/Yt > Q̂Y |X(τ |x), Xt = x) = m̂(x) + ĥ1/2(x)E(�t|�t > q̂(τ))(6)

To discussed the asymptotic properties of (6), the estimator of (3), to dothis we make the following basic assumptions:

2.1 Assumptions

A: Bandwidth

1. b −→ 0, as n −→∞

2. nb −→∞, as n −→∞

B: Kernel


1. K has compact support

2. K is symmetric

3. K is Lipschitz continuous

4. K is∫∞−∞K(u)du = 1 and

∫∞−∞ uK(u)du = 0 with µ2(K) =

∫∞−∞ u

2K(u)du

and R(K) =∫∞−∞K(u)

2du being the second moment (Variance) andRoughness of the kernel function respectively.

5. K is bounded and there is K̄ ∈ R, with K(u) ≤ K̄ < ∞ and K(u) ≥0,∀u ∈ R

3 Asymptotic Properties of the Estimator

To examine the asymptotic properties of our estimator, we quickly mentiontwo results of importance from [7, 8], see also [19] and [18]

In Theorem 1 of [8], under the assumptions of the aforementioned paper,

√nb[ĥ(x)− h(x)−Bias(ĥ(x))

] d−→ N (0, f−1(x)h2(x)λ2(x)∫ k2(u)du)where

λ2(x) = E[(�2 − 1)2|X = x

], � = Y−m(X)

h(X), µ2(k) =

∫u2K(u)du and

Bias(ĥ(x)) = b2

2µ2(k)h

′′(x)2

This means that

ĥ(x)d−→ h(x)

withE(ĥ(x)) = h(x)+ b2

2µ2(k)h

′′(x)2 = Mĥ, V ar(ĥ(x)) =1

nbf(x)R(k)h2(x)λ2(x) =

Vĥ

Hence

ĥ(x)∼N(Mĥ, Vĥ

)Also in [7], we have that

m̂(x)d−→ N

(Mm̂, Vm̂

)whereMm̂ = m(x) +

b2

2m′′(x)µ2(k), Vm̂ =

σ2(x)nbf(x)

∫∞−∞K

2(u)du = σ2(x)R(k)nbf(x)

and

Bias(m̂(x)) = b2

2m′′(x)µ2(k)


We want to show the mean and Variance of our estimator (6) using Slutsky’stheorem, but we considered the second term of (6):

ĥ(x)E(�t|�t > q̂(τ))d−→ h(x)E(�t|�t > q(τ))

Let

B = ĥ1/2(x)E(�t|�t > q̂(τ)) and E(�t|�t > q̂(τ)) > 0Now

E[B] = E[ĥ(x)E(�t|�t > q̂(τ))

]= E(�t|�t > q̂(τ))E

[ĥ(x)

]= E(�t|�t > q̂(τ))

[h(x) +

b2

2µ2(k)h

′′(x)2]

= E(�t|�t > q̂(τ))h(x) +E(�t|�t > q̂(τ))b2

2µ2(k)h

′′(x)2

u E(�t|�t > q(τ))h(x) +E(�t|�t > q(τ))b2

2µ2(k)h

′′(x)2

V ar[B] = V ar[ĥ(x)E(�t|�t > q̂(τ))

]= E(�t|�t > q̂(τ))2V ar

[ĥ(x)

]= E(�t|�t > q̂(τ))2

[ 1nbf(x)

R(k)h2(x)λ2(x)]

=E(�t|�t > q̂(τ))2R(k)h2(x)λ2(x)

nbf(x)

uE(�t|�t > q(τ))2R(k)h2(x)λ2(x)

nbf(x)

Hence by Slutsky’s theorem,

ĥ(x)E(�t|�t > q̂(τ))d−→ h(x)E(�t|�t > q(τ))

with mean and variance as given above.

