Portfolio Expected Return & Variance — Compute expected return and variance for a portfolio (intro) - Finance

SECTION 1

Historical Context & Motivation

Before the mid-twentieth century, investment decision-making was largely qualitative—security analysts evaluated companies one at a time and assembled portfolios based on intuition, industry familiarity, or simple rules of thumb such as "don't put all your eggs in one basket." There was no rigorous mathematical framework for quantifying how much risk a combination of assets actually carried, nor for demonstrating why spreading capital across multiple securities could reduce that risk without necessarily sacrificing return. Harry Markowitz changed the field permanently in 1952 when he published "Portfolio Selection" in The Journal of Finance, introducing the mean-variance framework that underpins virtually all modern portfolio theory.

Markowitz's insight was deceptively simple: an investor should care not only about each security's expected return but also about how each security's returns move relative to every other security in the portfolio—their covariances. By expressing portfolio risk as a function of individual variances and pairwise covariances, Markowitz demonstrated that diversification is not merely folk wisdom but a mathematically provable strategy for improving the risk-return trade-off. The timeline below traces the intellectual milestones that led to—and built upon—this breakthrough.

1952 • Markowitz's "Portfolio Selection"

Harry Markowitz introduces the mean-variance framework, demonstrating that rational investors should evaluate portfolios by their expected return and variance rather than selecting securities in isolation.

1958 • Tobin's Separation Theorem

James Tobin extends Markowitz's work by showing that every investor's optimal portfolio is a combination of a risk-free asset and a single risky portfolio, separating the investment decision from the financing decision.

1964 • Capital Asset Pricing Model (CAPM)

William Sharpe, John Lintner, and Jan Mossin independently develop CAPM, linking expected return to systematic risk (beta) and building directly on Markowitz's portfolio variance mechanics.

1990 • Nobel Prize in Economics

Markowitz, Sharpe, and Merton Miller share the Nobel Memorial Prize, validating decades of mean-variance research and cementing portfolio theory as a cornerstone of financial economics.

The central question this lesson addresses is straightforward yet foundational: given a set of assets with known expected returns, variances, and covariances, how do we compute the expected return and variance of the portfolio they form? Mastering these calculations is the prerequisite for efficient frontier construction, CAPM derivation, and virtually every advanced topic in investments.

SECTION 2

Core Principles & Definitions

Before diving into formulas, it is essential to anchor the discussion in a handful of foundational concepts. These definitions recur throughout portfolio theory and will serve as the building blocks for every calculation that follows.

Portfolio Weight (wᵢ)

The fraction of total portfolio value invested in asset i. Weights must sum to 1 (100 %). A weight of 0.40 means 40 % of capital is allocated to that asset.

Expected Return E(Rᵢ)

The probability-weighted average of all possible returns for asset i. It represents the return you anticipate receiving on average over many periods.

Variance (σ²) & Standard Deviation (σ)

Variance measures the dispersion of returns around the expected value. Standard deviation is its square root, expressed in the same units as return, making it easier to interpret.

Covariance (σᵢⱼ)

A measure of how two assets' returns move together. Positive covariance means they tend to rise and fall in tandem; negative covariance means they tend to move in opposite directions.

Correlation (ρᵢⱼ)

A standardized version of covariance, bounded between −1 and +1. It makes cross-asset dependence easy to compare regardless of the assets' individual volatilities.

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KEY TAKEAWAY

Think of a portfolio like a recipe. Each ingredient (asset) contributes flavor (return) in proportion to how much you add (weight). But the texture of the dish (risk) doesn't just depend on each ingredient alone—it depends on how they interact. Sugar and lemon juice together create a balanced sweet-tart profile neither achieves alone. Similarly, combining assets whose returns are not perfectly correlated can produce a portfolio with lower variance than any individual asset, even without reducing expected return.

SECTION 3

Visualizing Portfolio Construction

The diagram below illustrates how two individual assets—Stock A and Stock B—combine to form portfolios along a curve in risk-return space. The horizontal axis represents portfolio standard deviation (risk), and the vertical axis represents portfolio expected return. As the weight in each asset changes from 100 % A / 0 % B to 0 % A / 100 % B, the resulting portfolio traces out a curve whose shape depends critically on the correlation between the two assets. When correlation is less than +1, the curve bends to the left, demonstrating the diversification benefit—some combinations of A and B have lower risk than either asset alone.

The curved line shows portfolios formed by varying the weights in Stock A and Stock B when ρ < +1. The dashed red line shows the frontier when ρ = +1, where no diversification benefit exists. The green dot marks the minimum-variance portfolio—the combination with the lowest possible risk.

The curvature of the frontier is the visual manifestation of the core insight: because portfolio variance depends on covariance (not just individual variances), mixing assets with imperfect correlation creates risk-return combinations that fall to the left of the straight line connecting the two assets. The further the curve bends leftward, the greater the diversification benefit. This observation motivates the precise formulas we develop in the next section.

SECTION 4

Mathematical Framework

We now formalize the two key portfolio statistics—expected return and variance—beginning with the general n-asset case before specializing to the two-asset case that dominates introductory applications.

