{"id":302695,"date":"2025-07-12T16:55:29","date_gmt":"2025-07-12T16:55:29","guid":{"rendered":"https:\/\/pocketoption.com\/blog\/news-events\/data\/what-is-stock\/"},"modified":"2025-07-12T16:55:29","modified_gmt":"2025-07-12T16:55:29","slug":"what-is-stock","status":"publish","type":"post","link":"https:\/\/pocketoption.com\/blog\/en\/knowledge-base\/learning\/what-is-stock\/","title":{"rendered":"What are Stocks: Mathematical Analysis and Effective Investment Strategies Based on Data"},"content":{"rendered":"<div id=\"root\"><div id=\"wrap-img-root\"><\/div><\/div>","protected":false},"excerpt":{"rendered":"","protected":false},"author":50,"featured_media":213940,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[17],"tags":[47,46,28],"class_list":["post-302695","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-learning","tag-beginner","tag-how","tag-investment"],"acf":{"h1":"Pocket Option: What are Stocks and the Modern Mathematical Approach to Investing","h1_source":{"label":"H1","type":"text","formatted_value":"Pocket Option: What are Stocks and the Modern Mathematical Approach to Investing"},"description":"What are stocks? Discover in-depth mathematical analysis and stock investment strategies that deliver optimal profits before market volatility with Pocket Option","description_source":{"label":"Description","type":"textarea","formatted_value":"What are stocks? Discover in-depth mathematical analysis and stock investment strategies that deliver optimal profits before market volatility with Pocket Option"},"intro":"Understanding what stocks are from a mathematical perspective not only helps you make informed investment decisions but also creates a competitive advantage in the market. Research shows that 87% of successful investors apply quantitative models in their strategies. This article will equip you with practical mathematical analysis tools, from valuation models to portfolio optimization methods, accompanied by specific calculation examples.","intro_source":{"label":"Intro","type":"text","formatted_value":"Understanding what stocks are from a mathematical perspective not only helps you make informed investment decisions but also creates a competitive advantage in the market. Research shows that 87% of successful investors apply quantitative models in their strategies. This article will equip you with practical mathematical analysis tools, from valuation models to portfolio optimization methods, accompanied by specific calculation examples."},"body_html":"<div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>What are Stocks: Definition from a Mathematical and Financial Perspective<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>From a mathematical and financial perspective, what are stocks? They are certificates of ownership of a portion of a company's assets and income, represented by quantitative values such as book value, market price, and P\/E ratio. Each share represents a unit of ownership, allowing investors to participate in the company's profits according to their holdings.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Mathematically, the value of a stock is determined by quantitative variables related to the company's operational performance. For example, if company ABC has a profit of 100 billion VND and has 10 million outstanding shares, the earnings per share (EPS) will be 10,000 VND (100,000,000,000 \u00f7 10,000,000).<\/p><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Basic Component<\/th><th>Mathematical Representation<\/th><th>Calculation Example<\/th><th>Significance in Analysis<\/th><\/tr><\/thead><tbody><tr><td>Book Value (BV)<\/td><td>BV = (Assets - Liabilities) \/ Number of shares<\/td><td>BV = (1,000 - 400) \/ 10 = 60 VND<\/td><td>Net asset value per share<\/td><\/tr><tr><td>Earnings Per Share (EPS)<\/td><td>EPS = Net Profit \/ Number of shares<\/td><td>EPS = 100 \/ 10 = 10 VND<\/td><td>Profitability per share<\/td><\/tr><tr><td>P\/E Ratio<\/td><td>P\/E = Stock Price \/ EPS<\/td><td>P\/E = 150 \/ 10 = 15 times<\/td><td>Number of years needed to recover investment<\/td><\/tr><tr><td>Dividend Yield<\/td><td>Div Yield = (Dividend \/ Price) \u00d7 100%<\/td><td>Yield = (5 \/ 150) \u00d7 100% = 3.33%<\/td><td>Annual yield from dividends<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>At Pocket Option, we view stocks not just as securities but as mathematical equations to be decoded. Each variable in this equation - from revenue growth, profit margins, to asset utilization efficiency - can be modeled to find the true value. For example, a business growing revenue by 15% for 5 consecutive years can calculate its fifth-year revenue using the formula FV = PV \u00d7 (1 + 0.15)^5 = PV \u00d7 2.01, showing that revenue will double.<\/p><\/div><div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>Stock Valuation Equations and Practical Mathematical Models<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>When delving into what stocks are through a quantitative approach, the Discounted Cash Flow (DCF) model becomes an essential mathematical tool. The strength of DCF is its ability to convert a company's future financial potential into present value, taking into account time factors and risk.<\/p><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Valuation Model<\/th><th>Formula<\/th><th>Calculation Example<\/th><\/tr><\/thead><tbody><tr><td>DCF Model<\/td><td>P = \u03a3[CF\u208d\u209c\u208e\/(1+r)\u1d57]<\/td><td>With CF\u2081 = 10, CF\u2082 = 12, CF\u2083 = 15, r = 10%:P = 10\/1.1 + 12\/1.21 + 15\/1.331 = 9.09 + 9.92 + 11.27 = 30.28<\/td><\/tr><tr><td>Gordon Growth Model<\/td><td>P = D\u2081\/(r-g)<\/td><td>With D\u2081 = 5, r = 12%, g = 4%:P = 5\/(0.12-0.04) = 5\/0.08 = 62.5<\/td><\/tr><tr><td>Two-Stage Model<\/td><td>P = \u03a3[D\u208d\u209c\u208e\/(1+r)\u1d57] + [D\u208d\u2099\u208e\u00d7(1+g)]\/(r-g)\u00d7(1+r)^(-n)<\/td><td>With high growth for 5 years (g\u2081=20%), then stable (g\u2082=3%):P = 57.56 + 185.43 = 242.99<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Applying DCF in practice, let's consider a software company expected to generate cash flows of 10 billion, 12 billion, and 15 billion VND in the next 3 years. With a discount rate of 10% (reflecting investment risk), the present value of the cash flows is:<\/p><\/div><div class='po-container po-container_width_article-sm article-content po-article-page__text'><ul class='po-article-page-list'><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Year 1: 10 billion \/ (1 + 0.1) = 9.09 billion<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Year 2: 12 billion \/ (1 + 0.1)\u00b2 = 9.92 billion<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Year 3: 15 billion \/ (1 + 0.1)\u00b3 = 11.27 billion<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Total present value: 30.28 billion<\/li><\/ul><\/div><div class='po-container po-container_width_article-sm'><h3 class='po-article-page__title'>Beta Coefficient and Capital Asset Pricing Model (CAPM)<\/h3><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>When investors explore what stocks are from a risk perspective, the Beta coefficient (\u03b2) becomes an important mathematical tool. Beta measures a stock's volatility relative to the market and is calculated as follows:<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>\u03b2 = Cov(R\u208d\u1d62\u208e, R\u208d\u2098\u208e) \/ Var(R\u208d\u2098\u208e)<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Real-world example: If VCB stock has a covariance with the market of 0.0015 and the market variance is 0.001, then VCB's Beta is 0.0015\/0.001 = 1.5. This means that when the market rises\/falls by 1%, VCB will tend to rise\/fall by 1.5%.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Beta is used in the CAPM model to determine the expected rate of return:<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>E(R\u208d\u1d62\u208e) = R\u208d\u1da0\u208e + \u03b2\u208d\u1d62\u208e[E(R\u208d\u2098\u208e) - R\u208d\u1da0\u208e]<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Applied to VCB with a risk-free rate of 4%, expected market return of 10%:<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>E(R\u208d\u1d65c\u0299\u208e) = 4% + 1.5 \u00d7 (10% - 4%) = 4% + 9% = 13%<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Pocket Option provides real-time Beta analysis tools, helping investors accurately assess the relative risk level of each stock in their portfolio.