The Mathematical Formula for Excellent Investment in Stocks

Note: These mathematical principles are for informational purposes only and do not constitute investment or other type of advice.

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Discover the mathematical principles behind great stock investments – from risk-return analysis and the Sharpe ratio to compound interest and portfolio optimization. Learn how to apply proven formulas to invest smarter.

Introduction

Stock market investing combines logic, discipline, and timing. While market behavior often seems unpredictable, mathematics provides a rational framework that helps investors assess risk, measure returns, and determine long-term value. This article explores how key mathematical formulas can guide great investment decisions and create sustainable wealth.

Understanding the Risk-Return Relationship

At the heart of any investment is the balance between risk and reward. Investors need to quantify both before making decisions.

Expected Return Formula:

ER = Σ (p_i × r_i)

Where:

• pᵢ = probability of each outcome
• rᵢ = return for each outcome

This formula helps investors quantify potential gains relative to their likelihood. However, risk must also be measured. The standard deviation (σ) of returns serves as a key indicator of volatility:

σ = √Σ (r_i − ER)² / n

The power of “n” (the denominator) depends on what you are measuring:

• If you are measuring the entire population of returns (e.g., all possible outcomes), divide by n.
• If you are using a sample of returns (e.g., past monthly returns on an investment), then divide by (n − 1) instead of n.

Measuring Efficiency with the Sharpe Ratio

The Sharpe Ratio evaluates how efficiently an investment generates returns for its level of risk.

S = (R_p − R_f) / σ_p

Where:

• Rₚ = portfolio return
• R𝑓 = risk-free rate (e.g., U.S. Treasury yield)
• σₚ = portfolio volatility

The Power of Compounding

Compound growth is the quiet force behind long-term wealth creation.

Compound Interest Formula:

A = P(1 + r/n)^nt

Where:

• P = principal
• r = annual interest rate
• n = compounding periods per year
• t = number of years

Valuation Through the Price-to-Earnings Ratio

To judge whether a stock is fairly priced, investors often rely on the P/E ratio:

P/E = Market Price per Share / Earnings per Share (EPS)

A lower P/E may indicate value, while a higher P/E can signal growth expectations.
However, the “excellent” investor looks beyond the surface, comparing a company’s PEG ratio (P/E divided by earnings growth rate) to find stocks offering both value and growth potential.

Modern Portfolio Theory and Diversification

Developed by Harry Markowitz, Modern Portfolio Theory (MPT) mathematically defines how diversification can reduce overall risk.

Expected Portfolio Return:

E(R_p) = Σ w_i × E(R_i)

The expected return of the entire portfolio (E(Rp)) is the weighted average of the expected returns of each individual asset.

Symbol	Meaning	Explanation
E(Rp)	Expected portfolio return	The overall return you anticipate from your portfolio.
E(Ri)	Expected return of asset i	The expected (average) return of each stock or investment.
wi	Weight of asset i	The proportion of your total investment allocated to that asset (for example, if you invest 40% in Apple, then wi = 0.4).

Portfolio Variance:

σ_p² = ΣΣ w_iw_jCov(R_i, R_j)

Portfolio variance (σp²) measures the total risk (volatility) of your portfolio, considering how assets move together.

Symbol	Meaning	Explanation
σp²	Portfolio variance	The overall risk (volatility) of the portfolio.
wi, wj	Weights of assets i and j	How much of the portfolio is invested in each pair of assets.
Ri, Rj	Returns of assets i and j	The random returns (performance) of the two assets.
Cov(Ri, Rj)	Covariance between assets i and j	Measures how the two assets move together – if they both rise/fall together (positive) or in opposite directions (negative).

The Ultimate Formula for Excellent Investment

All principles can be summarized in one integrated expression:

Excellent Investment = (V × T) / R

Where:

• V = intrinsic value of the stock
• T = time horizon for compounding
• R = total risk (volatility, debt, uncertainty)

In simpler terms, excellence in investing emerges when a well-valued stock is held patiently through compounding cycles, under controlled risk.

Conclusion

There is no single, magic formula for perfect stock market success. Yet mathematics offers a compass. It quantifies risk, clarifies opportunities, and guides rational decision-making. From expected returns and Sharpe ratios to compounding and portfolio optimization, these formulas provide the framework for disciplined and intelligent investing.

In the long run, the most “excellent” investment isn’t just mathematical – it’s patient, consistent, and rooted in value.