risk assessment in commodity markets with semi-nonparametric speciﬁcations

MotivationModels and Methodology

ApplicationConclusion

Risk assessment in commodity markets withsemi-nonparametric specifications

Esther B. Del Brio, Andres Mora-Valencia, Javier Perote

I Network de Metodos Cuantitativos en Economıa - AFADECO, November 10,2017

Afadeco November 10,2017 Risk assessment in commodity markets



Outline

1 Motivation

2 Models and Methodology

3 Application

4 Conclusion




Introduction

Commodity prices experienced a boom in the mid-2000s.

With the boom of commodity prices, several ExchangeTraded Funds (ETFs) were created in order to track the priceof main commodities such as gold, silver and oil.

Financial institutions and regulators - e.g. Financial StabilityBoard (FSB) and Bank for International Settlements (BIS) -expressed their concerns about the potential systemicimpact of the ETFs industry in 2011.

Commodity assets may present higher volatilities than equityassets even in “relatively” calm periods.

Recently, there has been a sharp decline in the price ofthese assets affecting developed markets based on mining andenergy industries like Canada and Australia, based metalsbusinesses in Deutsche Bank and JP Morgan, among others .




Risk Measures

Value-at-risk (VaR) has been the standard market riskmeasure to assess regulatory capital requirements since 1996.

VaR has been criticized since they do not fulfill thesubadditivity property and its inability to accurately capturetail risk.

Expected Shortfall (ES) provides coherent risk measures.

Basel 2.5 (2012) proposes to replace VaR by ES.

ES is very sensitive to extreme events and may result inunstable capital numbers at high confidence levels.

ES does not satisfy the elicitability property, which generateddebate on whether ES can be backtestable.

VaR satisfies elicitability property and there are verywell-known backtesting methods for this measure.




Literature Review

Authors Asset (Variable) Best Model

Giot & Laurent (2003) Several commodity markets APARCH + skewed-t innovations

Hung et al. (2008) Energy commodities GARCH + heavy tail model

Chiu et al. (2010) Brent and WTI crude oil prices Hull-White model

Aloui & Mabrouk (2010) Oil and gas commodities prices FIAPARCH + skewed-t innovations

Youssef et al. (2015) Crude oil and gasoline market FIAPARCH + EVT

AndriosopoulosNomikos(2015)

8 spot energy commodities and SEI MC simulations

Steen et al. (2015) 19 commotity futures Quantile regression

Aloui & Jammazi (2015) Oil-exchange rate pairs Wavelet-based models

Lu et al. (2014) Crude oil futures and natural gas futures t-Copula & skewed-t for mgnls




VaR/ES-ARMA(1,1)-EGARCH(1,1)

Rt+1 = µt+1 + σt+1Zt+1:

µt+1 = µ+ φ1µt + θεt + εt+1,

εt+1 = Zt+1σt+1 Zt+1 ∼ G (0, 1),

logσ2t+1 = ω + α (|zt | − E [|zt |]) + γzt + β log σ2

t ,

VaR = µt+1 + σt+1qγ(Zt+1),

ES = µt+1 + σt+1ESγ(Zt+1).




Distributions (...for G)

1 Normal

2 Student’s t

3 Skewed t

4 GC Type A




Normal, Student’s t and Skewed-t

Normal:

φ (zt) = 1√2π

exp{− z2

t2

}.

Student’s t:

t (zt) =Γ( ν+1

2 )√π(ν−2)Γ( ν

2 )

(1 + z2

tν−2

)− ν+12,

Skewed t (Fernandez and Steel, 1998):

g (zt) =

−2

γ+ 1γ

t (γzt) zt < 0,

2γ+ 1

γ

t(ztγ

)zt ≥ 0,

.




Gram-Charlier Type A

GC Type A density:

f (zt ,d) =

(1 +

n∑s=1

dsHs (zt)

)φ (zt) ,

where:

φ (zt) is the normal pdf,

d = (d1, . . . , ds) ∈ Rs , and

Hs is the sth Hermite polynomial (HP) of order, which iscomputed in terms of the sth order derivative of the Gaussianpdf:

d sφ(zt)dzst

= (−1)s Hs (zt)φ (zt) .




GC Type A

First four HP:

H1 (zt) = zt

H2 (zt) = z2t − 1

H3 (zt) = z3t − 3zt

H4 (zt) = z4t − 6z2

t + 3

These polynomials form an orthonormal basis:∫Hs (zt)Hj (zt)φ (zt) dzt = 0 ∀s 6= j




Fitted Densities

0

0,005

0,01

0,015

0,02

0,025

Fig. 2. Fitted densities (left tails)

Histogram Normal

Gram-Charlier Student's t

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Fig. 1. Fitted densities

Histogram Normal

Gram-Charlier Student's t




GC Type A ES

ESα =

−φ(ϕ−1(α))α

[1 +

S∑s=2

ds{Hs

(ϕ−1 (α)

)+ sHs−2

(ϕ−1 (α)

)}]whereφ is the pdf of standard normal

andϕ−1 (α) is the α-quantile of the GC Type A distribution




Backtesting methods

We compute one-step ahead forecasts for VaR and ES through a rollingwindow of size T and compare the model performance according to:

Backtesting methods for VaR

1 Bernoulli coverage test

2 Relative comparison for VaR

Backtesting methods for ES

1 t-test

2 Relative comparison for ES




Commodity ETFs Prices

Sample: Daily prices form January 2007 to January 2016 for Gold, Silver,

Oil, Agriculture, Energy abd Broad Commodity ETFs




Commodity ETFs Returns

Return data exhibits: volatility clustering, leptokurtosis, skewness,leverage effects




Coverage test: 99%-VaR

Skewed-t and GC outperform the rest of the models. Result iscorroborated by the Diebold Mariano test for relative performance(pairwise comparison).




t-test: 97.5%-ES

Skewed-t and GC outperform the rest of the models. Result iscorroborated by the Diebold Mariano test for relative performance(pairwise comparison).




Conclusion

We applied backtesting methods for both VaR and ES todifferent Commodity ETFs.

We compare the performance of different parametric andsemi-nonparametric specifications both in univariateframework.

Coverage and relative performance tests show that the skewedt and Gram-Charlier outperform other more traditional densityspecifications.

We show that the Gram-Charlier distribution is very tractablefor empirical purposes and provide a closed expression for ESwith GC distribution.

We recommend the application of this distribution to mitigateregulation concerns about global financial stability andcommodities risk assessment.




Future Work

Applications of other risk measures such as median shortfall(it is also elicitable), spectral risk measures.

Compare results with other tests (Acerbi and Szekely, 2014).

Other commodity assets: Commodity Leveraged-ETFs.




Thank you!

Andres Mora-Valencia

[email protected]


risk assessment in commodity markets with semi-nonparametric speciﬁcations

Economy & Finance