This document provides an overview and application of the Black-Litterman portfolio optimization model. It summarizes the key steps of the Black-Litterman model, which combines an investor's subjective views on expected returns with an implied equilibrium to determine optimal portfolio weights. The document then applies the Black-Litterman model to 10 stocks from the Ho Chi Minh City stock exchange in Vietnam over a one-year period. It finds that Black-Litterman portfolios achieved significantly better return-to-risk performance than the traditional mean-variance approach.
Behavioral finance is an emerging field that combines psychology and financial decision making. It argues that markets are not always efficient in the short run and that people do not always make rational decisions. Behavioral finance provides insights into how emotions like greed and fear can influence investor behavior and make markets volatile. It aims to help investors make sane and safe decisions by restraining emotions and understanding crowd behavior and market sentiments.
Stock price prediction using k* nearest neighbors and indexing dynamic time w...Kei Nakagawa
The document proposes using k*-Nearest Neighbors and Indexing Dynamic Time Warping (IDTW) to predict stock prices based on past price fluctuations. IDTW measures the similarity between stock price movements over monthly periods while accounting for price levels. k*-NN then predicts future prices based on the k nearest past patterns weighted by their IDTW distance. An empirical study found IDTW-k*NN outperformed other methods like DTW-kNN in predicting major stock indices out-of-sample, providing evidence against the efficient market hypothesis.
Non-linear optimization applications in finance including volatility estimation with ARCH and GARCH models, line search methods, Newton's method, steepest descent method, golden section search method, and conjugate gradient method.
The document discusses pricing the Margrabe option using Monte Carlo simulation and an explicit closed-form solution. It begins by defining the Margrabe option and explaining its use. It then presents Margrabe's closed-form solution, which prices the option as a European call using a change of numeraire approach. Next, it analyzes the option's sensitivity to various parameters. Finally, it outlines different option pricing methods and focuses on Monte Carlo simulation and the change of numeraire approach.
This document discusses modelling techniques for a virtual non-life insurance company called Feldafinger Brandkasse using a deterministic and stochastic approach. It describes modelling the company's claims, reserves, assets, reinsurance, and underwriting risks. A DFA (dynamic financial analysis) model is created to simulate the company's economic results over time taking various risks into account. The model calculates required capital (RBC) and shows the company achieving a positive economic result of 5.9 million euros. Risks are aggregated and allocated to different business lines and the company level.
Este documento presenta modelos para estimar y modelar la volatilidad en datos financieros. Introduce los modelos ARCH y GARCH, que permiten que la varianza condicional del término de error dependa de los valores previos de los errores al cuadrado. El modelo GARCH especifica que la varianza corriente es una función de la varianza promedio a largo plazo, la información sobre volatilidad en el período previo, y la varianza ajustada del período previo. Finalmente, discute ejemplos de aplicación de modelos GARCH
The Capital Asset Pricing Model (CAPM) was developed by William Sharpe in 1970 to calculate the expected return of an asset based on its risk. It distinguishes between systematic risk that cannot be diversified away, such as market risk, and unsystematic risk that can be reduced through diversification. The CAPM formula states that the expected return of an asset is equal to the risk-free rate plus a risk premium that is proportional to the asset's systematic risk or beta. Beta measures how volatile an asset's returns are relative to the overall market. The CAPM makes simplifying assumptions about investors and markets. While widely used, some argue it may not perfectly predict returns in practice.
Modern Portfolio Theory (Mpt) - AAII Milwaukeebergsa
This document provides an overview of Modern Portfolio Theory (MPT) including its key concepts and measurements. MPT proposes that rational investors will use diversification to optimize their portfolios based on measures of expected return and risk. It defines risk as the standard deviation of returns and outlines how diversifying uncorrelated assets reduces non-systematic risk. The document then explains common MPT metrics like beta, alpha, the Sharpe Ratio, and R-squared and provides examples of how they are calculated and applied to funds.
Behavioral finance is an emerging field that combines psychology and financial decision making. It argues that markets are not always efficient in the short run and that people do not always make rational decisions. Behavioral finance provides insights into how emotions like greed and fear can influence investor behavior and make markets volatile. It aims to help investors make sane and safe decisions by restraining emotions and understanding crowd behavior and market sentiments.
Stock price prediction using k* nearest neighbors and indexing dynamic time w...Kei Nakagawa
The document proposes using k*-Nearest Neighbors and Indexing Dynamic Time Warping (IDTW) to predict stock prices based on past price fluctuations. IDTW measures the similarity between stock price movements over monthly periods while accounting for price levels. k*-NN then predicts future prices based on the k nearest past patterns weighted by their IDTW distance. An empirical study found IDTW-k*NN outperformed other methods like DTW-kNN in predicting major stock indices out-of-sample, providing evidence against the efficient market hypothesis.
Non-linear optimization applications in finance including volatility estimation with ARCH and GARCH models, line search methods, Newton's method, steepest descent method, golden section search method, and conjugate gradient method.
The document discusses pricing the Margrabe option using Monte Carlo simulation and an explicit closed-form solution. It begins by defining the Margrabe option and explaining its use. It then presents Margrabe's closed-form solution, which prices the option as a European call using a change of numeraire approach. Next, it analyzes the option's sensitivity to various parameters. Finally, it outlines different option pricing methods and focuses on Monte Carlo simulation and the change of numeraire approach.
This document discusses modelling techniques for a virtual non-life insurance company called Feldafinger Brandkasse using a deterministic and stochastic approach. It describes modelling the company's claims, reserves, assets, reinsurance, and underwriting risks. A DFA (dynamic financial analysis) model is created to simulate the company's economic results over time taking various risks into account. The model calculates required capital (RBC) and shows the company achieving a positive economic result of 5.9 million euros. Risks are aggregated and allocated to different business lines and the company level.
Este documento presenta modelos para estimar y modelar la volatilidad en datos financieros. Introduce los modelos ARCH y GARCH, que permiten que la varianza condicional del término de error dependa de los valores previos de los errores al cuadrado. El modelo GARCH especifica que la varianza corriente es una función de la varianza promedio a largo plazo, la información sobre volatilidad en el período previo, y la varianza ajustada del período previo. Finalmente, discute ejemplos de aplicación de modelos GARCH
The Capital Asset Pricing Model (CAPM) was developed by William Sharpe in 1970 to calculate the expected return of an asset based on its risk. It distinguishes between systematic risk that cannot be diversified away, such as market risk, and unsystematic risk that can be reduced through diversification. The CAPM formula states that the expected return of an asset is equal to the risk-free rate plus a risk premium that is proportional to the asset's systematic risk or beta. Beta measures how volatile an asset's returns are relative to the overall market. The CAPM makes simplifying assumptions about investors and markets. While widely used, some argue it may not perfectly predict returns in practice.
