Investment Management & Stock Market
Risk Management and Prediction Examples
Chapter 1: Investment Management Risk AnalysisIn Book 7: Risk Analysis and Prediction in Finance and Insurance
ISBN: 978-620-3-02745-7
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter presents a Risk Analysis to Investment Management for portfolio selection to achieve investment objectives controlling risk. Stochastic optimisation and Monte Carlo simulation are used to perform the risk analysis. Optimisation constructs the Efficient Frontier of optimal portfolios with an expected return in a predefined range with a determined increment. The simulation calculates and measures the portfolio return, Standard Deviation, Value at Risk (VAR), Sharpe Ratio, and Beta of Efficient Frontier portfolios; This analysis provides for the selection of the best Efficient Frontier portfolio with maximum Sharpe Ratio. Simulation sensitivity analysis identifies the riskiest asset.
Keywords: Risk Analysis, Investment Management, Market Risk, What-If Analysis; Sensitivity Analysis; Monte Carlo simulation; Optimisation; Stochastic model.
The Results
The simulation is run and the results are presented below. It is assumed that Asset Allocation and Selection are completed and Fund 1, Fund 2, Fund 3, and Fund 4 are selected. The Monthly Return (MR) of the four selected assets is available for the period from January 2009 to October 2015 (Ref. Appendix). Stochastic optimisation is used to resolve the Efficient Frontier of minimal mean-variance portfolios, which yields sufficient return to cover the liabilities and gain a maximum surplus with minimal risk.
The stochastic optimisation model is minimising the mean-variance of the portfolios subject to the following specific constraints: a) the expected portfolio return is from 9.1% to 9.5% in 0.1% increments, i.e., five Efficient Frontier portfolios; b) All the money, i.e., 100% of the available funds, are invested; and c) No short selling is allowed so all the fractions of the capital placed in each asset should be non-negative. Table 1 shows the Mean Return (µ) and the investment fractions of total funds invested in each asset.
Table 1: The Efficient Frontier Portfolios’ Investment Fractions
| µ | Fund 1 | Fund 2 | Fund 3 | Fund 4 |
| 0.091 | 0.29440 | 0.40323 | 0.30237 | 0 |
| 0.092 | 0.30565 | 0.37249 | 0.32186 | 0 |
| 0.093 | 0.31689 | 0.34175 | 0.34136 | 0 |
| 0.094 | 0.32814 | 0.31101 | 0.36085 | 0 |
| 0.095 | 0.33938 | 0.28027 | 0.38035 | 0 |
Table 2 shows the Mean Return (µ), Variance (V), Standard Deviation (σ), VAR, Sharpe Ratio (SR), and Beta (β).
Table 2: The Efficient Frontier Portfolios’ Details
| µ | V | σ | VAR | SR | β |
| 0.091 | 0.233 | 0.473 | -0.188 | 0.0776 | 0.723 |
| 0.092 | 0.242 | 0.491 | -0.225 | 0.0784 | 0.730 |
| 0.093 | 0.252 | 0.502 | -0.291 | 0.0787 | 0.740 |
| 0.094 | 0.267 | 0.517 | -0.375 | 0.0784 | 0.756 |
| 0.095 | 0.293 | 0.541 | -0.491 | 0.0767 | 0.777 |
The Efficient Frontier curve shows that an increase in the expected return of the portfolio causes an increase in the portfolio Standard Deviation. To emphasise, the Efficient Frontier gets flattered as expected. This shows that each additional unit of Standard Deviation (i.e., risk) allowed, increases the portfolio Mean Return by less and less. That is approximately the following points: 1. Main Return of 0.091 with Standard Deviation of 0.483; 1. Main Return of 0.091 with Standard Deviation of 0.483; 2. Main Return of 0.092 with Standard Deviation of 0.491; 3. Main Return of 0.093 with Standard Deviation of 0.502; 4. Main Return of 0.094 with Standard Deviation of 0.509; 5. Main Return of 0.095 with Standard Deviation of 0.542;
Chapter 3: Market Risk Analysis with Correlated Assets
In Book 7: Risk Analysis and Prediction in Finance and Insurance
ISBN: 978-620-3-02745-7
Author: Vojo Bubevski (Independent Researcher)
Abstract
A market risk analysis in a portfolio selection of correlated assets is presented in this chapter. The chapter elaborates on how to construct and select an optimal portfolio of correlated assets in order to control Value at Risk (VAR). Stochastic optimisation is used to construct the efficient frontier of minimal mean-variance investment portfolios with maximal return and minimal acceptable risk. Monte Carlo simulation is utilised to stochastically calculate and measure the Portfolio Return, VAR, and Sharpe Ratio of the efficient frontier portfolios. Simulation results are analysed and the optimal portfolio is selected from the Efficient Frontier based on the criteria of maximum Sharpe Ratio.
