- Year 1: 10 billion / (1 + 0.1) = 9.09 billion
- Year 2: 12 billion / (1 + 0.1)² = 9.92 billion
- Year 3: 15 billion / (1 + 0.1)³ = 11.27 billion
- Total present value: 30.28 billion
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.
What are Stocks: Definition from a Mathematical and Financial Perspective
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.
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 ÷ 10,000,000).
| Basic Component | Mathematical Representation | Calculation Example | Significance in Analysis |
|---|---|---|---|
| Book Value (BV) | BV = (Assets – Liabilities) / Number of shares | BV = (1,000 – 400) / 10 = 60 VND | Net asset value per share |
| Earnings Per Share (EPS) | EPS = Net Profit / Number of shares | EPS = 100 / 10 = 10 VND | Profitability per share |
| P/E Ratio | P/E = Stock Price / EPS | P/E = 150 / 10 = 15 times | Number of years needed to recover investment |
| Dividend Yield | Div Yield = (Dividend / Price) × 100% | Yield = (5 / 150) × 100% = 3.33% | Annual yield from dividends |
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 × (1 + 0.15)^5 = PV × 2.01, showing that revenue will double.
Stock Valuation Equations and Practical Mathematical Models
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.
| Valuation Model | Formula | Calculation Example |
|---|---|---|
| DCF Model | P = Σ[CF₍ₜ₎/(1+r)ᵗ] | With CF₁ = 10, CF₂ = 12, CF₃ = 15, r = 10%:P = 10/1.1 + 12/1.21 + 15/1.331 = 9.09 + 9.92 + 11.27 = 30.28 |
| Gordon Growth Model | P = D₁/(r-g) | With D₁ = 5, r = 12%, g = 4%:P = 5/(0.12-0.04) = 5/0.08 = 62.5 |
| Two-Stage Model | P = Σ[D₍ₜ₎/(1+r)ᵗ] + [D₍ₙ₎×(1+g)]/(r-g)×(1+r)^(-n) | With high growth for 5 years (g₁=20%), then stable (g₂=3%):P = 57.56 + 185.43 = 242.99 |
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:
Beta Coefficient and Capital Asset Pricing Model (CAPM)
When investors explore what stocks are from a risk perspective, the Beta coefficient (β) becomes an important mathematical tool. Beta measures a stock’s volatility relative to the market and is calculated as follows:
β = Cov(R₍ᵢ₎, R₍ₘ₎) / Var(R₍ₘ₎)
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%.
Beta is used in the CAPM model to determine the expected rate of return:
E(R₍ᵢ₎) = R₍ᶠ₎ + β₍ᵢ₎[E(R₍ₘ₎) – R₍ᶠ₎]
Applied to VCB with a risk-free rate of 4%, expected market return of 10%:
E(R₍ᵥcʙ₎) = 4% + 1.5 × (10% – 4%) = 4% + 9% = 13%
Pocket Option provides real-time Beta analysis tools, helping investors accurately assess the relative risk level of each stock in their portfolio.
Who Issues Stocks and Quantitative Analysis of the IPO Process
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.