Modern Portfolio Theory (Mpt) - AAII Milwaukeebergsa
This document provides an overview of Modern Portfolio Theory (MPT) including its key concepts and measurements. MPT proposes that rational investors will use diversification to optimize their portfolios based on measures of expected return and risk. It defines risk as the standard deviation of returns and outlines how diversifying uncorrelated assets reduces non-systematic risk. The document then explains common MPT metrics like beta, alpha, the Sharpe Ratio, and R-squared and provides examples of how they are calculated and applied to funds.
The project involved studying some of the popular filters and prediction algorithms used for stock market analysis. Based on that Moving Average Filter, Adaptive Kalman Filter, Multiple Linear Regression Filter, Bollinger Bands, and Chaikin Oscillator were developed and implemented in MATLAB. For carrying out the analysis, daily stock market data of 10 popular companies, over a period of 1 year was used. The overall project developed can be used as a complete package to carry out accurate and efficient stock market analysis and trend study.
This document discusses multiple linear regression analysis. It begins by defining a multiple regression equation that describes the relationship between a response variable and two or more explanatory variables. It notes that multiple regression allows prediction of a response using more than one predictor variable. The document outlines key elements of multiple regression including visualization of relationships, statistical significance testing, and evaluating model fit. It provides examples of interpreting multiple regression output and using the technique to predict outcomes.
Performance analysis and prediction of stock market for investment decision u...Hari KC
1) The document presents research on using regression techniques to analyze and forecast future stock prices of companies listed on the Nepalese stock exchange NEPSE.
2) The researcher aims to allow investors to make investment decisions with less risk by predicting stock market movements and stability.
3) Regression analysis and support vector machines will be used to fit linear and nonlinear models to stock price data and news sentiment to forecast prices.
A Powerpoint Presentation designed to provide beginners to MATLAB an introduction to the MATLAB environment and introduce them to the fundamentals of MATLAB including matrix generation and manipulation, Arrays, MATLAB Graphics, Data Import and Export, etc
This document discusses enterprise valuation, specifically the hybrid approach that combines discounted cash flow (DCF) analysis and relative valuation. It provides an overview of the two-step process for enterprise valuation, which involves valuing cash flows during the planning period using DCF analysis and estimating the terminal value using either DCF or a multiples approach. It then provides an example valuation of Tata Steel's acquisition of Corus, a steel company, where the planning period cash flows are valued using DCF and the terminal value is estimated using an EV/EBITDA multiple. Key details of the transaction, financing structure, and potential synergies are also summarized.
Time Series Analysis and Forecasting.pptssuser220491
This document discusses time series analysis and forecasting. It introduces time series data and examples. The main methods for forecasting time series are regression analysis and time series analysis (TSA), which examines past behavior to predict future behavior without causal variables. TSA involves analyzing trends, cycles, seasonality, and random variations. Forecasting accuracy is measured using techniques like mean absolute deviation and mean square error. Extrapolation models like moving averages, weighted moving averages, and exponential smoothing are discussed for forecasting, as well as approaches for stationary, additive seasonal, multiplicative seasonal, and trend data.
Este documento presenta el modelo de regresión lineal general como uno de los métodos más populares y aplicados en análisis cuantitativo. Explica los supuestos, estimación por mínimos cuadrados ordinarios, interpretación de los coeficientes, y aplicación del modelo bivariado y multivariado. Se detalla el proceso de estimación en Excel y se ilustran conceptos como la función de regresión poblacional, recta de regresión muestral, y error estándar de la estimación.
This document provides an introduction to mutual funds, including:
- It defines a mutual fund as an investment vehicle that pools money from investors to invest in securities like stocks and bonds. Returns are shared proportionally among investors.
- The advantages of mutual funds are listed as professional management, reduced costs through economies of scale, diversification, and choice of schemes. The disadvantages include loss of control over investments, fees and expenses, and costs not being controlled by investors.
- The structure of mutual funds in India is described as a three tier system with a sponsor, a trust, and professional managers (AMC), distributors, registrars and a custodian who perform various functions for the fund.
This document discusses various concepts related to file organization and data warehousing. It defines key terms like file, record, fixed and variable length records. It describes different types of single-level and multi-level indexes used for file organization, including B-trees. It also provides an overview of data warehousing concepts such as architecture and operations. The benefits of data warehousing for business analytics and insights are highlighted. Different file organization methods like sequential, heap, hash and indexed sequential access are also summarized.
This document provides an overview of time series analysis techniques including moving average (MA) models, exponential smoothing, and ARMA models. It describes the key components of MA models including the MA(q) notation and theoretical properties. Exponential smoothing is presented as a weighted moving average for smoothing and short-term forecasting. The ARMA model is introduced as combining autoregressive and moving average terms to model a time series.
Wang-Landau Monte Carlo simulation is a method for calculating the density of states function which can then be used to calculate thermodynamic properties like the mean value of variables. It improves on traditional Monte Carlo methods which struggle at low temperatures due to complicated energy landscapes with many local minima separated by large barriers. The Wang-Landau algorithm calculates the density of states function directly rather than relying on sampling configurations, allowing it to overcome barriers and fully explore the configuration space even at low temperatures.
Impact of Valuation Adjustments (CVA, DVA, FVA, KVA) on Bank's Processes - An...Andrea Gigli
The talk hold in London on September 10th at the 5th Annual XVA Forum on Funding, Capital and Valuation. It covered some implications of Valuation Adjustments like CVA, DVA, FVA and KVA (XVAs) in the Pricing of Derivatives, Data Model Definition, Risk Management, Accounting, Trade Workflow processing.
Security Analysis and Portfolio ManagementShrey Sao
Modern portfolio theory (MPT) provides a framework for constructing investment portfolios to maximize expected return based on a given level of market risk. MPT assumes investors aim to maximize returns for a given level of risk. It uses variance as a measure of risk and covariance to capture how asset returns move together. The efficient frontier graph shows the set of optimal portfolios that offer the highest expected return for a given level of risk. Individual investors select the portfolio on the efficient frontier that maximizes their utility based on their risk tolerance. MPT emphasizes diversification and the benefits of holding inefficiently priced assets.
Chapter2 International Finance ManagementPiyush Gaur
1. The document provides answers and explanations to questions about international monetary systems, including Gresham's Law, the gold standard, the Bretton Woods system, and exchange rate regimes.
2. It also answers questions about the European Monetary System, special drawing rights, criteria for a good international monetary system, and the prospects of the euro becoming a global reserve currency.