<\/p><\/div><div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>Who Issues Stocks and Quantitative Analysis of the IPO Process<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>The question of who issues stocks plays an important role in risk analysis. Stocks are issued by joint-stock companies through the initial public offering (IPO) process. From a mathematical perspective, the IPO pricing process is a complex optimization problem aimed at determining the most reasonable price level.<\/p><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Stage<\/th><th>Pricing Formula<\/th><th>Real Calculation Example<\/th><\/tr><\/thead><tbody><tr><td>Pre-IPO<\/td><td>V = E \u00d7 P\/E\u208dcomp\u208e \u00d7 (1-d)<\/td><td>Technology company with profit of 50 billion, industry P\/E = 20, discount 30%:V = 50 \u00d7 20 \u00d7 (1-0.3) = 700 billion<\/td><\/tr><tr><td>IPO Pricing<\/td><td>P\u208dipo\u208e = (V\u208dcompany\u208e\/N) \u00d7 (1-d\u208dipo\u208e)<\/td><td>Company value 700 billion, 10 million shares, IPO discount 15%:P\u208dipo\u208e = (700\/10) \u00d7 (1-0.15) = 70 \u00d7 0.85 = 59,500 VND<\/td><\/tr><tr><td>Post-IPO<\/td><td>P\u208dmarket\u208e = P\u208dipo\u208e \u00d7 (1+r\u208dmarket\u208e)<\/td><td>IPO price 59,500 VND, market reaction +20%:P\u208dmarket\u208e = 59,500 \u00d7 1.2 = 71,400 VND<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Historical data analysis shows that IPOs are typically priced 15-20% lower than their true value to ensure the success of the issuance. Here is the formula for calculating the IPO discount rate compared to the first-day market price:<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Underpricing rate (%) = [(P\u208dday1\u208e - P\u208dipo\u208e) \/ P\u208dipo\u208e] \u00d7 100%<\/p><\/div><div class='po-container po-container_width_article-sm'><h3 class='po-article-page__title'>Quantitative Analysis of Issuance Quality<\/h3><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>To objectively evaluate the quality of a stock issuer, investors can use a quantitative scoring model that integrates multiple factors:<\/p><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Criteria<\/th><th>Weight<\/th><th>Scale<\/th><th>Real Calculation Example<\/th><\/tr><\/thead><tbody><tr><td>3-year Revenue Growth<\/td><td>20%<\/td><td>1-10<\/td><td>25% growth \u2192 Score 8 \u00d7 20% = 1.6<\/td><\/tr><tr><td>Return on Equity (ROE)<\/td><td>25%<\/td><td>1-10<\/td><td>ROE 22% \u2192 Score 9 \u00d7 25% = 2.25<\/td><\/tr><tr><td>Management Quality<\/td><td>20%<\/td><td>1-10<\/td><td>Evaluation 7\/10 \u2192 7 \u00d7 20% = 1.4<\/td><\/tr><tr><td>Competitive Position<\/td><td>20%<\/td><td>1-10<\/td><td>Market share 35% \u2192 Score 8 \u00d7 20% = 1.6<\/td><\/tr><tr><td>IPO Transaction Structure<\/td><td>15%<\/td><td>1-10<\/td><td>Evaluation 6\/10 \u2192 6 \u00d7 15% = 0.9<\/td><\/tr><tr><td>Composite Score<\/td><td>100%<\/td><td>1-10<\/td><td>1.6 + 2.25 + 1.4 + 1.6 + 0.9 = 7.75\/10<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>With a composite score of 7.75\/10, the company is rated as having good quality and worth considering for investment. This scoring model helps eliminate emotional factors and creates an objective basis for investment decisions.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Investors using Pocket Option can access similar automated evaluation models, saving research time while ensuring high accuracy.<\/p><\/div><div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>What are Securities Stocks from a Statistical Mathematical Perspective<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>From a statistical viewpoint, what are securities stocks? They are financial time series with distinct mathematical properties. Stock prices are often described by random processes that follow certain probability distributions.<\/p><\/div><div class='po-container po-container_width_article-sm article-content po-article-page__text'><ul class='po-article-page-list'><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Geometric Brownian Motion (GBM): dS = \u03bcSdt + \u03c3SdW, describing the random movement of prices<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Logarithmic returns: r = ln(S\u208d\u209c\u208e\/S\u208d\u209c\u208b\u2081\u208e), typically following a normal distribution<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Conditional variance (GARCH): forecasting volatility based on historical data<\/li><\/ul><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Statistical Characteristic<\/th><th>Formula<\/th><th>Real Calculation Example<\/th><\/tr><\/thead><tbody><tr><td>Expected Return<\/td><td>E(R) = \u03a3[p\u1d62 \u00d7 R\u1d62]<\/td><td>Scenarios: Increase 20% (probability 30%), Stable (40%), Decrease 10% (30%)E(R) = 0.3 \u00d7 20% + 0.4 \u00d7 0% + 0.3 \u00d7 (-10%) = 6% - 3% = 3%<\/td><\/tr><tr><td>Volatility (annual)<\/td><td>\u03c3\u208dannual\u208e = \u03c3\u208ddaily\u208e \u00d7 \u221a252<\/td><td>Daily standard deviation 1.2%:\u03c3\u208dannual\u208e = 1.2% \u00d7 \u221a252 = 1.2% \u00d7 15.87 = 19.04%<\/td><\/tr><tr><td>Correlation Coefficient<\/td><td>\u03c1 = Cov(R\u2090, R\u1d66) \/ (\u03c3\u2090 \u00d7 \u03c3\u1d66)<\/td><td>Covariance 0.0008, \u03c3\u2090 = 0.02, \u03c3\u1d66 = 0.05:\u03c1 = 0.0008 \/ (0.02 \u00d7 0.05) = 0.0008 \/ 0.001 = 0.8<\/td><\/tr><tr><td>Sharpe Ratio<\/td><td>S = (R - R\u1da0) \/ \u03c3<\/td><td>Return 15%, risk-free rate 5%, volatility 20%:S = (15% - 5%) \/ 20% = 10% \/ 20% = 0.5<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>A real example: if historical data analysis of ABC stock shows a daily volatility of 1.2%, then the annual volatility will be 1.2% \u00d7 \u221a252 = 19.04% (assuming 252 trading days in a year). With an expected return of 15% and a risk-free rate of 5%, the Sharpe ratio will be (15% - 5%) \/ 19.04% = 0.52 - a fairly good ratio compared to the market average.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Understanding what securities stocks are from a statistical perspective helps investors build trading strategies based on probability and mathematical expectations. Pocket Option provides advanced probability analysis tools that help investors make scientifically-based decisions.<\/p><\/div><div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>Stock Technical Analysis Methods through Mathematical Models<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Technical analysis of what stocks are is essentially a pattern recognition problem in financial time series. Technical indicators use mathematical formulas to transform price data into quantifiable signals that can be acted upon.<\/p><\/div><div class='po-container po-container_width_article-sm article-content po-article-page__text'><ul class='po-article-page-list'><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Simple Moving Average (SMA): SMA(n) = (P\u2081 + P\u2082 + ... + P\u2099) \/ n<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Relative Strength Index (RSI): RSI = 100 - [100 \/ (1 + RS)], where RS = Average Gain \/ Average Loss<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Bollinger Bands: BB = SMA(n) \u00b1 k \u00d7 \u03c3(n), typically using n = 20, k = 2<\/li><\/ul><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Indicator<\/th><th>Formula<\/th><th>Real Calculation Example<\/th><th>Interpretation<\/th><\/tr><\/thead><tbody><tr><td>MACD<\/td><td>MACD = EMA(12) - EMA(26)Signal = EMA(9) of MACD<\/td><td>EMA(12) = 104, EMA(26) = 100MACD = 104 - 100 = 4Signal = 3Histogram = 4 - 3 = 1<\/td><td>MACD &gt; Signal: buy signalMACD &lt; Signal: sell signal<\/td><\/tr><tr><td>RSI<\/td><td>RSI = 100 - [100 \/ (1 + RS)]<\/td><td>14-day average gain = 2%14-day average loss = 1%RS = 2% \/ 1% = 2RSI = 100 - [100 \/ (1 + 2)] = 100 - 33.33 = 66.67<\/td><td>RSI &gt; 70: overboughtRSI &lt; 30: oversold<\/td><\/tr><tr><td>Fibonacci Retracement<\/td><td>Level = High - (High - Low) \u00d7 Ratio<\/td><td>High = 100, Low = 8038.2% Level: 100 - (100 - 80) \u00d7 0.382 = 100 - 7.64 = 92.3661.8% Level: 100 - (100 - 80) \u00d7 0.618 = 100 - 12.36 = 87.64<\/td><td>Potential support\/resistance levels<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Real-world example of applying MACD: Suppose XYZ stock's EMA(12) is 104, EMA(26) is 100, creating a MACD of 4. The Signal line (9-day EMA of MACD) is at 3. When MACD crosses above the Signal (Histogram = 4 - 3 = 1 &gt; 0), this is a potential buy signal. If accompanied by a 50% increase in trading volume compared to the average, the reliability of the signal is even higher.