| Stage | Pricing Formula | Real Calculation Example |
|---|---|---|
| Pre-IPO | V = E × P/E₍comp₎ × (1-d) | Technology company with profit of 50 billion, industry P/E = 20, discount 30%:V = 50 × 20 × (1-0.3) = 700 billion |
| IPO Pricing | P₍ipo₎ = (V₍company₎/N) × (1-d₍ipo₎) | Company value 700 billion, 10 million shares, IPO discount 15%:P₍ipo₎ = (700/10) × (1-0.15) = 70 × 0.85 = 59,500 VND |
| Post-IPO | P₍market₎ = P₍ipo₎ × (1+r₍market₎) | IPO price 59,500 VND, market reaction +20%:P₍market₎ = 59,500 × 1.2 = 71,400 VND |
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:
Underpricing rate (%) = [(P₍day1₎ – P₍ipo₎) / P₍ipo₎] × 100%
Quantitative Analysis of Issuance Quality
To objectively evaluate the quality of a stock issuer, investors can use a quantitative scoring model that integrates multiple factors:
| Criteria | Weight | Scale | Real Calculation Example |
|---|---|---|---|
| 3-year Revenue Growth | 20% | 1-10 | 25% growth → Score 8 × 20% = 1.6 |
| Return on Equity (ROE) | 25% | 1-10 | ROE 22% → Score 9 × 25% = 2.25 |
| Management Quality | 20% | 1-10 | Evaluation 7/10 → 7 × 20% = 1.4 |
| Competitive Position | 20% | 1-10 | Market share 35% → Score 8 × 20% = 1.6 |
| IPO Transaction Structure | 15% | 1-10 | Evaluation 6/10 → 6 × 15% = 0.9 |
| Composite Score | 100% | 1-10 | 1.6 + 2.25 + 1.4 + 1.6 + 0.9 = 7.75/10 |
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.
Investors using Pocket Option can access similar automated evaluation models, saving research time while ensuring high accuracy.
What are Securities Stocks from a Statistical Mathematical Perspective
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.
- Geometric Brownian Motion (GBM): dS = μSdt + σSdW, describing the random movement of prices
- Logarithmic returns: r = ln(S₍ₜ₎/S₍ₜ₋₁₎), typically following a normal distribution
- Conditional variance (GARCH): forecasting volatility based on historical data
| Statistical Characteristic | Formula | Real Calculation Example |
|---|---|---|
| Expected Return | E(R) = Σ[pᵢ × Rᵢ] | Scenarios: Increase 20% (probability 30%), Stable (40%), Decrease 10% (30%)E(R) = 0.3 × 20% + 0.4 × 0% + 0.3 × (-10%) = 6% – 3% = 3% |
| Volatility (annual) | σ₍annual₎ = σ₍daily₎ × √252 | Daily standard deviation 1.2%:σ₍annual₎ = 1.2% × √252 = 1.2% × 15.87 = 19.04% |
| Correlation Coefficient | ρ = Cov(Rₐ, Rᵦ) / (σₐ × σᵦ) | Covariance 0.0008, σₐ = 0.02, σᵦ = 0.05:ρ = 0.0008 / (0.02 × 0.05) = 0.0008 / 0.001 = 0.8 |
| Sharpe Ratio | S = (R – Rᶠ) / σ | Return 15%, risk-free rate 5%, volatility 20%:S = (15% – 5%) / 20% = 10% / 20% = 0.5 |
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% × √252 = 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.
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.
Stock Technical Analysis Methods through Mathematical Models
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.
- Simple Moving Average (SMA): SMA(n) = (P₁ + P₂ + … + Pₙ) / n
- Relative Strength Index (RSI): RSI = 100 – [100 / (1 + RS)], where RS = Average Gain / Average Loss
- Bollinger Bands: BB = SMA(n) ± k × σ(n), typically using n = 20, k = 2
| Indicator | Formula | Real Calculation Example | Interpretation |
|---|---|---|---|
| MACD | MACD = EMA(12) – EMA(26)Signal = EMA(9) of MACD | EMA(12) = 104, EMA(26) = 100MACD = 104 – 100 = 4Signal = 3Histogram = 4 – 3 = 1 | MACD > Signal: buy signalMACD < Signal: sell signal |
| RSI | RSI = 100 – [100 / (1 + RS)] | 14-day average gain = 2%14-day average loss = 1%RS = 2% / 1% = 2RSI = 100 – [100 / (1 + 2)] = 100 – 33.33 = 66.67 | RSI > 70: overboughtRSI < 30: oversold |
| Fibonacci Retracement | Level = High – (High – Low) × Ratio | High = 100, Low = 8038.2% Level: 100 – (100 – 80) × 0.382 = 100 – 7.64 = 92.3661.8% Level: 100 – (100 – 80) × 0.618 = 100 – 12.36 = 87.64 | Potential support/resistance levels |
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 > 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.