3. The final part presents a mini case about the potential for the United Kingdom to adopt the euro.
Time Series basic concepts and ARIMA family of models. There is an associated video session along with code in github: https://github.com/bhaskatripathi/timeseries-autoregressive-models
https://drive.google.com/file/d/1yXffXQlL6i4ufQLSpFFrJgymhHNXL1Mf/view?usp=sharing
An Empirical Investigation Of The Arbitrage Pricing TheoryAkhil Goyal
The study empirically tests the Arbitrage Pricing Theory (APT) developed by Ross in 1976 using daily stock return data from 1962-1972. It finds:
1) Factor analysis identifies 5 factors that explain stock returns within industry groups, supporting the APT.
2) Cross-sectional regressions show factor loadings can explain expected stock returns, as the APT predicts.
3) Adding total return variance to the regressions does not eliminate the explanatory power of factor loadings, supporting the APT over alternatives.
4) Tests across industry groups find no evidence factor structures differ, as the APT assumes consistent factors across stocks.
Presented "Random Walk on Graphs" in the reading group for Knoesis. Specifically for Recommendation Context.
Referred: Purnamrita Sarkar, Random Walks on Graphs: An Overview
The document discusses quaternions and their applications. It begins by providing a brief history of quaternions, defined by Sir William Rowan Hamilton in 1843. Quaternions represent rotations and orientations in 3D space and are used in computer graphics, control theory, physics, and other fields due to advantages like avoiding gimbal lock. The document then covers details of quaternion operations like addition, multiplication, and rotation representations. It provides examples of using quaternions to represent and perform rotations in three dimensions.
The document discusses portfolio diversification and asset allocation. It explains that asset allocation is the process of combining different asset classes like stocks, bonds, and cash to meet investment goals. Diversifying across asset classes can help lower risk and increase returns. The document provides examples showing how diversified portfolios performed better than non-diversified portfolios during market downturns.
- The chapter discusses portfolio theory and models for determining asset prices like the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT).
- Portfolio risk depends on the correlation and covariance of returns between assets. Diversification reduces unsystematic risk but not systematic market risk.
- CAPM suggests investors should hold a combination of the risk-free asset and the market portfolio. It provides a framework to determine required rates of return based on an asset's systematic risk or beta.
- APT assumes asset returns have predictable and unpredictable components related to macroeconomic factors. It provides an alternative model to CAPM for determining expected returns.
In this paper, the black-litterman model is introduced to quantify investor’s views, then we expanded
the safety-first portfolio model under the case that the distribution of risk assets return is ambiguous. When
short-selling of risk-free assets is allowed, the model is transformed into a second-order cone optimization
problem with investor views. The ambiguity set parameters are calibrated through programming
The project involved studying some of the popular filters and prediction algorithms used for stock market analysis. Based on that Moving Average Filter, Adaptive Kalman Filter, Multiple Linear Regression Filter, Bollinger Bands, and Chaikin Oscillator were developed and implemented in MATLAB. For carrying out the analysis, daily stock market data of 10 popular companies, over a period of 1 year was used. The overall project developed can be used as a complete package to carry out accurate and efficient stock market analysis and trend study.
This document discusses multiple linear regression analysis. It begins by defining a multiple regression equation that describes the relationship between a response variable and two or more explanatory variables. It notes that multiple regression allows prediction of a response using more than one predictor variable. The document outlines key elements of multiple regression including visualization of relationships, statistical significance testing, and evaluating model fit. It provides examples of interpreting multiple regression output and using the technique to predict outcomes.
Performance analysis and prediction of stock market for investment decision u...Hari KC
1) The document presents research on using regression techniques to analyze and forecast future stock prices of companies listed on the Nepalese stock exchange NEPSE.
2) The researcher aims to allow investors to make investment decisions with less risk by predicting stock market movements and stability.
3) Regression analysis and support vector machines will be used to fit linear and nonlinear models to stock price data and news sentiment to forecast prices.
A Powerpoint Presentation designed to provide beginners to MATLAB an introduction to the MATLAB environment and introduce them to the fundamentals of MATLAB including matrix generation and manipulation, Arrays, MATLAB Graphics, Data Import and Export, etc
This document discusses enterprise valuation, specifically the hybrid approach that combines discounted cash flow (DCF) analysis and relative valuation. It provides an overview of the two-step process for enterprise valuation, which involves valuing cash flows during the planning period using DCF analysis and estimating the terminal value using either DCF or a multiples approach. It then provides an example valuation of Tata Steel's acquisition of Corus, a steel company, where the planning period cash flows are valued using DCF and the terminal value is estimated using an EV/EBITDA multiple. Key details of the transaction, financing structure, and potential synergies are also summarized.
Time Series Analysis and Forecasting.pptssuser220491
This document discusses time series analysis and forecasting. It introduces time series data and examples. The main methods for forecasting time series are regression analysis and time series analysis (TSA), which examines past behavior to predict future behavior without causal variables. TSA involves analyzing trends, cycles, seasonality, and random variations. Forecasting accuracy is measured using techniques like mean absolute deviation and mean square error. Extrapolation models like moving averages, weighted moving averages, and exponential smoothing are discussed for forecasting, as well as approaches for stationary, additive seasonal, multiplicative seasonal, and trend data.
Este documento presenta el modelo de regresión lineal general como uno de los métodos más populares y aplicados en análisis cuantitativo. Explica los supuestos, estimación por mínimos cuadrados ordinarios, interpretación de los coeficientes, y aplicación del modelo bivariado y multivariado. Se detalla el proceso de estimación en Excel y se ilustran conceptos como la función de regresión poblacional, recta de regresión muestral, y error estándar de la estimación.
This document provides an introduction to mutual funds, including:
- It defines a mutual fund as an investment vehicle that pools money from investors to invest in securities like stocks and bonds. Returns are shared proportionally among investors.
- The advantages of mutual funds are listed as professional management, reduced costs through economies of scale, diversification, and choice of schemes. The disadvantages include loss of control over investments, fees and expenses, and costs not being controlled by investors.
- The structure of mutual funds in India is described as a three tier system with a sponsor, a trust, and professional managers (AMC), distributors, registrars and a custodian who perform various functions for the fund.
This document discusses various concepts related to file organization and data warehousing. It defines key terms like file, record, fixed and variable length records. It describes different types of single-level and multi-level indexes used for file organization, including B-trees. It also provides an overview of data warehousing concepts such as architecture and operations. The benefits of data warehousing for business analytics and insights are highlighted. Different file organization methods like sequential, heap, hash and indexed sequential access are also summarized.
This document provides an overview of time series analysis techniques including moving average (MA) models, exponential smoothing, and ARMA models. It describes the key components of MA models including the MA(q) notation and theoretical properties. Exponential smoothing is presented as a weighted moving average for smoothing and short-term forecasting. The ARMA model is introduced as combining autoregressive and moving average terms to model a time series.