Keywords: Risk Analysis, Investment Management, Market Risk, What-If Analysis; Sensitivity Analysis; Monte Carlo simulation; Optimisation; Stochastic model.
The Results
The stochastic optimisation model resolved the Efficient Frontier of two optimal portfolios. The simulation calculated the Mean Return, Standard Deviation, Variance, VAR, and Sharpe Ratio of the portfolios. Table 1 shows the Mean Return (µ), Variance (V), Standard Deviation (σ), VAR, and Sharpe Ratio (SR) of the Efficient Frontier’s two optimal portfolios.
Table 1: The Efficient Frontier Portfolios’ Results
| µ | µ | σ | VAR | SR |
| Portfolio 1 | 0.26076 | 0.45993 | -0.3248 | 0.26877 |
| Portfolio 2 | 0.15018 | 0.19305 | -0.1364 | 0.39980 |
Considering the Sharpe Ratio criterion, Portfolio 2 is the better portfolio that should be selected for execution. Also, the associated VAR and Standard Deviation are smaller, which confirms that the major risk factors are smaller. Thus, this is a better portfolio from the risk perspective.
It is established that considering the Sharpe Ratio criterion, Portfolio 2 is the better portfolio that should be selected for execution. Also, the associated VAR and Standard Deviation are smaller, which confirms that the major risk factors are smaller. Thus, this is a better portfolio from the risk perspective. Therefore, Portfolio 2 is selected for execution on the market, which is presented below.
The simulation calculated the Portfolio Return. The Return Mean is 0.1502 (i.e., 15.02%) with a Standard Deviation of 0.1933. There is a 5.0% probability that the Return will be below -0.136; a 90.0% probability that it will be in the range -0.136 – 0.494; and a 5.0% probability that it will be above 0.494.
Chapter 8: Forecasting Stock Market Direction
In Book 7: Risk Analysis and Prediction in Finance and Insurance
ISBN: 978-620-3-02745-7
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates the prediction of the stock market direction. A Neural Networks tool is used to predict unknown values of categorical dependent variables from known values of numeric and categorical independent variables. The purpose of a neural net is to predict the direction of the market in terms of Up or Down, based on monthly data on selected economic indicators. The stock market is notoriously difficult to predict. Some researchers defend the “random walk” hypothesis, which implies that such predictions are impossible. If successful predictions are possible, the selection of training data is crucial.
Keywords: Prediction, Risk Analysis, Stock Market, Market Up or Down Direction, Neural Networks.
The Results
Prediction Model results are presented in Table 1.
Table 1: Prediction Results for 12 Months in 2020
| Months in 2020 | Tag Used | Direction Prediction | Prediction Probability | Down Probability | Up Probability |
| 1 | predict | UP | 66.05% | 33.95% | 66.05% |
| 2 | predict | UP | 65.79% | 34.21% | 65.79% |
| 3 | predict | UP | 66.62% | 33.38% | 66.62% |
| 4 | predict | UP | 66.26% | 33.74% | 66.26% |
| 5 | predict | UP | 68.07% | 31.93% | 68.07% |
| 6 | predict | UP | 67.57% | 32.43% | 67.57% |
| 7 | predict | UP | 67.12% | 32.88% | 67.12% |
| 8 | predict | UP | 67.48% | 32.52% | 67.48% |
| 9 | predict | UP | 67.11% | 32.89% | 67.11% |
| 10 | predict | UP | 66.94% | 33.06% | 66.94% |
| 11 | predict | UP | 66.61% | 33.39% | 66.61% |
| 12 | predict | UP | 67.86% | 32.14% | 67.86% |
The prediction results are as follows. The Direction Prediction is “Up” for all twelve months in 2020 with associated reasonably high probability from 65.79% in Month 2, to 68.07% in Month 5. The “Down” probability is from 31.93% in Month 5 to 34.21% in Month 2, which is circa two times lower than the “Up” probability. The outcome is that the risk factor for the “Up” direction is circa two times smaller than the risk factor for the “Down” direction. This prediction is positive and will encourage investors to invest in the market in 2020.
