Chat with us, powered by LiveChat Asset price comparison models Asset pricing across the modern market involves different approaches that enhances the depiction of the relevant value ascribed to a certai | EssayAbode

Asset price comparison models Asset pricing across the modern market involves different approaches that enhances the depiction of the relevant value ascribed to a certai

I need help in thesis work. It is 14000 words. I attached my Proposal which I did. 

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Asset price comparison models

Student’s name



Asset price comparison models

Asset pricing across the modern market involves different approaches that enhances the depiction of the relevant value ascribed to a certain capital asset in comparison to other competing assets in the market. Capital asset pricing models depicts the relationship between systematic risks and the expected rate of return associated with a certain capital asset in the market. Therefore comparing the effectiveness of different asset pricing models is a critical procedure that would enhance the identification of the most relevant asset pricing model and enhance implementation of the model in the price depiction of the similar assets in the market. Moreover, estimation of the influential risks associated with different rates of return of particular capital asset influences conclusive decision making process that would influence profitability of an investment strategy. Therefore, this paper will conduct a credible research via mathematical models such as correlation studies, statistical arbitrage, hedging techniques, and Arch prediction models to compare their significances in reflecting the possible returns that would be realized from the different capital assets and eventually influence better investment decision making.

Background study

The capital asset pricing comparison study will be based on the previously published literatures on the effectiveness of different pricing models and utilize the data obtained to develop a credible research that will offer insight on the most effective methods to employ while pricing the capital assets in the current market. Since the modern market is associated with rapid market changes that affects the stock value and pricing through investigating the relevant aspects that affect pricing of capital assets this paper will avail reliable conclusion on the how pricing models affects the valuation and returns realized from the assets. Studies conducted on different financial modeling approaches that have been used to estimate the capital assets prices across the globe will be referred while coming up with the current paper.

Research question

· What are the significances of the different asset price comparison models?

· What is the most effective asset price comparison model?

· How could the identified asset price comparison model influence investment decision-making.

Objectives of the study

· To determine the significances of the different asset price comparison models.

· To establish the most effective asset price comparison model?

· To establish how the identified asset price comparison model influence investment decision-making.

Research methodology

The current study will be based on secondary sources of information which will offer the most credible capital asset price estimation models used in past instances and their significances to come up with reliable literatures. Moreover, the research will conduct mathematical computations on selected asset prices across the Hong Kong, New York stock exchange and London stock exchange to determine the significances of each pricing model on the investments decisions. Correlation studies, statistical arbitrage, hedging techniques, and arch prediction models will be used as the key price comparison model across the current paper and eventually offer reliable information on the significances of the approaches on the determination of asset value and related investor’s decision.


Guide to writing a dissertation

Sara Merino-Aceituno

This short guide has been specially designed for students in the MSc Financial Math- ematics and MSc Corporate and Financial Risk Management, University of Sussex.

1 Structure and contents of the dissertation

1.1 Structure

The dissertation must contain the following:

• Title (title-page)

• Abstract

• Table of contents

• Table of figures (where applicable)

• Introduction

– Description of the project (questions and objectives)

– Literature review and results

– Contributions/ personal work

– Structure of the document

• Main body

– Data-set: description (if you are using data)

– Methods (specially if you are doing numerics)

– Main results. Analysis of main results and their proofs

• Conclusions o What has been accomplished. Future works. Open questions.

• Appendices (if necessary)

• Bibliography

In the following sections we describe in more detail the content of each one of these parts.


1.2 Contents

1.2.1 Abstract

Describe succinctly the topic and goal of the project, the methods used and the results obtained. It should be a short summary of less than a page.

1.2.2 Introduction

The introduction must contain the following parts:

1. Description of the project. In this section include:

• description of the topic of the dissertation; • goals of the project; • motivation, i.e., why is the problem important/relevant (add references to jus-

tify your statements);

• which methodologies/strategies will be used to tackle the problem in consider- ation.

Do not assume that the reader has any previous knowledge on the subject.

2. Literature review. The goal of the literature review is to give a broad view of the topic on your dissertation and locate your work in the existing literature.

• Describe previous works and methods related to your problem, explain how they relate to your work (what do you do new or different with respect to these works? Do you reach new/different conclusions? Did they have different goals/used different methodologies?).

• You do not need to have read all the documents that you reference. You just need to know what they are about and what is their message (this typically amounts to reading the title, abstract and taking a quick look at the introduc- tion and body of the document).