Wang-Landau Monte Carlo simulation is a method for calculating the density of states function which can then be used to calculate thermodynamic properties like the mean value of variables. It improves on traditional Monte Carlo methods which struggle at low temperatures due to complicated energy landscapes with many local minima separated by large barriers. The Wang-Landau algorithm calculates the density of states function directly rather than relying on sampling configurations, allowing it to overcome barriers and fully explore the configuration space even at low temperatures.
Impact of Valuation Adjustments (CVA, DVA, FVA, KVA) on Bank's Processes - An...Andrea Gigli
The talk hold in London on September 10th at the 5th Annual XVA Forum on Funding, Capital and Valuation. It covered some implications of Valuation Adjustments like CVA, DVA, FVA and KVA (XVAs) in the Pricing of Derivatives, Data Model Definition, Risk Management, Accounting, Trade Workflow processing.
Security Analysis and Portfolio ManagementShrey Sao
Modern portfolio theory (MPT) provides a framework for constructing investment portfolios to maximize expected return based on a given level of market risk. MPT assumes investors aim to maximize returns for a given level of risk. It uses variance as a measure of risk and covariance to capture how asset returns move together. The efficient frontier graph shows the set of optimal portfolios that offer the highest expected return for a given level of risk. Individual investors select the portfolio on the efficient frontier that maximizes their utility based on their risk tolerance. MPT emphasizes diversification and the benefits of holding inefficiently priced assets.
Chapter2 International Finance ManagementPiyush Gaur
1. The document provides answers and explanations to questions about international monetary systems, including Gresham's Law, the gold standard, the Bretton Woods system, and exchange rate regimes.
2. It also answers questions about the European Monetary System, special drawing rights, criteria for a good international monetary system, and the prospects of the euro becoming a global reserve currency.
3. The final part presents a mini case about the potential for the United Kingdom to adopt the euro.
Time Series basic concepts and ARIMA family of models. There is an associated video session along with code in github: https://github.com/bhaskatripathi/timeseries-autoregressive-models
https://drive.google.com/file/d/1yXffXQlL6i4ufQLSpFFrJgymhHNXL1Mf/view?usp=sharing
An Empirical Investigation Of The Arbitrage Pricing TheoryAkhil Goyal
The study empirically tests the Arbitrage Pricing Theory (APT) developed by Ross in 1976 using daily stock return data from 1962-1972. It finds:
1) Factor analysis identifies 5 factors that explain stock returns within industry groups, supporting the APT.
2) Cross-sectional regressions show factor loadings can explain expected stock returns, as the APT predicts.
3) Adding total return variance to the regressions does not eliminate the explanatory power of factor loadings, supporting the APT over alternatives.
4) Tests across industry groups find no evidence factor structures differ, as the APT assumes consistent factors across stocks.
Presented "Random Walk on Graphs" in the reading group for Knoesis. Specifically for Recommendation Context.
Referred: Purnamrita Sarkar, Random Walks on Graphs: An Overview
The document discusses quaternions and their applications. It begins by providing a brief history of quaternions, defined by Sir William Rowan Hamilton in 1843. Quaternions represent rotations and orientations in 3D space and are used in computer graphics, control theory, physics, and other fields due to advantages like avoiding gimbal lock. The document then covers details of quaternion operations like addition, multiplication, and rotation representations. It provides examples of using quaternions to represent and perform rotations in three dimensions.
The document discusses portfolio diversification and asset allocation. It explains that asset allocation is the process of combining different asset classes like stocks, bonds, and cash to meet investment goals. Diversifying across asset classes can help lower risk and increase returns. The document provides examples showing how diversified portfolios performed better than non-diversified portfolios during market downturns.
- The chapter discusses portfolio theory and models for determining asset prices like the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT).
- Portfolio risk depends on the correlation and covariance of returns between assets. Diversification reduces unsystematic risk but not systematic market risk.
- CAPM suggests investors should hold a combination of the risk-free asset and the market portfolio. It provides a framework to determine required rates of return based on an asset's systematic risk or beta.
- APT assumes asset returns have predictable and unpredictable components related to macroeconomic factors. It provides an alternative model to CAPM for determining expected returns.
In this paper, the black-litterman model is introduced to quantify investor’s views, then we expanded
the safety-first portfolio model under the case that the distribution of risk assets return is ambiguous. When
short-selling of risk-free assets is allowed, the model is transformed into a second-order cone optimization
problem with investor views. The ambiguity set parameters are calibrated through programming
Improving Returns from the Markowitz Model using GA- AnEmpirical Validation o...idescitation
Portfolio optimization is the task of allocating the investors capital among
different assets in such a way that the returns are maximized while at the same time, the
risk is minimized. The traditional model followed for portfolio optimization is the
Markowitz model [1], [2],[3]. Markowitz model, considering the ideal case of linear
constraints, can be solved using quadratic programming, however, in real-life scenario, the
presence of nonlinear constraints such as limits on the number of assets in the portfolio, the
constraints on budgetary allocation to each asset class, transaction costs and limits to the
maximum weightage that can be assigned to each asset in the portfolio etc., this problem
becomes increasingly computationally difficult to solve, ie NP-hard. Hence, soft computing
based approaches seem best suited for solving such a problem. An attempt has been made in
this study to use soft computing technique (specifically, Genetic Algorithms), to overcome
this issue. In this study, Genetic Algorithm (GA) has been used to optimize the parameters
of the Markowitz model such that overall portfolio returns are maximized with the standard
deviation of the returns being minimized at the same time. The proposed system is validated
by testing its ability to generate optimal stock portfolios with high returns and low standard
deviations with the assets drawn from the stocks traded on the Bombay Stock Exchange
(BSE). Results show that the proposed system is able to generate much better portfolios
when compared to the traditional Markowitz model.
The document summarizes a study that tests the Capital Asset Pricing Model (CAPM) using two portfolios constructed from stocks on the Helsinki stock exchange. Daily return data from 2014-2015 on four large stocks was used to create an equal-weighted and value-weighted portfolio. The CAPM was tested to see if it could accurately price the returns of the two portfolios. The study found that the CAPM could precisely price the value-weighted portfolio but not the equal-weighted portfolio, suggesting some limitation of the CAPM in pricing assets.
The document discusses probability-based approaches for calculating expected returns and variance under uncertainty. It provides an example using return data for a stock to calculate the expected return of 9.25% and variance of 0.02%. It also discusses how portfolio return and variance depends on asset weights, the individual asset expected returns and variances, and the correlation between the assets. Assuming the two example assets are perfectly negatively correlated, it calculates the asset weights needed for a zero risk portfolio and the expected return of that portfolio as 25.36%. Finally, it discusses limits to diversification in practice, such as the inability to hold all securities and that only unsystematic risk can be reduced through diversification.
