University of Sydney
Business School
Discipline of Business Analytics

A Joint Value at Risk and Expected Shortfall Combination Framework and its
Applications in the Cryptocurrency Market

A dissertation submitted in partial fulfilment
of the requirements for the award of Honours
_Author:_ Zhengkun Li
450018729 _Supervisors:_
Prof. Richard Gerlach
Dr. Laurent Pauwels

### Certificate of Originality

I hereby declare that I have read and understood the Academic Honesty in
Coursework Policy 2015.

I understand that failure to comply with the above can lead to the University
commencing proceedings against me for potential student misconduct under
Chapter 8 of the University of Sydney By-Law 1999 (as amended). I declare that
this submission is my own work and to the best of my knowledge it contains no
materials previously published or written by another person, nor material
which to a substantial extent has been accepted for the award of any other
degree or diploma at University of Sydney or at any other educational
institution, except where due acknowledgement is made in the thesis.

Any contribution made to the research by others, with whom I have worked at
University of Sydney or elsewhere, is explicitly acknowledged in the thesis. I
also declare that the intellectual content of this thesis is the product of my
own work, except to the extent that assistance from others in the project’s
design and conception or in style, presentation and linguistic expression is
acknowledged.

Zhengkun LI

### Acknowledgement

First and foremost, I would like to express my gratitude to my supervisors,
Professor Richard Gerlach and Dr. Laurent Pauwels. Richard, you have
introduced me to the academic world of quantitative financial risk management
and time series modelling with your expert knowledge, patient guidance and
support. Laurent, the suggestions, guidance and time series knowledge from you
are invaluable. I greatly appreciate my supervisory team, as without their
unselfish attitude and profound knowledge, this thesis and other works would
not have been possible in this challenging honour year.

I wish to thank the BA honour coordinator Dr. Dmytro Matsypura and the
lecturer Professor Junbin Gao. Dmytro, thanks for your continuous support and
kind suggestions that have guided us through the honour year. Junbin, I cannot
understand most journal articles without the statistical knowledge from your
lectures, and the knowledge you shared has introduced me to the statistical
world. Additionally, I would like to thank the 2018 business school honour
coordinator Associate Professor Diane van den Broek: the knowledge and skills
from your lectures taught me how to conduct research as a new researcher, and
thank you for your support during the whole year.

I also wish to acknowledge the supports of all the staff in the discipline of
Business Analytics, especially the comments, suggestions and insights of
Associate Professor Artem Prokhorov, Associate Professor Daniel Oron and Dr.
Chao Wang. Your advice related to the fields of cryptocurrency and financial
time series modelling have guided me to improve my research quality.
Additionally, thanks American Journal Experts (AJE) for English language
editing.

Thanks also to the research lab crew with whom I have shared the majority of
my honour time. Firstly, to the 2018 BA cohort - Haonan, Jiarui, Kaiming, Siyu
and Zhenhao - it has been my pleasure to spend my honour year with this
incredible group of people. To the PhD candidates - Jessica Leung and Henry
Cheung - the enjoyable and humorous moments shared with you are unforgettable.

Finally, I would like to convey my gratefulness to my parents and girlfriend
Chen TAN for continuously supporting me in my studies and daily life, which
always motivates me to keep going. I owe all my sanity to you. Thank you.

### Abstract

Value at risk and expected shortfall are increasingly popular tail risk
measures in the financial risk management field. Both academia and financial
institutions are working to improve tail risk forecasts in order to meet the
requirements of the Basel Capital Accord; it states that one purpose of risk
management and measuring risk accuracy is, since extreme movements cannot
always be avoided, financial institutions can prepare for these extreme
returns by capital allocation, and putting aside the appropriate amount of
capital so as to avoid default in times of extreme price or index movements.
Forecast combination has drawn much attention, as a combined forecast can
outperform the individual forecasts under certain conditions. We propose two
methodology, one is a semiparametric combination framework that can jointly
produce combined value at risk and expected shortfall forecasts, another one
is a parametric regression framework named as Quantile-ES regression that can
produce combined expected shortfall forecasts. The favourability of the
semiparametric combination framework has been presented via an empirical study
- application in cryptocurrency markets with high-frequency data where the
necessity of risk management application increases as the cryptocurrency
market becomes more popular and mature. Additionally, the general framework of
the parametric Quantile-ES regression has been presented via a simulation
study, whereas it still need to be improved in the future. The contributions
of this work include but are not limited to the enabling of the combination of
expected shortfall forecasts and the application of risk management procedures
in the cryptocurrency market with high-frequency data.

###### Contents

  1. 1 Introduction
  2. 2 Literature Review
    1. 2.1 Cryptocurrency
    2. 2.2 Tail Risk Measures
      1. 2.2.1 Value at Risk
      2. 2.2.2 Expected Shortfall
    3. 2.3 Volatility and Tail Risk Modelling
    4. 2.4 Forecast Combination
    5. 2.5 Forecast Evaluation
  3. 3 Methodology
    1. 3.1 Individual Model Specification
      1. 3.1.1 Parametric Models
      2. 3.1.2 Nonparametric Models
      3. 3.1.3 Semiparametric Models
    2. 3.2 Forecast Evaluation
      1. 3.2.1 Violation Rate/Ratio
      2. 3.2.2 UC Test
      3. 3.2.3 CC Test
      4. 3.2.4 DQ Test
      5. 3.2.5 Quantile Loss Function
      6. 3.2.6 AL Log Score
      7. 3.2.7 Model Confidence Set
    3. 3.3 Proposed Joint Combination Framework
  4. 4 Empirical Study
    1. 4.1 Data Description and Analysis
    2. 4.2 Forecasting Study
      1. 4.2.1 Forecasting VaR
      2. 4.2.2 Forecasting ES
  5. 5 Simulation Study
  6. 6 Conclusion
  7. A Data Analysis
  8. B Forecast Plots

###### List of Figures

  1. 4.1 Hourly Prices from 27/10/2017 19:00 to 28/10/2017 19:00
  2. 4.2 Log Return Plots
  3. 4.3 95% VaR Forecasts for the BTC
  4. 4.4 Individual Weights of the Combined VaR Forecasts at the 95% Confidence Level for the BTC Market
  5. 4.5 99% VaR Forecasts for the BTC Market
  6. 4.6 Individual Weights of the Combined VaR Forecasts at the 99% Confidence Level for the BTC Market
  7. 4.7 95% ES Forecasts for the BTC Market
  8. 4.8 Individual Weights of the Combined ES Forecasts at the 95% Confidence Level for the BTC Market
  9. 4.9 99% ES Forecasts for the BTC Market
  10. 4.10 Individual Weights of the Combined ES Forecasts at the 99% Confidence Level for the BTC
  11. 5.1 Distribution of Simulated Return Process
  12. A.1 Log Return Distributions
  13. A.2 ACF and PACF of the Squared Returns
  14. B.1 95% VaR Forecasts for the BCH Market
  15. B.2 Individual Weights of the Combined VaR Forecasts at the 95% Confidence Level for the BCH Market
  16. B.3 95% VaR Forecasts for the ETC Market
  17. B.4 Individual Weights of the Combined VaR Forecasts at the 95% Confidence Level for the ETC Market
  18. B.5 95% VaR forecasts for the ETH Market
  19. B.6 Individual Weights of the Combined VaR Forecasts at the 95% Confidence Level for the ETH Market
  20. B.7 95% VaR Forecasts for the LTC Market
  21. B.8 Individual Weights of the Combined VaR Forecasts at the 95% Confidence Level for the LTC Market
  22. B.9 99% VaR forecasts for the BCH Market
  23. B.10 Individual Weights of the Combined VaR Forecasts at the 99% Confidence Level for the BCH Market
  24. B.11 99% VaR Forecasts for the ETC Market
  25. B.12 Individual Weights of the Combined VaR Forecasts at the 99% Confidence Level for the ETC Market
  26. B.13 99% VaR Forecasts for the ETH Market
  27. B.14 Individual Weights of the Combined VaR Forecasts at the 99% Confidence Level for the ECH Market
  28. B.15 99% VaR Forecasts for the LTC Market
  29. B.16 Individual Weights of the Combined VaR Forecasts at the 99% Confidence Level for the LTC Market
  30. B.17 95% ES Forecasts for the BCH Market
  31. B.18 Individual Weights of the Combined ES Forecasts at the 95% Confidence Level for the BCH Market
  32. B.19 95% ES Forecasts for the ETC Market
  33. B.20 Individual Weights of the Combined ES Forecasts at the 95% Confidence Level for the ETC Market
  34. B.21 95% ES Forecasts for the ETH Market
  35. B.22 Individual Weights of the Combined ES Forecasts at the 95% Confidence Level for the LTC Market
  36. B.23 95% ES Forecasts for the LTC Market
  37. B.24 Individual Weights of the Combined ES Forecasts at the 95% Confidence Level for the LTC Market
  38. B.25 99% ES Forecasts for the BCH Market
  39. B.26 Individual Weights of the Combined ES Forecasts at the 99% Confidence Level for the BCH Market
  40. B.27 99% ES Forecasts for the ETC Market
  41. B.28 Individual Weights of the Combined ES Forecasts at the 99% Confidence Level for the ETC Market
  42. B.29 99% ES Forecasts for the ETH Market
  43. B.30 Individual Weights of the Combined ES Forecasts at the 99% Confidence Level for the ETH Market
  44. B.31 99% ES Forecasts for the LTC Market
  45. B.32 Individual Weights of the Combined ES Forecasts at the 99% Confidence Level for the LTC Marlet

###### List of Tables

  1. 3.1 Nominal Levels for ES Forecasts
  2. 4.1 Descriptive Statistics Summary Table
  3. 4.2 Augmented Dickey-Fuller Test Results Summary Table
  4. 4.3 Forecast Comparison of the VaR Violation Ratio at the 95% Confidence Level
  5. 4.4 Counts of Model Rejections by the Statistical Tests for the 95% VaR Forecasts
  6. 4.5 Quantile Loss Function for the 95% VaR Forecasts
  7. 4.6 Forecast Comparison of the VaR Violation Ratios at the 99% Confidence Level
  8. 4.7 Counts of Model Rejections by the Statistical Tests for the 99% VaR Forecasts
  9. 4.8 Quantile Loss Function for the 99% VaR Forecasts
  10. 4.9 Summary of the Target ESRate for the Parametric Models
  11. 4.10 Forecast Comparison of the ES Violation Ratios at the 95% Confidence Level
  12. 4.11 Counts of Model Rejections by the Statistical Tests for the 95% ES Forecasts
  13. 4.12 AL Log Score for the 95% ES Forecasts
  14. 4.13 Summary Results of the MCS for the 95% ES Forecasts
  15. 4.14 Forecast Comparison of the ES Violation Ratios at the 99% Confidence Level
  16. 4.15 Counts of Model Rejections by Statistical Tests for the 99% ES Forecasts
  17. 4.16 AL Log Score for the 99% ES Forecasts
  18. 4.17 Summary Results of the MCS for the 99% ES Forecasts
  19. 5.1 Backtesting Results Summary for the Simulation Study
  20. A.1 Descriptive Statistics Summary Table for Student’s t-Distribution DOF
  21. A.2 DOF of the Skewed Student’s t-Distribution

##  Chapter 1 Introduction

Effective risk management is of paramount importance. Therefore, an accurate
risk measure is needed, as any unexpected movements of the financial market
can have a large impact on both investors and regulators. The advantage of
measuring risks in an acceptable way is that we can change our behaviour to
reduce or avoid risks. Value at risk (VaR) and expected shortfall (ES) are two
sensitive risk measures adopted and recommended by the Basel Capital Accord
for financial institutions to estimate the minimum capital requirements for
possible losses to protect their investments from extreme market events and
movements.

VaR, introduced by J.P. Morgan in their RiskMetrics publication (Morgan et
al., 1996), is formally defined as a threshold value such that, within a given
probability, the potential loss on an underlying asset or portfolio over a
given period will not exceed the threshold value. However, VaR is criticised
as it gives no information regarding any possible loss beyond the given
threshold value. Also, VaR is not a subadditive measure; that is, the overall
VaR of a well-diversified portfolio can be larger than the sum of the
individual VaRs, which is not consistent with finance literature. Those
shortcomings led to the proposal of ES (Artzner, 1997; Artzner et al., 1999),
which is defined as the expected loss over a given period conditional on the
loss being greater than the VaR threshold. Thus, ES is also called conditional
VaR (CVaR) and is a coherent risk measure that can overcome several
shortcomings of VaR, such as non-subadditivity. Financial institutions can
allocate their capital efficiently for extreme market risks by jointly using
VaR and ES as suggested by the Basel III Capital Accord and Yamai & Yoshiba
(2005). As a result, the need for highly accurate VaR and ES forecasts is
increasing dynamically.

To generate accurate VaR and ES forecasts, many models are proposed to
estimate tail risk. Those models can be mainly categorised into three groups:
parametric, nonparametric and semiparametric. Parametric models such as GARCH
family models are highly dependent on the distribution assumptions of the
underlying residual series; that is, an adaptable distribution assumption can
produce a large improvement in the corresponding forecast performances. On the
other hand, nonparametric (including semiparametric) models may be subject to
the bias-variance tradeoff, and dependent on whether the corresponding
measurement equation can be the true equation of the underlying time series.

The application of forecast combination has drawn much attention, as it can be
benefited by combining different individual forecasts and leveraging the
advantages from each forecast. The methodology of forecast combination in
point forecast has been well developed, especially in the forecasting of
economic indicators such as interest rates and inflation rates. For tail risk
measures, the methodology used to combine different VaR quantile forecasts is
well developed by adeptly using a series of quantile regression frameworks
such as quantile regression (Halbleib & Pohlmeier, 2012) or quantile LASSO
regression (Li & Zhu, 2008). However, there is a sparse amount (even no) of
literature focused on the ES combination puzzle; that is, how to combine
several ES forecasts is still questionable.

Recently, the emergence of cryptocurrency markets, especially the Bitcoin
market, has drawn much attention from academics, investors and regulators.
However, following the large impact of cryptocurrency markets on the financial
world, the debates on the maturity of cryptocurrency markets at different
levels are never-ending. Nevertheless, several studies, such as Chu et al.
(2017) and Dyhrberg et al. (2018), have focused on the investment and forecast
properties of cryptocurrency. Additionally, there is a research gap concerning
tail risk measures in cryptocurrency markets, as the quantile risk measure may
require a larger sample size than others and cryptocurrency markets began to
emerge in the middle of 2017.

Regarding those research gaps concerning how to combine ES forecasts
efficiently to generate accurate ES forecast series, this thesis proposes a
new semiparametric joint VaR and ES combination method that allows the
combination of a series of VaR forecasts and ES forecasts jointly. The
empirical study shows the favourability of the proposed method as well as the
application of several tail risk measure estimation methods in five selected
cryptocurrencies according to their market capitalisation. Additionally, this
thesis also extends the existing quantile regression frameworks to a new
parametric regression frameworks (named as Quantile-ES Regression) which can
combine different ES forecasts by treating them as VaR forecasts.
Consequently, the two proposed methods may start a new discussion on ES
forecast combination and can be extended in several aspects to further improve
the forecast accuracy.

The remainder of this thesis is organized as follows. Chapter 2 reviews the
literature related to the estimation and evaluation of VaR and ES forecasts as
well as the forecast combination puzzle. Chapter 3 illustrates the methodology
used in this thesis. Chapter 4 shows the application of several existing VaR
and ES estimation methods as well as the proposed method. Chapter 5 proposes
the parametric Quantile-ES regression with a simulation study. Concluding
remarks are in Chapter 6.

##  Chapter 2 Literature Review

###  2.1 Cryptocurrency

The emergence of the cryptocurrency market since Bitcoin was introduced by
Nakamoto (2008) has posed great challenges and opportunities for investors,
entrepreneurs, economists and policymakers. The popularity and the use of
cryptocurrencies did not grow substantially until 2017, and the analysis of
cryptocurrencies, especially Bitcoin, has received considerably attention from
both academia and industry for different purposes.

The total market capitalisation of cryptocurrency is over 230 billion USD,
with more than 1700 cryptocurrencies (CoinMarketCap.com accessed on 8th
August, 2018). Bitcoin has a market capitalisation of 112 billion USD, which
is nearly 50 percent of the entire market. Ethereum, Ripple, Bitcoin Cash and
EOS have the next largest market capitalisations, each with a market
capitalisation greater than 5 billion dollars.