Chapter 1: Risk Analysis of Asset Liability Management
In Book 9: Miscellaneous Risk Analysis
ISBN: 978-620-3-20060-7
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter presents a Risk Analysis of Asset Liability Management (ALM) that applies a stochastic DMAIC method. The presented stochastic models are suitable for ALM risk modelling under Solvency II and Basel III. This method combines Monte Carlo Simulation, Optimisation, and Six Sigma Define, Measure, Analyse, Improve, and Control (DMAIC) methodology. The method determines an optimally diversified minimal variance investment portfolio, which gains the desired range of return with minimal financial risk. Simulation and optimisation are conventionally applied to find the optimal portfolio to provide the required return. Also, the Six Sigma DMAIC methodology is used to measure and improve the portfolio management process to establish an optimally diversified portfolio. Applying Six Sigma DMAIC to the portfolio management process is an improvement in comparison with conventional stochastic ALM risk models. It offers financial institutions internal model options for Basel III and Solvency II, which can help them to reduce their capital requirements and Value-at-Risk (VAR) providing for higher business capabilities and increasing their competitive position, which is their ultimate objective.
Keywords: Risk Analysis, Asset Liability Management, Portfolio Optimisation – Minimal Variance, Financial Risk, What-If Analysis, Sensitivity Analysis, Monte Carlo simulation; Stochastic model, Six Sigma DMAIC; Basel III; Solvency II.
The Results
Stock 4 is replaced with Option 4 (i.e., the option on Stock 4). Thus, the investment fractions for this simulation model are In Stock 1, 28.6%; 0.7% in Stock 2; 28.5% in Stock 3; and 42.2% in Option 4.
The results comparison of the Initial Optimal Portfolio vs. Hedged Portfolio is shown in Table 5.
Table 5: Initial vs. Hedged Portfolio Results
| Portfolio | Mean Return | Standard Deviation | VAR |
| Initial | 0.0900 | 0.4758 | -0.2304 |
| Hedged | 0.0901 | 0.3848 | -0.0045 |
The hedged optimal portfolio is significantly better than the initial optimal portfolio. The mean return is 9% for both initial and hedged portfolios but the financial risk is considerably reduced by the hedged portfolio. That is i) Standard Deviation was reduced from 47.58% to 38.48%; ii) VAR was reduced from 23.04% to only 0.45%. These figures strongly suggest that the financial risk of the Hedged Portfolio is significantly reduced.
The simulation calculated the Hedged Portfolio Return. The Hedged Portfolio Return Mean is 0.0901 (9.01%) with a Standard Deviation of 0.3848 (38.48%). There is a 39.7% probability that the Mean will be negative; a 55.3% probability that it will be in the range of 0.0 – 0.707; and a 5.0% probability that it will be above 0.707.
Chapter 5: Risk Analysis of Investment Management
In Book 9: Miscellaneous Risk Analysis
ISBN: 978-620-3-20060-7
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter presents a Risk Analysis of Investment Management that applies a stochastic DMAIC method. The presented stochastic models are suitable for Investment risk modelling under Solvency II and Basel III. This method combines Monte Carlo Simulation, Optimisation, and Six Sigma Define, Measure, Analyse, Improve, and Control (DMAIC) methodology. The method determines an optimally diversified minimal variance investment portfolio, which gains the desired range of return with minimal financial risk. Simulation and optimisation are conventionally applied to find the optimal portfolio to provide the required return. Also, the Six Sigma DMAIC methodology is used to measure and improve the portfolio management process to establish an optimally diversified portfolio. Applying Six Sigma DMAIC to the portfolio management process is an improvement in comparison with conventional stochastic Investment risk models. It offers financial institutions internal model options for Basel III and Solvency II, which can help them to reduce their capital requirements and Value-at-Risk (VAR) providing for higher business capabilities and increasing their competitive position, which is their ultimate objective.
Keywords: Risk Analysis, Investment Management, Portfolio Optimisation – Minimal Variance, Financial Risk, What-If Analysis, Sensitivity Analysis, Monte Carlo simulation; Stochastic model, Six Sigma DMAIC; Basel III; Solvency II.
The Results
The overall results of all the portfolios on the Efficient Frontier are presented in Table 7 including the initial portfolios. Also, the associated hedged portfolios are shown in Blue-Italic. The table includes the Mean Return (µ), Variance (V), Standard Deviation (σ), VAR, Sharpe Ratio (SR), and Beta (β) of the optimal portfolios.