Machine Learning Applications in Technical Analysis
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.
| Algorithm | Operating Principle | Specific Application | Average Accuracy |
|---|---|---|---|
| Neural Networks (ANN) | y = f(Σ(wᵢxᵢ + b)) | Short-term price prediction based on 20 technical indicators | 58-65% |
| Random Forest | f = 1/n Σfᵢ(x) | Trend classification (up/down/sideways) | 65-72% |
| LSTM | Neural network with long-term “memory” capability | Complex time series analysis | 60-68% |
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.
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.
Building an Optimal Stock Portfolio Using Mathematics
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.
| Component | Formula | Real Calculation Example |
|---|---|---|
| Expected Portfolio Return | E(Rp) = Σ(wᵢ × E(Rᵢ)) | 2-stock portfolio: w₁ = 60%, E(R₁) = 12%; w₂ = 40%, E(R₂) = 8%E(Rp) = 0.6 × 12% + 0.4 × 8% = 7.2% + 3.2% = 10.4% |
| Portfolio Risk | σp² = Σi Σj (wᵢwⱼσᵢⱼ) | σ₁ = 20%, σ₂ = 15%, ρ₁₂ = 0.3σp² = (0.6)² × (20%)² + (0.4)² × (15%)² + 2 × 0.6 × 0.4 × 0.3 × 20% × 15%σp² = 0.0144 + 0.0036 + 0.00216 = 0.02016σp = √0.02016 = 14.2% |
| Sharpe Ratio | SR = (Rp – Rf) / σp | Rp = 10.4%, Rf = 4%, σp = 14.2%SR = (10.4% – 4%) / 14.2% = 6.4% / 14.2% = 0.45 |
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:
- Optimal weights (w₁, w₂) = (0.6; 0.4)
- Expected portfolio return = 0.6 × 12% + 0.4 × 8% = 10.4%
- Portfolio volatility = 14.2% (calculated using the formula above)
- Sharpe ratio = (10.4% – 4%) / 14.2% = 0.45
Quantitative Diversification Strategy
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:
| Number of Stocks | Reduction in Non-Systematic Risk | Real Example |
|---|---|---|
| 1 | 0% | 1-stock portfolio with σ = 30% |
| 5 | ~50% | 5-stock portfolio with average correlation 0.3:σ reduced from 30% to ~21% |
| 10 | ~65% | 10-stock portfolio with average correlation 0.3:σ reduced from 30% to ~18% |
| 20 | ~75% | 20-stock portfolio with average correlation 0.3:σ reduced from 30% to ~16.5% |
| 30+ | ~80% | 30+ stock portfolio with average correlation 0.3:σ reduced from 30% to ~15.5% |
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:
σp = √[n × (1/n)² × σ² + n × (n-1) × (1/n)² × ρ × σ²]
σp = √[10 × (0.1)² × (0.3)² + 10 × 9 × (0.1)² × 0.3 × (0.3)²]
σp = √[0.009 + 0.0243] = √0.0333 = 18.25%
This proves that diversification has helped reduce risk from 30% to 18.25% – a nearly 40% reduction without reducing expected returns.
Pocket Option provides automatic portfolio optimization tools, helping investors determine the optimal weight for each stock in their portfolio based on individual risk tolerance.
Fundamental Stock Analysis Using Quantitative Methods
Fundamental analysis when exploring who issues stocks focuses on intrinsic value based on quantitative financial factors. This method transforms financial reports into comparable metrics.