However, the cryptocurrency market was criticised for the existence of a
potential bubble underlying the prosperity gained (Fry & Cheah, 2016) as well
as for its level of maturity (Sontakke & Ghaisas, 2017). Many investors
continue to enter the market for the large potential profits caused by market
immaturity.

Nevertheless, Dyhrberg et al. (2018) examined the investibility of
cryptocurrencies by estimating the transaction costs and intraday trading
patterns, which suggested that cryptocurrencies, especially Bitcoin, are
investible. Thus, the application of the risk management procedures and
exploration of the volatility properties remain innovative due to the
existence of a large number of trading platforms and investors. A
comprehensive risk analysis would be a useful tool for both investors and
decision makers.

Currently, the literature on the volatility and other risk properties analysis
of cryptocurrencies is limited. GARCH family models were most commonly
employed in the existing papers to investigate the volatility properties of
cryptocurrencies, especially Bitcoin.

Dyhrberg (2016a, 2016b) explored the hedging properties of Bitcoin by
employing the GARCH family models with the daily return between 2010 and 2015.
She found that Bitcoin has some properties similar to those of gold and can
clearly be used as a hedge asset against American stocks and the dollar.
However, this result may not reflect the higher volatility exhibited by the
cryptocurrency market since the cryptocurrency market became active in the
middle 2017.

Chu et al. (2017) assessed the VaR of 7 cryptocurrencies by applying 12 GARCH
family models with the daily return data between 2014 and 2017. However,
cryptocurrencies exhibited much larger intraday volatility than traditional
financial stocks; therefore, the use of daily data may tend to underestimate
the risk properties of the given cryptocurrency.

Chan et al. (2017) conducted a statistical analysis of 7 cryptocurrencies by
fitting parametric models with several distribution assumptions to the daily
data from 2014 to 2017. They found that no single distribution can fit all the
cryptocurrencies. In other words, different cryptocurrencies have different
distribution behaviours.

Katsiampa (2017) explored the volatility of Bitcoin price with daily data from
2010 to 2016. Moreover, he applied a goodness-of-fit test to investigate the
powers of several GARCH family models. However, no additional evaluation
methods are employed to test the performance of each model.

Osterrieder & Lorenz (2017) provided an extreme value analysis of Bitcoin
daily returns and compared them to the traditional currency exchange rates.
They found that Bitcoin exhibits higher volatility and that the return
distribution is heavier tailed and has stronger non-normal characteristics.

Gkillas & Katsiampa (2018) studied the tail behaviour of the daily returns of
five major cryptocurrencies according to their market capitalisation by
applying extreme value theory. They found that the riskiest cryptocurrency is
Bitcoin Cash, whereas the least risky cryptocurrencies are Bitcoin and
Litecoin.

###  2.2 Tail Risk Measures

####  2.2.1 Value at Risk

Value at risk (VaR) is a tail quantile of a portfolio’s return distribution
and is defined as the maximum possible loss over a certain holding period at
the given confidence level. VaR has become a standard measure of market risk
since J. P. Morgan published the RiskMetrics system in 1994. Furthermore, VaR
has been widely used by financial institutions since the Basel Committee on
Banking Supervision (BCBS) chose it as an indicator for setting regulatory
capital requirements (Komitee, 1996).

Jorion (1996), Duffie & Pan (1997) and Dowd (1998) introduced the theoretical
framework for VaR, which can be defined as:

$\alpha=\mathbf{P}_{t+1}(r_{t+1}\leq\mathbf{VaR}_{t+1|t}(\alpha)|\mathcal{F}_{t})$
(2.1)

or

$\mathbf{VaR}_{t+1|t}(\alpha)\equiv F_{t}^{-1}(\alpha),$ (2.2)

where $\alpha\in(0,1)$ is the quantile or significance level, and
$\mathcal{F}_{t}$ denotes the information set available at time t. $F_{t}$ is
the conditional distribution, and $F_{t}^{-1}$ its inverse. Therefore, VaR can
be regarded as the $\alpha$-level quantile of a given portfolio’s loss
distribution.

Although VaR is an intuitive risk measure and has been widely used in both
industry and academia, the practicability of VaR was heavily criticised, as
the measure is incoherent and considers only the distribution quantile while
ignoring extreme loss beyond the given quantile level.

Artzner et al. (1999) provided results on the non-subadditivity of VaR; that
is, the VaR for a well-diversified portfolio can be even larger than the sum
of the individual VaRs under certain conditions. This scenario is inconsistent
with finance literature.

Additionally, the Bank of International Settlement Committee on the Global
Financial System argued that extreme market movement and the tail behaviour of
the financial asset’s return distribution cannot be captured and monitored by
VaR (Chen et al., 2012).

Furthermore, Ibragimov & Prokhorov (2016) provided new results on the non-
subadditivity of VaR which showed that diversification does not reduce VaR for
a large class of dependent extreme-tailed risks.

Consequently, another measure addressing those limitations, and that can also
be regarded as a complement to VaR, is expected shortfall.

####  2.2.2 Expected Shortfall

Artzner (1997) proposed the concept of ES, which gives the expected loss
conditional on returns exceeding the corresponding VaR threshold. Therefore,
ES is also known as conditional VaR (CVaR). Following Artzner et al. (1999),
ES can be defined as:

$\mathbf{ES}_{t+1|t}(\alpha)=\mathbf{E}_{t}[r_{t+1}|r_{t+1}<\mathbf{VaR}_{t+1|t}(\alpha)]$
(2.3)

ES combines aspects of VaR with more information about the returns
distribution in the tail; thus, it can alleviate the problems inherent in VaR
(Acerbi & Tasche, 2002). Additionally, a considerable advantage of ES over VaR
is the preservation of the convexity property, which allows ES to enter the
objective and constraints in optimisation problems under certain conditions
(Rockafellar & Uryasev, 2002).

Yamai & Yoshiba (2005) compared the performances of VaR and ES under fat-
tailed distributions and found that ES is a better risk measure than VaR in
terms of tail risk. They emphasised the facts that VaR can disregard losses
beyond the VaR level, which can cause serious problems in the real world, and
that the information provided by VaR can mislead the investors.

Both VaR and ES can be used as tail risk measures to provide a more
comprehensive risk quantification jointly because ES, in general, can be
regarded as the subproduct of VaR estimation, as noted by Yamai & Yoshiba
(2005) and Basel III Capital Accord.

###  2.3 Volatility and Tail Risk Modelling

According to Fama (1965), the distribution of financial returns should be fat-
tailed and negative skewed with conditional heteroscedasticity. This
observation is sufficiently supported by empirical results. On the basis of
these results, different models have been introduced to capture those
characteristics of financial time series.

Engle & Manganelli (2001) categorised VaR and ES estimation methods into three
groups: parametric, semiparametric and nonparametric. Additionally, they
noticed that the main difference between those groups is the method used to
address the return distributions (Engle & Manganelli, 2004).

Volatility is an important indicator in financial time series’ risk properties
and can be statistically defined as:

$\sigma_{t+1}=\sqrt{\mathbf{Var}_{t+1}|\mathcal{F}_{t}},$ (2.4)

where $\mathbf{Var}_{t+1}$ denotes the conditional variance of the given time
series at time t+1.

Some VaR and ES estimation approaches, especially the parametric class,
require modelling of volatility, which has drawn considerable attention from
both academia and industry for risk management purposes.

Engle (1982) proposed the autoregressive conditional heteroscedasticity (ARCH)
model, which has become the most popular and has been widely applied in
volatility modelling. The ARCH model allows the conditional variance to change
as a function of past residuals while leaving the unconditional variance
constant over time.

Bollerslev (1986) extended the ARCH model to the generalised ARCH (GARCH)
model, which allows a much more flexible lag structure. Until recently, the
ARCH and GARCH models are widely used and recognised as essential methods in
financial time series analysis because they can capture the changes in
variance over time. Furthermore, GARCH (1,1) is commonly used as a benchmark
in the literature.

A group of GARCH models are derived to capture the different volatility
features of financial time series. Hence, these models constitute the so-
called GARCH family models. Nelson (1991) introduced the exponential GARCH
(EGARCH) model to capture the leverage effect, that is, negative innovation
(bad news) tends to have a larger impact on the future volatility of the given
financial asset (Black, 1976). Furthermore, Glosten et al. (1993) proposed the
GJR-GARCH model to capture the leverage effect and other stylised features of
financial time series, such as volatility clustering. In addition, a large
group of extensions, such as the threshold GARCH (Zakoian, 1994) and the
double threshold GARCH (Li & Li, 1996), are proposed to capture different
features and overcome the drawbacks suffered by the GARCH model.

Because parametric modelling requires assumptions about the return
distributions, several distributions, such as the normal distribution,
Student’s t-distribution, asymmetric Laplace (AL) distribution (Chen et al.,
2012), and two-sided Weibull distribution (Chen & Gerlach, 2013), are employed
in parametric models to capture the tail behaviour of the return
distributions.

Nonparametric methods, such as the historical simulation approach, are widely
applied in industry as they do not make any distribution assumption and are
easy to implement. These methods are based on the assumption that past
information provides a complete representation of future expected returns,
that is, “history repeats itself”.

Semiparametric methods combine components of both parametric and nonparametric
methods. A commonly used semiparametric approach is conditional autoregressive
value at risk (CAViaR) proposed by Engle & Manganelli (2004), which provides a
framework to estimate VaR directly rather than based on the volatility
estimation. However, as the CAViaR model does not provide a framework for ES
estimation, Taylor (2017) extended it by incorporating the maximum likelihood
based on an AL density, which enables joint modelling of the VaR and ES.
Additionally, Taylor (2008) proposed the conditional autoregressive expectiles
(CARE) models, which incorporated the expectile theory into the VaR and ES
estimation.

###  2.4 Forecast Combination

No model is universally adequate for the estimation of VaR and ES; that is,
the best model is unknown and may vary by asset and over time. In this
situation, an alternative approach to selecting a specific risk model is to
combine the forecasts generated by several individual models.

Timmermann (2006) provided an overview of the forecast combination problem. He
provided three main arguments to support the belief that combined forecast may
be more favourable than individual forecasts. First, combined forecasts tend
to be robust to structural breaks. Second, combined forecasts benefit from
diversification gains from the combination of individual forecasts with
different assumptions, information sets, etc. Third, the impact of
misspecification from individual forecasts can be reduced by combining them.

The field of forecast combination has developed several methods for combining
point forecasts. Additionally, the empirical facts support the hypothesis that
the combined forecast can outperform individual forecasts under certain
conditions (see, e.g., Pauwels & Vasnev, 2012; Vasnev et al., 2013). The
existing well-established methods include simple average (Timmermann, 2006),
median forecast (Chan et al., 1999), Bayesian averaging (Liu & Maheu, 2009),
and exponentially weighted averaging (Yang, 2004).

However, less literature has focused on the combination of quantile forecasts.
Increasing attention has recently been devoted to the combination of quantile
forecasts to improve forecast accuracy, especially the estimation of VaR in
the financial risk management field. Moreover, no framework for how to combine
ES forecasts has been proposed in the literature.

Because true quantile prediction will always minimise the quantile loss
function, which is a linear piecewise function, Halbleib & Pohlmeier (2012)
combined VaR forecasts by employing the quantile regression introduced by
Koenker & Bassett Jr (1978). Additional extended quantile regression
frameworks were introduced to avoid overfitting and improve forecast accuracy
by incorporating different penalty terms, for example, quantile lasso
regression Li & Zhu (2008), quantile ridge regression and quantile elastic net
regression (Bayer, 2017). Those combination methods are called optimal
weighted averages as they were estimated by optimising the given tick loss
function.

###  2.5 Forecast Evaluation

Following the needs for accurate VaR and ES forecasts in financial risk
management, various commonly used tests are proposed to assess the VaR and ES
forecasts.

The evaluation procedures for VaR quantile forecasts are well-developed; VaR
forecasts can be directly assessed by applying the existing tests.

Kupiec (1995) proposed the violation ratio and unconditional coverage (UC)
test to assess whether the actual number of VaR violations over the testing
period equals the theoretically expected number.

Christoffersen (1998) extended the UC test and developed a likelihood ratios
statistic to assess the joint assumptions of UC and the independence of
failures, that is, whether the probability of an exception or violation in any
step depends on the outcome of the previous steps. This test, called the
conditional coverage (CC) test, can reject VaR quantile forecasts with either
more or less clustered violations.

Engle & Manganelli (2004) introduced the dynamic quantile (DQ) test. Similar
to the CC test, DQ test is used to assess the UC, CC and independence
properties. The DQ test has been proved to be more powerful than the CC and UC
tests. Moreover, the quantile loss function from Koenker & Bassett Jr (1978)
was regularly applied as a criterion because the best set of VaR quantile
forecasts should minimize the quantile loss function.

However, optimal assessment of ES forecasts remains an open issue under
investigation. Chen et al. (2012) illustrated how to treat ES forecasts as VaR
quantile forecasts, which allowed the ES forecasts to be directly assessed.

Fissler et al. (2016) introduced a series of joint score functions to evaluate
VaR and ES forecasts jointly based on the theoretical fact that the true ES
forecasts may not be able to minimise the quantile loss function or any other
loss function (Weber, 2006; Gneiting, 2011). Furthermore, Taylor (2017)
developed the AL log score function based on the joint framework and AL
distribution. Additionally, the model confidence set (MCS) proposed by Hansen
et al. (2011) can produce the model set that are statistically favoured based
on the given loss function.

##  Chapter 3 Methodology

###  3.1 Individual Model Specification

This section specifies the individual models that will be considered and
applied to generate the one-step-ahead forecasts of both VaR and ES. Moreover,
these models will be categorised into three groups: parametric, nonparametric
and semiparametric.

####  3.1.1 Parametric Models

##### Exponential Weighted Moving Average

The exponential weighted moving average (EWMA) method, developed by J. P.
Morgan (Morgan et al., 1996), computes the conditional variance at time $t+1$
by assigning unequal weights to past returns. The EWMA assumes that the
residuals of financial returns follow a normal distribution with the most
recent returns tending to have a larger impact on the return at time $t+1$ and
thus having higher weights. The weights exponentially decrease from recent
returns to returns that are far in the past. Thus, the conditional variance at
time $t+1$ is defined as:

$\displaystyle\hat{\sigma}^{2}_{t+1|t}=(1-\theta)r_{t}^{2}+\theta{\sigma}_{t}^{2},$

where $\theta$ is a decay factor, with recommended values of $0.94$ for daily
data and $0.97$ for monthly data. When $\theta$ is set to $1$, the weights are
assigned equally, and the EWMA method is equivalent to the normal distribution
method, which assigns equal weights to all past information.

##### GARCH (1,1)

According to Bollerslev (1986), the GARCH (1,1) model is defined as:

$\displaystyle\sigma_{t+1}^{2}=\omega+\alpha\epsilon^{2}_{t}+\beta\sigma_{t}^{2}$
$\displaystyle r_{t+1}=\mu_{t+1}+\epsilon_{t+1}$
$\displaystyle\epsilon_{t+1}=\sigma_{t+1}z_{t+1}$ $\displaystyle
z_{t+1}\sim\mathcal{D}(0,1)$ $\displaystyle\alpha,~{}\beta\geq 0,$

where $\\{z_{t}\\}$ is a sequence of independent and identically distributed
(i.i.d.) random variables. $\mu_{t+1}$ is the conditional mean at time t+1.

##### EGARCH (1,1)

In contrast to the GARCH model, the EGARCH model proposed by Nelson (1991)
makes no restriction on the parameters and considers the leverage effect. The
EGARCH (1,1) model is defined as:

$\displaystyle\ln({\sigma_{t+1}^{2}})=\alpha_{0}+\alpha_{1}\frac{|\epsilon_{t}|+\gamma_{1}\epsilon_{t}}{\sigma_{t}}+\beta_{1}\ln(\sigma^{2}_{t})$
$\displaystyle r_{t+1}=\mu_{t+1}+\epsilon_{t+1}$
$\displaystyle\epsilon_{t+1}=\sigma_{t+1}z_{t+1}$ $\displaystyle
z_{t}\sim\mathcal{D}(0,1),$

where the term $|\epsilon_{t-i}|+\gamma_{i}\epsilon_{t-i}$ is used to capture
the leverage effects in the financial market. When $\epsilon_{t-i}$ is
positive, which indicates positive innovation, the term for the positive
innovation is $1+\gamma_{i}$, whereas for a negative innovation, this term
will be $1-\gamma_{i}$.

