What Is a Conditional Value-at-Risk?

The first monograph on multi-objective CVaR model research

Multi-objective conditional value-at-risk theory

Chapter 1 Introduction ... ..................... 1

1.1 Risk Management Issues ............... ............ 1

1.1.1 Definition of risk ... ............ 1

1.1.2 Development of Risk Management ... ......... 2

1.1.3 Model of risk .................. ............ 4

1.2 The theory and development of VaR model ... ....... 7

1.2.1 The origin of the VaR model ... ......... 7

1.2.2 Summary of VaR model research ... ..... 9

1.3 Overview of CVaR Model Theory Development ............ ..11

1.3.1 The CVaR model is proposed ... ..... 12

1.3.2 CVaR research in portfolio investment ...

1.3.3 Comparison of CVaR and other risk measures ... 14

1.3.4 CVaR promotion and application research ... .. 15
Chapter 2 Single-Target VaR and CVaR Risk Models ... 18

2.1 VaR model ... .............. 18

2.1.1 Definition of VaR ... .......... 18

2.1.2 Calculation method of VaR ............................ ... 19

2.2 Single Objective CVaR Model .............................. ....... 20

2.2.1 Definition of single target CVaR model ... .. twenty one

2.2.2 The basic theorem of single target CVaR ... . twenty two

2.2.3 CVaR Optimal Portfolio ............... .... twenty three

2.3EVaR (entropicVaR) model ... . 25

2.4 WCVaR (worst-CVaR) model ... .. 28
Chapter 3 Discrete Single-Objective CVaR Model ... .. 32

3.1 Questions and Model Establishment ... ... 32

3.2 Portfolio Optimization for Discrete CVaR ...

3.3 Risk aversion for discrete CVaR ... ... 42

3.4 Summary ... ................... 46

Chapter 4 Discrete Multi-Objective CVaR Model ... .. 48

4.1 Questions raised and model establishment ... ... 48

4.2 Discrete multi-objective CVaR based on weights and confidence levels ...

4.3 Application of Securities Portfolio and Real Estate Portfolio ...

4.3.1 Application of Securities Portfolio .................. ... 57

4.3.2 Real estate investment portfolio application ... .... 60

4.4 Risk aversion in discrete multi-objective CVaR models ...

4.5 Summary of this chapter ... ............... 69
Chapter 5 Continuous Multi-Objective CVaR Model ... .. 71

5.1 Multi-objective CVaR model with multiple -VaR values ...

5.2 Multi-objective CVaR model based on weight reset letter level ...

5.3 Minimum -VaR loss value CVaR model based on weight reset letter level .. 78

5.4 CVaR model of maximum -VaR loss value based on weight reset letter level ...

5.5 Application of Portfolio Investment ............... ....... 85

5.6 Summary of this chapter ... ............... 91
Chapter 6 Multi-Stage Dynamic CVaR Model ... .... 93

6.1 Multi-stage CVaR model with deterministic state transition at multiple confidence levels ...

6.2 Multi-stage CVaR model with deterministic state transition at a single confidence level ...

6.3 Multi-stage dynamic CVaR model of Markov-type state transition ...

6.3.1 Finite-phase dynamic CVaR model ...

6.3.2 Infinite Phase Dynamic CVaR Model ...

6.3.3 Final loss dynamic CVaR model ...

6.4 CVaR control model under continuous time conditions ...

6.5 Summary of this chapter ... .............. 110
Chapter 7: Multi-cycle Multi-objective CVaR Model Based on Weights ...