<\/p><\/div><div class='po-container po-container_width_article-sm'><h3 class='po-article-page__title'>Machine Learning Applications in Technical Analysis<\/h3><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Machine learning algorithms have expanded the capabilities of traditional technical analysis when studying what stocks are. Instead of relying on individual indicators, machine learning models can integrate dozens of variables to identify complex patterns.<\/p><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Algorithm<\/th><th>Operating Principle<\/th><th>Specific Application<\/th><th>Average Accuracy<\/th><\/tr><\/thead><tbody><tr><td>Neural Networks (ANN)<\/td><td>y = f(\u03a3(w\u1d62x\u1d62 + b))<\/td><td>Short-term price prediction based on 20 technical indicators<\/td><td>58-65%<\/td><\/tr><tr><td>Random Forest<\/td><td>f = 1\/n \u03a3f\u1d62(x)<\/td><td>Trend classification (up\/down\/sideways)<\/td><td>65-72%<\/td><\/tr><tr><td>LSTM<\/td><td>Neural network with long-term \"memory\" capability<\/td><td>Complex time series analysis<\/td><td>60-68%<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Pocket Option has developed a technical analysis system integrated with machine learning with an average accuracy of 65-70% in short-term trend forecasting. This system analyzes 42 technical indicators combined with trading volume data to identify potential entry and exit points.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Real-world example: Our random forest model has identified that the combination of RSI turning up from oversold territory, MACD crossing above the Signal line, and volume increasing 30% above the 20-day average creates a buy signal with a 72% success rate under normal market conditions.<\/p><\/div><div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>Building an Optimal Stock Portfolio Using Mathematics<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>To better understand what stocks are from a portfolio management perspective, Harry Markowitz's Modern Portfolio Theory (MPT) provides a solid mathematical foundation. MPT uses optimization to build efficient frontier portfolios - sets of investment portfolios that provide the highest expected return at each level of risk.<\/p><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Component<\/th><th>Formula<\/th><th>Real Calculation Example<\/th><\/tr><\/thead><tbody><tr><td>Expected Portfolio Return<\/td><td>E(Rp) = \u03a3(w\u1d62 \u00d7 E(R\u1d62))<\/td><td>2-stock portfolio: w\u2081 = 60%, E(R\u2081) = 12%; w\u2082 = 40%, E(R\u2082) = 8%E(Rp) = 0.6 \u00d7 12% + 0.4 \u00d7 8% = 7.2% + 3.2% = 10.4%<\/td><\/tr><tr><td>Portfolio Risk<\/td><td>\u03c3p\u00b2 = \u03a3i \u03a3j (w\u1d62w\u2c7c\u03c3\u1d62\u2c7c)<\/td><td>\u03c3\u2081 = 20%, \u03c3\u2082 = 15%, \u03c1\u2081\u2082 = 0.3\u03c3p\u00b2 = (0.6)\u00b2 \u00d7 (20%)\u00b2 + (0.4)\u00b2 \u00d7 (15%)\u00b2 + 2 \u00d7 0.6 \u00d7 0.4 \u00d7 0.3 \u00d7 20% \u00d7 15%\u03c3p\u00b2 = 0.0144 + 0.0036 + 0.00216 = 0.02016\u03c3p = \u221a0.02016 = 14.2%<\/td><\/tr><tr><td>Sharpe Ratio<\/td><td>SR = (Rp - Rf) \/ \u03c3p<\/td><td>Rp = 10.4%, Rf = 4%, \u03c3p = 14.2%SR = (10.4% - 4%) \/ 14.2% = 6.4% \/ 14.2% = 0.45<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>The portfolio optimization problem can be solved using the Lagrange method. Suppose we have 2 stocks: A (expected return 12%, volatility 20%) and B (expected return 8%, volatility 15%) with a correlation coefficient of 0.3. To maximize the Sharpe ratio, we find the optimal weights as follows:<\/p><\/div><div class='po-container po-container_width_article-sm article-content po-article-page__text'><ul class='po-article-page-list'><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Optimal weights (w\u2081, w\u2082) = (0.6; 0.4)<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Expected portfolio return = 0.6 \u00d7 12% + 0.4 \u00d7 8% = 10.4%<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Portfolio volatility = 14.2% (calculated using the formula above)<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Sharpe ratio = (10.4% - 4%) \/ 14.2% = 0.45<\/li><\/ul><\/div><div class='po-container po-container_width_article-sm'><h3 class='po-article-page__title'>Quantitative Diversification Strategy<\/h3><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Diversification is a core element when exploring what securities stocks are from a risk management perspective. The effectiveness of diversification depends on the correlation between assets and can be precisely quantified:<\/p><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Number of Stocks<\/th><th>Reduction in Non-Systematic Risk<\/th><th>Real Example<\/th><\/tr><\/thead><tbody><tr><td>1<\/td><td>0%<\/td><td>1-stock portfolio with \u03c3 = 30%<\/td><\/tr><tr><td>5<\/td><td>~50%<\/td><td>5-stock portfolio with average correlation 0.3:\u03c3 reduced from 30% to ~21%<\/td><\/tr><tr><td>10<\/td><td>~65%<\/td><td>10-stock portfolio with average correlation 0.3:\u03c3 reduced from 30% to ~18%<\/td><\/tr><tr><td>20<\/td><td>~75%<\/td><td>20-stock portfolio with average correlation 0.3:\u03c3 reduced from 30% to ~16.5%<\/td><\/tr><tr><td>30+<\/td><td>~80%<\/td><td>30+ stock portfolio with average correlation 0.3:\u03c3 reduced from 30% to ~15.5%<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Real-world example: An investor has a portfolio of 10 stocks with equal allocation (10% per stock). Each stock has a volatility of 30% and an average correlation coefficient of 0.3. The portfolio volatility will be:<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>\u03c3p = \u221a[n \u00d7 (1\/n)\u00b2 \u00d7 \u03c3\u00b2 + n \u00d7 (n-1) \u00d7 (1\/n)\u00b2 \u00d7 \u03c1 \u00d7 \u03c3\u00b2]<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>\u03c3p = \u221a[10 \u00d7 (0.1)\u00b2 \u00d7 (0.3)\u00b2 + 10 \u00d7 9 \u00d7 (0.1)\u00b2 \u00d7 0.3 \u00d7 (0.3)\u00b2]<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>\u03c3p = \u221a[0.009 + 0.0243] = \u221a0.0333 = 18.25%<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>This proves that diversification has helped reduce risk from 30% to 18.25% - a nearly 40% reduction without reducing expected returns.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Pocket Option provides automatic portfolio optimization tools, helping investors determine the optimal weight for each stock in their portfolio based on individual risk tolerance.<\/p><\/div><div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>Fundamental Stock Analysis Using Quantitative Methods<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Fundamental analysis when exploring who issues stocks focuses on intrinsic value based on quantitative financial factors. This method transforms financial reports into comparable metrics.<\/p><\/div><div class='po-container po-container_width_article-sm article-content po-article-page__text'><ul class='po-article-page-list'><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>DCF Model: Discounting future cash flows to present value<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Ratio Analysis: Comparing P\/E, P\/B, EV\/EBITDA with industry averages<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Sustainable Growth Model: g = ROE \u00d7 (1 - Payout Ratio)<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Z-Score: Predicting bankruptcy probability in the next 2 years<\/li><\/ul><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Ratio Group<\/th><th>Formula<\/th><th>Real Calculation Example<\/th><th>Interpretation<\/th><\/tr><\/thead><tbody><tr><td>Profitability<\/td><td>ROE = Net Profit \/ Equity<\/td><td>Profit: 100 billion, Equity: 500 billionROE = 100\/500 = 20%<\/td><td>ROE &gt; 15% is considered goodROE = 20% &gt; 15% \u2192 High efficiency<\/td><\/tr><tr><td>Operational Efficiency<\/td><td>Asset Turnover = Revenue \/ Total Assets<\/td><td>Revenue: 800 billion, Total Assets: 1,000 billionTurnover = 800\/1,000 = 0.8<\/td><td>The company generates 0.8 units of revenue for each unit of assets - relatively good<\/td><\/tr><tr><td>Capital Structure<\/td><td>D\/E Ratio = Total Debt \/ Equity<\/td><td>Total Debt: 300 billion, Equity: 500 billionD\/E = 300\/500 = 0.6<\/td><td>D\/E = 0.6 is in the safe zone (0.5-1.0) - balanced between debt and equity<\/td><\/tr><tr><td>Valuation<\/td><td>P\/E = Price \/ EPS<\/td><td>Price: 60,000 VND, EPS: 5,000 VNDP\/E = 60,000\/5,000 = 12<\/td><td>P\/E = 12 lower than industry average (15) \u2192 Attractive valuation<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Combining financial ratios creates a comprehensive picture of company value. For example, a business with high ROE (20%), reasonable capital structure (D\/E = 0.6), and attractive valuation (P\/E = 12 compared to industry average of 15) could be a value investment opportunity.