##### GJRGARCH (1,1)

To capture other stylised characteristics of financial time series, such as
volatility clustering, Glosten et al. (1993) proposed an asymmetric GARCH
model named according to the authors’ initials, GJRGARCH. The model is
formally defined as:

$\displaystyle\sigma_{t+1}^{2}=\alpha_{0}+\alpha_{1}\epsilon_{t}+\beta_{1}\sigma^{2}_{t}+\gamma_{1}\mathbf{I}_{t}\epsilon_{t}^{2}$
$\displaystyle r_{t+1}=\mu_{t+1}+\epsilon_{t+1}$
$\displaystyle\epsilon_{t+1}=\sigma_{t+1}z_{t+1}$ $\displaystyle
z_{t}\sim\mathcal{D}(0,1)$

and:

$\mathbf{I}_{t}=\begin{cases}1&\text{if $\epsilon_{t}<0$}\\\ 0&\text{if
$\epsilon_{t}\geq 0$}\end{cases}$

where the indicator function, $\mathbf{I}_{t}$, is used to denote the leverage
effect. Clearly, a negative innovation will have a different impact on
conditional volatility from a positive innovation with the same magnitude
under the GJRGARCH framework with the indicator function.

##### VaR and ES Estimators

This section defines the corresponding estimators of both VaR and ES under
different distributions as the parametric models require an assumption on the
residuals distribution $\mathcal{D}$. This thesis will consider three commonly
used distributions: the standard normal distribution, the standardised
Student’s t-distribution and the standard skewed Student’s t-distribution.

###### Standard Normal Distribution

McNeil et al. (2005) show that the corresponding $1-\alpha$ confidence level
VaR and ES estimators are defined as:

$\begin{split}&\mathbf{VaR}_{t+1}(\alpha)=\mu_{t+1}+\sigma_{t+1}\Phi^{-1}(\alpha)\\\
&\mathbf{ES}_{t+1}(\alpha)=\mu_{t+1}+\sigma_{t+1}\frac{\phi(\Phi^{-1}(\alpha))}{\alpha},\end{split}$

where $\Phi^{-1}$ and $\phi$ are the inverse cumulative distribution function
(CDF) and probability density function (PDF) of the standard normal
distribution, respectively. $\mu_{t+1}$ and $\sigma_{t+1}$ are the
corresponding one-step-ahead conditional mean and standard deviation forecasts
from GARCH family models, respectively.

###### Standardised Student’s t-Distribution

According to McNeil et al. (2005), the corresponding $1-\alpha$ confidence
level VaR and ES estimators are defined as:

$\begin{split}&\mathbf{VaR}_{t+1}(\alpha)=\mu_{t+1}+\sigma_{t+1}t^{-1}_{v}(\alpha)\sqrt{\frac{v-2}{v}}\\\
&\mathbf{ES}_{t+1}(\alpha)=\mu_{t+1}+\sigma_{t+1}\frac{t_{v}(t_{v}^{-1}(\alpha)}{\alpha}\Big{(}\frac{v+(t_{v}^{-1}(\alpha))^{2}}{v-1}\Big{)}\sqrt{\frac{v-2}{v}},\end{split}$

where $v$ is the degree of freedom (DOF) of the standardised Student’s
t-distribution, which is estimated from the empirical sample distribution.
$t_{v}$ and $t_{v}^{-1}$ are the PDF and inverse CDF of the standardised
Student’s t-distribution, respectively.

###### Standard Skewed Student’s t-Distribution

Contino & Gerlach (2017) derive the VaR and ES estimators based on the
standard skewed Student’s t-distribution from Hansen (1994); the corresponding
$1-\alpha$ confidence level VaR and ES estimators are defined as:

$\begin{split}&\mathbf{VaR}_{t+1}(\alpha)=\mu_{t+1}+\sigma_{t+1}skt_{\alpha,v,\lambda}^{-1}\\\
&\mathbf{ES}_{t+1}(\alpha)=\mu_{t+1}+\sigma_{t+1}E[\epsilon|\epsilon<\mathbf{VaR}_{t+1}(\alpha)],\end{split}$

where:

$\begin{split}&E[\epsilon|\epsilon<\mathbf{VaR}_{t+1}(\alpha)]=\frac{c(1-\lambda)^{2}}{b\times
skt_{v,\lambda}(\mathbf{VaR}_{t+1}(\alpha))}\frac{v-2}{1-v}\Big{(}1+\frac{1}{v-2}\Big{(}\frac{b\mathbf{VaR}_{t+1}(\alpha)+a}{1-\lambda}\Big{)}^{2}\Big{)}^{\frac{1-v}{2}}-\frac{a}{b}\\\
\end{split}$

The term $skt_{v,\lambda}$ is the CDF of the standard skewed Student’s
t-distribution, which is defined as:

$skt(\epsilon|v,\lambda)=\left\\{\begin{array}[]{ll}(1+\lambda)t_{v}\Big{(}\sqrt{\frac{v}{v-2}}\Big{(}\frac{b\epsilon+a}{1-\lambda}\Big{)},\lambda\Big{)}&\text{if}~{}\epsilon<-\frac{a}{b}\\\
\frac{1-\lambda}{2}(1+\lambda)\Big{[}t_{v}\Big{(}\sqrt{\frac{v}{v-2}}\Big{(}\frac{b\epsilon+a}{1-\lambda}\Big{)},v\Big{)}-0.5\Big{]}&\text{if}~{}\epsilon\geq-\frac{a}{b}\end{array}\right.$

$v,\lambda$ are the kurtosis parameter with a range of $2<v<\infty$ and the
skewness parameter with a range of $-1<\lambda<1$, respectively. Furthermore,
a, b and c in the CDF function above are defined as:

$\begin{split}&a=4\lambda c(\frac{v-2}{v-1})\\\
&b=\sqrt{1+3\lambda^{2}-a^{2}}\\\
&c=\frac{\Gamma(\frac{v+1}{2})}{\sqrt{\pi(v-2)}\Gamma(\frac{v}{2})},\end{split}$

$\Gamma(*)$ denotes the gamma function.

####  3.1.2 Nonparametric Models

##### Historical Simulation

Instead of making a distribution assumption, the historical simulation (HS)
method simply takes the corresponding $\alpha$-level VaR quantile of the
empirical return distribution based on only the assumption that past
information can be a good representation of future returns. Similarly, the HS
method computes ES by taking the average of past empirical losses that exceed
the corresponding VaR threshold. Thus, the VaR and ES estimators are defined
as:

$\displaystyle\begin{split}&\mathbf{VaR}_{t+1}(\alpha)=\textbf{Quantile}\\{\\{r_{t}\\}_{t=1}^{n},\alpha\\}\\\
&\mathbf{ES}_{t+1}(\alpha)=\frac{1}{n}\sum_{t=1}^{n}r_{t}\mathbf{I}(r_{t}<\mathbf{VaR}_{t}(\alpha)),\end{split}$

where $\\{r_{t}\\}_{t=1}^{n}$ is the past return series, and the indicator
function $\mathbf{I}(r_{t}<\mathbf{VaR}_{t}(\alpha))$ is used to denote the
scenarios in which the losses exceed the corresponding VaR threshold.

####  3.1.3 Semiparametric Models

##### Conditional Autoregressive Value at Risk and Expected Shortfall

Engle & Manganelli (2004) proposed the CAViaR framework, which specifies the
evolution of the quantile over time by using an autoregressive process and
estimates the parameters with the regression quantiles introduced by Koenker &
Bassett Jr (1978). The general CAViaR specification is defined as:

$\displaystyle\mathbf{VaR}_{t}(\alpha)=\beta_{0}+\sum_{i=1}^{q}\beta_{i}\mathbf{VaR}_{t-i}(\alpha)+\sum^{r}_{j=1}\beta_{j}\bm{l}(\mathbf{x}_{t-j}),$

where $p=q+r+1$ is the dimension of parameter vector $\\{\bm{\beta}\\}$, and
the $\bm{l}(*)$ is a function of a finite number of lagged values of
observables. The second term, $\beta_{i}\mathbf{VaR}_{t-j}$, is used to ensure
that the quantile can change ‘smoothly’ over time, and the function
$\bm{l}(\mathbf{x}_{t-j})$ is used to connect the quantile to the past
observable variables $\mathbf{x}_{t-j}$ that belong to the information set
$\mathcal{F}_{t-j}$.

Furthermore, with the general CAViaR specification, Engle & Manganelli (2004)
introduced four examples of CAViaR processes defined as follows:

Adaptive (ADA):

$\begin{split}\mathbf{VaR}_{t}(\alpha)=\mathbf{VaR}_{t-1}(\alpha)+\beta_{1}\Big{\\{}\Big{[}1+\text{exp}\Big{(}G[r_{t-1}-\mathbf{VaR}_{t-1}(\alpha)]\Big{)}\Big{]}^{-1}-\alpha\Big{\\}},\end{split}$

where G is a positive finite number. The ADA model incorporates the following
rule: whenever the empirical losses exceed the corresponding VaR threshold,
the model immediately increases the VaR, whereas the VaR is slightly decreased
when the empirical losses do not exceed the threshold. The purpose of this
strategy is to reduce the probability of sequences of exceedances. Moreover,
the ADA model has a unit coefficient the lagged VaR. Additionally, three other
alternatives are available and are defined as:

Symmetric Absolute Value (SAV):

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{VaR}_{t}(\alpha)=\beta_{1}+\beta_{2}\mathbf{VaR}_{t-1}(\alpha)+\beta_{3}|r_{t-1}|,$

Asymmetric Slope (AS):

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{VaR}_{t}(\alpha)=\beta_{1}+\beta_{2}\mathbf{VaR}_{t-1}(\alpha)+\beta_{3}(r_{t-1})^{+}+\beta_{4}(r_{t-1})^{-},$

where $(x)^{+}=\text{max}(x,0)$ and $(x)^{-}=-\text{min}(x,0)$.

Indirect GARCH(1, 1) (IG):

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{VaR}_{t}(\alpha)=\sqrt{\beta_{1}+\beta_{2}\mathbf{VaR}_{t-1}^{2}(\alpha)+\beta_{3}r^{2}_{t-1}},$

where the parameters set $\\{\bm{\beta}\\}$ should be restricted to be $>0$ to
ensure a positive value under the square root. The SAV and IG models respond
symmetrically to the past returns, and the AS model responds differently to
the past returns.

CAViaR Parameters Estimation

The parameters of CAViaR models were estimated by incorporating the regression
quantiles by Koenker & Bassett Jr (1978). They extended the notion of the
sample quantile to the linear regression model. The parameter set
$\\{\bm{\beta}\\}$ can be estimated by solving the following optimisation
problem:

$\underset{\beta}{\text{min}}\frac{1}{T}\sum^{T}_{t=1}[\alpha-\mathbf{I}(r_{t}<\mathbf{VaR}_{t}(\alpha))][r_{t}-\mathbf{VaR}_{t}(\alpha)]$

One special case of the regression quantiles is the least absolute deviation
(LAD), which is a more robust estimator than the ordinary least squares (OLS)
estimator, especially when the errors have a fat-tailed distribution.

However, CAViaR models do not provide a framework for ES estimation, which
limits the use of CAViaR models for financial risk management purposes.
Nevertheless, Taylor (2017) extended the CAViaR framework by incorporating the
AL log score function and adding the connection between VaR and ES.

Based on the CAViaR framework, Taylor specified the additional ES component by
considering the crossing between VaR and ES, as by definition, ES must be
lower than the corresponding VaR threshold. Taylor introduced two examples to
connect the ES component to the corresponding VaR from the CAViaR framework.

The first formulation models the ES component as the product of the VaR
quantile and a constant multiplicative factor (Gourieroux et al., 2012). To
avoid crossing, the constant was restricted to be greater than 1 by expressing
it in terms of an exponential function. The connection function is defined as:

$\mathbf{ES}_{t}(\alpha)=(1+\text{exp}(\gamma_{0}))\mathbf{VaR}_{t}(\alpha)$
(3.1)

Expression (3.1) correctly describes the relationship between ES and VaR but
may excessively restrict the ES component as ES will have the same dynamics as
the corresponding VaR under this restriction. Therefore, an alternative
approach that incorporates the autoregressive process is defined as:

$\displaystyle\begin{split}&\mathbf{ES}_{t}(\alpha)=\mathbf{VaR}_{t}(\alpha)-x_{t}\\\
&x_{t}=\left\\{\begin{array}[]{cc}\gamma_{0}+\gamma_{1}(\mathbf{VaR}_{t-1}(\alpha)-r_{t-1})+\gamma_{2}x_{t-1}&\text{if}~{}r_{t-1}\leq\mathbf{VaR}_{t-1}(\alpha)\\\
x_{t-1}&\text{otherwise}\end{array}\right.\end{split}$

where the parameter set $\\{\bm{\gamma}\\}$ is restricted to be nonnegative to
ensure the estimated VaR quantile and ES do not cross. Additionally, the use
of autoregressive expression can essentially smooth the magnitude of
exceedances beyond the corresponding VaR threshold.

CAViaR-ES Parameters Estimation

Taylor extended the CAViaR models to the CAViaR-ES framework, which allows the
joint estimation of VaR and ES. The parameters are estimated by minimising a
new scoring function based on the AL density. The new scoring function, AL log
score, is also considered as an evaluation function to test the model
performance; the details of this score function will be presented in Section
3.2 later. According to Taylor (2017), the AL log score is defined as:

$\mathcal{S}(\mathbf{VaR}_{t}(\alpha),\mathbf{ES}_{t}(\alpha),r_{t})=-\ln\Big{(}\frac{\alpha-1}{\mathbf{ES}_{t}(\alpha)}\Big{)}-\frac{(r_{t}-\mathbf{VaR}_{t}(\alpha)(\alpha-\mathbf{I}(r_{t}\leq\mathbf{VaR}_{t}(\alpha))}{\alpha\mathbf{ES}_{t}(\alpha)},$

where the expressions of $\mathbf{VaR}_{t}(\alpha)$ and
$\mathbf{ES}_{t}(\alpha)$ can be any combination of the CAViaR models and the
additional ES components from Taylor’s joint framework.

##### Conditional Autoregressive Expectiles

Taylor (2008) proposed the conditional autoregressive expectiles (CARE)
framework, which incorporates expectile theory into the estimation of VaR and
ES.

Expectile Theory

The concept of expectiles was originally introduced by Aigner et al. (1976) to
construct an estimator that assigns different weights to positive and negative
residuals in the classic regression model estimation process. In their work,
for any univariate continuous random variable $Y$, the corresponding
$\tau$-level expectile $\mu_{\tau}$ is defined as:

$\mu_{\tau}=\underset{\mu_{\tau}}{\text{argmin}}~{}E\Big{[}|\tau-\mathbf{I}[Y<\mu_{\tau}]|(Y-\mu_{\tau})^{2}\Big{]},$
(3.2)

where the $\tau$-level expectile $\mu_{\tau}$ can be estimated by minimising
the expectation function (3.2) with $\tau\in(0,1)$. Note that $\mu_{0.5}=E(Y)$
because the weights are equally assigned to values greater and less than the
corresponding expectile $\mu_{\tau}$.

Therefore, given the sample returns $r_{1},\cdots,r_{n}$ on $Y$, with a fixed
$\tau$, the constant $\tau$-level expectile on $Y$, $\mu_{\tau}$, can be
estimated by minimising the asymmetric sum of squares function, denoted as:

$\sum^{n}_{t=1}\Big{[}|\tau-\mathbf{I}[r_{t}<\mu_{\tau}]|(r_{t}-\mu_{\tau})^{2}\Big{]}$
(3.3)

CARE Modelling

Taylor took the CAViaR framework from Engle & Manganelli (2004) and replaced
the dynamic quantile terms with dynamic expectile terms and thus introduced
three models, CARE-SAV, CARE-AS and CARE-IG, where the model expressions are
defined by the following specifications:

CARE-SAV:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mu_{t}=\beta_{1}+\beta_{2}+\beta_{3}|r_{t-1}|,$

where the parameter set $\\{\bm{\beta}\\}$ is unrestricted.