7.1 Questions and definitions .................. ......... 111

7.2 Multi-objective CVaR calculation in a single cycle ...

7.3 Multi-cycle multi-objective CVaR model for the entire period ...

7.4 Power supply and loan cycle issues ... ..... 119

7.4.1 Multi-cycle production (power supply) issues ... ... 119

7.4.2 Multi-period loan issues ... ....... 124
Chapter 8 Two-layer multi-objective CVaR model ... ... 134

8.1 General two-layer multi-objective CVaR model ...

8.1.1 Two-level multi-objective CVaR model without weights ...

8.1.2 Two-layer multi-objective CVaR model with weights ...

8.2 Weight-based class I two-layer multi-objective CVaR model ...

8.3 Weight-based Class two-layer multi-objective CVaR model.

8.4 Application of product procurement and pricing in the supply chain ...

8.4.1 Product purchase and pricing with single repurchase strategy ...

8.4.2 Multi-product procurement and pricing issues with repurchase strategies ...

8.5 Coordination model of supply and marketing risk for multi-product production and supply chain ...

8.5.1 Supply chain risk coordination model ... .... 164

8.5.2 The establishment of specific models ... ....... 167

8.5.3 Discussion ... .............. 175

references................................................ ............. 176

index................................................. ................ 186

Chapter 1 IntroductionThis chapter first introduces the definition of risk, risk management issues, and risk management models, then explains the origin of VaR and a summary of theoretical research, and finally gives the background of the CVaR model and a summary of theoretical research.
1.1 Risk management issues 1.1.1 Definition of risk Risk refers to a loss or gain that people suffer when they engage in certain activities, and risks include characteristics such as uncertainty, objectivity, and gains and losses. When people engage in any activity, The impact of environment, conditions, and various human factors, resulting in unpredictable losses, can be attributed to risk. The greatest feature of risk is uncertainty. This uncertainty can be divided into objective risk and subjective risk. Subjective risk is often It is caused by people's uncertain behavior. Although they know the cause of the risk, they cannot be overcome, and objective risk is an unpredictable factor. In fact, when people engage in activities, there are often two types of risks coexist.
For the risk of general things, we can use probability to describe the probability of their occurrence, and measure the loss of things, that is, the risk of things can be measured.This measure is a predicted value, but There is a big problem with the accuracy of this measure. The risk measure refers to a certain value that promotes and increases the frequency or severity of losses. It is a numerical description of the losses caused by an accident. Therefore, the risk factor is used as an explanation when describing risks. Variable, and loss as an explanatory variable, as in financial investment and commodity procurement, demand is considered to be the main uncertainty factor.
Risks can be divided into social risks and individual risks. Social risks are losses caused by non-individuals, or at least the factors that individuals often cannot prevent, and such losses usually spread over a large range. Once social risks occur, any particular social individual It is difficult to stop the spread of losses. For example, social and political-related wars, unemployment, strikes, and natural disasters such as earthquakes and floods are social risks. Social risks not only affect a group or a group, but also affect a lot of people. A large part of the population, even the entire human society. Individual risk refers to the risk of loss of certain individuals or certain families caused by social individuals. For example, the risk of personal property loss due to fire, explosion, theft, etc. The risk of an individual's legal liability for loss of property or physical injury to others is an individual risk.
Risk can be divided into system risk and market risk, which people often encounter in engaging in business activities. System risk refers to risk factors that people cannot overcome or avoid in activities, such as social risk. Market risk is that people can predict in activities And try to avoid risks, such as individual risks. People can avoid risks through preset rules, such as fire prevention measures to avoid fires.
At present, the risks that can be avoided (managed or controlled) in academic circles are roughly divided into four types: environmental risks, market risks, information risks, and business risks. Environmental risks are caused by factors such as natural disasters, social conflicts, and economic fluctuations; Information risk is the risk caused by people's insufficient knowledge of things. Business risk refers to the risk of loss caused by people taking wrong actions. Among them, environmental risk is the most difficult to overcome and avoid. Therefore, we are dealing with risk issues. Try to consider possible losses caused by all risk factors.
1.1.2 The development of risk management As a scientific management, risk management originated in the early 20th century. With the birth of the machinery industry revolution, the risk management of productive enterprises has become more and more important. French scientific management master Fayol in the theory of business management The risk management idea was first proposed in China.From 1929 to 1933, a major economic crisis occurred in many countries around the world. This crisis brought about the Great Depression in the United States. The United States suffered from economic recession, factory closures, worker unemployment and social wealth. At the time of the huge loss, people began to gradually pay attention to the importance of risk management.
In 1930, the American Management Association (AMA) and many universities proposed the importance and urgency of risk management research. After 1945, the world's political and economic situation had undergone profound changes, and scientific and technological progress had promoted the second industrial revolution. The risk management of production, logistics and inventory has attracted the attention of almost all enterprises, especially the high socialization of production and the increasing complexity of enterprise scale and asset value. Product competition between enterprises has become more intense, resulting in social conflicts, The turbulence in the international situation has led to the frequent occurrence of various risk events, and the chain and spread of disaster accidents in the supply chain have also increased the risk losses of business operations.This has also led to the risk management becoming the most vigorous in the world after 1950 One of the important theories.
After entering the 21st century, in a highly competitive market environment, manufacturing companies in the supply chain must bear various risks and uncertainties brought by complex supply chain networks. Market risks are caused by changes in consumer demand Information risk is caused by information asymmetry between suppliers, producers, and retailers. As supply chain networks become more complex and supply chain members are expanding, manufacturing companies are facing unprecedented fluctuations, so the more they face More and more attention is paid to the risk factors in the continuously growing supply chain management, and an attempt is made to use an integrated supply chain risk management model to solve unpredictable market demand risks.From the analysis of the current literature, people are paying more and more attention to procurement, Risk decision-making models for production and sales, and even focus on the integrated risk issues of procurement, production and sales.
Since the second half of 2008, the global financial crisis triggered by the U.S. subprime mortgage crisis has swept across countries around the world, and has spread from the financial industry to various areas of the economy such as manufacturing and the real economy. Under this impact, many manufacturing companies are on the verge of bankruptcy. Or it has already closed down. How to reduce the negative impact of the financial crisis on the company and seek survival in the crisis is the main problem facing the company. Under this severe situation, maximize profit as the sole goal of business management Can no longer meet the needs of the current situation, an important way for enterprises to survive is to strengthen risk control and reduce risks.
The current risk management issues are caused by various factors such as uncertainty, complexity, diversification, systematization and socialization. 1.1 Risk management issues 3
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And target composition. There are a large number of risk issues in various fields. Below we discuss several types of typical risk management issues.
(1) Financial portfolio (loan) investment and real estate portfolio investment issues. This type of problem is measured by considering risk factors such as price (interest rate), demand, etc., while using overall return rate, opportunity loss rate, and investment surplus rate as risk targets to measure. And control the selection of portfolio investment to find the portfolio investment that makes each risk objective reach the optimal. This is a multi-objective risk decision problem. If you do not consider risk objectives such as investment surplus rate, you only consider the single risk objective to determine the portfolio. If you invest, there may be a risk of overinvestment or underinvestment.

