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ScienceDirect Procedia Economics and Finance 24 (2015) 228 – 236

International Conference on Applied Economics, ICOAE 2015, 2-4 July 2015, Kazan, Russia

Comparison of nonparametric methods for estimating the level of risk in finance a, Katarína Frajtová-Michalíková *, Tomáš Klieštikb, Hussam Musac a

Faculty of Operation and Economics of Transport and Communications, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia Faculty of Operation and Economics of Transport and Communications, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia c Faculty of Economics, Matej Bel University in Banská Bystrica, Tajovského 10, Banská Bystrica 975 90, Slovakia

b

Abstract The present paper is intended to selected aspects of approximation, respectively quantification of market risk exposure of financial instruments. Featured approximation methods or quantifying the level of risk are first described in theory and then applied to real data. Specifically, the shares of Oracle, Coca Cola and Pfizer term in January 2012 to February 2015. The price development is on a daily basis. An integral part of the paper is a graphical representation of the results and their comparison. The distribution function of the random variable representing the risk is not known and therefore it must be estimated. In the present paper we will assume that the theoretical probability distribution of losses is not known to us and we have to estimate, i.e. apply called nonparametric estimates. © B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors.Published PublishedbybyElsevier Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of the Organizing Committee of ICOAE 2015. Selection and/or peer-review under responsibility of the Organizing Committee of ICOAE 2015.

Keywords: Value at Risk, Conditional Value at Risk, risk, Simulation, Comparison, Risk measure.

* Corresponding author. Tel.: +421-41513-3246. E-mail address: [email protected]

2212-5671 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Selection and/or peer-review under responsibility of the Organizing Committee of ICOAE 2015. doi:10.1016/S2212-5671(15)00653-X

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Katarína Frajtová-Michalíková et al. / Procedia Economics and Finance 24 (2015) 228 – 236

1. Introduction When the late 80s of last century management Risk Management at JP Morgan Investment Bank assessed the possibility of full reinsurance decide between two investment choices Morgan (1996). The first option constituted an investment in bonds that banks will generate steady income, but at the same time the bank will risk volatility values. The second option was to cash investment in which is maintained as the market value of the property. In other words, professional managers should decide the value risks and earnings risks. In this case, choosing the first option and the methodology respectively approach was named as Value at Risk. So far, kept the controversy, who wrote the following entry: "At the end of each trading day, we should know what are the risks to which we are exposed to market in all types of business they are in, and in all places where we do business so proverbial reports 4:15 pm ". Whether it was Till Guldimann, head of research at J. P. Morgan or Sir Dennis Weatherstone Chairman of J.P. Morgan. Currently, this method is already widely used as in banking, where it created, as well as the insurance, investment funds. In addition to its wide application in the financial markets, it can be applied even in non-financial institutions Boda and Kanderová (2013). Its relative simplicity and also easy interpretation it to shaping. We must not forget its versatility (possibility to use different types of risk), and in particular that the result is a single number that quantifies the risk of the entire portfolio Gavlakova et al. (2014). Of course since its inception has been VaR constantly modified and adapted to the new conditions in the financial world. And as it happens, and it has been subjected to criticism. It should be noted that in the context of the still ongoing financial crisis, criticism eligible. However, in the case of a normal, stable development of the markets, the use of VaR still relevant. 2. Value at Risk VaR is mathematically expressed as one-sided percentile of the distribution of profits and losses of the financial instrument or portfolio (for example 1%) during the period (for example 1 day), determined on the basis of the historical period (for example 5 years). The basic guideline in choosing the period, the frequency of revaluation instruments or the frequency of reports on changes in the portfolio. Generally, the risk management is important to the short-term (daily horizons). Conversely, for regulatory reports is preferable longer time horizon (monthly and yearly horizons). Jorion (2007). When quantifying VaR is often assumed normal distribution of income and losses. Proceeds X | q N P , V 2 according to the assumption can be written as the density of the normal distribution Alexander (2009) :

f ( x)

1 2SV

x P 2

e

2V 2

(1)

where P is the mean value of the distribution and V the standard deviation. Where the yield is in the domain of