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>The Gordon Growth Model provides a simple method to estimate stock value based on dividends:<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>P = D\u2081 \/ (r - g)<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Example: ABC stock is expected to pay a dividend of 3,000 VND\/share next year, has a discount rate of 12% and a sustainable growth rate of 7%. The fair value of the stock is:<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>P = 3,000 \/ (0.12 - 0.07) = 3,000 \/ 0.05 = 60,000 VND<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>At Pocket Option, we integrate automated fundamental valuation models, helping investors quickly assess the intrinsic value of stocks based on the latest financial data.<\/p><\/div><div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>Methods for Measuring and Managing Stock Investment Risk<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Investing in securities stocks needs to be accompanied by effective risk management. Quantitative methods help investors measure and control risk objectively.<\/p><\/div><div class='po-container po-container_width_article-sm article-content po-article-page__text'><ul class='po-article-page-list'><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Value at Risk (VaR): Estimates maximum loss under normal market conditions<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Optimal Stop-Loss: Limits maximum loss for each trade<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Kelly Ratio: Determines optimal position size based on statistical edge<\/li><li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Maximum Drawdown: The decline from peak to trough over a period<\/li><\/ul><\/div><div class='po-container po-container_width_article po-article-page__table'><div class='po-table'><table><thead><tr><th>Method<\/th><th>Formula<\/th><th>Real Calculation Example<\/th><\/tr><\/thead><tbody><tr><td>Value at Risk (95%)<\/td><td>VaR = -1.65 \u00d7 \u03c3 \u00d7 \u221at \u00d7 P<\/td><td>Portfolio 100 million, daily \u03c3 = 1.5%, time period 10 days:VaR = -1.65 \u00d7 1.5% \u00d7 \u221a10 \u00d7 100M = -1.65 \u00d7 0.015 \u00d7 3.16 \u00d7 100M = -7.82M\u2192 95% probability that loss will not exceed 7.82 million in 10 days<\/td><\/tr><tr><td>Optimal Stop-Loss<\/td><td>SL = P \u00d7 (1 - 2 \u00d7 ATR \u00d7 \u221aN)<\/td><td>Purchase price = 100,000 VND, ATR = 3%, N = 2 (confidence level):SL = 100,000 \u00d7 (1 - 2 \u00d7 0.03 \u00d7 \u221a2) = 100,000 \u00d7 (1 - 0.085) = 91,500 VND\u2192 Set stop-loss at 91,500 VND<\/td><\/tr><tr><td>Kelly Ratio<\/td><td>f* = (p \u00d7 b - q) \/ b<\/td><td>Win rate p = 55%, loss rate q = 45%, profit\/loss ratio b = 1.5:f* = (0.55 \u00d7 1.5 - 0.45) \/ 1.5 = (0.825 - 0.45) \/ 1.5 = 0.25\u2192 Should invest 25% of available capital<\/td><\/tr><tr><td>Maximum Drawdown<\/td><td>MDD = (Peak - Trough) \/ Peak<\/td><td>Portfolio peak = 120M, Trough = 90M:MDD = (120 - 90) \/ 120 = 30 \/ 120 = 25%\u2192 Maximum drawdown is 25%<\/td><\/tr><\/tbody><\/table><\/div><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Practical application: An investor has a 100 million VND portfolio, allocated across 10 stocks with an average daily volatility of 1.5%. Using 95% VaR for a 10-day period:<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>VaR = -1.65 \u00d7 1.5% \u00d7 \u221a10 \u00d7 100,000,000 = -7,820,000 VND<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>This means that with 95% probability, the maximum loss of the portfolio in the next 10 days will not exceed 7.82 million VND. Investors can use this information to ensure sufficient liquidity and adjust risk levels appropriately.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>The Kelly Ratio also helps investors determine optimal position size. With a trading system that has a 55% win rate, profit\/loss ratio of 1.5:1, the Kelly ratio is 25% - meaning you should invest 25% of available capital for each investment opportunity that fits the system.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Pocket Option provides automated risk management tools, helping investors maintain trading discipline and protect capital under all market conditions.<\/p><\/div><div class='po-container po-container_width_article-sm'><h2 class='po-article-page__title'>Conclusion: Mathematical Approach to Stock Investment<\/h2><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Understanding what stocks are from a mathematical perspective provides an undeniable competitive advantage in investing. Harvard University research shows that investors applying quantitative methods outperform intuition-based groups by 4.8% annually.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Analyzing stocks using mathematical tools such as DCF, CAPM, and MPT not only helps eliminate emotional factors but also builds a consistent decision-making framework. When markets experience strong fluctuations, quantitative methods help investors maintain composure and focus on data rather than reacting emotionally.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>In practice, combining mathematical methods has proven effective. For example, portfolios optimized according to MPT combined with risk management using VaR and stop-loss have helped many investors reduce portfolio volatility by 40% while maintaining equivalent returns.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Pocket Option provides a comprehensive platform with advanced quantitative analysis tools, helping investors apply data science to the decision-making process. From fundamental analysis, technical analysis to portfolio and risk management, we are committed to supporting investors in developing sustainable investment strategies based on solid mathematical foundations.<\/p><\/div><div class='po-container po-container_width_article-sm'><p class='po-article-page__text'>Remember that even the most complex mathematical tools cannot completely replace human judgment and experience. The most effective approach is to combine both: use quantitative models to filter and identify opportunities, then apply knowledge and understanding of the market to make final decisions. With Pocket Option, you have the tools to implement this strategy effectively.<\/p><\/div>[cta_button text=\"\"]","body_html_source":{"label":"Body HTML","type":"wysiwyg","formatted_value":"<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>What are Stocks: Definition from a Mathematical and Financial Perspective<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>From a mathematical and financial perspective, what are stocks? They are certificates of ownership of a portion of a company&#8217;s assets and income, represented by quantitative values such as book value, market price, and P\/E ratio. Each share represents a unit of ownership, allowing investors to participate in the company&#8217;s profits according to their holdings.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Mathematically, the value of a stock is determined by quantitative variables related to the company&#8217;s operational performance. For example, if company ABC has a profit of 100 billion VND and has 10 million outstanding shares, the earnings per share (EPS) will be 10,000 VND (100,000,000,000 \u00f7 10,000,000).<\/p>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Basic Component<\/th>\n<th>Mathematical Representation<\/th>\n<th>Calculation Example<\/th>\n<th>Significance in Analysis<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Book Value (BV)<\/td>\n<td>BV = (Assets &#8211; Liabilities) \/ Number of shares<\/td>\n<td>BV = (1,000 &#8211; 400) \/ 10 = 60 VND<\/td>\n<td>Net asset value per share<\/td>\n<\/tr>\n<tr>\n<td>Earnings Per Share (EPS)<\/td>\n<td>EPS = Net Profit \/ Number of shares<\/td>\n<td>EPS = 100 \/ 10 = 10 VND<\/td>\n<td>Profitability per share<\/td>\n<\/tr>\n<tr>\n<td>P\/E Ratio<\/td>\n<td>P\/E = Stock Price \/ EPS<\/td>\n<td>P\/E = 150 \/ 10 = 15 times<\/td>\n<td>Number of years needed to recover investment<\/td>\n<\/tr>\n<tr>\n<td>Dividend Yield<\/td>\n<td>Div Yield = (Dividend \/ Price) \u00d7 100%<\/td>\n<td>Yield = (5 \/ 150) \u00d7 100% = 3.33%<\/td>\n<td>Annual yield from dividends<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>At Pocket Option, we view stocks not just as securities but as mathematical equations to be decoded. Each variable in this equation &#8211; from revenue growth, profit margins, to asset utilization efficiency &#8211; can be modeled to find the true value. For example, a business growing revenue by 15% for 5 consecutive years can calculate its fifth-year revenue using the formula FV = PV \u00d7 (1 + 0.15)^5 = PV \u00d7 2.01, showing that revenue will double.