- DCF Model: Discounting future cash flows to present value
- Ratio Analysis: Comparing P/E, P/B, EV/EBITDA with industry averages
- Sustainable Growth Model: g = ROE × (1 – Payout Ratio)
- Z-Score: Predicting bankruptcy probability in the next 2 years
| Ratio Group | Formula | Real Calculation Example | Interpretation |
|---|---|---|---|
| Profitability | ROE = Net Profit / Equity | Profit: 100 billion, Equity: 500 billionROE = 100/500 = 20% | ROE > 15% is considered goodROE = 20% > 15% → High efficiency |
| Operational Efficiency | Asset Turnover = Revenue / Total Assets | Revenue: 800 billion, Total Assets: 1,000 billionTurnover = 800/1,000 = 0.8 | The company generates 0.8 units of revenue for each unit of assets – relatively good |
| Capital Structure | D/E Ratio = Total Debt / Equity | Total Debt: 300 billion, Equity: 500 billionD/E = 300/500 = 0.6 | D/E = 0.6 is in the safe zone (0.5-1.0) – balanced between debt and equity |
| Valuation | P/E = Price / EPS | Price: 60,000 VND, EPS: 5,000 VNDP/E = 60,000/5,000 = 12 | P/E = 12 lower than industry average (15) → Attractive valuation |
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.
The Gordon Growth Model provides a simple method to estimate stock value based on dividends:
P = D₁ / (r – g)
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 = 3,000 / (0.12 – 0.07) = 3,000 / 0.05 = 60,000 VND
At Pocket Option, we integrate automated fundamental valuation models, helping investors quickly assess the intrinsic value of stocks based on the latest financial data.
Methods for Measuring and Managing Stock Investment Risk
Investing in securities stocks needs to be accompanied by effective risk management. Quantitative methods help investors measure and control risk objectively.
- Value at Risk (VaR): Estimates maximum loss under normal market conditions
- Optimal Stop-Loss: Limits maximum loss for each trade
- Kelly Ratio: Determines optimal position size based on statistical edge
- Maximum Drawdown: The decline from peak to trough over a period
| Method | Formula | Real Calculation Example |
|---|---|---|
| Value at Risk (95%) | VaR = -1.65 × σ × √t × P | Portfolio 100 million, daily σ = 1.5%, time period 10 days:VaR = -1.65 × 1.5% × √10 × 100M = -1.65 × 0.015 × 3.16 × 100M = -7.82M→ 95% probability that loss will not exceed 7.82 million in 10 days |
| Optimal Stop-Loss | SL = P × (1 – 2 × ATR × √N) | Purchase price = 100,000 VND, ATR = 3%, N = 2 (confidence level):SL = 100,000 × (1 – 2 × 0.03 × √2) = 100,000 × (1 – 0.085) = 91,500 VND→ Set stop-loss at 91,500 VND |
| Kelly Ratio | f* = (p × b – q) / b | Win rate p = 55%, loss rate q = 45%, profit/loss ratio b = 1.5:f* = (0.55 × 1.5 – 0.45) / 1.5 = (0.825 – 0.45) / 1.5 = 0.25→ Should invest 25% of available capital |
| Maximum Drawdown | MDD = (Peak – Trough) / Peak | Portfolio peak = 120M, Trough = 90M:MDD = (120 – 90) / 120 = 30 / 120 = 25%→ Maximum drawdown is 25% |
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:
VaR = -1.65 × 1.5% × √10 × 100,000,000 = -7,820,000 VND
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.
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.
Pocket Option provides automated risk management tools, helping investors maintain trading discipline and protect capital under all market conditions.
Conclusion: Mathematical Approach to Stock Investment
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.
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.
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.
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.
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.
FAQ
What are stocks and how to assess their intrinsic value?
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 = D₁/(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.
Who issues stocks and how does the issuance process work?
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.
How to apply mathematics in technical analysis of stocks?
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)±2×σ 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.
How to optimize a stock portfolio based on mathematics?
Portfolio optimization uses Markowitz theory (MPT) by finding stock weights that maximize the Sharpe ratio SR=(Rp-Rf)/σp. 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.
What tools does Pocket Option provide for quantitative stock analysis?
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.