P kV , P kV , then the probability does not depend only on the parameter k, but on the basis of definition equation (1) the likelihood can be expressed as follows: P ^P kV x P kV `

P kV

³ f ( x)dR P ³ V P V k

If true VaR

P ^ X xD `

P kV

k

1 2SV

x P 2

e

2V 2

dR

(2)

xD , then looking for such a population loss for a given confidence level D to which it applies: xD

³

f

f ( x)dx

x P½ P ®Z D ¾ 1D V ¿ ¯

(3)

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The variable Z is a random variable standardized normal distribution N (0,1) for which the inverse function applies.

^

P Z I 1 (D )` D

xD P

I 1 (D )

V

(4)

VaR corresponding to the quantile, which depends on the selected confidence level D symmetrical in mean and therefore the quantile xD shall apply: xD

P I 1 (D )V

P I 1 (1 D )V

. Normal distribution is (5)

Or, where the VaR express the loss, so given the fact we can formally rewrite: VaR

xD

I 1 (1 D )V P

(6)

Density function – Gain/Loss

1% probability

Equation (6) describes the relative VaR, i.e. express in percentage. The nominal VaR is calculated similarly, only the entire right side of the equation (6) multiplied by the current market price of the financial instrument, respectively the entire portfolio. Confidence level may have different values, but most often calculated on 95 to 99 per cent level of significance. Rule, the higher level of accuracy is selected, the higher the expected loss. Not then, that a higher level of confidence is more accurate estimate of the loss. This fact shows that the VaR is trying to describe almost all variance losses. Graphically, the Value at Risk may describe as shown in Figure 1. The figure but a distinct lack of substantial VaR method. Represents is the assumption that changes in the value of the portfolio, respectively changes in values of financial instruments follow exactly normal distribution Sollis (2009) and Kollar and Bartosova (2014). And it is this assumption in practice is not always true. However, if it exceeds the VaR, which should be as shown in Figure 1 only occur with a probability of 1%, then VaR is silent on maximum losses, which in this case can be enormous. This significant weakness VaR at least partially removed CVaR.

VAR 0,99

Gain/Loss

Fig. 1 Graphical representation of the maximum losses under the concept of VaR

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3. Conditional Value at Risk As mentioned Value at Risk the maximum possible loss which the entity expects with a given probability. This information but in itself does not contain any other information on emergency losses in excess of VaR, which may be many times more. Conditional Value at Risk CVaR, respectively in the literature we can meet also called Expected Tail Loss (ETL) is trying to at least partially describe the maximum loss Sollis (2009). This is not a precise calculation, but rather an estimate of the average loss in case of exceeding the VaR Spuchlakova et al. (2014). In historical simulation or Monte Carlo simulation we can easily calculate the CVaR and is also readily observable from the histogram, as regards the averaging of values exceeding VaR simulated result. It is different in linear models, where any actual or approximate distribution of losses is not. CVaR is then, according to Alexander (2009) may be defined:

CVaRD

E ( X X xD )

(7)

CVaR is then according to equation (7) conditionally expectation is equal to the share of probability weighted x P( X xD ) . At the same time must be average of the random variables X and smaller than D the probability that P( X xD ) D . Therefore, we can CVaR expressed by the integral: allowed xD

CVaRD ( X )

D 1 ³ xf ( x)dx

(8)

f

Z | q N (0,1) follows the standard normal Where the function f is a function of the density. If the random variable distribution, which describes the equation (1). Then, on the basis of equation (8) we can derive CVaR for variable Z as follows: ) 1 (D )

CVaRD ( Z )

D 1

³

) 1 (D )

zM ( z )dz

( 2SD ) 1

f

ª 1 D « e ¬ 2S 1

where

³

ze

1 z 2 dz 2

f 1 z2 2

) 1 (D )

º » ¼ f

(9)

D M ª¬) (D ) º¼ 1

1

M is the density function and I the distribution function of the standard normal distribution. Member in