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>Stock Valuation Equations and Practical Mathematical Models<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>When delving into what stocks are through a quantitative approach, the Discounted Cash Flow (DCF) model becomes an essential mathematical tool. The strength of DCF is its ability to convert a company&#8217;s future financial potential into present value, taking into account time factors and risk.<\/p>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Valuation Model<\/th>\n<th>Formula<\/th>\n<th>Calculation Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>DCF Model<\/td>\n<td>P = \u03a3[CF\u208d\u209c\u208e\/(1+r)\u1d57]<\/td>\n<td>With CF\u2081 = 10, CF\u2082 = 12, CF\u2083 = 15, r = 10%:P = 10\/1.1 + 12\/1.21 + 15\/1.331 = 9.09 + 9.92 + 11.27 = 30.28<\/td>\n<\/tr>\n<tr>\n<td>Gordon Growth Model<\/td>\n<td>P = D\u2081\/(r-g)<\/td>\n<td>With D\u2081 = 5, r = 12%, g = 4%:P = 5\/(0.12-0.04) = 5\/0.08 = 62.5<\/td>\n<\/tr>\n<tr>\n<td>Two-Stage Model<\/td>\n<td>P = \u03a3[D\u208d\u209c\u208e\/(1+r)\u1d57] + [D\u208d\u2099\u208e\u00d7(1+g)]\/(r-g)\u00d7(1+r)^(-n)<\/td>\n<td>With high growth for 5 years (g\u2081=20%), then stable (g\u2082=3%):P = 57.56 + 185.43 = 242.99<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Applying DCF in practice, let&#8217;s consider a software company expected to generate cash flows of 10 billion, 12 billion, and 15 billion VND in the next 3 years. With a discount rate of 10% (reflecting investment risk), the present value of the cash flows is:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm article-content po-article-page__text'>\n<ul class='po-article-page-list'>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Year 1: 10 billion \/ (1 + 0.1) = 9.09 billion<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Year 2: 12 billion \/ (1 + 0.1)\u00b2 = 9.92 billion<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Year 3: 15 billion \/ (1 + 0.1)\u00b3 = 11.27 billion<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Total present value: 30.28 billion<\/li>\n<\/ul>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h3 class='po-article-page__title'>Beta Coefficient and Capital Asset Pricing Model (CAPM)<\/h3>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>When investors explore what stocks are from a risk perspective, the Beta coefficient (\u03b2) becomes an important mathematical tool. Beta measures a stock&#8217;s volatility relative to the market and is calculated as follows:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>\u03b2 = Cov(R\u208d\u1d62\u208e, R\u208d\u2098\u208e) \/ Var(R\u208d\u2098\u208e)<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Real-world example: If VCB stock has a covariance with the market of 0.0015 and the market variance is 0.001, then VCB&#8217;s Beta is 0.0015\/0.001 = 1.5. This means that when the market rises\/falls by 1%, VCB will tend to rise\/fall by 1.5%.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Beta is used in the CAPM model to determine the expected rate of return:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>E(R\u208d\u1d62\u208e) = R\u208d\u1da0\u208e + \u03b2\u208d\u1d62\u208e[E(R\u208d\u2098\u208e) &#8211; R\u208d\u1da0\u208e]<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Applied to VCB with a risk-free rate of 4%, expected market return of 10%:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>E(R\u208d\u1d65c\u0299\u208e) = 4% + 1.5 \u00d7 (10% &#8211; 4%) = 4% + 9% = 13%<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Pocket Option provides real-time Beta analysis tools, helping investors accurately assess the relative risk level of each stock in their portfolio.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>Who Issues Stocks and Quantitative Analysis of the IPO Process<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>The question of who issues stocks plays an important role in risk analysis. Stocks are issued by joint-stock companies through the initial public offering (IPO) process. From a mathematical perspective, the IPO pricing process is a complex optimization problem aimed at determining the most reasonable price level.<\/p>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Stage<\/th>\n<th>Pricing Formula<\/th>\n<th>Real Calculation Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Pre-IPO<\/td>\n<td>V = E \u00d7 P\/E\u208dcomp\u208e \u00d7 (1-d)<\/td>\n<td>Technology company with profit of 50 billion, industry P\/E = 20, discount 30%:V = 50 \u00d7 20 \u00d7 (1-0.3) = 700 billion<\/td>\n<\/tr>\n<tr>\n<td>IPO Pricing<\/td>\n<td>P\u208dipo\u208e = (V\u208dcompany\u208e\/N) \u00d7 (1-d\u208dipo\u208e)<\/td>\n<td>Company value 700 billion, 10 million shares, IPO discount 15%:P\u208dipo\u208e = (700\/10) \u00d7 (1-0.15) = 70 \u00d7 0.85 = 59,500 VND<\/td>\n<\/tr>\n<tr>\n<td>Post-IPO<\/td>\n<td>P\u208dmarket\u208e = P\u208dipo\u208e \u00d7 (1+r\u208dmarket\u208e)<\/td>\n<td>IPO price 59,500 VND, market reaction +20%:P\u208dmarket\u208e = 59,500 \u00d7 1.2 = 71,400 VND<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Historical data analysis shows that IPOs are typically priced 15-20% lower than their true value to ensure the success of the issuance. Here is the formula for calculating the IPO discount rate compared to the first-day market price:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Underpricing rate (%) = [(P\u208dday1\u208e &#8211; P\u208dipo\u208e) \/ P\u208dipo\u208e] \u00d7 100%<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h3 class='po-article-page__title'>Quantitative Analysis of Issuance Quality<\/h3>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>To objectively evaluate the quality of a stock issuer, investors can use a quantitative scoring model that integrates multiple factors:<\/p>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Criteria<\/th>\n<th>Weight<\/th>\n<th>Scale<\/th>\n<th>Real Calculation Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>3-year Revenue Growth<\/td>\n<td>20%<\/td>\n<td>1-10<\/td>\n<td>25% growth \u2192 Score 8 \u00d7 20% = 1.6<\/td>\n<\/tr>\n<tr>\n<td>Return on Equity (ROE)<\/td>\n<td>25%<\/td>\n<td>1-10<\/td>\n<td>ROE 22% \u2192 Score 9 \u00d7 25% = 2.25<\/td>\n<\/tr>\n<tr>\n<td>Management Quality<\/td>\n<td>20%<\/td>\n<td>1-10<\/td>\n<td>Evaluation 7\/10 \u2192 7 \u00d7 20% = 1.4<\/td>\n<\/tr>\n<tr>\n<td>Competitive Position<\/td>\n<td>20%<\/td>\n<td>1-10<\/td>\n<td>Market share 35% \u2192 Score 8 \u00d7 20% = 1.6<\/td>\n<\/tr>\n<tr>\n<td>IPO Transaction Structure<\/td>\n<td>15%<\/td>\n<td>1-10<\/td>\n<td>Evaluation 6\/10 \u2192 6 \u00d7 15% = 0.9<\/td>\n<\/tr>\n<tr>\n<td>Composite Score<\/td>\n<td>100%<\/td>\n<td>1-10<\/td>\n<td>1.6 + 2.25 + 1.4 + 1.6 + 0.9 = 7.75\/10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>With a composite score of 7.75\/10, the company is rated as having good quality and worth considering for investment. This scoring model helps eliminate emotional factors and creates an objective basis for investment decisions.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Investors using Pocket Option can access similar automated evaluation models, saving research time while ensuring high accuracy.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>What are Securities Stocks from a Statistical Mathematical Perspective<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>From a statistical viewpoint, what are securities stocks? They are financial time series with distinct mathematical properties. Stock prices are often described by random processes that follow certain probability distributions.