• Don’t just copy-paste a sequence of abstracts: (a) first, it is plagiarism;

(b) second, it shows no critical thinking (no capacity to assimilate new infor- mation, process it, compare it with your own work, and express it in a coherent way);

(c) third, it is impossible for the reader to understand as there is no logical underlying structure in the document.

• Tools: I recommend to use Google Scholar for your search of research papers (and then look at the introduction of relevant papers to find further references). It has also the advantage that it gives you directly the citation.


– Don’t just pick the first 10 papers that are listed on Google Scholar, re- member: show critical thinking! Choose which papers are more relevant than others. Go to those papers and look at their introduction as it already should contain a literature review.

• You can get inspired on the introduction of the paper ’Continuum dynamics of the intention field under weakly cohesive social interaction’ by Degond et. al. Beware though! The introduction of a paper is always shorter than a literature review, so take this example just to understand the style, but your literature review should be longer and more detailed.

• Aim to have at least 25 references. If you have less, it typically means that your literature search is poor. But beware!, do not pick just 25 papers just vaguely related to your subject just to get 25 references.

3. Contributions/ personal work. Any paper or report is always a mixture of known things and new things. It is important that you clearly separate which are the new things from the rest, stating them in a clear and concise way. In this section you can add:

(a) What have you produced that is new (e.g., analysis of a particular set of data with a given method; comparison of different methods to analyse a particular data set; developing new code; reaching different conclusions from the existing ones in the literature or reinforcing the existing results;…).

(b) What have you learned (e.g., you have learned a new method that was not explained in class; or you mastered one method explained in class by taking it into practice; or you had to learn a new theory,…).

(c) What have been the challenges of the project (what difficulties did you have to overcome).

State clearly the contributions in the introduction using bullet points and referring to the corresponding sections in the report (i.e., in which section of the document appears that contribution).

4. Structure of the document. Describe clearly the structure of the document: enumer- ate each section of the main body and explain what is covered in that section.

1.2.3 Main body

The main body must contain the following information:

1. Description of the data-set (where applicable).

• Presentation of the data. Do not add large tables of raw data (it will be incomprehensible), represent your data using graphs.

• Critical analysis of the data. Describe the data-set and analyse how com- plete/good/relevant is the data-set in relation to the question that you want to investigate. Would extra data be desirable?, in that case what type of data? May there be biases in the data?…


2. Methods. Explain the methods used and the strategy to investigate the ques- tion/problem of your dissertation.

• Show that you understand the advantages and limitations of each method and on which assumptions they are build. Why are they good to analyse the data available and study the question at hand?

• Do not assume previous knowledge by the reader – Derive (shortly) the formulas that you use (specially if not learned at

lectures) or give a reference.

– Give the definition of any mathematical object or tool before using them, even if they are basic (e.g. the definition of VaR, the student-t distribution, etc.).

• State the software and libraries used. Describe the code developed.

3. Main results. The conclusions of your data analysis or investigations must be clearly stated. Do not just write the output of the code, you must make an analysis of the output and explain the results. This critical thinking is crucial. What do the results mean? Which conclusions can be reached? Which is the range of validity of these results? How conclusive are these results and how could they be improved…).

1.2.4 Figures/graphs/tables

If you add a figure/graph/table, it must have an explanatory caption (describing the meaning of the figure) and must be referred in the main text. It must also appear in the list of figures.

Also, make sure that graphs and plot have axes properly labelled and with units. Make sure that they are readable (as sometimes the font may be too small).

1.3 Once again: show critical thinking!

Your capacity for critical thinking is crucial (and what employers look for). You must show critical thinking in the following sections:

• Literature review (selection of relevant papers and comparison between them and your work).

• Analysis of the data set (i.e., how good/ incomplete is the data).

• Methodology/strategy used to investigate the topic of your dissertation (explain your choices, advantages, limitations and assumptions of the methods).

• Analysis of the results (what can be gathered from the graphs and tables that you present and the outcome of your project?).


2 Some golden rules

• Writing takes A LOT of time: do not leave it for the last two weeks, start writing immediately, keeping track of your work. Writing and editing do not happen at the same time. You have to polish your text many times to make it good.

• Use a spelling checker.

• All statements must be precise and justified (by references or data).

• Use bold and italics to make important information stand out.

• Plagiarism is a serious offence. Avoid it!

• For coherence and clarity:

– Label each section and refer to it explicitly. Example: In Sec. 3…, do not write ’in the section above’.

– Establish a notation before starting to type. Keep notation coherent and con- sistent. For example, do not use the same symbol for two different concepts. Or two different symbols for the same concept.