M ª) 1 (D ) º¼ square brackets represents D the quantile of the standard normal distribution function ¬ representing the value at this point. We can convert the transformation of a random variable Z to X. X | q N (P , V 2 ) : X

ZV P

(10)

If the transformation (10) is substituted into (9) we obtain the final equation for the Conditional Value at Risk: CVaRD ( X )

CVaRD (Z )V P

D 1M ª¬I 1 (D ) º¼ V P

(11)

4. Data, discussion on empirical results As the name implies contribution, we apply the so-called nonparametric methods, i.e. distribution losses L is not known and therefore we have to determine. The most commonly used two methods for the estimation: an estimate based on historical data and Monte Carlo simulation. An example of the calculation of VaR and CVaR

232


using those methods. It will include shares of Oracle, Coca Cola and Pfizer term in January 2012 to February 2015. The price development is on a daily basis. Figures 2 to 4 show the evolution of prices and yields of the shares.

Fig. 2. (a) Development of price Oracle; (b) Development of yield Oracle

Fig. 3. (a) Development of price Coca cola; (b) Development of yield Coca cola

Fig. 4. (a) Development of price Pfizer; (b) Development of yield Pfizer


233

It is further approximated by the loss of the nuclear density estimation. The density of the x value is estimated as follows: 1 n § x xi · (12) ¦k n h i 1 ¨© h ¸¹ where n is the number of observations, k ( x) the buffer core function of the parameter h is the width of the window. The higher the larger h is, the smoother estimate. In the following text, we will refer to estimate nuclear comma. 4.1. Estimate based on historical data Based on the empirical estimate of the distribution function to prices of financial instruments in the past and we expect that the distribution of further losses is consistent with the allocation of losses in the past. The basic disadvantage of historical simulation is entirely dependent on the type, scope and accessibility of data, which are chosen for the calculation. The second significant disadvantage is the assumption that past data say what will happen in the future. But time series can be so short that makes it past as market failure with low temporal frequency. Figure 5 shows the empirical density loss of individual stocks and Table 1 lists the kurtosis (γ - kurtosis, γ'- nuclear kurtosis estimated from the relation (11).

60 Oracle

50 40

CocaCola

30 Pfizer

20 10 0 0.06

0.04

0.02 0.00

0.02

0.04

0.06

Fig. 5. Density distribution losses – historical estimate Table 1. Kurtosis of each stock on the basis of historical data J

J´

Oracle

13,628924559495205

12,789709876266253

Coca cola

7,702162047015071

7,352679485119721

Pfizer

4,577376415618715

4,4496036004357045

VaR´

CVaR

CVaR´

Table 2. Oracle – historical estimate. D VaR 0,9

0,014457396422062178

0,013752455795677854

0,022995431293091367

0,022540799064655743

0,95

0,019870187122843483

0,019422310756971983

0,029099359563303128

0,028721334674778776

234


0,99

0,029672853005697386

Table 3. Coca Cola – historical estimate. D VaR

0,028461784457946893

0,04999201608889922

0,05178566437271211

VaR´

CVaR

CVaR´

0,9

0,01026182864922744

0,009965831435079564

0,016403143914173277

0,016131743786735032

0,95

0,014249997434633685

0,014132278123233366

0,020754085776401143

0,020524872055514884

0,99

0,024495541235178584

0,024907260201378012

0,03315407169885499

0,03395630831836801

VaR´

CVaR

CVaR´

Table 4. Pfizer – historical estimate. D VaR 0,9

0,010367396603694436

0,010292164674634896

0,016384206726990785

0,016014300430776635

0,95

0,014131915521464075

0,013418530351437807

0,02075328067291467

0,020467550342149067

0,99

0,02472587342532055

0,025237746891002333

0,030968417808310618

0,03132111538283929

4.2. Monte Carlo simulation Monte Carlo simulation is in practice quite widespread and popular method. Simulation may be based on historical data, which is the standard distribution governed by historical data. Or directly model the "brand new" probability distribution and historical data are used little or not at all. In this case, the model not only the development of yield, but also the development of standard deviations and correlations. Simulation, we say that is likely to behave risk factor in the near term, which is unlimited. Valid but, the longer horizon, the decreasing accuracy. The result of the simulation is then a random distribution of gains and losses.