CARE-AS:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mu_{t}=\beta_{1}+\beta_{2}\mu_{t-1}+\Big{(}\beta_{3}\mathbf{I}(r_{t-1}>0)+\beta_{4}\mathbf{I}(r_{t-1}<0)\Big{)}|r_{t-1}|,$

where the conditional expectile responds to positive and negative shocks
asymmetrically, which is similar to the CAViaR-AS model.

CARE-IG:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mu_{t}=-\sqrt{\beta_{1}+\beta_{2}\mu_{t-1}^{2}+\beta_{3}r_{t-1}^{2}},$

where the parameter set $\\{\bm{\beta}\\}$ is restricted to be positive to
ensure positivity under the square root.

CARE Parameters Estimation

The parameters of CARE models are estimated by minimising the asymmetric sum
of squares function, as shown in expression (3.3). The estimation process of
the parameter set $\\{\bm{\beta}\\}$ can be expressed as:

$\\{\bm{\beta}\\}=\underset{\\{\bm{\beta}\\}}{\text{argmin}}~{}\sum^{n}_{t=1}\Big{[}|\tau-\mathbf{I}[r_{t}-\mu_{\tau}]|(r_{t}-\mu_{\tau})^{2}\Big{]}$

CARE VaR and ES Estimators

For the VaR, Taylor used the $\tau$-level expectile as the estimator of the
corresponding VaR quantile at the same level. This one-to-one mapping from
expectiles to quantiles is supported by the theoretical works of Jones (1994),
Abdous & Remillard (1995), etc.

To connect the expectiles to the corresponding ES, Taylor showed that the ES
estimator is a simple function of the expectile based on the theoretical work
from Newey & Powell (1987). The one-to-one connection function that links the
expectile and ES is defined as:

$\mathbf{ES}_{t}(\alpha)=\Big{(}1+\frac{\tau}{(1-2\tau)\alpha_{\tau}}\Big{)}\mu_{\tau},$

where the expectile $\mu_{\tau}$ is equivalent to the corresponding
$\alpha$-level quantile of the given return series.

###  3.2 Forecast Evaluation

The violation ratio, UC test, CC test, DQ test and quantile loss function are
employed to test the forecast accuracy of the VaR estimation. These tests are
now standard. For the ES estimation, the quantile loss function is replaced by
the AL log score function. In addition, the model confidence set that
incorporates the AL log score function is applied to compare the joint VaR and
ES forecast performances.

####  3.2.1 Violation Rate/Ratio

The violation rate is an informal indicator for comparing the VaR quantile and
ES forecast performances by testing whether the empirical violation rate
equals the theoretical expectation. We define the corresponding VaR and ES
violation rates as Expressions (3.4) and (3.5), respectively.

$\textbf{VRate}=\frac{\sum^{T+m}_{t=T+1}\mathbf{I}(r_{t}<\mathbf{VaR}_{t}(\alpha))}{m}$
(3.4)
$\textbf{ESRate}=\frac{\sum^{T+m}_{t=T+1}\mathbf{I}(r_{t}<\mathbf{ES}_{t}(\alpha))}{m},$
(3.5)

where $m$ is the forecasting period, and $T$ is the estimation sample size,
in-sample size or initial window size.

The use of the violation rate is slightly different between VaR and ES because
of their mathematical properties. For the $\alpha$-level VaR, which can be
treated like the $\alpha$-level quantile, the best model should be able to
produce the VaR forecast with an $\alpha$ violation rate. For example, the
$95\%$ confidence level VaR should have a violation ratio of $5\%$. A model
with a higher or lower violation rate should be rejected since it is not
consistent with the theoretical expectation, even if the model performs well
in terms of the other evaluation criteria.

The concept of violation ratio will be employed to visualise the violation
rate and make it easy to compare. The violation ratio is defined as:

$\begin{split}&\textbf{VRatio}=\frac{\textbf{VRate}}{\alpha}\\\
&\textbf{ESRatio}=\frac{\textbf{ESRate}}{\delta_{\alpha}},\end{split}$

where $\alpha$ is the quantile level, and $\delta_{\alpha}$, which is slightly
different between each model and depends on the distribution assumption,
denotes the corresponding nominal level of the ES forecasts.

For the ES forecasts, the denominator of the violation ratio is not the
corresponding significance level $\alpha$ because the ES forecasts cannot be
treated as $\alpha$-level quantile forecasts. Nevertheless, Chen et al. (2012)
and Gerlach & Chen (2014) provided recommendations for testing ES forecasts by
treating them as quantile forecasts at specific quantile levels. They further
proved that such levels appear to be robust to the underlying conditional
distributions. Table 3.1 summarises the corresponding denominator
$\delta_{\alpha}$ that should be used to calculate the violation ratio for
parametric models with different distribution assumptions (Gerlach & Chen,
2014). The single number in each bracket is the DOF of the corresponding
Student’s t or skewed Student’s t-distribution. Furthermore, for the
nonparametric and semiparametric models, Gerlach & Chen (2014) proposed that
the $1\%$ and $5\%$ ES values will occur close to $0.36\%$ and $1.8\%$,
respectively, for daily return data. For example, see Gerlach & Wang (2016).
Therefore, the target violation ratio of each VaR and ES forecast is $1$, with
the corresponding denominator used in the calculation. Additionally, a
violation ratio $<1$ indicates that the risk and loss forecasts are higher
than the true values (conservative); alternatively, when a violation ratio
$>1$ indicates that risk and loss forecasts are lower than the true values
(anti-conservative). Financial institutions must allocate sufficient capital
and funds to cover future potential losses; thus, overestimating the risk
level is preferred to underestimating it, that is, a violation ratio $=0.95$
is preferred to a violation ratio $=1.05$. For example, see Gerlach et al.
(2011).

Table 3.1: Nominal Levels for ES Forecasts | Nominal Levels $\delta_{\alpha}$
---|---
$\alpha$ | N(0,1) | AL | t*(10) | t*(6) | t*(4) | Sk-t*(6) | Sk-t*(4) | TW
0.01 | 0.0038 | 0.0037 | 0.0036 | 0.0034 | 0.0032 | 0.0034 | 0.0032 | 0.0035-0.0037
0.05 | 0.0196 | 0.0184 | 0.0184 | 0.0175 | 0.0164 | 0.0175 | 0.0164 | 0.0184-0.0187

####  3.2.2 UC Test

The UC test from Kupiec (1995) is a formal statistical procedure to assess the
accuracy of VaR quantile forecasts by testing whether the violation rates are
equal to the theoretical expectation. The ES forecasts are tested using the
corresponding quantile level, as shown in Table 3.1. Mathematically, the null
and alternative hypothesis are defined as:

$\displaystyle H_{0}:\frac{N}{m}=\alpha$ $\displaystyle
H_{1}:\frac{N}{m}\neq\alpha,$

where $N$ is the number of violations during the forecasting period $m$. The
corresponding likelihood test statistic is defined as:

$\displaystyle
t_{uc}=2\ln\left[\left(1-\frac{N}{m}\right)^{T-m}\left(\frac{N}{m}\right)^{N}\right]-2\ln\left[(\alpha)^{m-N}(1-\alpha)^{N}\right]\sim\chi^{2}(1)$
(3.6)

The UC test follows the Chi-squared distribution with DOF $=1$, that is, the
corresponding critical values are 3.84 and 6.63 at the 5% and 1% significance
levels, respectively. The UC test can reject VaR or ES forecasts with too many
or too few violations.

####  3.2.3 CC Test

The CC test is another hypothesis test proposed by Christoffersen (1998) to
jointly investigate the following:

  * $-$

Whether the number of violations is statistically consistent with the
theoretically expected value (unconditional coverage property).

  * $-$

Whether the violations are independently distributed, e.g., no violation
clustering.

  * $-$

Whether the violation clustering is within allowable limits (conditional
coverage property).

The corresponding likelihood test statistic is defined as:

$\displaystyle
t_{cc}=-2\ln[(\alpha)^{m-N}(1-\alpha)^{N}]+2\ln[(1-\pi_{01})^{\pi_{00}}\pi_{01}^{n_{01}}(1-\pi_{11})^{n_{10}}\pi_{11}^{n_{11}}]\sim\chi^{2}(2),$
(3.7)

where:

  * $-$

$n_{00}$ denotes the number of cases in which no failure is followed by no
failure.

  * $-$

$n_{10}$ denotes the number of cases in which failure is followed by no
failure.

  * $-$

$n_{01}$ denotes the number of cases in which no failure is followed by
failure.

  * $-$

$n_{11}$ denotes the number of cases in which failure is followed by failure.

  * $-$

$\pi_{01}$ denotes the probability of having a failure at time $t$ given that
no failure occurred at time $t-1$.

  * $-$

$\pi_{10}$ denotes the probability of no failure at time $t$ given that a
failure occurred at time $t-1$.

  * $-$

$\pi_{11}$ denotes the probability of having a failure at time $t$ given that
a failure occurred at time $t-1$.

  * $-$

$\pi_{00}$ denotes the probability of having no failure at time $t$ given that
a failure occurred at time $t-1$.

The CC test asymptotically follows a Chi-squared distribution with DOF $=2$,
that is, the corresponding critical values are 5.99 and 9.21 at the 5% and 1%
significance levels, respectively.

####  3.2.4 DQ Test

Similar to the UC and CC tests, the DQ test proposed by Engle & Manganelli
(2004) is a joint test for the coverage and independence properties of the VaR
and ES forecasts. The DQ test is more powerful than the UC and CC tests
(Berkowitz et al., 2011). Both in-sample and out-of-sample tests were
introduced by Engle & Manganelli (2004), but we focus on only the out-of-
sample test because the DQ test is employed to assess the VaR and ES forecasts
with rolling windows. To begin, at the given significance level $\alpha$, a
series of ‘hits’ are calculated. The variable H is defined as:

$H_{t}=\mathbf{I_{t}}(\mathbf{VaR}_{t}(\alpha)<r_{t})-\alpha,$

where $t\in m$ is the out-of-sample testing period, which must belong to the
forecasting window $m$. Statistically, the expectation of hits,
$E(\textbf{H})$, is zero. Therefore, the null hypothesis of the DQ test is
that the series of hits H is composed of i.i.d. variables with the hit rate
equal to the significance level $\alpha$.

Furthermore, the DQ test constructs a regression as follows:

$H_{t}=\beta_{0}+\sum^{K}_{k=1}\beta_{k}H_{t-k}+\epsilon_{t},$ (3.8)

where k is the lag number. With the regression (3.8), the DQ test jointly
assesses the null hypothesis, which is defined as:

$\displaystyle H_{0}:\beta_{0}=\beta_{1}=\dots=\beta_{k}=0,$ (3.9)

where $\beta_{0}=0$ indicates $E(\textbf{H})=0$, and
$\beta_{1}=\dots=\beta_{k}=0$ indicates the independence property. The lag
value k was set to $1$ or $4$, as recommended by Engle & Manganelli (2004),
since Chen et al. (2012) illustrated that the test outcome is not sensitive to
the choice of lag value $k$.

The corresponding test statistic is defined as:

$DQ=\frac{\mathbf{H}^{\prime}\mathbf{W}(\mathbf{W}^{\prime}\mathbf{W})^{-1}\mathbf{W}^{\prime}\mathbf{H}}{\alpha(1-\alpha)},$

where for $k=1$,
$W_{t}^{\prime}=\Big{(}1,H_{t-1},\mathbf{VaR}_{t}(\alpha)\Big{)}$, and $k=4$,
$W_{t}^{\prime}=\Big{(}H_{t-1},\dots,H_{t-4},\mathbf{VaR}_{t}(\alpha)\Big{)}$.
Asymptotically, the DQ test follows a Chi-squared distribution with DOF = k+2,
that is, the corresponding critical values are 7.82 and 11.34 at the 5% and 1%
significance levels, respectively, for a lag value equal to 1. The
corresponding critical values are 12.59 and 16.81 at the 5% and 1%
significance levels, respectively, for a lag value equal to 4.

####  3.2.5 Quantile Loss Function

The CC test, UC test and DQ test are all statistical tests that do not
consider differences in magnitude; that is, the tests have less power to
distinguish between VaR forecasts with very similar dynamics and violation
ratios. The quantile loss function, proposed by Koenker & Bassett Jr (1978),
can be used as a criterion function to assess VaR forecasts by accounting for
the violations and corresponding magnitude; the most accurate VaR quantile
forecast should minimise the quantile loss function. The quantile loss
function is defined as:

$\text{QLF}=\sum^{T+m}_{t=T+1}\Big{(}r_{t}-\mathbf{VaR}_{t}(\alpha)\Big{)}\Big{(}\alpha-\mathbf{I}(r_{t}<\mathbf{VaR}_{t}(\alpha))\Big{)}$
(3.10)

The quantile loss function expressed as Equation (3.10) is a piecewise
function that can be used to assess VaR quantile forecasts by treating
violation and non-violation scenarios in different ways. For example, for a
95% confidence level VaR forecast, if a violation occurs, 95% of the violation
magnitude is penalised; by contrast, 5% of the non-violation magnitude is
penalised in non-violation cases. Therefore, the VaR quantile forecasts with
the lowest loss function value are preferred. However, for ES forecasts, the
best forecasts may not minimise the quantile loss function or other loss
functions, a condition that is called not elicitable (Fissler et al., 2016).

####  3.2.6 AL Log Score

The term elicitable is used to describe whether the correct forecast of a
certain measure can uniquely minimise the expectation of at least one loss or
scoring function (Fissler et al., 2016). Gneiting (2011) showed that the ES
measure is not elicitable. Nevertheless, Fissler et al. (2016) illustrated
that although ES is not elicitable, the VaR and ES measures are jointly
elicitable; for example, the mean and variance are jointly elicitable, even
through the variance is not elicitable itself. They presented the form of the
strictly consistent scoring functions used for jointly testing the VaR and ES
forecasts:

$\begin{split}\mathcal{S}(\mathbf{VaR}_{t}(\alpha),\mathbf{ES}_{t}(\alpha),r_{t})&=\Big{(}\mathbf{I}(r_{t}\leq\mathbf{VaR}_{t}(\alpha))-\alpha\Big{)}G_{1}\Big{(}\mathbf{VaR}_{t}(\alpha)\Big{)}-\mathbf{I}(r_{t}\leq\mathbf{VaR}_{t}(\alpha))G_{1}(r_{t})\\\
&+G_{2}\Big{(}\mathbf{ES}_{t}(\alpha)\Big{)}\Big{(}\mathbf{ES}_{t}(\alpha)-\mathbf{VaR}_{t}(\alpha)+\mathbf{I}(r_{t}\leq\mathbf{VaR}_{t}(\alpha))\\\
&\times(\mathbf{VaR}_{t}(\alpha)-r_{t})/\alpha\Big{)}-\zeta_{2}\Big{(}\mathbf{ES}_{t}(\alpha)\Big{)}+a(r_{t}),\end{split}$
(3.11)

where $G_{1},G_{2}$ and $\zeta_{2}$ are functions that must satisfy certain
conditions, namely, $G_{2}=\zeta_{2}^{\prime}$, $G_{1}$ is increasing, and
$\zeta_{2}$ is increasing and convex. Taylor (2017) proposed the AL log score
function based on Expression (3.11), as well as the AL log-likelihood
function, which is defined as:

$\text{AL}=\frac{1}{m}\sum^{T+m}_{t=T+1}\Big{[}-\ln\Big{(}\frac{\alpha-1}{\mathbf{ES}_{t}(\alpha)}\Big{)}-\frac{(r_{t}-\mathbf{VaR}_{t}(\alpha))(\alpha-\mathbf{I}(r_{t}\leq\mathbf{VaR}_{t}(\alpha)))}{\alpha\mathbf{ES}_{t}(\alpha)}\Big{]}$
(3.12)

The AL log score function is strictly consistent and can be used as a loss
function to jointly assess the accuracy of VaR and ES forecasts (Gneiting &
Raftery, 2007). Models that can minimise the AL log score function are
preferred.

####  3.2.7 Model Confidence Set

The model confidence set (MCS) proposed by Hansen et al. (2011) is further
employed to compare the ES forecast performances between each applied method
and identify the set of the best models. The MCS summarises the relative
performance of an entire set of models by selecting the models that are
statistically superior at a given confidence level.