Hence,

ĈES(X)τ = m̂(x) + ĥ(x)E(�t|�t > q̂(τ))= m̂(x) +B

= A+B

Now,


E[ĈES(X)τ

]= E[A+B]

= E[A] + E[B]

=[m(x) +

b2

2m′′(x)µ2(k)

]+ E(�t|�t > q̂(τ))

[h(x) +

b2

2µ2(k)h

′′(x)2]

= m(x) +b2

2m′′(x)µ2(k) + E(�t|�t > q̂(τ))h(x) +

E(�t|�t > q̂(τ))b2

2µ2(k)h

′′(x)2

= m(x) + E(�t|�t > q̂(τ))h(x) +b2

2µ2(k)

[m′′(x) + E(�t|�t > q̂(τ))h′′(x)2

]︸︷︷︸=Bias

u m(x) + E(�t|�t > q(τ))h(x) +b2

2µ2(k)

[m′′(x) + E(�t|�t > q(τ))h′′(x)2

]︸︷︷︸=Bias

So that,

Bias(ĈES(X)τ

)=b2

2µ2(k)

[m′′(x) + E(�t|�t > q̂(τ))h′′(x)2

]ub2

2µ2(k)

[m′′(x) + E(�t|�t > q(τ))h′′(x)2

]and

V ar(ĈES(X)τ

)= V ar(A+B)

= V ar(A) + V ar(B), Cov(A,B) = 0

=σ2(x)R(k)

nbf(x)+

E(�t|�t > q̂(τ))2R(k)h2(x)λ2(x)nbf(x)

=R(k)

nbf(x)

[σ2(x) + E(�t|�t > q̂(τ))2h2(x)λ2(x)

]u

R(k)

nbf(x)

[σ2(x) + E(�t|�t > q(τ))2h2(x)λ2(x)

]=⇒ ĈES(X)τ

d−→ CES(X)τ , with mean and variance given above (usingSlutsky’s theorem).

Where

ĈES(X)τ := E(Yt/Yt > Q̂Y |X(τ |x), Xt = x) = m̂(x) + ĥ1/2(x)E(�t|�t > q̂(τ))(7)

Hence, our estimator for CES(X)τ , see [17] for its prediction intervals.


4 Smoothing Parameter (Bandwidth) Selec-

tion

In Nonparametric methods, the choice of an optimal smoothing parameter cannot be over emphasized. We choose the smoothing parameter that minimizesthe Asymptotic Mean Square Error (AMSE) below:

AMSE(ĈES(X)τ

)= E

[(ĈES(X)τ − CES(X)τ

)2]= E

[(ĈES(X)τ − E

[ĈES(X)τ

]+Bias

(ĈES(X)τ

))2]= E

[(ĈES(X)τ − E

[ĈES(X)τ

])2]+Bias

(ĈES(X)τ

)× E

[ĈES(X)τ − E

[ĈES(X)τ

]]+Bias2

(ĈES(X)τ

)= V ar

(ĈES(X)τ

)+Bias2

(ĈES(X)τ

)=

R(k)

nbf(x)

[σ2(x) + E(�t|�t > q̂(τ))2h2(x)λ2(x)

]+

[b2

2µ2(k)

[m′′(x) + E(�t|�t > q̂(τ))h′′(x)2

]]2=

R(k)

nbf(x)

[σ2(x) + E(�t|�t > q̂(τ))2h2(x)λ2(x)

]+b4

4µ22(k)

[m′′(x) + E(�t|�t > q̂(τ))h′′(x)2

]2u

R(k)

nbf(x)

[σ2(x) + E(�t|�t > q(τ))2h2(x)λ2(x)

]+b4

4µ22(k)

[m′′(x) + E(�t|�t > q(τ))h′′(x)2

]2(8)

Therefore,

bopt = argminb>0

AMSE(ĈES(X)τ

)and hence, d

dbAMSE

(ĈES(X)τ

)= 0

which gives

bopt =

{R(k)

[σ2(x) + E(�t|�t > q(τ))2h2(x)λ2(x)

]µ22(k)f(x)

[m′′(x) + E(�t|�t > q(τ))h′′(x)2

]2} 1

5

× n−15 (9)


Table 1: Summary of the bootstrap resultsMin. 1st Qu. Median Mean 3rd Qu. Max.