60 Oracle

50 40

Coca cola

30 Pfizer

20 10 0 0.06

0.04

0.02 0.00

0.02

0.04

0.06

Fig. 6. Density distribution losses – Monte Carlo Simulation

Table 5. Kurtosis individual stocks based on Monte Carlo simulation. J

J´

Oracle

12.783542303307575

12.662821877999438

Coca cola

7.4972952982662795

7.443564385643973

Pfizer

4.422138024545934

4.4037767452212115


Table 6. Oracle – Monte Carlo Simulation. D VaR

VaR´

CVaR

CVaR´

0,9

0,01461531360221495

0,014542550535111286

0,023211780461360274

0,023141953123902204

0,95

0,020030467847850092

0,01996783262073988

0,029352899625178567

0,02927320078642977

0,99

0,029816281317728406

0,029648073597087543

0,050733157489000924

0,050626199070938856

VaR´

CVaR

CVaR´

Table 7. Coca Cola – Monte Carlo Simulation. D VaR 0,9

0,010276831287277872

0,010228097031949993

0,01641348106087121

0,016367319136354214

0,95

0,014265790564397402

0,014229014813495063

0,020768519045571365

0,020720364280256025

0,99

0,024442965092209886

0,02443622187513923

0,03324146799863603

0,033201704867445815

Table 8. Pfizer – Monte Carlo Simulation. D VaR

VaR´

CVaR

CVaR´

0,9

0,010446085538112458

0,01037909227714203

0,016507788070611568

0,016448396660721877

0,95

0,014257415607188696

0,014181763761456185

0,020912356754240705

0,02086029466984165

0,99

0,024936461992142035

0,024867413974650025

0,031104527305078976

0,031051694310908914

235

5. Conclusions According to the results indicated in Tables 2-4 and 6-8, and of course on the basis of the comparison, we can conclude that both nonparametric methods i.e. Monte Carlo simulation and estimation based on historical data generated almost identical and therefore in our view, be left to the choice of a particular user. Acknowledgements The contribution is an output of the science project VEGA 1/0656/14- Research of Possibilities of Credit Default Models Application in Conditions of the SR as a Tool for Objective Quantification of Businesses Credit Risks. References Alexander, C., 2009. Market Risk Analysis, Value at Risk Models (Volume IV), Publisher: Wiley. Alexander, G.J.; Baptista, A.M., 2004. A comparison of VaR and CVaR constraints on portfolio selection with the mean-variance model, Management Science, 50, Issue 9, 1261-1273 Boda, M., Kanderova, M., 2013. Usability of economic capital and its concept in the financial management of non-financial firms, 9th international scientific conference, Financial management of firms and financial institutions, Ostrava, Czech Republic, 44-53. Bollerslev, T., 1986. Generalized autoregressive heteroskedasticity. Journal of Econometrics, 31, 307-327. Buc, D,. Kliestik, T., 2013. Aspects of Statistics in Terms of Financial Modelling and Risk, In: 7th International Days of Statistics and Economics, Prague, Czech Republic, 215-224, Dengov, V.V., Melnikova, E.P., 2012. Adverse selection on credit markets: the analysis of the empirical research results, Proceedings of the 7th international conference on currency, banking and international finance. 27 and 28 September 2012, Bratislava, Slovenska Republika., 38-52. Engle, R. F., 1982. Autoregresive, conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987-1007. Gavlakova, P.; Valaskova, K.; Dengov, V., 2014. Credit Risk and Its Evaluation, In: 2nd International Conference on Economics and Social Science (ICESS), Shenzen, China, Advances in Education Research, 61, 104-108, Jaros, J., Melichar, V., Svadlenka, L., 2014. Impact of the Financial Crisis on Capital Markets and Global Economic Performance, In: 18th International Conference on Transport Means, Kaunas Lithuania, 431-434. Jorion, P., 2007. Value at risk: the new benchmark for managing financial risk, Publisher: McGraw-Hill

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