Table 7: The Efficient Frontier Portfolios
| µ | V | σ | VAR | SR | β |
| 0.088 | 0.223 | 0.472 | -0.204 | 0.073 | 0.723 |
| 0.088 | 0.174 | 0.417 | -0.036 | 0.083 | 0.655 |
| 0.090 | 0.232 | 0.482 | -0.228 | 0.076 | 0.730 |
| 0.090 | 0.181 | 0.425 | -0.066 | 0.086 | 0.661 |
| 0.092 | 0.247 | 0.497 | -0.283 | 0.077 | 0.740 |
| 0.092 | 0.193 | 0.439 | -0.124 | 0.088 | 0.671 |
| 0.094 | 0.266 | 0.515 | -0.362 | 0.079 | 0.756 |
| 0.094 | 0.217 | 0.466 | -0.212 | 0.087 | 0.691 |
| 0.096 | 0.295 | 0.543 | -0.454 | 0.078 | 0.777 |
| 0.096 | 0.252 | 0.502 | -0.326 | 0.085 | 0.724 |
| 0.098 | 0.332 | 0.576 | -0.537 | 0.077 | 0.798 |
| 0.098 | 0.298 | 0.546 | -0.446 | 0.082 | 0.761 |
| 0.100 | 0.378 | 0.615 | -0.625 | 0.076 | 0.820 |
| 0.100 | 0.348 | 0.590 | -0.565 | 0.079 | 0.799 |
| 0.102 | 0.429 | 0.655 | -0.732 | 0.074 | 0.848 |
| 0.102 | 0.410 | 0.640 | -0.687 | 0.076 | 0.840 |
The Efficient Frontier of the optimal portfolios is resolved. The Efficient Frontier shows that an increase in the expected return of the portfolio causes an increase in the portfolio Standard Deviation. Also, the Efficient Frontier gets flattered as expected. This shows that each additional unit of Standard Deviation allowed, increases the portfolio Mean Return by less and less. To emphasise, the Standard Deviation (i.e., the risk) of Hedged portfolios for a given return is less than the risk of the original portfolios. Also, the risk of Hedged portfolios asymptotically converges with the risk of the original portfolios for higher returns.
Chapter 5: Predicting Stock Market Direction
In Book 12: Operations Research Applications for Decision Analysis and Prediction
ISBN: 978-620-5-49772-2
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter illustrates the prediction of the stock market direction. A Neural Networks tool is used to predict unknown values of categorical dependent variables from known values of numeric and categorical independent variables. The purpose of a neural net is to predict the direction of the market in terms of Up or Down, based on monthly data on selected economic indicators. The stock market is notoriously difficult to predict. Some researchers defend the “random walk” hypothesis, which implies that such predictions are impossible. If successful predictions are possible, the selection of training data is crucial.
Keywords: Prediction, Risk Analysis, Stock Market, Market Up or Down Direction, Neural Networks.
The Results
Table 1: Prediction Results for 12 Months in 2020
| Months in 2020 | Tag Used | Direction Prediction | Prediction Probability | Down Probability | Up Probability |
| 1 | predict | UP | 66.05% | 33.95% | 66.05% |
| 2 | predict | UP | 65.79% | 34.21% | 65.79% |
| 3 | predict | UP | 66.62% | 33.38% | 66.62% |
| 4 | predict | UP | 66.26% | 33.74% | 66.26% |
| 5 | predict | UP | 68.07% | 31.93% | 68.07% |
| 6 | predict | UP | 67.57% | 32.43% | 67.57% |
| 7 | predict | UP | 67.12% | 32.88% | 67.12% |
| 8 | predict | UP | 67.48% | 32.52% | 67.48% |
| 9 | predict | UP | 67.11% | 32.89% | 67.11% |
| 10 | predict | UP | 66.94% | 33.06% | 66.94% |
| 11 | predict | UP | 66.61% | 33.39% | 66.61% |
| 12 | predict | UP | 67.86% | 32.14% | 67.86% |
Prediction Model results are presented in Table 1. The prediction results are as follows. The Direction Prediction is “Up” for all twelve months in 2020 with an associated reasonably high probability from 65.79% in Month 2, to 68.07% in Month 5. The “Down” probability is from 31.93% in Month 5 to 34.21% in Month 2, which is circa two times lower than the “Up” probability. The outcome is that the risk factor for the “Up” direction is circa two times smaller than the risk factor for the “Down” direction. This prediction is positive and will encourage investors to invest in the market in 2020.