– Define immediately every new symbol/notation that you introduce. Or, in other words, you should not write a formula containing unexplained symbols.

– Always refer to which equation you are talking about: do not write ’on the right hand side of the equation we have…’ or ’we encounter this kind of equation’; write instead ’on the right hand side of Eq. (4) we have…’, ’we encounter equations of the form (X)’.

• Avoid giving an evaluation of your own work, i.e, things like ’It is remarkable…’, ’It is interesting….’, ’It is ugly…’, ’it proves to be quite significant…’. Write directly WHY it is remarkable/interesting/ugly/significant (without saying that it is, that is for the reader to judge).

• Do not write a reasoning that takes place in your head, sort of ’Now I want to see this…, ok, then this does not seem a bad idea,…., so now I will try to check that’. Structure your text as a sequence of logical steps and always explain where are we heading. Otherwise the text lacks structure, it looks too informal, it becomes messy and you lose the reader. In summary, do not try to write the mental process by which you got to the result. Instead, show the result and justify it directly, concisely and clearly. Do not write the document as if you were doing a transcript of an oral explanation or conversation. Naturally, in your report you will want to add all the failed attempts and paths tried. I suggest that you present the final results obtained immediately and that you present every path tried as a way to justify modeling choices or methodology. For example, you can say that in your model you consider an assumption X because if you consider the assumption Y as an alternative, then something fails; and then you can use your attempt to show why it fails.


• Explain always what you are going to do/explain/proof, specially at the beginning of each section. Do not keep the reader guessing.

• Be concise. Re-read your text several times; if you can erase words without changing the meaning of the sentence, erase them. If you can use one word instead of three, make that change. Save space and go to the point.

3 For non-English speakers

Here are some expressions that can be useful:

• To recast an equation into.

• This paper is devoted to

• The starting point of our mathematical analysis is

• Our main result is

• The goal of this paper is to investigate

• XXX is a very challenging and largely open problem (reference)

• The particular framework we are interested in was made popular by . (reference)

• These models have been widely studied since… (reference)

• Our focus will be on…

• This function/equation/model is introduced in (reference)

• Before going further, a discussion seems in order concerning…

• This is of course not surprising since…

• (A given statement) is still very much under debate.

• It is important to stress that

• XXX serves as our starting point of our rigorous study.

• (A given mathematical expression) leads to… Ex: the kinetic equation leads to a free transport equation.

• A complete review of this subject/discussion is beyond the scope of this paper, and we refer the interested reader to (references).

• There is also an important literature devoted to…

• The approach in (reference) is…


• Going back to the subject of the present paper…

• Our work is greatly indebted to… (reference)

• (A fact) indicates…

• This paper is organized as follows: in Section

• We will not attempt here to derive/proof/show…

• We recall the results from/ the following result…

• The properties of XXX will be further investigated in section XXX

• Recalling that (such and such property), we write/conclude/deduce…

• We drop a term (not ignore or neglect)

• The fact that (statement) will need to be addressed very carefully in the rigorous proof.

• XXX plays a significant role because…

• (Some methodology or proofs) is fairly classical.

• The main difficulty here is…

• More precisely… / To be more precise…

• (Such and such property) gives rise to (the following conclusions or equation).

• We note that… (some property or fact).

• To lack a property or feature (not to miss it).

• To prove that XXX, we need to…

• The claim follows from…

• By Proposition XX, we have that…

• We have thus showed that…

• We can thus improved (expression XXX) as follows:

• We deduce/ consequently / this last expression is consequence of…/ which implies / implying / (property XXX) follows / so XXX holds / Using the fact that XXX, we have YYY

• We refer to (References) for further details on (something in particular).

• We establish (some results)


• The main tool in deriving XXX is…

• Proceeding similarly we have…

• (Property XXX) relies on the following (lemma/proposition/bound)

• In order to establish (expression XXX), we…

• We need to show that…


  • Structure and contents of the dissertation
    • Structure
    • Contents
      • Abstract
      • Introduction
      • Main body
      • Figures/graphs/tables
    • Once again: show critical thinking!
  • Some golden rules
  • For non-English speakers


Rag0531-01 sample/Sample01.docx

School of Mathematical and Physical Sciences

Department of Mathematics


Course: Corporate and Financial Risk Management

Candidate Number: 176950

Title: Portfolio Optimization Model Based on VaR and CVaR

Supervisor: Max Jensen

Date: 2018.08.02

Number of Words: 13798 words

In making this submission I declare that the information contained on this cover sheet is correct and that the content of this dissertation is my own work.