Expection.CES 5.109 5.367 5.43 5.433 5.495 5.762Variance.CES 0.7324 1.4806 1.7466 2.0631 2.2274 10.5496Bias.CES 0.6774 0.8821 0.9293 0.9371 0.9832 1.322AMSE.CES 0.8558 1.2168 1.3216 1.4031 1.4924 3.248

5 Simulation Study

To examine the performance of our estimators, we conducted a simulationstudy considering the following data generating location-scale model

Yt = m(Yt−1) + h(t)1/2�t, t = 1, 2, ..., n (10)

wherem(Yt−1) = sin(0.5Yt−1), �t ∼ t(ν = 3), h(t) = hi(Yt−1) + θh(Yt−1),

i = 1, 2and h1(Yt−1) = 1+0.01Y

2t−1 +0.5sin(Yt−1), h2(Yt−1) = 1−0.9exp

(−2Y 2t−1

)Yt and h(t) are set to zero (0) initially, then Yt is generated recursively from

(10) above. The data generating process was also used by [11].

6 Application to financial market data

Using historical daily series {Yt} on the log returns of the closing prices forthe period between January 02, 1997 to December 29, 2017 trading days ofTOTAL company quoted on the Nigerian Stock Exchange, we illustrate theapplicability of the estimator.

In Figure 2 and Figure 3, the estimation of the location function andthe estimation of the scale function using local linear regression are presentedrespectively. In Figure 1, the time series plot of the simulated daily returnsis presented, Figure 4 shows the 95% conditional expected shortfall (CES)using the proposed estimator. We performed bootstrapping on the proposedestimator to verify its properties, the summary of the results is presented inTable 1. As can be seen in Table 1, the bias and asymptotic mean squareerror (AMSE) are relatively small, indicating a good performance of the esti-mator. Figure 5 and Figure 6 shows respectively the time series plot of theTOTAL’s historical market data, quoted on the Nigerian Stock Exchange andthe 95% conditional expected shortfall (CES) for the TOTAL data.


Simulated Daily Returns plot

Time

Dai

ly R

etur

ns

0 500 1000 1500 2000 2500

−10

010

20

Figure 1: Plot of the simulateddata showing its evolution.

−10 0 10 20

−10

010

20

Fitted and Predicted mean function for Simulated returns

x

y

Non−param. m(x)Predicted m(x)

Figure 2: Estimation of the mean func-tion, the red colored line showing thetrue mean function while the greencolor dotted line shows the predictedmean function

−10 0 10 20

010

020

030

040

0

Fitted and Predicted variance function for Simulated returns

x

Squ

aed

Res

idua

ls

Non−param. h(x)Predicted h(x)

Figure 3: Estimation of thevariance function, the red col-ored line showing the true vari-ance function while the greencolor dotted line shows the pre-dicted variance function

CES for Simulated Daily Returns

Time

Dai

ly R

etur

ns

0 500 1000 1500 2000 2500

−10

010

20 CES (95%)

Figure 4: Estimation of the 95% CES,shown by the red colored thick line


Evolution of TOTAL's Daily Log Difference Returns plot

Time

Logd

iff o

f Dai

ly R

etur

ns

14000 15000 16000 17000

−0.

10−

0.05

0.00

0.05

0.10

Figure 5: Plot of the TOTALcompany data showing its evo-lution.

CES for TOTAL's daily Log Difference returns Data

Time

logd

iff(T

OTA

L)

0 500 1000 1500 2000 2500

−0.

10−

0.05

0.00

0.05

0.10 CES (95%)

Figure 6: Estimation of the 95% CESusing the TOTAL data, shown by thered colored thick line

7 Conclusion

In this paper, we have proposed a nonparametric method of estimation forconditional expected shortfall assuming that the returns on an asset or port-folio admits a location-scale model. We have verified the properties of theestimator through simulation using bootstrap method, the variance, bias, andasymptotic mean square error (AMSE) are relatively small, meaning that theestimator performs well. Financial market data from the Nigerian Stock Ex-change was used for illustration.

Acknowledgements. The first author wishes to thank African Union,Pan African University, Institute for Basic Sciences Technology and InnovationNairobi, Kenya and Japan International Cooperation (JICA) for supportingthis research.

Disclosure statement. The authors declare that there is no conflict ofinterest regarding the publication of this paper.

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Received: October 7, 2019; Published: October 23, 2019

nonparametric estimation of conditional expected shortfall … · 2019. 10. 24. · 1050 e. torsen,...

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