According to Hansen et al. (2011), the starting point of the MCS is a finite
set of forecasting models, denoted as $\mathcal{M}_{0}$. Next, the MCS aims to
identify the best model set $\mathcal{M}^{*}$ through a sequence of
significance tests defined as:

$\mathcal{M^{*}}=\\{i\in\mathcal{M}_{0}:u_{i,j}\leq 0\quad\text{for all}\quad
j\in\mathcal{M}_{0}\\}$

where $u_{i,j}=\mathbf{E}(d_{ij,t})$ is the expected loss differential between
each model pair and $d_{ij,t}=L_{i,t}-L_{j,t}$ is the model loss differential.
We set the loss function $L$ as the AL log score during the ES evaluation
process.

Therefore, when given an initial set of forecasting models $\mathcal{M}_{0}$,
the MCS algorithm searches for the set of models that cannot be rejected
statistically at the given confidence level. The corresponding null hypothesis
is defined as:

$H_{0,\mathcal{M}}:u_{ij}=0,\quad\text{for all}\quad i,j\in\mathcal{M}.$

The t-statistic associated with the null hypothesis of the MCS is defined as:

$t_{ij}=\frac{\bar{d}_{ij}}{\sqrt{\hat{var}(\bar{d}_{ij})}},$
$T_{\mathcal{M}}=\max_{i,j\in\mathcal{M}}|t_{ij}|$

where $\bar{d}_{ij}=n^{-1}\sum^{n}_{t=1}d_{ij,t}$ measures the relative loss
between model i and model j. Further, the MCS employs a stationary bootstrap
to address the issue that the asymptotic distribution of the test statistic is
non-standard.

Simply, the MCS detects the best set of models based on the following
algorithm:

  * $-$

Step 1. Enter the model set $\mathcal{M}=\mathcal{M}_{0}$, which contains all
the forecast models.

  * $-$

Step 2. Test the null hypothesis $H_{0,\mathcal{M}}$ at the given significance
level.

  * $-$

Step 3. If the null hypothesis $H_{0,\mathcal{M}}$ is rejected, the MCS
eliminates an object or a model from $\mathcal{M}$ and repeats Step 2 until
the null hypothesis is accepted at the given significance level. In the final
result, there is no significance difference between each model included in MCS
$\mathcal{M}$.

###  3.3 Proposed Joint Combination Framework

In this section, we propose a new methodology that allows the joint VaR and ES
combinations from a group of models. The idea was inspired and motivated by
Taylor (2017), who developed a VaR and ES joint forecast framework based on
the proposed AL score function. Similarly, our new methodology combines the
VaR and ES forecasts and obtains the optimal weights by minimising the AL
score function.

If a number of VaR and ES forecasts are available at time t, the combined VaR
and ES forecasts are defined as:

$\mathbf{VaR}_{t}(\alpha)=\sum^{n}_{i=1}\beta_{t}^{i}\mathbf{VaR}_{t}^{i}(\alpha)$
(3.13)

where $n$ is the number of VaR forecast models available at time t, and
$\beta_{t}^{i}$ is the weight assigned to model $i$ at time t. Expression
(3.13) returns the combined VaR forecast at time t.

$\mathbf{ES}_{t}(\alpha)=\sum^{h}_{j=1}\gamma_{t}^{j}\mathbf{ES}_{t}^{j}(\alpha)$
(3.14)

where $h$ is the number of ES forecast models available at time t, and
$\gamma_{t}^{j}$ is the weight assigned to model $j$ at time t. Expression
(3.14) returns the combined ES forecast at time t.

Clearly, two sets of weights must be estimated under the proposed combination
framework. The corresponding weight estimators are defined as:

$\begin{split}\\{\bm{\beta},\bm{\gamma}\\}&=\underset{\bm{\beta},\bm{\gamma}}{\text{argmin}}~{}E\Big{[}-\ln\Big{(}\frac{\alpha-1}{\mathbf{ES}_{t}(\alpha)}\Big{)}-\frac{(r_{t}-\mathbf{VaR}_{t}(\alpha))(\alpha-\mathbf{I}(r_{t}\leq\mathbf{VaR}_{t}(\alpha)))}{\alpha\mathbf{ES}_{t}(\alpha)}\Big{]}\\\
&=\underset{\bm{\beta},\bm{\gamma}}{\text{argmin}}~{}\frac{1}{m}\sum^{T+m}_{t=T+1}\Big{[}-\ln\Big{(}\frac{\alpha-1}{\mathbf{ES}_{t}(\alpha)}\Big{)}-\frac{(r_{t}-\mathbf{VaR}_{t}(\alpha))(\alpha-\mathbf{I}(r_{t}\leq\mathbf{VaR}_{t}(\alpha)))}{\alpha\mathbf{ES}_{t}(\alpha)}\Big{]}\end{split}$
(3.15)

where $\mathbf{VaR}_{t}(\alpha)$ and $\mathbf{ES}_{t}(\alpha)$ are the
combined VaR and ES forecasts, which are defined as Expressions (3.13) and
(3.14), respectively.

Additionally, the following constraints are set to increase the convexity
because in the forecast combination literature, the convexity restriction on
the combined weights often helps to improve the forecast performance. For
example, see Timmermann (2006). The convex weights are nonnegative and sum to
one, which is denoted as:

$\begin{split}&\sum^{n}_{i=1}\beta^{i}_{t}=1,\quad\sum^{h}_{j=1}\gamma_{t}^{j}=1\\\
&\beta^{i}_{t}\geq 0,\quad\gamma^{j}_{t}\geq 0\end{split}$ (3.16)

Furthermore, one additional inequality constraint is enforced to avoid the
crossing between the combined VaR and ES. We avoid the crossing by employing a
straightforward function, as shown in Expression (3.17).

$\sum^{h}_{j=1}\gamma_{t}^{j}\mathbf{ES}_{t}^{j}(\alpha)\leq\sum^{n}_{i=1}\beta_{t}^{i}\mathbf{VaR}_{t}^{i}(\alpha)$
(3.17)

Therefore, the proposed method combines multiple VaR and ES forecasts by
constructing a convex optimisation problem, and the weights are estimated by
solving the convex optimisation problem.

##  Chapter 4 Empirical Study

###  4.1 Data Description and Analysis

Five hourly cryptocurrency market prices are analysed: Bitcoin (BTC); Bitcoin
Cash (BCH); Ethereum Classic (ETC); Ethereum (ETH) and Litecoin (LTC). Hourly
price data from 27 October 2017 19:00 to 03 May 2018 09:00 are obtained from
Binance, one of the largest cryptocurrency trading platforms. One reason we
use data from this period is that the data before this period may not reflect
the recent characteristics and risks of the cryptocurrency market.

Furthermore, we use hourly data to analyse the risk characteristics of the
given cryptocurrencies since the high-frequency data contains more information
than do daily data. Moreover, the cryptocurrency market may present higher
intraday volatility than traditional financial stock markets. Figure 4.1 plots
the intraday prices for the five cryptocurrencies from 27 October 2017 19:00
to 28 October 2017 19:00 (the first 25 observations of each data set).
Fluctuating price movements are clearly observable, and the volatility is
substantial.

With the hourly price data, the percentage log return series are defined and
generated as:

$y_{t}=\Big{(}\ln(P_{t})-\ln(P_{t-1})\Big{)}\times 100$

where $P_{t}$ is the (closing) price in hour t.

Figure 4.1: Hourly Prices from 27/10/2017 19:00 to 28/10/2017 19:00

Table 4.1 summarises the descriptive statistics of the log return series for
the five specified cryptocurrencies. Each series has the same amount of sample
data, that is, $T+m=4502$, where T is the estimation sample and m is the
forecasting period, which were defined in Section 3. The values of $T$ and $m$
are specified in the next section, Forecasting Study. The last two indicators,
skewness and kurtosis, reflect the distribution behaviour. The log return
series of all the cryptocurrencies exhibit positive skewness (but skewness
values of less than 1), which indicates that the tail on the right side of the
distribution is slightly longer than that on the left side. Moreover, the five
cryptocurrencies also exhibit high kurtosis values, which indicates that the
corresponding distributions tend to have heavy tails or many outliers. For
example, the kurtosis value of ETH is 17.010, indicating that the distribution
of the ETH log return series has heavy tails compared to those other
cryptocurrencies and the normal distribution (with a kurtosis value $\approx
3$). Moreover, the plots of the autocorrelation function (ACF) and partial
autocorrelation function (PACF) for the squared returns show that the squared
return (defined as $((r_{t}/r_{t-1})-1)^{2}$) series of all five
cryptocurrencies exhibit significant autocorrelation (see Appendix A.2). This
result is consistent with our expectation, as autocorrelation is a
characteristic of financial time series (Fama, 1965). Therefore,
cryptocurrencies exhibit the characteristics that similar to those of
traditional financial time series, except cryptocurrencies exhibit positive
skewness while most traditional financial assets, such as stock returns,
exhibit negative skewness.

Table 4.1: Descriptive Statistics Summary Table | BTC | BCH | ETC | ETH | LTC
---|---|---|---|---|---
count | 4502.000 | 4502.000 | 4502.000 | 4502.000 | 4502.000
mean | 0.010 | 0.033 | 0.017 | 0.020 | 0.023
std | 1.518 | 2.630 | 2.472 | 1.715 | 2.085
min | -14.550 | -25.412 | -21.504 | -19.060 | -19.547
25% | -0.595 | -1.127 | -1.063 | -0.627 | -0.849
50% | 0.000 | 0.000 | 0.000 | 0.030 | 0.000
75% | 0.638 | 1.065 | 1.103 | 0.729 | 0.877
max | 14.377 | 20.909 | 20.320 | 20.041 | 21.853
skewness | 0.610 | 0.068 | 0.361 | 0.548 | 0.484
kurtosis | 13.650 | 11.109 | 8.831 | 17.010 | 14.014

A unit root test called the augmented Dickey-Fuller (ADF) test, proposed by
Dickey & Fuller (1981), are employed to assess the stationary property of the
given log return series. The ADF test is relatively robust to sample size
compared to other unit root tests (Fedorová, 2016). The null hypothesis of the
ADF test is that the given time series has a unit root, that is, the time
series exhibits some time-dependent structure and is not stationary. Thus, a
stationary time series will reject the ADF test. The critical values of the
ADF test depend on the sample size and trend, and the critical values are all
$>-5$. We applied the ADF test to the five log return series; the test results
are summarised in Table 4.2. The null hypothesis was rejected in each time
series, which indicates the hourly log return series of the five
cryptocurrencies are stationary.

Table 4.2: Augmented Dickey-Fuller Test Results Summary Table | BTC | BCH | ETC | ETH | LTC
---|---|---|---|---|---
ADF Statistic | -12.6124 | -12.1315 | -13.0000 | -12.9536 | -12.9536
ADF p-value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Test Result | Reject | Reject | Reject | Reject | Reject

###  4.2 Forecasting Study

The one-step-ahead VaR and ES forecasts, incorporating a rolling window with
initial window size $T=2000$, are generated for each hour in the forecasting
sample with size $m=2502$ in each series, using a range of competing models,
including the nonparametric, semiparametric and parametric models specified in
Section 3. Figure 4.2 plots the log return series for the five
cryptocurrencies, where the red vertical line in each series splits the data
sets into the first estimation period (or window) and the forecast period.

Figure 4.2: Log Return Plots

Notes: The red vertical line splits each data set into an estimation sample
and forecast sample for the individual models. The black vertical line splits
the data set on the right side of the red line into an estimation sample and
forecast sample for the forecast combination methods.

We employed the HS method, denoted as HS-168, as the nonparametric method. The
estimation sample size used for the HS method is 168, the last seven trading
days (7*24 = 168 hours). This sample size is different from those of the other
methods, which use an estimation sample size of 2000.

Taylor’s CAViaR-ES and CARE frameworks are used as the semiparametric methods.
We employed the CAViaR-ES-SAV-AR and CAViaR-ES-AS-EXP models; that is, the VaR
forecasts are generated by the CAViaR-SAV and CAViaR-AS models, and the AR and
AS formulas are employed to connect the ES forecasts with the corresponding
VaR forecasts. For the CARE frameworks, we employed the CARE-SAV and CARE-AS
models. All the models were specified in Section 3.

The EWMA, GARCH, GJRGARCH and EGARCH models are applied as the parametric
methods to generate the one-step-ahead VaR and ES forecasts for the five
cryptocurrencies. Furthermore, as these parametric methods require a
distribution assumption for the residuals, we employed the standard normal
distribution, standardised Student’s t-distribution and standard skewed
Student’s t-distribution. Those models are denoted as EWMA, GARCH-T, GARCH-
SKT, GJRGARCH-T, GJRGARCH-SKT, EGARCH-N, EGARCH-T and EGARCH-SKT.

A total of 2502 VaR forecasts and 2502 ES forecasts are generated (for HS-168,
we simply take the last 2502 forecasts for consistency). We employed two
standard forecast combination approaches as benchmarks: the simple average and
the median forecast methods. Furthermore, the proposed joint combination
method (denoted as proposed method) is applied in each data set by
incorporating the rolling windows approach with a window size $n=1251$ (50% of
2502 forecasts), resulting in $1251$ combined VaR forecasts and $1251$
combined ES forecasts produced jointly.

To compare the performance of each method, we applied the evaluation
procedures specified in Section 3.2 to assess the final 1200 VaR forecasts and
1200 ES forecasts of each method. We tested the last 1200 forecasts instead of
the last 1251 forecasts to obtain integer violation numbers that are easy to
compare. For example, for the forecasts at 95% and 99% confidence levels, the
expected violation numbers are 60 ($1200\times 5\%$) and 12 ($1200\times
1\%$), respectively.

Moreover, the GARCH family models with the normal distribution and Student’s
t-distribution specifications were implemented in Python by using the ARCH
package developed by Kevin Sheppard. The skewed Student’s t-distribution and
the other specified models, including the proposed joint combination
framework, were all developed by the author because corresponding Python
packages are not available. Additionally, for the evaluation procedures, all
the tests were developed by the author and implemented in Python. Those python
codes will be merged to the ARCH package in the future as one contribution of
this thesis.

####  4.2.1 Forecasting VaR

##### VaR Forecasts at the 95% Confidence Level

Table 4.3: Forecast Comparison of the VaR Violation Ratio at the 95%
Confidence Level

Model | BTC | BCH | ETC | ETH | LTC | Mean | Median | RMSE
---|---|---|---|---|---|---|---|---
EWMA | 0.250 | 0.700 | 0.400 | 0.517 | 0.317 | 0.437 | 0.400 | 0.563
HS-168 | 0.883 | 0.850 | 1.150 | 1.200 | 1.250 | 1.067 | 1.150 | 0.173
GARCH-T | 0.833 | 1.017 | 0.867 | 1.000 | 0.867 | 0.917 | 0.867 | 0.090
GARCH-SKT | 0.833 | 1.017 | 0.883 | 0.900 | 0.883 | 0.903 | 0.883 | 0.103
GJRGARCH-T | 0.867 | 1.100 | 0.900 | 0.950 | 0.900 | 0.943 | 0.900 | 0.097
GJRGARCH-SKT | 0.850 | 1.050 | 0.883 | 0.883 | 0.883 | 0.910 | 0.883 | 0.110
EGARCH-N | 0.667 | 0.800 | 0.800 | 0.833 | 0.700 | 0.760 | 0.800 | 0.240
EGARCH-T | 0.883 | 1.017 | 0.883 | 0.983 | 0.900 | 0.933 | 0.900 | 0.073
EGARCH-SKT | 0.867 | 1.000 | 0.933 | 0.867 | 0.900 | 0.913 | 0.900 | 0.087
CARE-SAV | 0.900 | 1.783 | 1.217 | 1.217 | 1.167 | 1.257 | 1.217 | 0.297
CARE-AS | 0.767 | 1.833 | 1.217 | 1.167 | 1.133 | 1.223 | 1.167 | 0.317
CAViaR-ES-SAV-AR | 0.333 | 0.933 | 0.450 | 0.700 | 0.517 | 0.587 | 0.517 | 0.413
CAViaR-ES-AS-EXP | 0.417 | 1.033 | 0.500 | 0.633 | 0.600 | 0.637 | 0.600 | 0.377
Simple Average | 0.683 | 1.000 | 0.783 | 0.817 | 0.750 | 0.807 | 0.783 | 0.193
Median Forecast | 0.783 | 0.983 | 0.850 | 0.900 | 0.833 | 0.870 | 0.850 | 0.130
Proposed Joint Method | 0.883 | 1.000 | 0.900 | 1.017 | 0.917 | 0.943 | 0.917 | 0.063

Notes: Boxes indicate the best model for each currency, red shading indicates
the 2nd best model for each currency, bold indicates the worst model for each
currency. ‘RMSE’ stands for the square root of the average squared difference
between the violation ratio for each currency and the target value of 1.