Questions and Answers At Least 75 Words each.Please answer th.docxmakdul
Questions and Answers: At Least 75 Words each.
Please answer the following questions.
1. What are the differences and similarities between samples and populations?
2. What are the measures of Central Tendency assumptions?
3. What are measures of Dispersion used for and what are the assumptions for each?
4. Define collaboration and how you will apply it in Statistics? (100 Words)
The Capital Asset Pricing Model:
Theory and Evidence
Eugene F. Fama and Kenneth R. French
T he capital asset pricing model (CAPM) of William Sharpe (1964) and JohnLintner (1965) marks the birth of asset pricing theory (resulting in aNobel Prize for Sharpe in 1990). Four decades later, the CAPM is still
widely used in applications, such as estimating the cost of capital for firms and
evaluating the performance of managed portfolios. It is the centerpiece of MBA
investment courses. Indeed, it is often the only asset pricing model taught in these
courses.1
The attraction of the CAPM is that it offers powerful and intuitively pleasing
predictions about how to measure risk and the relation between expected return
and risk. Unfortunately, the empirical record of the model is poor—poor enough
to invalidate the way it is used in applications. The CAPM’s empirical problems may
reflect theoretical failings, the result of many simplifying assumptions. But they may
also be caused by difficulties in implementing valid tests of the model. For example,
the CAPM says that the risk of a stock should be measured relative to a compre-
hensive “market portfolio” that in principle can include not just traded financial
assets, but also consumer durables, real estate and human capital. Even if we take
a narrow view of the model and limit its purview to traded financial assets, is it
1 Although every asset pricing model is a capital asset pricing model, the finance profession reserves the
acronym CAPM for the specific model of Sharpe (1964), Lintner (1965) and Black (1972) discussed
here. Thus, throughout the paper we refer to the Sharpe-Lintner-Black model as the CAPM.
y Eugene F. Fama is Robert R. McCormick Distinguished Service Professor of Finance,
Graduate School of Business, University of Chicago, Chicago, Illinois. Kenneth R. French is
Carl E. and Catherine M. Heidt Professor of Finance, Tuck School of Business, Dartmouth
College, Hanover, New Hampshire. Their e-mail addresses are �[email protected]
edu� and �[email protected]�, respectively.
Journal of Economic Perspectives—Volume 18, Number 3—Summer 2004 —Pages 25– 46
legitimate to limit further the market portfolio to U.S. common stocks (a typical
choice), or should the market be expanded to include bonds, and other financial
assets, perhaps around the world? In the end, we argue that whether the model’s
problems reflect weaknesses in the theory or in its empirical implementation, the
failure of the CAPM in empirical tests implies that most applications of the model
are invalid.
We begin by outlining the logic of t ...
This document summarizes the capital asset pricing model (CAPM). It begins by outlining the logic and key assumptions of the CAPM, including that all investors hold the same market portfolio which must lie on the efficient frontier. It then states that the CAPM predicts the expected return of an asset is determined by its beta, or non-diversifiable risk relative to the market. However, the document notes that empirical tests have found the CAPM performs poorly in applications. It concludes the CAPM's failings indicate applications based on the model are invalid, challenging researchers to develop alternative models.
Testing and extending the capital asset pricing modelGabriel Koh
This paper attempts to prove whether the conventional Capital Asset Pricing Model (CAPM) holds with respect to a set of asset returns. Starting with the Fama-Macbeth cross-sectional regression, we prove through the significance of pricing errors that the CAPM does not hold. Hence, we expand the original CAPM by including risk factors and factor-mimicking portfolios built on firm-specific characteristics and test for their significance in the model. Ultimately, by adding significant factors, we find that the model helps to better explain asset returns, but does still not entirely capture pricing errors.
This document provides an overview of an upcoming presentation on asset pricing models. The presentation will cover capital market theory, the capital market line, security market line, capital asset pricing model, and diversification. It will discuss the assumptions and formulas for the capital market line and security market line. The capital market line shows expected returns based on portfolio risk, while the security market line shows expected individual asset returns based on systematic risk. The capital asset pricing model uses the concept of beta to calculate the expected return of an asset based on its risk relative to the market.
The document summarizes the capital asset pricing model (CAPM) and reviews early empirical tests of the model. It begins by outlining the logic and key assumptions of the CAPM, including that the market portfolio must be mean-variance efficient. However, empirical tests found problems with the CAPM's predictions about the relationship between expected returns and market betas. Specifically, cross-sectional regressions did not find intercepts equal to the risk-free rate or slopes equal to the expected market premium. To address measurement error, later tests examined portfolios rather than individual assets. In general, the early empirical evidence revealed shortcomings in the CAPM's ability to explain returns.
This document summarizes the Capital Asset Pricing Model (CAPM). It begins by outlining the key assumptions and logic behind the CAPM. The CAPM builds on Harry Markowitz's portfolio choice model by adding assumptions of a risk-free rate and market clearing prices. This implies that the market portfolio must be mean-variance efficient. The CAPM then predicts that an asset's expected return is determined by its beta, or non-diversifiable risk relative to the market. However, the document notes that empirical tests have found the CAPM performs poorly in validating these predictions. It concludes that while theoretical or implementation issues may be to blame, the CAPM's failure in empirical tests means its applications are generally invalid.
This document provides an overview and definitions of key concepts in asset pricing models, including the capital asset pricing model (CAPM). It discusses the assumptions of capital market theory, defines a risk-free asset and its characteristics, and explains how combining a risk-free asset with risky portfolios affects expected return and standard deviation. It also defines the market portfolio, systematic and unsystematic risk, the capital market line (CML), and how diversification eliminates unsystematic risk.
Modern portfolio theory (MPT) is a theory of finance that aims to construct portfolios that offer the maximum expected return for a given level of risk or the minimum risk for a given level of expected return. MPT uses diversification and asset allocation to reduce portfolio risk. It assumes investors are rational and markets are efficient. MPT models asset returns as normally distributed and defines risk as standard deviation of returns. It seeks to minimize total portfolio variance by combining assets whose returns are not perfectly correlated. The efficient frontier shows the optimal risk-return tradeoff and the capital allocation line incorporates a risk-free asset into the analysis. MPT is widely used but also faces criticisms around its assumptions.
what do you want to do is you can do, if only you are willing to do....right? business it not only for our own selves, but also for everybody good also.
This document provides an overview of portfolio theory and the Capital Asset Pricing Model (CAPM). It defines key concepts like the efficient frontier, market portfolio, capital market line (CML), beta, and the security market line (SML). The CAPM holds that an asset's expected return is determined by its non-diversifiable risk as measured by its beta. Beta measures how an asset's returns co-vary with the market portfolio. The document provides examples of estimating betas and calculating expected returns using the CAPM framework. It concludes by noting the CAPM is a useful but not perfect model of the risk-return relationship.