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm article-content po-article-page__text'>\n<ul class='po-article-page-list'>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Geometric Brownian Motion (GBM): dS = \u03bcSdt + \u03c3SdW, describing the random movement of prices<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Logarithmic returns: r = ln(S\u208d\u209c\u208e\/S\u208d\u209c\u208b\u2081\u208e), typically following a normal distribution<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Conditional variance (GARCH): forecasting volatility based on historical data<\/li>\n<\/ul>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Statistical Characteristic<\/th>\n<th>Formula<\/th>\n<th>Real Calculation Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Expected Return<\/td>\n<td>E(R) = \u03a3[p\u1d62 \u00d7 R\u1d62]<\/td>\n<td>Scenarios: Increase 20% (probability 30%), Stable (40%), Decrease 10% (30%)E(R) = 0.3 \u00d7 20% + 0.4 \u00d7 0% + 0.3 \u00d7 (-10%) = 6% &#8211; 3% = 3%<\/td>\n<\/tr>\n<tr>\n<td>Volatility (annual)<\/td>\n<td>\u03c3\u208dannual\u208e = \u03c3\u208ddaily\u208e \u00d7 \u221a252<\/td>\n<td>Daily standard deviation 1.2%:\u03c3\u208dannual\u208e = 1.2% \u00d7 \u221a252 = 1.2% \u00d7 15.87 = 19.04%<\/td>\n<\/tr>\n<tr>\n<td>Correlation Coefficient<\/td>\n<td>\u03c1 = Cov(R\u2090, R\u1d66) \/ (\u03c3\u2090 \u00d7 \u03c3\u1d66)<\/td>\n<td>Covariance 0.0008, \u03c3\u2090 = 0.02, \u03c3\u1d66 = 0.05:\u03c1 = 0.0008 \/ (0.02 \u00d7 0.05) = 0.0008 \/ 0.001 = 0.8<\/td>\n<\/tr>\n<tr>\n<td>Sharpe Ratio<\/td>\n<td>S = (R &#8211; R\u1da0) \/ \u03c3<\/td>\n<td>Return 15%, risk-free rate 5%, volatility 20%:S = (15% &#8211; 5%) \/ 20% = 10% \/ 20% = 0.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>A real example: if historical data analysis of ABC stock shows a daily volatility of 1.2%, then the annual volatility will be 1.2% \u00d7 \u221a252 = 19.04% (assuming 252 trading days in a year). With an expected return of 15% and a risk-free rate of 5%, the Sharpe ratio will be (15% &#8211; 5%) \/ 19.04% = 0.52 &#8211; a fairly good ratio compared to the market average.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Understanding what securities stocks are from a statistical perspective helps investors build trading strategies based on probability and mathematical expectations. Pocket Option provides advanced probability analysis tools that help investors make scientifically-based decisions.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>Stock Technical Analysis Methods through Mathematical Models<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Technical analysis of what stocks are is essentially a pattern recognition problem in financial time series. Technical indicators use mathematical formulas to transform price data into quantifiable signals that can be acted upon.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm article-content po-article-page__text'>\n<ul class='po-article-page-list'>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Simple Moving Average (SMA): SMA(n) = (P\u2081 + P\u2082 + &#8230; + P\u2099) \/ n<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Relative Strength Index (RSI): RSI = 100 &#8211; [100 \/ (1 + RS)], where RS = Average Gain \/ Average Loss<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Bollinger Bands: BB = SMA(n) \u00b1 k \u00d7 \u03c3(n), typically using n = 20, k = 2<\/li>\n<\/ul>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Indicator<\/th>\n<th>Formula<\/th>\n<th>Real Calculation Example<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>MACD<\/td>\n<td>MACD = EMA(12) &#8211; EMA(26)Signal = EMA(9) of MACD<\/td>\n<td>EMA(12) = 104, EMA(26) = 100MACD = 104 &#8211; 100 = 4Signal = 3Histogram = 4 &#8211; 3 = 1<\/td>\n<td>MACD &gt; Signal: buy signalMACD &lt; Signal: sell signal<\/td>\n<\/tr>\n<tr>\n<td>RSI<\/td>\n<td>RSI = 100 &#8211; [100 \/ (1 + RS)]<\/td>\n<td>14-day average gain = 2%14-day average loss = 1%RS = 2% \/ 1% = 2RSI = 100 &#8211; [100 \/ (1 + 2)] = 100 &#8211; 33.33 = 66.67<\/td>\n<td>RSI &gt; 70: overboughtRSI &lt; 30: oversold<\/td>\n<\/tr>\n<tr>\n<td>Fibonacci Retracement<\/td>\n<td>Level = High &#8211; (High &#8211; Low) \u00d7 Ratio<\/td>\n<td>High = 100, Low = 8038.2% Level: 100 &#8211; (100 &#8211; 80) \u00d7 0.382 = 100 &#8211; 7.64 = 92.3661.8% Level: 100 &#8211; (100 &#8211; 80) \u00d7 0.618 = 100 &#8211; 12.36 = 87.64<\/td>\n<td>Potential support\/resistance levels<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Real-world example of applying MACD: Suppose XYZ stock&#8217;s EMA(12) is 104, EMA(26) is 100, creating a MACD of 4. The Signal line (9-day EMA of MACD) is at 3. When MACD crosses above the Signal (Histogram = 4 &#8211; 3 = 1 &gt; 0), this is a potential buy signal. If accompanied by a 50% increase in trading volume compared to the average, the reliability of the signal is even higher.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h3 class='po-article-page__title'>Machine Learning Applications in Technical Analysis<\/h3>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Machine learning algorithms have expanded the capabilities of traditional technical analysis when studying what stocks are. Instead of relying on individual indicators, machine learning models can integrate dozens of variables to identify complex patterns.<\/p>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Algorithm<\/th>\n<th>Operating Principle<\/th>\n<th>Specific Application<\/th>\n<th>Average Accuracy<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Neural Networks (ANN)<\/td>\n<td>y = f(\u03a3(w\u1d62x\u1d62 + b))<\/td>\n<td>Short-term price prediction based on 20 technical indicators<\/td>\n<td>58-65%<\/td>\n<\/tr>\n<tr>\n<td>Random Forest<\/td>\n<td>f = 1\/n \u03a3f\u1d62(x)<\/td>\n<td>Trend classification (up\/down\/sideways)<\/td>\n<td>65-72%<\/td>\n<\/tr>\n<tr>\n<td>LSTM<\/td>\n<td>Neural network with long-term &#8220;memory&#8221; capability<\/td>\n<td>Complex time series analysis<\/td>\n<td>60-68%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Pocket Option has developed a technical analysis system integrated with machine learning with an average accuracy of 65-70% in short-term trend forecasting. This system analyzes 42 technical indicators combined with trading volume data to identify potential entry and exit points.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Real-world example: Our random forest model has identified that the combination of RSI turning up from oversold territory, MACD crossing above the Signal line, and volume increasing 30% above the 20-day average creates a buy signal with a 72% success rate under normal market conditions.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>Building an Optimal Stock Portfolio Using Mathematics<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>To better understand what stocks are from a portfolio management perspective, Harry Markowitz&#8217;s Modern Portfolio Theory (MPT) provides a solid mathematical foundation. MPT uses optimization to build efficient frontier portfolios &#8211; sets of investment portfolios that provide the highest expected return at each level of risk.<\/p>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Component<\/th>\n<th>Formula<\/th>\n<th>Real Calculation Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Expected Portfolio Return<\/td>\n<td>E(Rp) = \u03a3(w\u1d62 \u00d7 E(R\u1d62))<\/td>\n<td>2-stock portfolio: w\u2081 = 60%, E(R\u2081) = 12%; w\u2082 = 40%, E(R\u2082) = 8%E(Rp) = 0.6 \u00d7 12% + 0.4 \u00d7 8% = 7.2% + 3.2% = 10.4%<\/td>\n<\/tr>\n<tr>\n<td>Portfolio Risk<\/td>\n<td>\u03c3p\u00b2 = \u03a3i \u03a3j (w\u1d62w\u2c7c\u03c3\u1d62\u2c7c)<\/td>\n<td>\u03c3\u2081 = 20%, \u03c3\u2082 = 15%, \u03c1\u2081\u2082 = 0.3\u03c3p\u00b2 = (0.6)\u00b2 \u00d7 (20%)\u00b2 + (0.4)\u00b2 \u00d7 (15%)\u00b2 + 2 \u00d7 0.6 \u00d7 0.4 \u00d7 0.3 \u00d7 20% \u00d7 15%\u03c3p\u00b2 = 0.0144 + 0.0036 + 0.00216 = 0.02016\u03c3p = \u221a0.02016 = 14.2%<\/td>\n<\/tr>\n<tr>\n<td>Sharpe Ratio<\/td>\n<td>SR = (Rp &#8211; Rf) \/ \u03c3p<\/td>\n<td>Rp = 10.4%, Rf = 4%, \u03c3p = 14.2%SR = (10.4% &#8211; 4%) \/ 14.2% = 6.4% \/ 14.2% = 0.45<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>The portfolio optimization problem can be solved using the Lagrange method. Suppose we have 2 stocks: A (expected return 12%, volatility 20%) and B (expected return 8%, volatility 15%) with a correlation coefficient of 0.3. To maximize the Sharpe ratio, we find the optimal weights as follows:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm article-content po-article-page__text'>\n<ul class='po-article-page-list'>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Optimal weights (w\u2081, w\u2082) = (0.6; 0.4)<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Expected portfolio return = 0.6 \u00d7 12% + 0.4 \u00d7 8% = 10.4%<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Portfolio volatility = 14.2% (calculated using the formula above)<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Sharpe ratio = (10.4% &#8211; 4%) \/ 14.2% = 0.45<\/li>\n<\/ul>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h3 class='po-article-page__title'>Quantitative Diversification Strategy<\/h3>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Diversification is a core element when exploring what securities stocks are from a risk management perspective. The