(2) The problem of multi-product portfolio procurement and inventory management in the supply chain. Similar to the problem of portfolio investment, because risk factors such as market risk are difficult to overcome, the principle of risk dispersion is the main method used in risk management. Purchase and inventory of various products to reduce the sales risk caused by uncertain demand, that is, price, demand, etc. are the main risk factors, and the overall target is return rate, stock-out rate, overbooking rate, etc., to find the optimal product The combined ordering strategy makes each risk objective optimal. The formulation of management strategies related to product procurement and inventory issues is one of the most important tasks for current enterprises. All along, the enterprise's procurement plan is usually carried out as a single product, mainly due to calculations. The complexity makes it very difficult to formulate multi-product portfolio purchasing and inventory strategies.

(3) The problem of reasonable power supply in urban power supply. Due to the large power consumption and tight energy supply in the city, the power supply department must not only increase its own revenue, but also reasonably meet the power consumption of various industries and urge power-consuming units to save power, that is, use electricity The main market risk factors are price and power demand, and the return rate, power shortage rate, and excess power supply rate are the risk targets. The proportion of power supply under different periods is determined so that each risk target is optimal to achieve the best returns and The purpose of saving electricity.

(4) The problem of rational combination development and utilization of land resources is also a multi-objective risk management problem. In recent years, urbanization has developed rapidly, and a large amount of land has been developed into residential, commercial, and industrial areas, with the price and demand of land as risk factors. Taking return rate, investment failure rate, etc. as risk targets, and selecting the optimal development ratio to achieve optimal risk targets is an issue that needs to be solved at present. The solution of this problem is of great significance for the government to formulate urban development plans.


