Portfolio Theory Basics: Risk and Return Explained

Portfolio theory, founded by Harry Markowitz in 1952, answers a simple question with mathematics: how should an investor combine risky assets so that expected return is as high as possible for a given level of risk? Risk is measured as the variability of returns, and the key insight is that combining assets that do not move in lockstep reduces overall variability.

This matters because it turned the folk wisdom of diversification into an optimization problem. It explains why a portfolio of two volatile stocks can be calmer than either stock alone, and it is the framework behind mutual funds, pension asset allocation, and robo-advisors.

Key Concepts

• Expected return E[R]: the probability-weighted average return of an asset or portfolio. Variance (σ²) and standard deviation (σ): measures of how far returns spread around the expectation; the standard quantitative definitions of risk.
• Portfolio weights (w_i): the fraction of wealth in each asset; they sum to 1.
• Covariance and correlation (ρ): measures of how two assets move together; ρ ranges from −1 to +1, and diversification works best when it is low.
• Systematic vs unsystematic risk: market-wide risk that diversification cannot remove versus firm-specific risk that it can.
• Efficient frontier: the set of portfolios offering the highest expected return for each risk level; rational investors choose from this curve.

Formulas

Portfolio expected return: E[R_p] = Σ w_i E[R_i]. Portfolio variance: σ_p² = Σ_i Σ_j w_i w_j σ_ij, where σ_ij is the covariance between assets i and j. For two assets: σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂. The analysis assumes returns are summarized adequately by mean and variance and that weights sum to one; it does not assume normality.

Worked Example

Asset A has expected return 10% with standard deviation 15%; asset B has expected return 6% with standard deviation 8%; their correlation is ρ = 0.2. Build a portfolio with 60% in A and 40% in B. Expected return: E[R_p] = 0.6 × 10% + 0.4 × 6% = 6% + 2.4% = 8.4%. Variance: σ_p² = (0.6)²(0.15)² + (0.4)²(0.08)² + 2(0.6)(0.4)(0.2)(0.15)(0.08). Compute term by term: 0.36 × 0.0225 = 0.0081; 0.16 × 0.0064 = 0.001024; the cross term is 0.48 × 0.2 × 0.012 = 0.001152. The sum is 0.010276, so σ_p = √0.010276 = 10.14%. The weighted average of the two standard deviations would be 0.6 × 15% + 0.4 × 8% = 12.2%, so diversification cut risk by about 2 percentage points while keeping the 8.4% expected return. If correlation were +1, no reduction would occur; if it were −1, risk could be eliminated entirely.

Common Misconceptions

• Diversification eliminates all risk. It removes only firm-specific risk; market-wide systematic risk remains no matter how many assets you hold.
• Adding more stocks always helps equally. Most diversification benefit arrives with the first 20 to 30 stocks; beyond that, extra holdings add little.
• Low correlation means the assets are unrelated. Correlation measures only linear co-movement; assets can be nonlinearly linked yet show low correlation.

Real-World Applications

Portfolio theory drives mutual fund construction, pension and endowment asset allocation, robo-advisor portfolios, and risk budgeting in banks; the efficient frontier is the standard picture in every investment policy statement.

Portfolio theory proves that risk depends on how assets move together, not just on each asset alone, so a well-chosen mix can deliver the same expected return with less volatility than any single holding.