Tabel 3.4 summarises the violation ratios for the 95% VaR forecasts, as well
as the mean and median of the ratios across the five cryptocurrency markets,
for each model. As illustrated in Section 3.2, the target value of the VaR
violation ratio is 1. For the BTC market, all the models produce violation
ratios less than 1, that is, all 16 methods overestimate the risk exhibited by
the BTC market. The violation ratios from the parametric models, except for
EGARCH-N (0.667), are similar. This result is not surprising because EGARCH-N
is the only GARCH family model that incorporates the standard normal
distribution, which may not be consistent with the true distribution of the
BTC market (see Table 4.1 and Figure A.1 in Appendix A). Furthermore, the
CARE-SAV model (0.900) produces the best forecast in terms of the violation
ratio while the EWMA method (0.250) produces the worst forecast, as it also
assumes that the underlying financial asset follows a standard normal
distribution. HS-168, EGARCH-T and the proposed method produce the second best
forecasts with a violation ratio of 0.883. Additionally, the CAViaR-ES joint
models tend to highly overestimate the risk exhibited by the market. Recall
that HS-168 produces the 2nd best forecast with a violation ratio of 0.883.
One possible reason is that it uses information from the past 168 hours (1
week), which can provide a better representation of the current risk level of
the BTC market than can the past 2000 hours, which are used by the other
methods.

For the BCH market, the violation ratios produced by each model are much
better (closer to the target value of 1) than they are in the BTC market,
except for the CARE-SAV and CARE-AS. The CARE-SAV and CARE-AS models produce
VaR forecasts that are highly anti-conservative, with violation ratios equal
to 1.783 and 1.833, respectively. Both models tend to underestimate the risk
exhibited by the market. The CARE-AS model has the worst performance, with a
violation ratio of 1.833. EGARCH-N also tends to overestimate the risk
exhibited by the market, with a violation ratio of 0.800. Moreover, EGARCH-
SKT, simple average and the proposed method produce the best VaR forecasts,
with violation ratios exactly equal to the target ratio of 1. The median
forecast method produces the second best forecast, with a violation ratio of
0.983.

For the ETC market, every model, except HS-168, CARE-SAV and CARE-AS, is
conservative; that is, they tend to overestimate the risk exhibited by the
market with a violation ratio less than 1. EGARCH-SKT generates the best VaR
forecast, with a corresponding violation ratio of 0.933, whereas the EWMA
method performs the worst, with a violation ratio of 0.517. The proposed
method produces the second best VaR forecast, with a violation ratio of 0.900.

For the ETH market, GARCH-T produces the best VaR forecast, with a violation
ratio equal to the target value of 1. EGARCH-T (0.983) produces the second
best VaR forecast, and the EWMA method (0.517) produces the worst forecast and
tends to overestimate the risk level. Although the proposed method produces
the third best forecast, with a violation ratio of 1.017, only a slight
difference exists between the proposed method and EGARCH-T (violation ratio of
0.983). Moreover, GARCH-T, GJRGARCH-T and EGARCH-T all produce relatively
favourable forecasts, which suggests that Student’s t-distribution fits the
ETH market well and is a good representation of the true distribution of the
market.

For the LTC market, the proposed method produces the best VaR forecast, with a
violation ratio of 0.917. The GJRGARCH-T, EGARCH-T and EGARCH-SKT models
produce the second best VaR forecasts, with violation ratios all equal to
0.900. Furthermore, all the applied methods, except HS-168, CARE-SAV and CARE-
AS, tend to overestimate the risk exhibited by the market.

Therefore, for the 95% VaR forecasts, EWMA, EGARCH-N, CAViaR-ES-SAV-AR and
CAViaR-ES-AS-EXP models are all clearly inaccurate at the 95% confidence level
in terms of the violation ratio, with a target value of 1, across all five
cryptocurrency markets. The mean, median and RMSE show the same results. For
example, the EWMA method has the highest RMSE (0.563), that is, the average
distance between the violation ratios and the target value of 1 is the
largest, indicating the high inaccuracy of the VaR forecasts of the EWMA
method. By contrast, the proposed method has the lowest RMSE (0.063), which
indicates the relatively high accuracy of the VaR forecast. EGARCH-T has the
second lowest RMSE (0.073), as EGARCH-T is quite stable across the five
markets. Generally, the proposed method produces relatively stable and
accurate VaR forecasts at the 95% confidence level: the proposed method
performs the best in two markets, the second best in two markets and the third
best in one market. Furthermore, the proposed method always performs the best
in terms of the mean, median and RMSE.

However, a model with a violation ratio close to the target value of 1 is a
necessary but not sufficient condition to guarantee the forecast accuracy, as
it only ensures that the forecasts are consistent with the theoretical
expectation of the violation ratio. Therefore, the results of the application
of four standard tests and one loss function are summarised in Table 4.4 and
Table 4.5 to further assess the forecast performances.

Figure 4.3: 95% VaR Forecasts for the BTC

Regarding the four standard statistical tests conducted at the 5% significance
level, the VaR forecasts produced by the EWMA method were rejected the most
times by each test, across the five markets. This result is consistent with
expectations, as those forecasts perform the worst in terms of the violation
ratio. Moreover, the CAViaR-ES joint models also perform poorly as the
forecasts were rejected more than 3 times by each test and were rejected more
times in each market. For example, the CAViaR-ES-SAV-AR model is rejected 4,
4, 4 and 3 times by the UC, CC, DQ1 and DQ4 tests, respectively. In general,
more than one-half of the models perform well in the UC and CC tests, whereas
fewer models perform well in the DQ1 and DQ4 tests because the DQ test is more
powerful than the UC and CC tests. Both the proposed method and the GJRGARCH-T
model perform well: they pass all the tests across all five markets. The
second best models are the GJRGARCH-SKT model and the median forecast method,
which are rejected only 1 time by the DQ test.

Table 4.4: Counts of Model Rejections by the Statistical Tests for the 95% VaR
Forecasts

Model | UC | CC | DQ1 | DQ4 | Total
---|---|---|---|---|---
EWMA | 5 | 5 | 4 | 4 | 5
HS-168 | 0 | 0 | 2 | 1 | 2
GARCH-T | 0 | 0 | 1 | 1 | 2
GARCH-SKT | 0 | 0 | 1 | 1 | 2
GJRGARCH-T | 0 | 0 | 0 | 0 | 0
GJRGARCH-SKT | 0 | 0 | 1 | 0 | 1
EGARCH-N | 1 | 0 | 3 | 1 | 3
EGARCH-T | 0 | 0 | 2 | 2 | 3
EGARCH-SKT | 0 | 0 | 1 | 1 | 2
CARE-SAV | 0 | 1 | 2 | 2 | 2
CARE-AS | 0 | 1 | 2 | 2 | 2
CAViaR-ES-SAV-AR | 4 | 4 | 4 | 3 | 5
CAViaR-ES-AS-EXP | 4 | 4 | 5 | 3 | 5
Simple Average | 2 | 1 | 1 | 0 | 2
Median Forecast | 0 | 0 | 0 | 1 | 1
Proposed Joint Method | 0 | 0 | 0 | 0 | 0

Notes: Boxes indicate the best model, red shading indicates the 2nd best
model, and bold indicates the worst model. ‘Total’ indicates the number of
markets in which the model was rejected by at least one test.

The quantile loss function is employed to assess the accuracy of the VaR
forecasts by considering the magnitude of the difference between the VaR
forecasts and the corresponding return series. Table 4.5 summarises the
quantile loss function values of the VaR forecasts from each model across all
five markets. For the BTC market, the EWMA method produces the worst VaR
forecasts with the highest loss function value of 161.148, and the CAViaR-ES
joint models perform the second worst, with corresponding loss function values
of 155.95 and 153.328. This result is consistent with the violation ratios and
statistical tests results, where the EWMA method is rejected either 4 or 5
times by the four formal tests across the five markets. Additionally, Figure
4.3 plots several VaR forecasts of the test period (the last 1000 observations
of log returns) and clearly explains why the EWMA method performs worse under
those evaluation procedures. The VaR forecasts produced by the EWMA method
(red line) are far from the return series, which results in a low violation
ratio and high loss function value. The risk level exhibited in the BTC market
is highly overestimated by the EWMA method and the CAViaR-ES joint models.
However, the CAViaR-ES joint models are the only models that attempt to
capture extreme returns. Moreover, all the other individual models, except
EGARCH-N, produce VaR forecasts with similar loss function values of
approximately 138.00 to 140.00. The proposed method performs the best, with
the lowest quantile loss function value of 137.542, followed by HS-168 and
CARE-AS, which have the 2nd and 3rd best performance, respectively.

For the BCH market, the individual model performances are similar to one
another in terms of the loss function values. The CARE-AS model performs the
worst, with a loss function value of 275.337. Similarly, the other three
semiparametric models all have loss function values greater than 265. The
three forecast combination approaches perform well: the proposed method has
the lowest loss function value of 250.108, followed by the simple average
(253.653) and median forecast (255.024).

For the ETC market, the proposed method has the lowest loss function value of
227.437, followed by GARCH-SKT (227.486) and GARCH-T (227.666). The CAViaR-ES
joint models and the EWMA method perform considerably worse than the other
approaches, as indicated by the higher loss function values. The simple
average and median forecast approaches are both ranked among the top
performers.

For the ETH market, the semiparametric and parametric models produce VaR
forecasts with similar loss function values, and the EWMA method performs the
worst with a loss function value of 201.509. The three forecast combination
methods perform well: the simple average method has the lowest loss function
value of 178.625, followed by the median forecast (180.118) and the proposed
method (180.207).

For the LTC market, similar to the performances in the ETC market, the
proposed method performs the best, with a loss function value of 179.087,
followed by GARCH-SKT and GARCH-T, with loss function values of 179.539 and
179.647, respectively. The EWMA method has the highest loss function value of
210.186, and the CAViaR-ES joint models perform as poorly as they do in other
market.

Table 4.5: Quantile Loss Function for the 95% VaR Forecasts

Model | BTC | BCH | ETC | ETH | LTC
---|---|---|---|---|---
EWMA | 161.148 | 267.943 | 255.850 | 201.509 | 210.186
HS-168 | 138.006 | 263.544 | 242.394 | 194.412 | 189.966
GARCH-T | 140.942 | 257.341 | 227.666 | 181.259 | 179.647
GARCH-SKT | 141.258 | 257.177 | 227.486 | 182.149 | 179.539
GJRGARCH-T | 139.431 | 261.688 | 229.253 | 180.455 | 179.719
GJRGARCH-SKT | 139.926 | 261.797 | 229.437 | 181.737 | 179.830
EGARCH-N | 143.987 | 258.109 | 231.661 | 181.897 | 184.690
EGARCH-T | 138.752 | 256.870 | 230.621 | 180.817 | 181.008
EGARCH-SKT | 139.062 | 256.579 | 230.405 | 181.777 | 180.858
CARE-SAV | 138.916 | 272.917 | 232.933 | 183.957 | 185.268
CARE-AS | 138.136 | 275.337 | 232.261 | 187.989 | 187.206
CAViaR-ES-SAV-AR | 155.995 | 266.213 | 253.110 | 185.514 | 198.067
CAViaR-ES-AS-EXP | 153.328 | 268.959 | 249.299 | 188.684 | 197.811
Simple Average | 140.156 | 253.653 | 228.460 | 178.625 | 181.886
Median Forecast | 138.842 | 255.024 | 227.780 | 180.118 | 180.375
Proposed Method | 137.542 | 250.108 | 227.437 | 180.207 | 179.087

Notes: Boxes indicate the best model in each market, red shading indicates the
2nd best model and blue shading indicates the 3rd best model. Bold indicates
the worst model.

In summary, for the VaR forecasts at the 95% confidence level, the EWMA method
always performs the worst, except in the BCH market. The reason for the poor
performance is straightforward: the EWMA method assumes that the return series
of the underlying currencies follow a standard normal distribution, which is
inconsistent with reality. Additionally, it estimates the volatility with the
empirical data instead of forecasting the true volatility. Therefore, the
method is hard capture the tailed behaviour of the underlying currencies.
Similarly, the HS-168 method is not competitive. This result is consistent
with expectations, as HS-168 uses the empirical distribution instead of
estimating the true distribution of the given return series and relies
strongly on the choice of window size. For example, HS-168 performs well in
the BTC market but performs poorly in the other four markets, which indicates
that the window size of 186 hours is representative of the current market risk
level of the BTC market but not for the other four markets. Moreover, for the
performances of the four semiparametric models are not as competitive as those
of the GARCH family models (parametric models), possibly because the
distribution assumption in the GARCH family models fits the true distributions
of the given currencies well, resulting in low forecast errors and quantile
loss function values. For example, the models that incorporate Student’s-t and
skewed Student’s t-distributions always perform better than EGARCH-N, which
assumes the residual series follow a normal distribution. The semiparametric
methods may benefit from the lack of distribution restriction in some cases
but are highly subject to the bias-variance tradeoff and dependent on the
fitness of measurement equation. For example, Figure 4.3 plots the 95%
confidence level VaR forecasts for one semiparametric model (CAViaR-ES-AS-EXP)
and one nonparametric model (HS-168). Clearly, both HS-168 and CAViaR-ES-AS-
EXP generate series of VaR forecasts with high forecast errors (far away from
the return series) and, consequently, high loss function values. Additionally,
although CAViaR-ES-AS-EXP tends to capture the dynamics of the return series,
the measurement equation may not be the true equation of the time series
process and produces worse forecast performances.

Figure 4.4: Individual Weights of the Combined VaR Forecasts at the 95%
Confidence Level for the BTC Market

Furthermore, the forecast performances of the applied forecast combination
approaches are relatively favourable and stable across each market. The simple
average and median forecast approaches are stable and robust to the different
currency markets, although they may not perform the best. This result is
consistent with McNeil et al. (2005), who suggest that the forecast
combination approaches, especially the median forecast method, performed
better during the global financial crisis. In contrast to the simple average
and median forecast, which combine the individual forecasts by either taking a
simple average or taking the 50th percentile after reordering the individual
forecasts at each point, the proposed method combines the individual forecasts
by assigning the optimal weight to each individual forecast, where the weights
are estimated by optimising the tick loss function (AL log score). Figure 4.4
plots the weights assigned to individual forecasts to produce the combined
forecast under the proposed method in the BTC market. Clearly, the weights
assigned to individual models are not constant over time. The EWMA method
(black line) sometimes has a weight of 20%, even though it performs the worst
in the BTC market. These weights help to smooth the combined VaR series (see
the black line in Figure 4.3): both the dynamic variance and forecast error of
the proposed method decrease, leading to a lower loss function value. HS-168
(red line) and CARE-AS (blue line) have large weights across time as they are
the best and second best individual models, with the lowest and second lowest
loss function values, respectively. The high weights assigned to those two
models guarantee the forecast accuracy of the combined VaR forecasts and
reduce the loss function value. The weight plots and VaR series plots of the
other four markets have been placed in the Appendix to save space (see
Appendix B).