The above-mentioned financial loan portfolio investment, real estate portfolio investment, multi-product portfolio purchase and inventory issues in the supply chain, rational power supply in urban power supply, and rational portfolio development of land resources are all multi-objective risk decision problems, and they have similar risk factors , Risk objectives, and decision variables. Among them, portfolio investment and real estate investment problems can be studied using the conditional value-at-risk (CVaR) model. The following chapters of this book will establish corresponding multi-objective CVaR for the above-mentioned multi-objective risk decision problems. Models to provide some theoretical tools for solving these problems.
For a long time, the methods of using quantitative model theory to study risk management problems have been mainly concentrated in the financial field. Risk management research methods in other fields are mainly based on qualitative analysis. Most of the few quantitative model studies are also transplanted or borrowed from financial risk models. There is no relatively comprehensive risk model theory applicable to various industries such as finance, energy, manufacturing and retail. In recent years, the industries of finance, real estate, energy, manufacturing and retail account for an increasing proportion of the entire economic development, so The effective management of the risks involved in these industries has become a very urgent task at the moment. If one of these industries develops in parallel, the development of other industries will be affected. For example, the continued growth of land prices in the country makes the real estate bubble The risks appear to be increasing, and it is very important to measure the risk of bank credit losses caused by real estate investment; for example, the risks faced by the procurement and inventory of multiple product portfolios in the supply chain are extremely important for companies to improve profitability and reduce the risk of losses ; Reasonable power supply and determination of urban power supply system Great significance for energy conservation and economic development. Likewise, if a company or a risk facing the industry is not well managed and prevention, easily lead to bankruptcy, a large number of job losses, and eventually become important issues affecting social development.
1.1.3 Risk Models As early as the 1930s, scholars began to study uncertain risk management models. For example, the inventory management model is a typical risk decision model. A factory and a store cannot be guaranteed without the necessary inventory. Normal production activities and sales activities. Insufficient inventory may cause the factory to stop working, the store will be out of stock, and lose sales opportunities; and if the inventory is too large, it will accumulate working capital, increase storage costs, and reduce corporate profits; etc. Issues related to storage volume require people to make decisions. After long-term practice, scholars have researched and established many inventory models and formed a risk management branch called storage theory, which is used for production and sales of enterprises and commercial operation departments. .
In order to reduce the unreasonable occupation of funds, rational and scientific use of funds, and accelerated capital turnover, storage theory has provided many effective methods for various production enterprises and stores. For specific or random demand types, appropriate methods can be used to replenish inventory To improve the level of business management.
Among them, the single-period random storage model is a typical risk model, such as the newsboy model: This model is often used to study the demand for perishable products. In the model, if the product in this period is not used up, the price of the product will be reduced in the next period. , Profit decreases, and even becomes worthless; and if the current product can not meet the demand, it will cause losses due to lack of stock or lost sales opportunities, whether it is oversupply or in short supply, there will be losses, the model is to study how much the order can make expectations The total loss is the lowest or the total profit is the largest. There are a lot of problems in ordering such products in the real world, such as bookstores ordering books and magazines, bakery production of bread, and supermarket purchase of fresh products.
The model assumptions are as follows:
(1) Make an order decision at the beginning of the cycle, set the order quantity to Q, and the supply can be realized instantly;

(2) The demand r in a period is a non-negative random variable, and its distribution function and density function are known;

(3) The initial inventory is zero, and the fixed subscription fee is also zero (or constant).