Portfolio selection refers to the allocation of wealth to different assets in order to achieve the purpose of diversifying risks and ensuring returns. In 1952, Markowitz used variance to quantify the risk of stock returns and proposed a mean-variance analysis method for portfolio selection. Opened the prelude to modern finance research.

VaR and CVaR risk measurement methods have many applications, such as measurement of credit risk, determination of internal risk capital, capital allocation, financial supervision, etc. VaR and CVaR as a new method to measure portfolio risk have developed rapidly in recent years. At present, many scholars have compared VaR and CVaR from the aspects of definition, nature, calculation, etc. and compared the differences between mean-VaR and mean-CVaR models. Based on the existing research results, this paper empirically studies the portfolio optimization model based on VaR and CVaR constraints.

This paper uses a combination of theoretical research and empirical research. Firstly, introduce the definition and properties of VaR and CVaR risk measurement methods, and the corresponding mean-VaR model and mean-CVaR model. Secondly, discuss the boundary and the efficient frontier of the mean-VaR model and mean-CVaR model. Analysis of the difference between VaR and CVaR constraints under different confidence levels. Finally, empirical research is carried out according to the model discussed above, under the same confidence level, comparative analysis of each model. Under different confidence levels, the change of the effective frontier of the mean-CVaR model was studied, and the effect of adding risk-free assets to the efficient frontier of the mean-CVaR model was studied. The results show that CVaR can better measure risk, especially when the return on assets does not satisfy the normal distribution. The CVaR value increases with the confidence level increase, which indicates that the CVaR method has higher accuracy for measuring the tail loss. Overall, the mean-CVaR model has the better adaptability to the risk-measure and risk control of the portfolio than the mean-VaR model, both in terms of accuracy and breadth.

key word: portfolio VaR CVaR the mean-VaR model the mean-CVaR model



1.Intoduction 3

2. Literature review 8

2.1 Value at Risk 8

2.2 Conditional Value at Risk 9

3. Methodology 11

3.1 Value at Risk 11

3.1.1 The definition of VaR 11

3.1.2 The calculation method of VaR 13

3.1.3 The advantages and disadvantages of VaR 15

3.2 Conditional Value at Risk 20

3.2.1 The definition of CVaR 20

3.2.2 Risk composition of CVaR in the portfolio 21

3.2.3 The calculation method of CVaR 26

3.3 The Characteristics of CVaR 28

3.3.1 The parameter selection of CVaR 28

3.3.2 The nature of CVaR 32

3.3.3 The application of CVaR 34

3.4 The comparative study of CVaR and VaR 38

3.5 The mean-VaR model and mean-CVaR model 43

3.6 Comparisons and characteristics of the mean-VaR model and mean-CVaR model 44

3.6.1 Mean-VaR model and mean-CVaR model Boundaries 44

3.6.2 Mean-VaR model and mean-CVaR model efficient frontiers 46

3.6.3 Portfolio selection under VaR and CVaR constraints 49

3.7 The difference between VaR and CVaR constraints under different confidence levels 50

3.8 The mean-CVaR model of adding risk-free assets 53

4.Empirical analysis 58

4.1 Data description 58

4.2 Calculation of related variables 59

4.3 Comparison of mean-CVaR model with mean-variance model 61

4.4 Comparison of mean-CVaR model with mean- VaR model 65

4.5 Influence of the confidence level on the efficient frontier of the mean-CVaR model 67

4.6 Analysis of the effective frontier of the mean-CVaR model of adding risk-free assets 70

5.Conclusion 73

1. Introduction

With the development of the capital market, more and more institutional investors and individual investors have joined the financial market. How to successfully build a portfolio of securities has become the key to successful investor investment. Therefore, the research of portfolio theory and method is becoming more and more important. The theory of portfolio selection is that investors allocate wealth to different assets to achieve the purpose of spreading risks and increasing returns. At present, this theory has been widely used in investment and financial management as a major analytical tool.

The key issue of portfolio theory is the risk measurement, to carry out effective financial market risk management and optimization of investment portfolios, accurate measurement of risks becomes the foundation and core. Whether it is risk hedging, risk dispersion, risk transfer or risk compensation to manage risks, the first thing is to measure the magnitude of the risk and the possibility of the loss occurring. The effectiveness of financial market risk management depends to a large extent on the quality of the risk measurement. In short, if there is no accurate risk measurement method and the market risk measurement is not accurate, then market risk management and portfolio management cannot be realized.