The document discusses the arbitrage pricing theory (APT), which relates a security's expected return to multiple common risk factors. It provides examples of how the APT can be used to model returns based on factors like inflation, GDP growth, and exchange rates. The APT assumes perfect capital markets, homogeneous investor expectations, and allows short selling and arbitrage opportunities. It implies a linear relationship between expected returns and factor sensitivities similar to the capital asset pricing model. Empirical tests provide some support for the APT but also have limitations.
ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS.pptPankajKhindria
The Arbitrage Pricing Theory (APT) proposes that the expected return of a financial asset can be modeled as a linear function of various macroeconomic factors where sensitivity to changes in each factor is represented by a factor-specific beta coefficient. In contrast to the Capital Asset Pricing Model which relies on a single market factor, the APT allows for multiple common factors that influence asset returns. Empirical tests of the APT have been inconclusive due to difficulty in identifying a set of factors that consistently explains security returns.
This document provides an update on capital structure arbitrage strategies. It begins with an overview of Merton's structural model for pricing debt and equity. It then discusses the CreditGrades model, which builds on Merton's framework. The document reviews literature on using structural models in capital structure arbitrage trading strategies, and replicates Yu's 2006 strategy from 2004-2014. It proposes periodically recalibrating the model to match market spreads, finding this improves performance over keeping parameters fixed. In conclusion, structural models can provide a basis for capital structure arbitrage strategies but require adjustments to align implied and market spreads.
The Capital Asset Pricing Model (CAPM) was developed in the 1960s as a way to determine the expected return of an asset based on its risk. CAPM assumes that investors will be compensated only based on an asset's systematic or non-diversifiable risk as measured by its beta. The model builds on Markowitz's portfolio theory and introduces the security market line, which plots the expected return of an asset against its beta. According to CAPM, the expected return of an asset is equal to the risk-free rate plus a risk premium that is proportional to the asset's beta.
This document summarizes the key points of the article "Risk, Return, and Equilibrium: Empirical Tests" by Eugene F. Fama and James D. MacBeth. The article tests the relationship between average stock returns and risk using the two-parameter portfolio model. It finds that the data is consistent with investors holding efficient portfolios as the model predicts, and that stock prices fully reflect available information as implied by an efficient market. Specifically:
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1. JOHN VON NEUMANN INSTITUTE
VIETNAM NATIONAL UNIVERSITY HOCHIMINH CITY
BLACK-LITTERMAN PORTFOLIO
OPTIMIZATION
Hoang Hai Nguyen
nguyen.hoang@jvn.edu.vn
HCM City, July- 2012
1
2. Abstract: In practice, mean-variance optimization results in non-intuitive and extreme
portfolio allocations, which are highly sensitive to variations in the inputs. Generally,
efficient frontiers based on historical data lead to highly concentrated portfolios. The
Black-Litterman approach overcomes, or at least mitigates, these problems to a large
extent. The highlight of this approach is that it enables us to incorporate investment
views (which are subjective in nature). These aspects make the Black-Litterman model a
strong quantitative tool that provides an ideal framework for strategic/tactical asset
allocation. In this project, we will apply the Black-Litterman model for the context of
VietNam equity markets. To represent the VietNam equity markets, we select top K = 10
most market capitalization stocks of the Ho Chi Minh city stock exchange with historical
data of at least 1 year for tactical asset allocation. As extension part, we empirically
compare the performance of the two approaches. The study found that the BlackLitterman efficient portfolios achieve a significantly better return-to-risk performance
than the mean-variance optimal approach/strategy.
2
3. 1. Introduction
Since publication in 1990, the Black-Litterman asset allocation model has gained wide
application in many financial institutions. As developed in the original paper, the BlackLitterman model provides the flexibility of combining the market equilibrium with
additional market views of the investor. The Black-Litterman approach may be contrasted
with the standard mean-variance optimization in which the user inputs a complete set of
expected returns 1 and the portfolio optimizer generates the optimal portfolio weights.
Because there is a complex mapping between expected returns and the portfolio weights,
and because there is no natural starting point for the expected return assumptions, users
of the standard portfolio optimizers often find their specification of expected returns
produces output portfolio weights which do not seem to make sense. In the BlackLitterman model the user inputs any number of views, which are statements about the
expected returns of arbitrary portfolios, and the model combines the views with
equilibrium, producing both the set of expected returns of assets as well as the optimal
portfolio weights.
Although Black and Litterman concluded in their 1992 article [Black and Litterman,
1992]:
“. . . our approach allows us to generate optimal portfolios that start at a set of neutral
weights and then tilt in the direction of the investor’s views.”
they did not discuss the precise nature of that phenomenon. As we demonstrate here, the
optimal portfolio for an unconstrained investor is proportional to the market equilibrium
portfolio plus a weighted sum of portfolios reflecting the investor’s views. Now the
economic intuition becomes very clear. The investor starts by holding the scaled market
equilibrium portfolio, reflecting her uncertainty on the equilibrium, then invests in
portfolios representing her views. The Black-Litterman model computes the weight to put
on the portfolio representing each view according to the strength of the view, the
covariance between the view and the equilibrium, and the covariances among the views.
We show the conditions for the weight on a view portfolio to be positive, negative, or
zero. We also show that the weight on a view increases when the investor becomes more
bullish on the view, and the magnitude of the weight increases when the investor
becomes less uncertain about the view.
The rest of the article is organized as follows. In Section 2, we review the basics of the
Black-Litterman asset allocation model. In Section 3, we present our empirical findings
of the study and data description. Then we present our main results in Section 4
3
4. 2. The Black-Litterman model
The Black and Litterman (1990, 1991, 1992) asset allocation model is a sophisticated
asset allocation and portfolio construction method that overcomes the drawbacks of
traditional mean-variance optimization. The Black-Litterman model uses a Bayesian
approach to combine the subjective views of investors about the expected return of assets.
The practical implementation of the Black-Litterman model was discussed in detail in the
context of global asset allocation (Bevan and Winkelmann, 1998), sector allocation
(Wolfgang, 2001) and portfolio optimization (He and Litterman, 1999). In order to
incorporate the subjective views of investors, the Black-Litterman model combines the
CAPM (Sharpe, 1964), reverse optimization (Sharpe, 1974), mixed estimation (Theil,
1971, 1978), the universal hedge ratio/Black‟s global CAPM (Black and Litterman
1990, 1991, 1992; Litterman, 2003), and mean-variance optimization (Markowitz,
1952). The Black-Litterman model creates stable and intuitively appealing mean-variance
efficient portfolios based on investors‟ subjective views and also eliminates the input
sensitivity of the mean-variance optimization. The most important input in mean-variance
optimization is the vector of expected returns. The model starts with the CAPM
equilibrium market portfolio returns starting point for estimation of asset returns, unlike
previous similar models started with the uninformative uniform prior distributions. The
CAPM equilibrium market portfolio returns are more intuitively connected to market and
reverse optimization of the same will generate a stable distribution of return estimations.