Suppose that during a period T, the demand r is a non-negative random variable, and ri (i = 1,2,) corresponds to ...
Take the value, the corresponding probability P (ri) is known, and the optimal storage strategy is to minimize the expected value of the total cost within the time T or maximize the return. Let a be the total unit product cost (storage cost and purchase price) when supply exceeds demand, and b The total cost per unit product (out-of-stock cost) when demand exceeds supply. Use the minimum expected loss model to determine:
1.1 Risk Management Issues 5
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min a. (Q. ri) P (ri) + b. (ri. Q) P (ri).
Q ri.Qri> Q
The first term in the above formula represents the loss of oversupply, and the second term represents the loss of oversupply, and the minimum demand Q (integer), it is not difficult to derive that the optimal order quantity should meet the following inequality:
Q.1Q.P (ri) a + b
i = 1i = 1
In recent decades, many risk management models have been established.Here are four types of risk management models currently commonly used in financial engineering, supply chain management and other fields.
1. The model of minimum expected loss early The risk model in the early days basically followed the criterion of minimum expected loss. This criterion was widely used in the inventory model and gradually extended to the supply chain risk model. The model is generally described as: set the loss function F (x, ), where x Rn is the decision variable, X is the set of admissible decisions, risk factor Rm is a random variable, and the corresponding probability distribution density function is p (z), which is determined according to the minimum expected loss criterion The best decision problem is defined as follows:
minE (f (x, )).
xX
For the above problem, the optimal decision can only be obtained if the probability distribution of the random variable is determined. In fact, it is difficult to determine the probability distribution of risk factors in advance, such as securities investment, and this phenomenon is common in financial risk. Expected loss value It is the possible average of future losses caused by risk factors, that is, the average expected loss minimum model is to find a strategy to minimize the average future loss.
2. The minimum variance model of expected loss describes the other statistical index of risk as variance. The variance indicates the degree of deviation of the actual value of the random variable from its mathematical expectation.
minE [f (x, ). E (f (x, ))] 2.
xX
The smaller the variance of the expected loss of the portfolio, the smaller the change in the actual loss around the expected loss, that is, the smaller the overall risk level of the portfolio.
In 1952, Markowitz created a mean-variance model using mean-variance as a risk measure for portfolio investment. Let x Rn be the n stocks of the investment, and be a random vector that represents the n-stock return and the mean loss function. Defined as f (x, ) =. XT, the decision maker wants to find a portfolio with the smallest variance when the given return is . The general model is as follows:
minE [.x T. E (.x T)] 2,
stE (.x T) = ,
x1 + x2 ++ xn = 1, xi.0, i = 1,2,, n.
······
The original intention of this model is to diversify risk through portfolio investment. Even if the returns of individual stocks are reduced, the overall return can reach a given return requirement. Minimization of variance is very suitable for risk aversion, but for some risk issues in the financial field For example, loans, the quantile of loss becomes more important. At this time, it is obviously not accurate to use only the mean or variance to measure the risk. The value-at-risk (VaR) model introduced below considers a given level of probability confidence. Maximum possible loss value.
3. VaR minimum value VaR value is the maximum value of risk loss at a given level of probability confidence. Definition (x, y) = p (z) dz, y R,
f (x, z) .y
The above formula shows that the loss function is less than the probability distribution of y. Obviously, the smaller the y, the smaller the probability. Given the confidence level (0,1) and the decision variable x, define y (x) = min (y R: (x , y) .)
It is an -VaR loss value for decision x at the confidence level . At the confidence level and decision x, the possible loss value of the loss function caused by risk factors is y (x) at the maximum.
The VaR model is different from the mean and variance model in describing risk.It calculates the strategy adopted when the risk loss is the largest in the case of high probability, which can help monitor the position of the observation point of the loss level of many financial institutions when investing. There is a fatal weakness in the VaR model. When the stochastic risk distribution has fat tails, the cumulative loss is very large, but the VaR value cannot be reflected. Of course, the mean and variance models cannot simultaneously reflect this random risk. We need to calculate the quantile point. Cumulative loss, Conditional Risk Value (CVaR) just solves this problem. The calculation of VaR is difficult, which also limits its wide application. Regarding the VaR model, we will introduce it in detail in section 1.2.