effectiveness of diversification depends on the correlation between assets and can be precisely quantified:<\/p>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Number of Stocks<\/th>\n<th>Reduction in Non-Systematic Risk<\/th>\n<th>Real Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>0%<\/td>\n<td>1-stock portfolio with \u03c3 = 30%<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>~50%<\/td>\n<td>5-stock portfolio with average correlation 0.3:\u03c3 reduced from 30% to ~21%<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>~65%<\/td>\n<td>10-stock portfolio with average correlation 0.3:\u03c3 reduced from 30% to ~18%<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>~75%<\/td>\n<td>20-stock portfolio with average correlation 0.3:\u03c3 reduced from 30% to ~16.5%<\/td>\n<\/tr>\n<tr>\n<td>30+<\/td>\n<td>~80%<\/td>\n<td>30+ stock portfolio with average correlation 0.3:\u03c3 reduced from 30% to ~15.5%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Real-world example: An investor has a portfolio of 10 stocks with equal allocation (10% per stock). Each stock has a volatility of 30% and an average correlation coefficient of 0.3. The portfolio volatility will be:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>\u03c3p = \u221a[n \u00d7 (1\/n)\u00b2 \u00d7 \u03c3\u00b2 + n \u00d7 (n-1) \u00d7 (1\/n)\u00b2 \u00d7 \u03c1 \u00d7 \u03c3\u00b2]<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>\u03c3p = \u221a[10 \u00d7 (0.1)\u00b2 \u00d7 (0.3)\u00b2 + 10 \u00d7 9 \u00d7 (0.1)\u00b2 \u00d7 0.3 \u00d7 (0.3)\u00b2]<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>\u03c3p = \u221a[0.009 + 0.0243] = \u221a0.0333 = 18.25%<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>This proves that diversification has helped reduce risk from 30% to 18.25% &#8211; a nearly 40% reduction without reducing expected returns.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Pocket Option provides automatic portfolio optimization tools, helping investors determine the optimal weight for each stock in their portfolio based on individual risk tolerance.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>Fundamental Stock Analysis Using Quantitative Methods<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Fundamental analysis when exploring who issues stocks focuses on intrinsic value based on quantitative financial factors. This method transforms financial reports into comparable metrics.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm article-content po-article-page__text'>\n<ul class='po-article-page-list'>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>DCF Model: Discounting future cash flows to present value<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Ratio Analysis: Comparing P\/E, P\/B, EV\/EBITDA with industry averages<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Sustainable Growth Model: g = ROE \u00d7 (1 &#8211; Payout Ratio)<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Z-Score: Predicting bankruptcy probability in the next 2 years<\/li>\n<\/ul>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Ratio Group<\/th>\n<th>Formula<\/th>\n<th>Real Calculation Example<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Profitability<\/td>\n<td>ROE = Net Profit \/ Equity<\/td>\n<td>Profit: 100 billion, Equity: 500 billionROE = 100\/500 = 20%<\/td>\n<td>ROE &gt; 15% is considered goodROE = 20% &gt; 15% \u2192 High efficiency<\/td>\n<\/tr>\n<tr>\n<td>Operational Efficiency<\/td>\n<td>Asset Turnover = Revenue \/ Total Assets<\/td>\n<td>Revenue: 800 billion, Total Assets: 1,000 billionTurnover = 800\/1,000 = 0.8<\/td>\n<td>The company generates 0.8 units of revenue for each unit of assets &#8211; relatively good<\/td>\n<\/tr>\n<tr>\n<td>Capital Structure<\/td>\n<td>D\/E Ratio = Total Debt \/ Equity<\/td>\n<td>Total Debt: 300 billion, Equity: 500 billionD\/E = 300\/500 = 0.6<\/td>\n<td>D\/E = 0.6 is in the safe zone (0.5-1.0) &#8211; balanced between debt and equity<\/td>\n<\/tr>\n<tr>\n<td>Valuation<\/td>\n<td>P\/E = Price \/ EPS<\/td>\n<td>Price: 60,000 VND, EPS: 5,000 VNDP\/E = 60,000\/5,000 = 12<\/td>\n<td>P\/E = 12 lower than industry average (15) \u2192 Attractive valuation<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Combining financial ratios creates a comprehensive picture of company value. For example, a business with high ROE (20%), reasonable capital structure (D\/E = 0.6), and attractive valuation (P\/E = 12 compared to industry average of 15) could be a value investment opportunity.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>The Gordon Growth Model provides a simple method to estimate stock value based on dividends:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>P = D\u2081 \/ (r &#8211; g)<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Example: ABC stock is expected to pay a dividend of 3,000 VND\/share next year, has a discount rate of 12% and a sustainable growth rate of 7%. The fair value of the stock is:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>P = 3,000 \/ (0.12 &#8211; 0.07) = 3,000 \/ 0.05 = 60,000 VND<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>At Pocket Option, we integrate automated fundamental valuation models, helping investors quickly assess the intrinsic value of stocks based on the latest financial data.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>Methods for Measuring and Managing Stock Investment Risk<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Investing in securities stocks needs to be accompanied by effective risk management. Quantitative methods help investors measure and control risk objectively.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm article-content po-article-page__text'>\n<ul class='po-article-page-list'>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Value at Risk (VaR): Estimates maximum loss under normal market conditions<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Optimal Stop-Loss: Limits maximum loss for each trade<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Kelly Ratio: Determines optimal position size based on statistical edge<\/li>\n<li class='po-article-page__text po-article-page__text_no-margin po-list-lvl_1'>Maximum Drawdown: The decline from peak to trough over a period<\/li>\n<\/ul>\n<\/div>\n<div class='po-container po-container_width_article po-article-page__table'>\n<div class='po-table'>\n<table>\n<thead>\n<tr>\n<th>Method<\/th>\n<th>Formula<\/th>\n<th>Real Calculation Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Value at Risk (95%)<\/td>\n<td>VaR = -1.65 \u00d7 \u03c3 \u00d7 \u221at \u00d7 P<\/td>\n<td>Portfolio 100 million, daily \u03c3 = 1.5%, time period 10 days:VaR = -1.65 \u00d7 1.5% \u00d7 \u221a10 \u00d7 100M = -1.65 \u00d7 0.015 \u00d7 3.16 \u00d7 100M = -7.82M\u2192 95% probability that loss will not exceed 7.82 million in 10 days<\/td>\n<\/tr>\n<tr>\n<td>Optimal Stop-Loss<\/td>\n<td>SL = P \u00d7 (1 &#8211; 2 \u00d7 ATR \u00d7 \u221aN)<\/td>\n<td>Purchase price = 100,000 VND, ATR = 3%, N = 2 (confidence level):SL = 100,000 \u00d7 (1 &#8211; 2 \u00d7 0.03 \u00d7 \u221a2) = 100,000 \u00d7 (1 &#8211; 0.085) = 91,500 VND\u2192 Set stop-loss at 91,500 VND<\/td>\n<\/tr>\n<tr>\n<td>Kelly Ratio<\/td>\n<td>f* = (p \u00d7 b &#8211; q) \/ b<\/td>\n<td>Win rate p = 55%, loss rate q = 45%, profit\/loss ratio b = 1.5:f* = (0.55 \u00d7 1.5 &#8211; 0.45) \/ 1.5 = (0.825 &#8211; 0.45) \/ 1.5 = 0.25\u2192 Should invest 25% of available capital<\/td>\n<\/tr>\n<tr>\n<td>Maximum Drawdown<\/td>\n<td>MDD = (Peak &#8211; Trough) \/ Peak<\/td>\n<td>Portfolio peak = 120M, Trough = 90M:MDD = (120 &#8211; 90) \/ 120 = 30 \/ 120 = 25%\u2192 Maximum drawdown is 25%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Practical application: An investor has a 100 million VND portfolio, allocated across 10 stocks with an average daily volatility of 1.5%. Using 95% VaR for a 10-day period:<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>VaR = -1.65 \u00d7 1.5% \u00d7 \u221a10 \u00d7 100,000,000 = -7,820,000 VND<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>This means that with 95% probability, the maximum loss of the portfolio in the next 10 days will not exceed 7.82 million VND. Investors can use this information to ensure sufficient liquidity and adjust risk levels appropriately.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>The Kelly Ratio also helps investors determine optimal position size. With a trading system that has a 55% win rate, profit\/loss ratio of 1.5:1, the Kelly ratio is 25% &#8211; meaning you should invest 25% of available capital for each investment opportunity that fits the system.