The Black-Litterman model converts these CAPM equilibrium market portfolio returns to
implied return vector as a function of risk-free return, market capitalization, and
covariance with other assets. Implied returns are also known as CAPM returns, market
returns, consensus returns, and reverse optimized returns. Equilibrium returns are the set
of returns that clear the market if all investors have identical views.
The following is the Black-Litterman formula (Equation 1) along with detailed
description of each of its components. In this project, K represents the number of views
and N represents the number of assets in the model.
[ ] = ( ∑)
+ ′Ω
( ∑) ∏ + ′Ω
(1)
where,
E[R] is the new (posterior) combined return vector (N × 1 column vector);
τ, a scalar;
Σ, the covariance matrix of excess returns (N × N matrix);
P, a matrix that identifies the assets involved in the views (K × N matrix or 1 × N
4
5. row vector in the special case of 1 view);
Ω, a diagonal covariance matrix of error t erms from the expressed views
representing the uncertainty in each view (K × K matrix);
∏, the implied equilibrium return vector (N × 1 column vector);
Q, the View Vector (K x 1 column vector)
The Black-Litterman model uses the equilibrium returns as a starting point and the
equilibrium returns of the assets are derived using a reverse optimization method using
Equation 2
∏ =
∑
(2)
where,
∏, is the implied equilibrium excess return vector;
, a risk aversion coefficient;
∑, the covariance matrix, and
, is the market capitalization weight of the assets.
The risk aversion coefficient characterizes the expected risk-return tradeoff and it acts as
a scaling factor for the reverse optimization. The risk aversion coefficient can be
calculated using equation 3
=
(3)
The implied equilibrium return vector is nothing but the market capitalization-weighted
portfolio. In the absence of views, investors should hold the market portfolio. However,
Black-Litterman model allows investors to incorporate their subjective views on the
expected return of some of the assets in a portfolio, which may differ from the implied
equilibrium returns. The subjective views of investors can be expressed in either absolute
or relative terms.
where, Q, the view vector, which is k × 1 dimension; k, the number of views, either
absolute or relative. The uncertainty of views results in a random, unknown,
independently, normally distributes error term vector ( ) with mean 0 and covariance
matrix Ω. Thus a view has the form Q+
5
6. Q+ =
:
:
:
+ :
(4)
Investor views on the market and their confidence level on the views form the basis for
arriving at new combined expected return vector. With respect to investor views, we
need to consider the following aspects while developing the Black-Litterman model:
1. Each view should be unique and uncorrelated with the other.
2. While constructing the views, we need to ensure that the sum of views is either
0 or 1, which ensures that all the views are fully invested.
The investor view matrix (P) was constructed differently by various authors. He and
Litterman (1999) and Izorek (2005) used a market capitalization weighted scheme.
However, market capitalization weighted scheme is applicable only in relative views.The
expected return on the views is organized as a column vector (Q) expressed as Kx1
vector.
Omega, the covariance matrix of views, is a symmetric matrix with non-diagonal
elements as 0s. For calculating it, we have assumed that the variance of the views will be
proportional to the variance of the asset returns, just as the variance of the prior
distribution is. This method has been used by He and Litterman (1999) and Meucci
(2006). Using these expected return, risk aversion coefficient (λ) and covariance matrix
(∑), new asset weights can be allocated using equation 5.
= ( ∑)
* E[R]
(5)
Before we attempt to detail the empirical examination of the Black-Litterman model, it
might be useful to give an intuitive description of the major steps, which are presented in
Figure 1
6
7. Figure 1: Major steps behind the Black-Litterman model.
3. Empirical findings of the study and Data description
Data description
The current study is based on various stocks constructed and maintained by the Ho chi
minh city stock exchange (HSE), VietNam. We select top K = 10 most market
capitalization of Ho chi minh city stock exchange with historical and data of at least 1
year and use daily closing prices from January 1st, 2011 to January 31st, 2012.
List of 10 stocks are selected such as:
No.
Stocks
Code
1
Baoviet Holdings
PetroVietnam Fertilizer and
Chemicals Company
Vietnam export import Bank
FPT Corporation
Hoang Anh Gia Lai JSC
Masan Group Corporation
Saigon Securities Inc
Sai Gon Thuong Tin Bank
Vingroup
Vinamilk Corp
BVH
Market
capitalization
(billion VND)
46,272
DPM
12,160
4.61%
EIB
FPT
HAG
MSN
SSI
STB
VIC
VNM
15,523
10,610
13,645
64,409
7,549
14,040
35,595
47,095
5.89%
4.02%
5.17%
24.39%
1.79%
5.31%
13.48%
17.84%
2
3
4
5
6
7
8
9
10
Proportion
17.51%
7
8. Empirical findings of the study
As VietNam is an emerging economy that could withstand the after-effects of global
financial meltdown, several foreign institutional investors are keen on parking their
investments in the country. Each of them has different long-term and short-term views on
different sectors of the VietNam equity market. This has motivated to empirically
examine the tactical asset allocation across different sectors of VietNam equity market
through Black-Litterman approach.
The study has considered the monthly closing price of ten stocks of HSE from January
1st, 2011 to January 31st, 2012. The daily closing price of stocks has been taken to
compute the continuous compounded return of daily these stocks by taking the natural
logarithmic of price difference. This is represented as follows:
= ln( ) − ln(
)
where,
is the return at time t
, price at time t, and
, price at time t-1
A risk-return profile of 10 stocks over a one years, from 1st, 2011 to January 31st, 2012,
is presented in the Table 1 and Figure 2.
Table 1 and Figure 2 indicate the risk-return profiles of ten stocks of HSE.
Table 1.Historical risk-return profile of different sectors
(1st, 2011 to January 31st, 2012)
No.