4. The minimal model of conditional risk value Uryasev and Rockafellar proposed a VaR correction method in 1999: conditional value-at-risk (CVaR). The CVaR value is the cumulative loss value under the calculation of VaR value.
(x) = (1.) .1. f (x, z) p (z) dz.
f (x, z) .y (x)
1.2 VaR Model Theory and Development 7
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VaR refers to the maximum expected loss value of a financial asset or portfolio of securities at a certain level of confidence and holding period, while CVaR refers to the cumulative expected loss value of the amount of loss exceeding the VaR. VaR values can be calculated at the same time when calculating CVaR, This makes the CVaR risk value simultaneously characterize the quantile loss. The key is that the calculation of CVaR is much more convenient than VaR, which makes CVaR quickly applied in the fields of finance, real estate, power, and supply chain.
The above four types of models are the more popular models in current risk management.This book will mainly introduce the content of the fourth type of risk model. Other types of risk models are not given here.
1.2 The theory and development of the VaR model 1.2.1 The origin of the VaR model Finance has a history of more than 100 years in the West, but modern finance as a scientific theory has been developed since the 1950s. Before that, Traditional theories are mainly descriptive summaries of experience. For example, the proverb "Do not put eggs in a basket" reflects the idea of modern decentralized investment. However, since the above theories are all qualitative descriptions, they cannot be guided in practice. Standardized investment decisions.
In 1952, Harry M. Markowitz published a paper entitled PortfolioSelection in Journal of Finance, marking the beginning of modern finance and laying the foundation for the development of modern investment theory. This document discusses the determination of securities returns and risks. The main principle of the method is to establish the framework of the mean-variance model and clarify the method for determining the effective frontier of the portfolio. However, when there are a large number of securities to choose from, this method is difficult to calculate due to the large number of parameters that need to be calculated at the time. Implementation. In 1963, Markowitz's student William F. Sharpe proposed a single-index model for simplified calculations.This model assumes that the return on assets is only related to the overall return of the market, which greatly reduces the amount of calculations and enables modern investment. The theory can be applied to the investment practice where a large number of securities exist.
The mean-variance model and index model are based on the measurement of risk by the variance of the rate of return, and the variance measures the degree of deviation of the asset's return from the expected value. It considers the return above and below the expected value as Potential risks. In practice, investors do not consider the possibility of higher returns than expected as unfavorable results. Since then, scholars have tried to use different risk metrics and build their relevant models.
Markowitz considers using semi-variance instead of variance as a measure of risk.Half-variance does not consider the case where the return rate is higher than the expected return, but only calculates the case where the return rate is lower than its expected value, that is, only the negative deviation is regarded as the risk. But there are many difficulties in the calculation of half-variance. For a given portfolio x0, only those observations with negative bias are considered when calculating the half-variance. If the weight of the portfolio is changed slightly, a new combination x1 is obtained. The negative deviations of the observations in x0 may become positive and vice versa. Therefore, the observations with positive and negative changes in these deviations must be added to or removed from the semivariance of the combination x1. Because the observation set to be considered when calculating the combined semivariance Is a function of portfolio weights, which makes its calculation more difficult than variance.
Although it is easy to perform mathematical processing when analyzing the mean-variance model, it is computationally difficult to use it for portfolio optimization, because the quadratic programming problem must be solved. Konno, Konno, and Yama-zaki proposed using the mean absolute deviation (mean -absolutedeviation (MAD) to measure risk. Because MAD is a piecewise linear convex function of portfolio positions, if you use MAD as a risk metric, you can quickly perform an efficient portfolio optimization process through linear programming. Ogryczak and Ruszcznski prove that The portfolio on the effective frontier is consistent with the effective portfolio based on second-order stochastic dominance. Harlow uses low partial moments (LPMs) to measure risk by considering only the left tail of the distribution of returns. Different risk measures and portfolio selection models from different perspectives Reflecting the relationship between investors' investment behavior and preferences, the distribution of assets in the optimal portfolio may also be different.