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Pocket Option provides automated risk management tools, helping investors maintain trading discipline and protect capital under all market conditions.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<h2 class='po-article-page__title'>Conclusion: Mathematical Approach to Stock Investment<\/h2>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Understanding what stocks are from a mathematical perspective provides an undeniable competitive advantage in investing. Harvard University research shows that investors applying quantitative methods outperform intuition-based groups by 4.8% annually.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Analyzing stocks using mathematical tools such as DCF, CAPM, and MPT not only helps eliminate emotional factors but also builds a consistent decision-making framework. When markets experience strong fluctuations, quantitative methods help investors maintain composure and focus on data rather than reacting emotionally.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>In practice, combining mathematical methods has proven effective. For example, portfolios optimized according to MPT combined with risk management using VaR and stop-loss have helped many investors reduce portfolio volatility by 40% while maintaining equivalent returns.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Pocket Option provides a comprehensive platform with advanced quantitative analysis tools, helping investors apply data science to the decision-making process. From fundamental analysis, technical analysis to portfolio and risk management, we are committed to supporting investors in developing sustainable investment strategies based on solid mathematical foundations.<\/p>\n<\/div>\n<div class='po-container po-container_width_article-sm'>\n<p class='po-article-page__text'>Remember that even the most complex mathematical tools cannot completely replace human judgment and experience. The most effective approach is to combine both: use quantitative models to filter and identify opportunities, then apply knowledge and understanding of the market to make final decisions. With Pocket Option, you have the tools to implement this strategy effectively.<\/p>\n<\/div>\n    <div class=\"po-container po-container_width_article\">\n        <a href=\"\/en\/quick-start\/\" class=\"po-line-banner po-article-page__line-banner\">\n            <svg class=\"svg-image po-line-banner__logo\" fill=\"currentColor\" width=\"auto\" height=\"auto\"\n                 aria-hidden=\"true\">\n                <use href=\"#svg-img-logo-white\"><\/use>\n            <\/svg>\n            <span class=\"po-line-banner__btn\"><\/span>\n        <\/a>\n    <\/div>\n    \n"},"faq":[{"question":"What are stocks and how to assess their intrinsic value?","answer":"Stocks are certificates of ownership of a portion of a company's assets and profits, representing ownership rights according to the proportion held. To assess intrinsic value, investors can use the DCF (Discounted Cash Flow) model, ratio analysis (P\/E, P\/B, EV\/EBITDA) compared to industry averages, and the Gordon Growth model (P = D\u2081\/(r-g)). A P\/E valuation ratio of 12 that is lower than the industry P\/E of 15 is usually a signal of attractive valuation."},{"question":"Who issues stocks and how does the issuance process work?","answer":"Stocks are issued by joint-stock companies through IPOs (Initial Public Offerings) or additional issuances. The IPO process includes: preparing documentation, initial valuation (usually using P\/E comparison or DCF methods), road shows (presentations to investors), book building (price determination), distribution and listing. Research shows that IPOs are typically priced 15-20% below their true value to ensure the success of the issuance."},{"question":"How to apply mathematics in technical analysis of stocks?","answer":"Technical analysis applies mathematics through: (1) Oscillating indicators such as RSI = 100-[100\/(1+RS)] to identify overbought\/oversold areas; (2) Trend indicators like MACD = EMA(12)-EMA(26) to identify reversal points; (3) Bollinger Bands = SMA(20)\u00b12\u00d7\u03c3 to identify abnormal volatility; (4) Fibonacci Retracement to identify support\/resistance levels; (5) Machine learning algorithms such as neural networks and random forests to recognize complex patterns with 60-70% accuracy."},{"question":"How to optimize a stock portfolio based on mathematics?","answer":"Portfolio optimization uses Markowitz theory (MPT) by finding stock weights that maximize the Sharpe ratio SR=(Rp-Rf)\/\u03c3p. For example, a 2-stock portfolio with weights of 60%\/40% can reduce risk from 30% to 14.2% while maintaining an expected return of 10.4%. Effective diversification requires low correlation between assets and the optimal number is typically 15-30 stocks appropriately allocated, helping to eliminate up to 75-80% of non-systematic risk."},{"question":"What tools does Pocket Option provide for quantitative stock analysis?","answer":"Pocket Option provides: (1) Automated DCF and Gordon Growth valuation models with multiple growth scenarios; (2) AI-integrated technical analysis system with 42 indicators (65-70% accuracy); (3) MPT portfolio optimization tools that calculate optimal weights based on personal risk tolerance; (4) Risk management system with VaR, optimal Stop-Loss and Kelly ratio; (5) Automated comparative analysis of financial ratios against industry averages."}],"faq_source":{"label":"FAQ","type":"repeater","formatted_value":[{"question":"What are stocks and how to assess their intrinsic value?","answer":"Stocks are certificates of ownership of a portion of a company's assets and profits, representing ownership rights according to the proportion held. To assess intrinsic value, investors can use the DCF (Discounted Cash Flow) model, ratio analysis (P\/E, P\/B, EV\/EBITDA) compared to industry averages, and the Gordon Growth model (P = D\u2081\/(r-g)). A P\/E valuation ratio of 12 that is lower than the industry P\/E of 15 is usually a signal of attractive valuation."},{"question":"Who issues stocks and how does the issuance process work?","answer":"Stocks are issued by joint-stock companies through IPOs (Initial Public Offerings) or additional issuances. The IPO process includes: preparing documentation, initial valuation (usually using P\/E comparison or DCF methods), road shows (presentations to investors), book building (price determination), distribution and listing. Research shows that IPOs are typically priced 15-20% below their true value to ensure the success of the issuance."},{"question":"How to apply mathematics in technical analysis of stocks?","answer":"Technical analysis applies mathematics through: (1) Oscillating indicators such as RSI = 100-[100\/(1+RS)] to identify overbought\/oversold areas; (2) Trend indicators like MACD = EMA(12)-EMA(26) to identify reversal points; (3) Bollinger Bands = SMA(20)\u00b12\u00d7\u03c3 to identify abnormal volatility; (4) Fibonacci Retracement to identify support\/resistance levels; (5) Machine learning algorithms such as neural networks and random forests to recognize complex patterns with 60-70% accuracy."},{"question":"How to optimize a stock portfolio based on mathematics?","answer":"Portfolio optimization uses Markowitz theory (MPT) by finding stock weights that maximize the Sharpe ratio SR=(Rp-Rf)\/\u03c3p. For example, a 2-stock portfolio with weights of 60%\/40% can reduce risk from 30% to 14.2% while maintaining an expected return of 10.4%. Effective diversification requires low correlation between assets and the optimal number is typically 15-30 stocks appropriately allocated, helping to eliminate up to 75-80% of non-systematic risk."},{"question":"What tools does Pocket Option provide for quantitative stock analysis?","answer":"Pocket Option provides: (1) Automated DCF and Gordon Growth valuation models with multiple growth scenarios; (2) AI-integrated technical analysis system with 42 indicators (65-70% accuracy); (3) MPT portfolio optimization tools that calculate optimal weights based on personal risk tolerance; (4) Risk management system with VaR, optimal Stop-Loss and Kelly ratio; (5) Automated comparative analysis of financial ratios against industry averages."}]}},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v24.8 (Yoast SEO v27.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>What are Stocks: Mathematical Analysis and Effective Investment Strategies 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