##### VaR Forecasts at the 99% Confidence Level

Table 4.6: Forecast Comparison of the VaR Violation Ratios at the 99%
Confidence Level

Model | BTC | BCH | ETC | ETH | LTC | Mean | Median | RMSE
---|---|---|---|---|---|---|---|---
EWMA | 0.333 | 0.833 | 0.583 | 0.583 | 0.500 | 0.567 | 0.583 | 0.433
HS-168 | 0.750 | 0.900 | 1.417 | 2.583 | 1.500 | 1.430 | 1.417 | 0.570
GARCH-T | 1.000 | 0.833 | 0.667 | 1.750 | 1.083 | 1.067 | 1.000 | 0.267
GARCH-SKT | 1.000 | 0.750 | 0.750 | 1.583 | 1.167 | 1.050 | 1.000 | 0.250
GJRGARCH-T | 1.167 | 0.833 | 0.833 | 1.917 | 1.000 | 1.150 | 1.000 | 0.283
GJRGARCH-SKT | 1.167 | 0.750 | 0.833 | 1.250 | 1.000 | 1.000 | 1.000 | 0.167
EGARCH-N | 1.750 | 1.250 | 1.500 | 2.083 | 1.417 | 1.600 | 1.500 | 0.600
EGARCH-T | 1.167 | 0.667 | 0.917 | 1.250 | 1.083 | 1.017 | 1.083 | 0.183
EGARCH-SKT | 1.167 | 0.667 | 1.083 | 1.000 | 1.083 | 1.000 | 1.083 | 0.133
CARE-SAV | 0.417 | 2.667 | 1.583 | 1.833 | 1.500 | 1.600 | 1.583 | 0.833
CARE-AS | 0.333 | 2.833 | 1.917 | 1.667 | 1.583 | 1.667 | 1.667 | 0.933
CAViaR-ES-SAV-AR | 0.250 | 0.583 | 0.333 | 0.500 | 0.750 | 0.483 | 0.500 | 0.517
CAViaR-ES-AS-EXP | 0.250 | 0.917 | 0.417 | 0.417 | 0.667 | 0.533 | 0.417 | 0.467
Simple Average | 0.583 | 1.000 | 0.750 | 1.000 | 1.083 | 0.883 | 1.000 | 0.150
Median Forecast | 1.000 | 0.833 | 0.750 | 1.167 | 1.083 | 0.967 | 1.000 | 0.133
Proposed Method | 0.900 | 1.083 | 1.083 | 1.000 | 1.000 | 1.013 | 1.000 | 0.053

Notes: Boxes indicate the best model for each currency, red shading indicates
the 2nd best model for each currency, and bold indicates the worst model for
each currency. ‘RMSE’ stands for the square root of the average squared
difference between the violation ratio for each currency and the target value
of 1.

Table 4.6 summarises the violation ratios for the 99% VaR forecasts across the
five cryptocurrency markets with a target ratio of 1. For the BTC market, the
GARCH-T, GARCH-SKT and median forecast models perform the best as they produce
VaR forecasts with violation ratios exactly equal to 1. The proposed method is
the second performing model, with a violation ratio of 0.9; that is, it
slightly overestimates the risk exhibited by the BTC market. The other
parametric models tend to underestimate the risk exhibited by the BTC market
as they all produce VaR forecasts with violation ratios greater than 1.
Similar to the performance at the 95% confidence level, EGARCH-N, which
assumes that the underlying asset’s residuals follow a standard normal
distribution, is the worst performing method: it produces a VaR forecast with
a violation ratio of 1.75 which is highly anti-conservative. The four
semiparametric models perform as poorly as they do at the 95% confidence
level: they are highly conservative (all less than 0.42) and tend to
overestimate the risk level.

For the BCH market, simple average is the best method, with a violation ratio
exactly equal to 1. EGARCH-T is the second best method, with a violation ratio
of 0.917, and the proposed method is the third best method, with a violation
ratio of 1.083. The worst method is the CARE-AS model, with a violation ratio
of 1.917. All the parametric and nonparametric models, except EGARCH-N, tend
to overestimate the risk level exhibited by the BCH market as they all produce
VaR forecasts with violation ratios less than one.

For the ETC market, EGARCH-T is the best model, with a violation ratio of
0.917. EGARCH-SKT and the proposed method are the second best models, with
violation ratios of 1.083. By contract, the CARE-AS model is the worst model,
with a violation ratio equals to 1.917, which is highly anti-conservative with
91.7% more VaR violations than expected. Another CARE family model, CARE-SAV,
also tends to underestimate the risk level. The CARE-SAV model is also highly
anti-conservative, with 58.3% more VaR violations than expected. By contrast,
CAViaR-ES-SAV-AS and CAViaR-ES-AS-EXP are highly conservative, with violation
ratios of 0.333 and 0.417, respectively. Moreover, the two formal forecast
combination approaches produce VaR forecast series with violation ratios equal
to 0.75.

For the ETH market, EGARCH-SKT, simple average and the proposed method are the
best approaches, with corresponding violation ratios all equal to the target
ratio of 1. Median forecast is the second best method with a violation ratio
of 1.167. By contrast, the worst model is HS-168, which is highly anti-
conservative, with 158.3% more violations than expected. Moreover, the
parametric models tend to underestimate the risk level exhibited by the ETH
market as they are all anti-conservative.

For the LTC market, GJRGARCH-T, GJRGARCH-SKT and the proposed method are the
best methods; they produce VaR forecast series with violation ratios exactly
equal to 1. The GARCH-T, EGARCH-T, EGARCH-SKT, simple average and median
forecast approaches are the second best methods, as they all produce VaR
forecasts with identical violation ratios of 1.083. The two CARE family models
perform the worst and are highly anti-conservative with violation ratios are
$\geq 1.500$. The parametric models all tend to underestimate the risk level
exhibited by the LTC market, and the VaR forecast series produced by the
parametric models all have violation ratios greater than 1.

Therefore, the proposed method is the most favoured model in terms of the
violation ratio: the proposed method performs the best in the ETH and ETC
markets, the second best in the BTC and ETC markets and the third best in the
BCH market. Although the proposed method is not the best in terms of the
violation ratio in some markets, the corresponding VaR forecasts produced by
the proposed method have stable forecast performances across the five
cryptocurrency markets. Furthermore, the proposed method performs well in
terms of the mean, median and RMSE, which also indicates the stability of the
proposed method.

Table 4.7: Counts of Model Rejections by the Statistical Tests for the 99% VaR
Forecasts

Model | UC | CC | DQ1 | DQ4 | Total
---|---|---|---|---|---
EWMA | 5 | 5 | 4 | 4 | 5
HS-168 | 0 | 0 | 2 | 1 | 2
GARCH-T | 0 | 0 | 1 | 1 | 2
GARCH-SKT | 0 | 0 | 1 | 1 | 2
GJRGARCH-T | 0 | 0 | 0 | 0 | 0
GJRGARCH-SKT | 0 | 0 | 1 | 0 | 1
EGARCH-N | 1 | 0 | 3 | 1 | 3
EGARCH-T | 0 | 0 | 2 | 2 | 3
EGARCH-SKT | 0 | 0 | 1 | 1 | 2
CARE-SAV | 0 | 1 | 2 | 2 | 2
CARE-AS | 0 | 1 | 2 | 2 | 2
CAViaR-ES-SAV-AR | 4 | 4 | 4 | 3 | 5
CAViaR-ES-AS-EXP | 4 | 4 | 5 | 3 | 5
Simple Average | 2 | 1 | 1 | 0 | 2
Median Forecast | 0 | 0 | 0 | 1 | 1
Proposed Method | 0 | 0 | 0 | 0 | 0

Notes: Boxes indicate the best model, red shading indicates the 2nd best
model, and bold indicates the worst model. ‘Total’ indicates the number of
markets in which the model was rejected by at least one test.

Table 4.7 summarises the results of four applied statistical tests. Clearly,
GJRGARCH-T and the proposed method are the best methods for obtaining the 99%
VaR forecasts, as they were not rejected by any of the four tests across the
five markets. The CAViaR-ES joint models and the EWMA method perform the
worst, with the largest number of rejections.

Table 4.8: Quantile Loss Function for the 99% VaR Forecasts

Model | BTC | BCH | ETC | ETH | LTC
---|---|---|---|---|---
EWMA | 49.755 | 80.686 | 71.638 | 59.248 | 58.112
HS-168 | 47.742 | 82.142 | 76.140 | 71.354 | 56.788
GARCH-T | 49.188 | 83.505 | 67.966 | 60.980 | 55.797
GARCH-SKT | 49.268 | 83.745 | 67.519 | 60.413 | 55.797
GJRGARCH-T | 48.112 | 81.818 | 67.961 | 59.974 | 55.882
GJRGARCH-SKT | 48.115 | 82.346 | 67.938 | 59.375 | 55.971
EGARCH-N | 48.466 | 81.018 | 69.081 | 61.231 | 57.109
EGARCH-T | 48.531 | 83.537 | 70.335 | 59.601 | 56.142
EGARCH-SKT | 48.561 | 83.937 | 70.059 | 60.011 | 56.126
CARE-SAV | 47.843 | 88.142 | 70.689 | 59.009 | 62.088
CARE-AS | 47.070 | 92.964 | 72.342 | 59.092 | 62.039
CAViaR-ES-SAV-AR | 52.876 | 88.274 | 77.476 | 60.943 | 63.859
CAViaR-ES-AS-EXP | 50.062 | 86.517 | 72.312 | 59.368 | 64.213
Simple Average | 47.232 | 80.845 | 67.346 | 57.037 | 56.404
Median Forecast | 47.577 | 81.315 | 66.977 | 57.990 | 55.978
Proposed Method | 46.959 | 80.447 | 67.102 | 57.316 | 55.556

Notes: Boxes indicate the best model in each market, red shading indicates the
2nd best model, and the blue shading indicates the 3rd best model. Bold
indicates the worst model.

We also applied the quantile loss function to further assess the model
performances across markets. Table 4.8 summarises the loss function values of
the models across the five cryptocurrency markets. The model with the lowest
loss function value is preferred. The three most preferred and the least
preferred models in each market are highlighted. For the BTC market, the
proposed method is the best method, with a quantile loss function of 46.959.
The CARE-AS model is the second best method, with a loss function of 47.070.
By contrast, CAViaR-ES-AS-EXP performs the worst, as it has the highest loss
function value of 52.876. Additionally, the three forecast combination
approaches are competitive, with lower loss function values.

For the BCH market, the proposed method produces the best VaR forecasts, with
the lowest quantile loss function value of 80.447. The EWMA method is the
second best method, with a loss function value of 80.686; however, the
corresponding violation ratio is 0.833, which is different from the target
ratio of 1 and inconsistent with the theoretical expectation. The CARE-AS
model performs the worst, with the highest loss function value of 92.964. In
general, the three forecast combination approaches perform better than the
individual models, with lower loss function values.

For the ETC market, the three forecast combination approaches perform well and
rank as the top 3 methods. Median forecast has the lowest loss function value
of 66.977, followed by the proposed method and simple average, with loss
function values of 67.102 and 67.346, respectively. By contrast, HS-168
performs the worst, with the highest loss function value of 76.140.
Additionally, the parametric models generally perform better than the
nonparametric models (including the semiparametric models).

For the ETH market, the three forecast combination approaches also perform
significantly better than the individual models. The simple average method has
the lowest loss function value of 57.037, followed by the proposed method and
median forecast, with the loss function values of 57.316 and 57.990,
respectively. The loss function values of all the other methods, except
HS-168, are approximately 60.000. HS-168 has the highest loss function value
of 71.354, which is substantially higher than those of the other methods.

For the LTC market, the proposed method (55.556) performs the best in terms of
the quantile loss function. The GARCH family models, which incorporate
Student’s t and skewed Student’s t-distributions, perform better than the
other methods, with corresponding loss function values of approximately
56.000. The four semiparametric models perform the worst, with loss function
values all greater than 62.000. Finally, CAViaR-ES-AS-EXP has the highest loss
function value of 64.213.

Figure 4.5: 99% VaR Forecasts for the BTC Market

In summary, the proposed method can generate accurate forecasts for the 99%
VaR quantile. Clearly, the three forecast combination approaches have
considerable advantages over the individual models. For example, the proposed
method can generate stable and relatively accurate 99% VaR forecasts across
the five cryptocurrency markets in terms of the violation ratio, statistical
tests and quantile loss function. The VaR series produced by the proposed
method across the five markets all have corresponding violation ratios near
the target ratio of 1, and they are not rejected by any of the four applied
statistical tests. Moreover, in terms of the quantile loss function, the
proposed method has the lowest loss function values in three markets (BTC, BCH
and LTC) and the second lowest loss function values in two markets (ETC and
ETH). Additionally, Figure 4.5 plots the 99% VaR forecasts for several methods
and the proposed method in the BTC market. The advantages of the proposed
method over the other methods can be observed in this plot. For example, the
VaR series produced by the proposed method (black line) is smoother than that
produced by CAViaR-ES-AS-EXP (green line) and has lower forecast error (closer
to the return series). Furthermore, GJRGARCH-SKT provides relatively
competitive VaR forecasts compared to the individual models, with a better
violation ratio, test results and loss function values.

Figure 4.6: Individual Weights of the Combined VaR Forecasts at the 99%
Confidence Level for the BTC Market

Figure 4.6 plots the optimal weights assigned to the individual models over
time for the BTC market. Clearly, the weights are vary considerably at each
forecast point. The proposed method generates more accurate VaR series by
assigning optimal weights to different individual forecasts over time. For
example, HS-168 (red line) and EWMA (black line) always take some weights to
smooth the combined VaR series. Additionally, different weights are assigned
to the other models to improve the forecast accuracy of the combined VaR
series.

####  4.2.2 Forecasting ES

The same set of models is applied to generate the one-step-ahead 95% and 99%
ES forecasts for the five given cryptocurrency markets. However, the
procedures that used to assess the ES forecasts are slightly different. We
still employ the violation ratio and four statistical tests to assess the
performances of the parametric models by treating them as VaR forecasts with
the corresponding target violation rates (Table 4.9), whereas for the other
models, the best way to assess the ES forecasts is via the joint VaR-ES loss
function (AL log score) and the MCS.

##### ES Forecasts at 95% Confidence Level

Table 4.9: Summary of the Target ESRate for the Parametric Models Model | 95% | 99% |
---|---|---|---
EWMA | 1.96% | 0.38% |
GARCH-T | 1.64% | 0.32% |
GARCH-SKT | 1.64% | 0.32% |
GJRGARCH-T | 1.64% | 0.32% |
GJRGARCH-SKT | 1.64% | 0.32% |
EGARCH-N | 1.96% | 0.38% |
EGARCH-T | 1.64% | 0.32% |
EGARCH-SKT | 1.64% | 0.32% |

Recall that Section 3.2 illustrated that the ES forecasts with daily return
data can be treated as VaR forecasts to be assessed. However, for hourly data,
this technique may not work. Nevertheless, we can still treat the parametric
models as VaR forecasts by fitting the corresponding distribution into each
estimation window and estimating the corresponding distribution parameters.
Table 4.9 summarises the target violation rates used for each method. We fit
the Student’s t and skewed Student’s t-distributions into the last 1200
estimation windows (with window size = 2000) to compute the corresponding DOF
for each market. The results are summarised in Table A.1 and Table A.2 in
Appendix A. The mean DOF across each window for each market is approximately 4
for both.

Table 4.10: Forecast Comparison of the ES Violation Ratios at the 95%
Confidence Level

Model | BTC | BCH | ETC | ETH | LTC | Mean | Median | RMSE
---|---|---|---|---|---|---|---|---
EWMA | 0.2551 | 1.0204 | 0.5952 | 0.5952 | 0.3827 | 0.5697 | 0.5952 | 0.4384
GARCH-T | 0.8638 | 0.5589 | 0.6606 | 1.5244 | 0.8638 | 0.8943 | 0.8638 | 0.3154
GARCH-SKT | 0.7622 | 0.5589 | 0.6606 | 1.3720 | 0.8638 | 0.8435 | 0.7622 | 0.3053
GJRGARCH-T | 0.9654 | 0.7622 | 0.7114 | 1.4736 | 1.0163 | 0.9858 | 0.9654 | 0.2102
GJRGARCH-SKT | 0.8638 | 0.7114 | 0.7114 | 1.3211 | 0.9146 | 0.9045 | 0.8638 | 0.2240
EGARCH-N | 1.1054 | 0.9779 | 1.1054 | 1.3180 | 0.9779 | 1.0969 | 1.1054 | 0.1146
EGARCH-T | 1.1179 | 0.5081 | 0.6098 | 1.2703 | 0.9146 | 0.8841 | 0.9146 | 0.2711
EGARCH-SKT | 1.1179 | 0.5081 | 0.6606 | 1.1687 | 0.8638 | 0.8638 | 0.8638 | 0.2508

Notes: Boxes indicate the best model for each currency, red shading indicates
the 2nd best model for each currency, and bold indicates the worst model for
each currency. ‘RMSE’ stands for the square root of the average squared
difference between the violation ratio for each currency and the target value
of 1.

Table 4.10 summarises the violation ratios for each 95% ES forecast from the
parametric models across all five cryptocurrency markets. Clearly, no single
model can adapt to all five markets well. In general, GJRGARCH-T performs well
as it has the best violation ratios in the BTC and LTC markets. Moreover,
GJRGARCH-T is also favourable in terms of the mean and median violation ratios
across the five markets. By contrast, the EWMA method always produces the
worst ES forecasts in terms of the violation ratio.