The VaR model for managing and controlling risk was first proposed by William J. Baumol in 1963. He published an article in ManagementScience and proposed a portfolio selection model considering the confidence level of expected returns. VaR refers to a certain probability (confidence level) The maximum possible loss value expected for a certain financial asset or portfolio during a certain holding period. For more than 30 years, William J. Baumol's VaR ideas failed to get the attention of the financial community. In the 1990s, as capital The trend of securitization and the increase in the share of derivative financial instruments in the financial market, various financial regulators, financial consulting organizations and private consortia around the world have paid more attention to methods for assessing, monitoring and controlling financial market risks.
In July 1993, the Group of Thirty (advisory group consisting of senior bankers, financiers and academics from major industrial countries) published a research report entitled `` Practices and Rules of Derivative Products '' and recommended the introduction of The VaR method is used to value assets and assess financial risks, and its derivative trading team further gives specific parameters for risk capital analysis.
In April 1993, the Bank for International Settlements of the Basel Banking Regulatory Commission (by Belgium, Canada, France, Germany, Italy, Japan, Luxembourg, the Netherlands, Sweden, Switzerland, the United Kingdom, and the United States) established a The standard model based on the modular method.First calculate the VaR of the investment portfolio in the face of interest rate risk, exchange rate risk, equity risk, and commodity risk. The total VaR value of the bank can be obtained by adding the VaR values of these 4 aspects. .
From October 1994, the world-renowned financial consortium JPMorgan Bank started to publish freely the information required to calculate VaR on the Internet and established an information system risk matrix. In the initial stage, this system was 300 financial instruments in 14 countries (1995 Expanded to 23 countries and regions in November: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Sweden, Switzerland, United Kingdom , The United States, the European Community, and Hong Kong, China) provide risk measurement methods. It provides a variance-covariance matrix and some related values that change over time. Users can use a computer to combine the risk matrix system with their position status. Build your own VaR system.
In April 1995, the Basel Committee launched a proposal for an extension of the market risk model, allowing banks to use their own VaR models to determine their capital requirements.
1.2 VaR Model Theory and Development 9
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In December 1995, the U.S. Securities and Exchange Commission (SEC) recommended that publicly traded U.S. companies should use VaR as an important indicator when calculating their information, calculate the VaR value during the reporting period, and compare this result with actual changes in market value. Compare.
In January 1996, the Basel Commissioner in the "Regulatory Framework of the Internal Model Method for Testing Market Risk Capital Requirements Using the" Post Test "Method, proposed the use of the" failure rate "to test model errors. This further established the VaR method in The use status in the financial industry, reports of financial or financial risks will be gradually released on the basis of VaR.
Now, the International Swap and Derivatives Association (ISDA), the International Settlement Bank and the Basel Banking Supervision Committee have all recommended the use of the VaR system to value market positions and evaluate financial risks. The VaR method is widely used in addition to financial institutions and financial regulatory agencies. It is also gradually adopted by some large non-financial institutions, such as Siemens, IBM, etc.VaR is a widely used, easy to understand and master method for calculating and controlling financial market risks, which makes the VaR model at the theoretical level and the application level Both have significant research topics.
1.2.2 Overview of VaR model research In recent years, the VaR method has become one of the mainstream financial risk measurement methods in the world because of its simple concept, easy to understand, and provides a unified risk measurement framework for investment portfolios. A lot of work has been done on the theoretical and practical issues of portfolio optimization under VaR metrics.
Linsmeier and Pearson, Du.e, and Pan introduced the concept and calculation method of VaR in a complete and complete way. Alexander and Baptista compared the two means-variance effective combinations, and pointed out that the combination with higher variance may have a smaller VaR. The portfolio that minimizes VaR globally may not exist.The mean-VaR effective set is a subset of the mean-variance effective set and may be empty.For non-normal distributions, use the mean-VaR method and the expected utility maximization model. Approximately the same.Sentana shows how fund managers make investment decisions under the condition of a mean-variance framework and satisfying VaR constraints.