Chapter 2: Comprehensive Investment Risk Assessment
In Book 13: Financial Risk Assessment and Management with Six Sigma DMAIC Methods
ISBN: 978-620-5-49700-5
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter presents the Six Sigma DMAIC application to improve Risk Management in Investment Portfolios. The objective is to select the optimal portfolio which achieves investment objectives with acceptable risk. Optimisation constructs an Efficient Frontier of optimal portfolios with an expected return in a predefined range with determined increment. Simulation stochastically calculates and measures Expected Return, Variance, Standard Deviation, and the major risk factors. Six Sigma capability metrics of Expected Return, Value at Risk, Sharpe Ratio, and Beta are calculated versus desired and specified limits. The analysis allows for the selection of the best Efficient Frontier portfolio with maximum Sharpe Ratio. Simulation sensitivity analysis identifies the riskiest asset. Portfolio revision considers options to improve the portfolio and replaces the asset with an option to reduce risk. Portfolio execution implements the revised portfolio. Ongoing portfolio management evaluates portfolio performance on a regular basis and if required, revises the portfolio considering changes in the market and the investor’s position.
Keywords: Investment Management, Market Risk Assessment, Basel III, Six Sigma DMAIC, Stochastic Optimisation, Monte Carlo Simulation
The Results
The overall results of all the portfolios on the Efficient Frontier including the initial and the associated revised portfolios (shown in Blue) are presented in Table 5 showing the Mean Return (µ), Variance (V), Standard Deviation (σ), VAR, Sharpe Ratio (SR) and Beta (β) of the optimal portfolios.
Table 5: The overall results
| µ | V | σ | VAR | SR | β |
| 0.088 | 0.223 | 0.472 | -0.204 | 0.073 | 0.723 |
| 0.088 | 0.174 | 0.417 | -0.036 | 0.083 | 0.655 |
| 0.090 | 0.232 | 0.482 | -0.228 | 0.076 | 0.730 |
| 0.090 | 0.181 | 0.425 | -0.066 | 0.086 | 0.661 |
| 0.092 | 0.247 | 0.497 | -0.283 | 0.077 | 0.740 |
| 0.092 | 0.193 | 0.439 | -0.124 | 0.088 | 0.671 |
| 0.094 | 0.266 | 0.515 | -0.362 | 0.079 | 0.756 |
| 0.094 | 0.217 | 0.466 | -0.212 | 0.087 | 0.691 |
| 0.096 | 0.295 | 0.543 | -0.454 | 0.078 | 0.777 |
| 0.096 | 0.252 | 0.502 | -0.326 | 0.085 | 0.724 |
| 0.098 | 0.332 | 0.576 | -0.537 | 0.077 | 0.798 |
| 0.098 | 0.298 | 0.546 | -0.446 | 0.082 | 0.761 |
| 0.100 | 0.378 | 0.615 | -0.625 | 0.076 | 0.820 |
| 0.100 | 0.348 | 0.590 | -0.565 | 0.079 | 0.799 |
| 0.102 | 0.429 | 0.655 | -0.732 | 0.074 | 0.848 |
| 0.102 | 0.410 | 0.640 | -0.687 | 0.076 | 0.840 |
Chapter 4: Price Evolution with Markov Chain Monte Carlo
In Book 13: Financial Risk Assessment and Management with Six Sigma DMAIC Methods
ISBN: 978-620-5-49700-5
Author: Vojo Bubevski (Independent Researcher)
Abstract
This chapter presents a generic model for Price Evolution with the Markov Chain Monte Carlo (MCMC) method. The Markov Chain Monte Carlo is a stochastic simulation process observed through a time where the probability distribution of the next stage of the process, given the current state, is independent of the past states. The Markov Chain simulation model is governed by the initial state at time zero and a one-step transition probability matrix. Each row of this matrix shows the probability distribution of the next state, given the current state for that row. An additional model illustrates the evolution of prices through time. It assumes that there is an underlying process, such as the state of the economy, that follows a Markov Chain with three states. Then for any given state, the price change is simulated with a probability distribution with parameters that depend on the state. Three simulations generate states and prices through time, starting with a given price, i.e., one simulation for any of the three possible starting states. The presented model is generic and applicable to all sorts of prices, including stock market prices, prices of goods in marketing, etc.
Keywords: Risk & Decision Analysis, Price Evolution, Markov Chain Monte Carlo, Financial Risk, What-If Analysis; Sensitivity Analysis; Monte Carlo simulation; Stochastic model.
Simulation 1 probability distribution of Price at Time 45 is calculated. Price at Time 45 Mean is $1,882.96 with a Standard Deviation of $1,305.07. There is a 5.0% probability that the Mean will be below $473; a 90.0% probability that it will be in the range of $473 – $4,254; and a 5.0% probability that it will be above $4,254.