Table 4.11: Counts of Model Rejections by the Statistical Tests for the 95% ES
Forecasts

Model | UC | CC | DQ1 | DQ4 | Total
---|---|---|---|---|---
EWMA | 4 | 2 | 3 | 4 | 5
GARCH-T | 2 | 0 | 1 | 0 | 2
GARCH-SKT | 1 | 0 | 1 | 0 | 2
GJRGARCH-T | 1 | 0 | 1 | 0 | 1
GJRGARCH-SKT | 0 | 0 | 0 | 0 | 0
EGARCH-N | 0 | 0 | 1 | 0 | 1
EGARCH-T | 1 | 2 | 0 | 0 | 2
EGARCH-SKT | 1 | 2 | 1 | 0 | 3

Notes: Boxes indicate the best model, red shading indicates the 2nd best
model, and bold indicates the worst model. ‘Total’ indicates the number of
markets in which the model was rejected by at least one test.

Four statistical tests are applied with their target violation rates (not ES
level) as the input. The test results are summarised in Table 4.11. Clearly,
the GJRGARCH-SKT model has the best performances in the statistical tests as
it is not rejected by any test in any market. By contrast, the EWMA method has
the worst performances and is rejected by at least one test in each market.

Table 4.12: AL Log Score for the 95% ES Forecasts

Model | BTC | BCH | ETC | ETH | LTC
---|---|---|---|---|---
EWMA | 2.0389 | 2.5503 | 2.5077 | 2.2602 | 2.3105
HS-168 | 1.9369 | 2.5477 | 2.4802 | 2.2272 | 2.2574
GARCH-T | 1.9045 | 2.5177 | 2.3723 | 2.1667 | 2.1492
GARCH-SKT | 1.9029 | 2.5093 | 2.3707 | 2.1650 | 2.1487
GJRGARCH-T | 1.8871 | 2.5229 | 2.3751 | 2.1530 | 2.1490
GJRGARCH-SKT | 1.8871 | 2.5173 | 2.3768 | 2.1532 | 2.1504
EGARCH-N | 1.9287 | 2.4977 | 2.3860 | 2.1732 | 2.1718
EGARCH-T | 1.8923 | 2.5248 | 2.3988 | 2.1539 | 2.1580
EGARCH-SKT | 1.8904 | 2.5127 | 2.3959 | 2.1561 | 2.1574
CARE-SAV | 1.9472 | 2.5997 | 2.4338 | 2.2018 | 2.2294
CARE-AS | 1.9585 | 2.5953 | 2.4303 | 2.2484 | 2.2444
CAViaR-ES-SAV-AR | 2.0203 | 2.5539 | 2.5110 | 2.1732 | 2.2640
CAViaR-ES-AS-EXP | 1.9920 | 2.5605 | 2.4890 | 2.1852 | 2.2593
Simple Average | 1.9043 | 2.4929 | 2.3915 | 2.1342 | 2.1725
Median Forecast | 1.8877 | 2.4977 | 2.3718 | 2.1459 | 2.1539
Proposed Method | 1.8791 | 2.4999 | 2.3717 | 2.1472 | 2.1486

Notes: Boxed indicates the most favoured model in each market, red shading
indicates the 2nd favoured model and the blue shading indicates the 3rd
favoured model. While bold indicates the least favoured model.

For the hourly data, we can treat the ES forecasts as VaR quantile forecasts
according to Table 4.9 only if they are produced by parametric models, that
is, we need to know the distribution assumptions. The AL log score function
and MCS are employed to further assess the forecast performances and compare
each model. Table 4.12 summarises the AL log scores for each model, and Table
4.13 summarises the 75% and 90% MCS results.

In terms of the AL log score, the model with the lowest score is preferred.
The proposed method has the lowest score (1.8791) in the BTC market and is
thus the most favoured model. GJRGARCH-T and GJRGARCH-SKT are the second most
favoured method, with scores of 1.8871. Median forecast is the third most
favoured method, with a score of 1.8877. Similar to finding for the VaR
forecast performance, the EWMA method has the highest AL score and is
therefore the least favoured. The semiparametric models are not as competitive
as the parametric models in general. However, the three forecast combination
approaches are competitive.

For the BCH market, the three forecast combination methods are competitive and
rank as the top three methods with similar scores. The semiparametric models
perform as poorly as observed before, and CARE-SAV has the highest score of
2.5997.

For the ETC market, GARCH-SKT has the lowest score (2.3707) and is therefore
the most favoured model, followed by the proposed method (2.3717) and median
forecast (2.3718), which are the second and third best models, respectively.
Additionally, GARCH-SKT, GJRGARCH-T and GJRGARCH-SKT also provide competitive
ES forecasts, with AL scores (approximately 2.375) similar to those of the top
three models.

For the ETH market, the top three models are simple average (2.1342), median
forecast (2.1459) and the proposed method (2.1472). In general, the parametric
models perform better than the nonparametric models (including the
semiparametric models).

For the LTC market, the proposed method performs the best, with a score of
2.1486, followed by GARCH-SKT (2.1487) and GJRGARCH-T (2.1490), which are the
second and third best models, respectively. By contrast, the EWMA method
provides the worst ES forecasts, with the highest score of 2.3105. In general,
similar to the findings for most of the other markets, the parametric models
perform much better than the nonparametric (including semiparametric) models.

Table 4.13: Summary Results of the MCS for the 95% ES Forecasts

Model | BTC | BCH | ETC | ETH | LTC | Mean | Total
---|---|---|---|---|---|---|---
EWMA | 0.0000 | 0.1530 | 0.0010 | 0.0340 | 0.0000 | 0.0376 | 0
HS-168 | 0.0090 | 0.1530 | 0.0000 | 0.2470 | 0.0000 | 0.0818 | 0
GARCH-T | 0.6200 | 0.0070 | 0.6860 | 0.5500 | 0.8820 | 0.5490 | 4
GARCH-SKT | 0.7260 | 0.1060 | 1.0000 | 0.2720 | 0.9990 | 0.6206 | 4
GJRGARCH-T | 1.0000 | 0.0150 | 0.7630 | 0.7620 | 0.9990 | 0.7078 | 4
GJRGARCH-SKT | 0.9990 | 0.0870 | 0.5490 | 0.7620 | 0.3520 | 0.5498 | 4
EGARCH-N | 0.0570 | 0.8390 | 0.6680 | 0.2720 | 0.1090 | 0.3890 | 3
EGARCH-T | 0.9150 | 0.0000 | 0.1070 | 0.7620 | 0.7550 | 0.5078 | 3
EGARCH-SKT | 0.9970 | 0.1530 | 0.2020 | 0.3740 | 0.7810 | 0.5014 | 3
CARE-SAV | 0.1260 | 0.0150 | 0.0110 | 0.0890 | 0.0000 | 0.0482 | 0
CARE-AS | 0.0240 | 0.0300 | 0.0110 | 0.0050 | 0.0000 | 0.0140 | 0
CAViaR-ES-SAV-AR | 0.0000 | 0.0660 | 0.0000 | 0.1350 | 0.0000 | 0.0402 | 0
CAViaR-ES-AS-EXP | 0.0000 | 0.0870 | 0.0010 | 0.0840 | 0.0000 | 0.0344 | 0
Simple Average | 0.7260 | 1.0000 | 0.5490 | 1.0000 | 0.1090 | 0.6768 | 4
Median Forecast | 0.9990 | 0.5880 | 0.8750 | 0.7620 | 0.8420 | 0.8132 | 5
Proposed Method | 0.9990 | 0.8390 | 0.6930 | 0.5710 | 1.0000 | 0.8204 | 5

Notes: Models with blue shading are included by the 75% MCS; models with
either red shading or blue shading are included by the 90% MCS. ‘Total’
indicates the number of markets in which the corresponding model is included
by the 75% MCS. Boxes indicate the best model.

With the MCS, we can further assess the ES forecast performances of the
models. Table 4.13 summarises the test results of the MCS for the 95% ES
forecasts. We implement both the 75% and 90% confidence levels for the MCS.
Blue shading in Table 4.13 highlights the models included by the 75% MCS while
red shading and blue shading jointly highlight the models included by the 90%
MCS. We also compute the mean p-value for each model across the five markets.
A higher average p-value indicates greater preferability of the corresponding
model. Moreover, in the last column, we highlight the number of times the
model was included by the 75% MCS, which is more strict than the 90% MCS.

Clearly, the proposed method and median forecast are the only two models
included by the 75% MCS across all five markets; the proposed method has a
higher average p-value (0.8204). GARCH-T, GARCH-SKT, GJRGARCH-T and GJRGARCH-
SKT are included by the 75% MCS in four markets. Moreover, the three EGARCH
family models are included by the 75% MCS in three markets. The other six
models are not included by the 75% MCS in any market. Furthermore, GARCH-SKT,
EGARCH-SKT, simple average, median forecast and the proposed method are
included by the 90% MCS in all five markets. By contrast, CARE-AS and CAViaR-
ES-AS-EXP are not included by the 90% MCS in any market. CARE-AS and CAViaR-
ES-SAV-AR are included by the 90% MCS only once, in the BTC market and the ETH
market, respectively.

Figure 4.7: 95% ES Forecasts for the BTC Market

In summary, in the case of the 95% ES forecasts, the proposed method is the
best approach in terms of the AL log score and MCS. Similar to findings for
the VaR forecasts, the proposed method can generate relatively accurate and
stable ES forecasts by assigning the optimal weights to individual forecasts
and obtaining the benefits of combination. Figure 4.8 presents the weights
assigned to each individual forecast to generate the combined ES forecast at
each time point for the BTC market. Clearly, different optimal weights are
assigned to each individual forecast over time to generate the combined ES
series. Additionally, the advantage of the ES series from the proposed method
over the ES series from the individual forecasts can be roughly observed in
Figure 4.7: EGARCH-T generates the ES series that is close to the return
series, which results in a higher violation ratio (1.1179); thus, EGARCH-T
tends to underestimate the risk level exhibited by the BTC market. CAViaR-ES-
AS-EXP tends to capture the dynamics of the return series, whereas the
measurement equation may not be consistent with the true dynamics of the time
series process, which results in an increased forecast errors of the
corresponding ES series. Both forecast combination approaches, the simple
average method and the proposed method, obtain advantages from averaging
individual forecasts. The difference between the methods is that simple
average is an equal-weighted approach, whereas the proposed method is an
optimal-weighted approach. The performance of simple average is subject to the
individual model specifications, as it combines all the individual models. By
contrast, the optimal-weighted approach may assign zero weights to some
individual models during the combination process. Therefore, the combined
forecasts from these two approaches perform differently. The median forecast
method also produces accurate and stable forecasts across the five markets,
which is consistent with McNeil et al. (2005). Moreover, similar to their VaR
quantile forecast performances, the GARCH family models incorporating
Student’s t and skewed Student’s t-distributions are accurate compared to the
other individual models.

Figure 4.8: Individual Weights of the Combined ES Forecasts at the 95%
Confidence Level for the BTC Market

##### ES Forecasts at the 99% Confidence Level

We also conduct a 99% ES forecasts study with the same model set and data set.
The same evaluation procedures are employed to assess the forecast accuracy.

Table 4.14: Forecast Comparison of the ES Violation Ratios at the 99%
Confidence Level

Model | BTC | BCH | ETC | ETH | LTC | Mean | Median | RMSE
---|---|---|---|---|---|---|---|---
EWMA | 0.6579 | 0.8772 | 0.8772 | 1.0965 | 0.6579 | 0.8333 | 0.8772 | 0.2053
GARCH-T | 0.5208 | 0.2604 | 0.2604 | 1.5625 | 0.2604 | 0.5729 | 0.2604 | 0.6521
GARCH-SKT | 1.8229 | 0.5208 | 0.7813 | 2.3438 | 2.3438 | 1.5625 | 1.8229 | 0.8417
GJRGARCH-T | 0.5208 | 0.2604 | 0.2604 | 1.5625 | 0.2604 | 0.5729 | 0.2604 | 0.6521
GJRGARCH-SKT | 1.8229 | 0.5208 | 0.7813 | 2.0833 | 2.0833 | 1.4583 | 1.8229 | 0.7375
EGARCH-N | 2.1930 | 1.9737 | 2.6316 | 3.0702 | 2.6316 | 2.5000 | 2.6316 | 1.5000
EGARCH-T | 0.5208 | 0.2604 | 0.0000 | 1.0417 | 0.0000 | 0.3646 | 0.2604 | 0.6521
EGARCH-SKT | 2.3438 | 0.7813 | 0.7813 | 2.0833 | 2.0833 | 1.6146 | 2.0833 | 0.7896

Notes: Boxed indicates the favoured model for each currency, red shading
indicates the 2nd favoured model for each currency, bold indicates the least
favoured model for each currency. ‘RMSE’ stands for the square root of the
average squared difference between the five violation ratios and their target
value of 1.

Table 4.14 summarises the corresponding ES violation ratios for each model
across the five markets. The 99% violation ratios are computed by dividing the
violation rate by the target ES violation rate, which is defined in Table 4.9.
In terms of the ES violation ratio, the EWMA method is best approach in the
BTC market, with a violation ratio of 0.6579. However, EWMA is still highly
conservative and tends to overestimate the risk level exhibited by the BTC
market. The three parametric models, GARCH-T, GJRGARCH-T and EGARCH-T, which
incorporate Student’s t-distribution, generate the second best ES forecasts,
with violation ratios of 0.5208. By contrast, EGARCH-SKT is highly anti-
conservative, as it has 134% more violations than expected and it highly
underestimates the risk level.

For the BCH market, the EWMA method produces the best ES forecasts with
respect to the violation ratio. EGARCH-SKT has the second best violation ratio
of 0.7813. By contrast, EGARCH-N has the worst performance, with 97% more
violations than expected. Moreover, EGARCH-N is the only model that
underestimates the risk level exhibited by the BCH market.

For the ETC market, the EWMA method has the best performance in terms of the
violation ratio. GARCH-SKT, GJRGARCH-SKT and EGARCH-SKT produce the second
best ES forecasts, with violation ratios equal to 0.7813. Unsurprisingly,
EGARCH-N, which incorporates the standard normal distribution, has the worst
performance and is highly anti-conservative, with 163% more violations than
expected.

For the ETH market, EGARCH-T produces the best ES forecasts, with a violation
ratio of 1.0417, which is close to the expected ratio. The EWMA method has the
second best performance, with a violation ratio of 1.0965. By contrast,
EGARCH-N is highly anti-conservative, with 207% more violations than expected,
and highly underestimates the risk level. Additonally, all the models tend to
underestimate the risk level exhibited by the ETH market: they all have
corresponding violation ratios greater than 1.

For the LTC market, the best model is still EWMA, and the worst model is
EGARCH-N. However, all the methods tend to highly overestimate or
underestimate the risk level exhibited by the market, as they are either
highly conservative or highly anti-conservative. For example, the second best
models are GARCH-T and GJRGARCH-T, both of which have corresponding violation
ratios of 0.2604, which is highly conservative and overestimates the risk
level. Moreover, in terms of the mean, median and RMSE, the EWMA method is the
best approach, whereas EGARCH-N is the worst.

Table 4.15: Counts of Model Rejections by Statistical Tests for the 99% ES
Forecasts

Model | UC | CC | DQ1 | DQ4 | Total
---|---|---|---|---|---
EWMA | 0 | 0 | 0 | 1 | 1
GARCH-T | 0 | 0 | 0 | 0 | 0
GARCH-SKT | 2 | 2 | 2 | 1 | 3
GJRGARCH-T | 0 | 0 | 0 | 0 | 0
GJRGARCH-SKT | 0 | 1 | 2 | 1 | 3
EGARCH-N | 4 | 4 | 5 | 3 | 5
EGARCH-T | 2 | 0 | 0 | 0 | 2
EGARCH-SKT | 1 | 2 | 3 | 0 | 4

Notes: Boxed indicates the most favoured model, red shading indicates the 2nd
favoured model and bold indicates the least favoured model. ‘Total’ indicates
the number of markets in which the model was rejected by at least one test.

We also employ the four statistical tests to assess the forecast performances
of the parametric models. GARCH-T and GJRGARCH-T achieve the best test results
and are not rejected by any test across the five markets. The EWMA method is
the second best approach. By contrast, EGARCH-N has the worst performance and
is rejected more than three times by each test and at least one time in each
market. These results are consistent with the results of the violation ratio.