In 1952, Markowitz measured the risk of the portfolio with the variance of the return on investment and proposed the mean-variance portfolio selection model. Although this model seems simple at present, it expresses the core idea that investors not only need to care about the expected return on investment but also need to consider the risk of investment. The model laid the theoretical foundation for portfolio selection. However, the mean-variance model has some defects. The variance only measures the overall risk of the portfolio, while ignoring the contribution of individual assets or risk factors to the overall risk of the portfolio. At the same time, the traditional mean-variance model is a theoretical model and does not consider the actual investment and transaction characteristics, such as cardinality constraints and minimum holdings constraints. On the other hand, for portfolios with asymmetric returns, The variance does not reflect the risk characteristics of the portfolio well. These disadvantages weaken the practicality of the mean-variance model.

In the financial market, the research of portfolio theory has been more than half a century. The P-factor method, sensitivity analysis, and the mean-variance model are not widely used because it is quite different from theoretical expectations in the application of investment practice and produces an irrelevant optimal portfolio. So researchers have proposed a series of methods to compensate for some theoretical shortcomings of the mean-variance model, such as: utility function optimization; non-variance risk measurement.

J.P. Morgan (1994) proposed the VaR method and was widely used in various fields of the financial economy. Later economists applied it to risk capital management, For example, in the risk measurement of portfolio theory, VaR is used instead of variance, and replace the efficient frontier of the mean-VaR model with the efficient frontier of the mean-variance model and so on. However, VaR has defects in the premise that asset returns do not satisfy the normal distribution. It means that VaR does not satisfy the sub-additive and consistency axioms, and the tail risk measurement is not enough. Therefore, the researchers propose a correction method for the value at risk method-conditional value at risk(CVaR).

Rockafellar and Stanislav Uryasev (1999) published the Optimization of Condition Value-at-Risk, this is the first article introducing CVaR, which gave the basic concepts of CVaR, and then several articles about CVaR basically revolved around this model. But the study of the mean-CVaR model is not deep enough. Since the concept and production of CVaR is rooted in VaR, understanding the concept and calculation of VaR is a prerequisite for understanding CVaR.

This article combines existing research results and first introduces the concept and calculation of VaR, and points out the defects of VaR, then expounds the concept and thought of CVaR, and introduce the superiority of CAR over VaR. this thesis mainly studies the theoretical discussion of the portfolio optimization model based on VaR and CVaR constraints and uses 10 stocks for empirical analysis. The characteristics and innovations of this paper are mainly reflected in the following aspects.

1. The boundary and the efficient frontier of the mean-VaR model and the mean-CVaR model.

2. The difference between VaR constraint and CVaR constraint under different confidence levels.

3. Impact of adding risk-free assets on the efficient frontier of the mean-CVaR model.

The full text is divided into five sections. In section1, talk about the reason why choosing this topic. In section2, through reading a lot of literature, introduced the research results that related scholars have made on portfolio optimization with VaR and CVaR. Section3 is the main body of this article, Firstly, the definition of VaR and several commonly used methods are introduced. The advantages and disadvantages of VaR are summarized through the relevant literature. The main flaws in VaR risk measurement methods are analyzed. (1) VaR does not satisfy the coherent axiom; (2) VaR tail loss measurement insufficient, the mean-VaR model is proposed and the shortcomings of the model are pointed out. After analyzing the shortcomings of VaR, the definition of the improved CVaR and the corresponding calculation method are proposed, the selection of relevant parameters of CVAR in the calculation process, and the application of CVaR in other aspects are analyzed. Establish the mean-VaR model and the mean-CVaR model, and exploring how to choose a portfolio under VaR and CVaR constraints. In section4, according to the method of section3, combining stock data to compare and analyze portfolios selection under VaR and CVaR constraints.

2. Literature Review

2.1 Research status of VaR

Since Markowitz's portfolio theory model proposed in 1952, the research of risk measures and financial capital allocation model has always been one of the hot topics in financial investment research. However, this model being based on the ideal assumption, it is difficult to accurately reflect the actual situation of the capital market. For this reason, many scholars have conducted a series of model improvement and theoretical perfection research work based on this model.

In 1994, the risk managers of J.P. Morgan Bank proposed a new method of risk measures: Value at Risk (VaR). Which is the maximum possible loss of capital in a certain period of time in the future under a certain probability level. It can fuse different market factors and risks into one, and can measure potential losses caused by different risk sources and their interactions. It has the advantage of measuring comprehensiveness. Artzner et al (1999), analyzed the basic requirements of risk measures, firstly tested the utility maximization and portfolio o

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