The concept of VaR is simple, but its measurement is a challenging statistical problem.Western scholars have conducted in-depth discussions on VaR measurement.
Bouchand proposed how to use the non-Gaussian characteristics of financial asset fluctuations to simply calculate the VaR of complex nonlinear combinations; Berkowitz proposed a new method for evaluating VaR; Li proposed a semi-parametric method to estimate VaR. This method The upper and lower limits of the VaR confidence interval can be constructed by calculating the skewness, kurtosis, mean, and variance of the yield series without making any distribution assumptions; Dowd [16,17] and Longin proposed a method for calculating the extreme value of VaR; Glasserman et al. Proposed a more effective MonteCarlo simulation method for calculating VaR, that is, using techniques such as importance sampling and stratified sampling to improve the calculation accuracy; Danielsson pointed out that although VaR is simple and easy to use, its results are easy Be manipulated maliciously; Cumperayot et al. And Danielsson et al. Pointed out that no matter how high the confidence level is taken, there is still the possibility of unpredictable losses in the future, so the VaR value cannot fully estimate the risks faced by financial assets;
Antonelli and Gabriella provided the calculation process of Monte Carlo simulation method; Andrey proposed the framework of dynamic VaR measurement; David and Ismael applied VaR to the risk measurement of oil price. The research of Chinese scholars on VaR method started as early as 1997 Niu Ang published a paper "New Methods of Bank Risk Management" in international financial research. Its development process can be roughly divided into two stages: the first stage is to understand the learning stage, and the research in this stage focuses on the concept of VaR , Introduction of methods [26.29].
Niu Ang and Zheng Wentong introduced the background of VaR method, calculation method, the use of VaR method, and the necessity of introducing China; Yao Gang introduced the VaR value of the asset portfolio in addition to the definition of VaR, and introduced linear assets. Pricing models and non-linear asset pricing models are specifically illustrated.
The second phase of research was not limited to the introduction of concepts and general calculation methods, but began to introduce VaR
The method is applied in China's financial supervision, futures market and securities market to conduct theoretical and empirical research, and then to improve the VaR method [30.63].
Yang Chunpeng and Cui Yuanmin studied the risk measurement indicator VaR of asymmetric financial derivatives. The VaR model of asymmetric financial derivatives such as options was obtained through the geometric Brownian motion model. The shortcomings of the VaR value, and a Monte Carlo VaR calculation method based on Markov chains are given; Sun Miqiang, Yang Zhongzhi, Yu Suhong, and Song Jun discuss the calculation of VaR based on a stochastic wave model.
Some scholars have applied VaR to China's financial supervision. Liu Yufei has analyzed the application of VaR model in China's financial supervision and its significance, and specifically pointed out how to test the VaR model by using the international "post-test" method. Huang Zhimeng , Cao Junhua, and Wu Chongfeng established a decision model for financial institutions in the context of the VaR supervision method, and analyzed and explained the motivation of financial institutions to obtain as much profitable opportunities as possible in the goal decision process of minimizing the opportunity cost. , Will report the underestimated VaR amount to the regulator to reduce the margin, thereby reducing the total cost of opportunistic investment throughout the risk regulatory period. From a regulatory perspective, this is an excessive speculation and also illustrates the Basel Committee The current regulatory regulations on market risk are not strict enough.Although it provides a mechanism to prevent banks from excessive risk-taking constraints, this constraint mechanism is not quantitative enough to effectively prevent it from adopting excessive speculative investment strategies. Xu Longbing and Lu Rong discussed the significance of VaR method to China's financial risk management and proposed corresponding policy suggestion.
Tao Hairong briefly described how to use the VaR model for risk management in China's insurance industry; Bao Wenbin and Gu Haibing discussed how to use the VaR model for risk management in China's futures market; Zhang Guangyi, Hang Jing,
Zhengnan Lu discussed the analysis of Hong Kong Hang Seng Index futures based on VaR technology; Chi Guotai, Jiang Dazhi, Ying Yang and Lin Jianhua discussed the decision-making model of loan portfolio optimization based on VaR yield constraints; Jinlong Chen and Wei Zhang discussed the market VaR Risk measurement model and its application.
There are relatively many literatures on how to apply VaR to China's securities market. Yan Taihua et al. Respectively proposed the application of VaR method to manage market risks in investment banks and China's securities market; Fan Ying under the premise that the yield rate follows the normal distribution of independent heteroscedasticity Calculate the VaR value of the Shenzhen Stock Exchange Composite Index and conduct "post hoc inspection"