1
2
3
4
5
6
7
8
9
10
Stocks Risk (%)
BVH
DPM
EIB
FPT
HAG
MSN
SSI
STB
VIC
VNM
52.61%
35.56%
18.52%
29.26%
41.33%
48.04%
40.62%
25.00%
42.67%
27.32%
Return (%)
10%
12%
15%
15%
16%
15%
18%
20%
15%
25%
8
9. 30%
VNM
25%
STB
Return
20%
HAG
FPT
EIB
15%
SSI
VIC
10%
MSN
DPM
BVH
5%
0%
0%
10%
20%
30%
40%
50%
60%
Risk
Figure 1. Scatter plot of risk-return profile of different sector
(1st, 2011 to January 31st, 2012)
Traditional mean variance optimization often leads to highly concentrated, undiversified
asset allocations. When developing an opportunity set, one should select non-overlapping
mutually exclusive asset classes that reflect the investors‟ investable universe. In this
project, we have presented two types of graphs – efficient frontier graphs and efficient
frontier asset allocation area graphs. Efficient frontier displays returns on the vertical axis
and the risk (standard deviation) of returns on the horizontal axis. Efficient frontier is the
locus of points, which represents the different combination of risk and return on an
efficient asset allocation, where an efficient asset allocation is one that maximizes return
per unit of risk. This is presented in Figure 3.
35%
Assets
Implied_EF
30%
Return
25%
20%
15%
10%
5%
0%
10%
20%
30%
40%
50%
60%
Risk
Figure 2: Efficient frontier, historical return versus risk.
9
10. Efficient frontier asset allocation area graphs complement the efficient frontier graphs.
They display the asset allocations of the efficient frontier across the entire risk spectrum.
Efficient frontier area graphs display risk on the horizontal axis. The efficient frontier
area graph displays all the asset allocation on the efficient frontier. This is helpful to
visualize the efficient frontier graphs and the efficient frontier asset allocation area graphs
together because one can simultaneously see the asset allocations associated with the
respective risk-return point on the efficient frontier, and vice versa.
To avoid the limitation of efficient frontiers based on historical data leads to highly
concentrated portfolios in the mean variance approach of Markowitz‟ s theory, the BlackLitterman model (1992) proposed a better solution. This was further researched and
emphasized by Von Neumann, Morgenstern and James Tobin. A rich literature on this
was well documented by Sharpe (1964, 1974), respectively. The pivotal point of BlackLitterman model is implied returns. Implied returns (otherwise known as equilibrium
returns) are the set of sectoral indices returns that clear the market if all investors have
identical views. This means the market follows the strong form efficiency of the efficient
market hypothesis or leads to a perfect competitive market. To compute the equilibrium
returns of the sectoral indices, we need an input parameter, that is, risk aversion
coefficient. The risk aversion coefficient characterizes the risk-return trade off. Risk
aversion coefficient is the ratio of risk-return and variance of the benchmark portfolio.
The mathematical representation of risk aversion coefficient (denoted by λ) is as follows:
=
−
where,
is the return on benchmark;
, the risk free rate, and
, is the variance of the benchmark.
This project considered HSE as the benchmark index to compute the risk aversion
coefficient. We have considered the risk free rate to be 8%. By computing the ratio of
risk premium and variance of HSE, we have calculated the risk aversion coefficient (λ) at
4.2%. The risk aversion coefficient characterizes the risk return trade off. From the daily
return series of stocks, we have generated the covariance matrix. This is represented in
Table 2
10
12. No.
1
2
3
4
5
6
7
8
9
10
Stocks
BVH
DPM
EIB
FPT
HAG
MSN
SSI
STB
VIC
VNM
Risk (%)
52.61%
35.56%
18.52%
29.26%
41.33%
48.04%
40.62%
25.00%
42.67%
27.32%
Total implied
return* (%)
44.84%
24.51%
5.25%
12.84%
27.04%
42.38%
22.22%
3.13%
21.51%
12.97%
Table 4.Implied return (∏ = λ∑
) of stocks - risk profile
(January 1st, 2011 to January 31st, 2012).
*Total implied return = implied excess return + risk free rate
After generating the implied return and risk of the stocks, we have generated the
optimized portfolio efficient frontier. Here, it is understood that implied returns are
considered as the E[R] of the respective stocks.
These implied returns are the starting point for the Black-Litterman model. However, it
has been observed that most investors stop thinking beyond this point while selecting the
optimal portfolio. If investors or market participants do not agree with implied returns,
the Black-Litterman model provides an effective framework for combining the implied
returns with the investor’s unique views or perception regarding the markets, which result
in well diversified portfolios reflecting their views.
To implement the Black-Litterman approach, an asset manager has to express his or her
views in terms of probability distribution. Black-Litterman assumes that the investor has
two kinds of views absolute and relative. For now, we assume that the investor has k
different views on linear combinations of E[R] of the n assets. This is explained in details
as an equation (Equation 1) in the methodology section.
In this project, we have considered the combination of one absolute and one relative view
on list of our stocks. These views are expressed as follows:
Absolute view
View 1
VNM will generate an absolute return of 10%.
Relative view
12
13. Views 2
MSN outperform HAG by 8%.
These two views are expressed as follows:
µVNM = 0.1
strong view:
= 0.0019
µMSN - µHAG= 0.08 weaker view:
Thus P =
0 0
0 0
0
0
0
0 0 0 0
0 −1 1 0 0
0
0
= 0.0065
1
0.1
0.0019
,q=
and Ω =
0
0.08
0
0
0.0065
Applying formula (1) to compute E[R], we get
E[R]
BVH
DPM
EIB
43.56% 23.97% 5.25%
FPT
HAG
MSN
SSI
STB
12.55% 27.77% 39.43% 22.01% 3.04%
VIC
VNM
21.60% 11.46%
Set up the quadratic problems for portfolion optimization:
min
¸
μ x≥R
Ax = 1
x≥0
where,
x: weight vector of portfolio
H: covariance matrix of our stocks
μ: new combined return vector
R: expected return contraint of portfolio
A: unity vector
13
15. Figure 3. Efficient Frontier and the Composition of Efficient Portfolios
using the Black-Litterman approach
Extension part: comparision of two approaches
Figure 4 plots the efficient frontier generated by implied return and Black Litterman
return. It can be concluded that Black-Litterman model provides the optimal portfolio
with maximum return and minimum risk in comparison to implied return based and mean
variance based portfolio optimization.
40%
35%
30%
Return
25%
20%
15%
10%
Implied_EF
5%
Black-Litterman EF
0%
10%
15%
20%
25%
30%
35%
40%
Risk
Figure 4. Efficient Frontier: Black-Litterman versus implied return.
15
16. REFERENCES
[1]
The intuition behind Black-Litterman model portfolios - Guangliang He,
[2]
A step-by-step guide to the Black-Litterman model - Thomas M. Idzorek (2005),
[3]
Exercises in Advanced Risk and Portfolio Management – A. Meucci (with code)
[4]
Optimization Methods in Finance - Gerard Cornuejols (2005),
[5]
An equilibrium approach for tactical asset allocation: Assessing Black-Litterman
model to Indian stock market - Alok Kumar Mishra (2011),
16