Portfolio Expected Return

PORTFOLIO EXPECTED RETURN (N ASSETS)

E(Rₚ) = Σᵢ wᵢ × E(Rᵢ) = w₁E(R₁) + w₂E(R₂) + … + wₙE(Rₙ)

where wᵢ = weight of asset i in the portfolio, and E(Rᵢ) = expected return of asset i. The expected return of a portfolio is simply a weighted average—no interaction terms appear.

Note the elegant simplicity: expected return is linear in weights. If you invest 60 % in Asset A yielding 8 % and 40 % in Asset B yielding 12 %, your portfolio's expected return is 0.60 × 8 % + 0.40 × 12 % = 9.6 %. No surprises, no interaction effects—just a straight blend.

Portfolio Variance — Two-Asset Case

PORTFOLIO VARIANCE (TWO ASSETS)

σ²ₚ = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁₂

where σ₁² and σ₂² are the variances of assets 1 and 2, and σ₁₂ is the covariance between assets 1 and 2. Equivalently, σ₁₂ = ρ₁₂ × σ₁ × σ₂.

PORTFOLIO VARIANCE — CORRELATION FORM

σ²ₚ = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂

This equivalent form substitutes correlation ρ₁₂ for raw covariance, which is often more intuitive because ρ is bounded between −1 and +1.

Unlike expected return, portfolio variance is not simply a weighted average of individual variances. The cross-product term (2w₁w₂ρ₁₂σ₁σ₂) is the mathematical engine of diversification. When ρ₁₂ < 1, this term reduces the portfolio's variance below the weighted sum of individual variances. When ρ₁₂ = −1, it is theoretically possible to construct a zero-variance portfolio—a result that underscores why correlation is the single most important input in portfolio construction.

PORTFOLIO STANDARD DEVIATION

σₚ = √(σ²ₚ)

Standard deviation is reported in the same percentage units as returns, making it the more practical risk measure for communication and comparison.

SECTION 5

How Correlation Shapes Portfolio Risk

The correlation coefficient ρ₁₂ is the lever that determines how much diversification benefit a two-asset portfolio captures. To build intuition, the diagram below plots portfolio standard deviation against the weight in Asset A for three correlation scenarios: ρ = +1, ρ = 0, and ρ = −1. Both assets are assumed to have a standard deviation of 20 % so that differences in the curves arise solely from correlation.

When ρ = +1 (red), portfolio σ is a simple weighted average—no diversification benefit. When ρ = 0 (amber), the minimum-variance portfolio achieves σ ≈ 14.1 %, a 30 % reduction. When ρ = −1 (green), risk can be eliminated entirely at w₁ = 0.50.

Correlation (ρ₁₂)	Diversification Effect	Minimum σₚ (equal σ case)
+1	None — portfolio σ equals the weighted average of individual σ values	20 % (no reduction possible)
+0.5	Moderate — curve bends left of the straight line	≈ 17.3 %
0	Significant — substantial risk reduction at the optimum mix	≈ 14.1 %
−0.5	Large — portfolio risk drops well below either asset's σ	≈ 10.0 %
−1	Maximum — risk can be completely eliminated	0 %

Impact of correlation on portfolio risk when both assets have σ = 20 %

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Why ρ = −1 Is Rare in Practice

In real financial markets, perfectly negative correlation between two risky assets is essentially nonexistent. Most equity pairs exhibit positive correlations, often in the range of 0.3 to 0.7. Commodities, real estate, and certain fixed-income instruments may offer lower or modestly negative correlations versus equities, but the theoretical ρ = −1 case serves primarily as a boundary illustration rather than a practical investment scenario.

SECTION 6

Worked Example — Two-Stock Portfolio

Suppose an investor allocates 60 % of her wealth to Stock X and 40 % to Stock Y. The following data are given: E(R_X) = 10 %, σ_X = 15 %, E(R_Y) = 14 %, σ_Y = 25 %, and ρ_XY = 0.30. We seek the portfolio's expected return, variance, and standard deviation.

Computing E(Rₚ), σ²ₚ, and σₚ

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Step 1 — Identify Given Values

w_X = 0.60, w_Y = 0.40, E(R_X) = 0.10, E(R_Y) = 0.14, σ_X = 0.15, σ_Y = 0.25, ρ_XY = 0.30. Verify weights sum to 1: 0.60 + 0.40 = 1.00 ✓
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Step 2 — Compute Portfolio Expected Return

E(Rₚ) = w_X × E(R_X) + w_Y × E(R_Y) = (0.60)(0.10) + (0.40)(0.14) = 0.060 + 0.056 = 0.116

E(Rₚ) = 11.6 %
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Step 3 — Compute Portfolio Variance

σ²ₚ = w²_Xσ²_X + w²_Yσ²_Y + 2w_Xw_Yρ_XYσ_Xσ_Y = (0.36)(0.0225) + (0.16)(0.0625) + 2(0.60)(0.40)(0.30)(0.15)(0.25) = 0.00810 + 0.01000 + 0.00540 = 0.02350

σ²ₚ = 0.02350
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Step 4 — Compute Portfolio Standard Deviation

σₚ = √(0.02350) = 0.15330…

σₚ ≈ 15.33 %
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Step 5 — Interpret the Results

The portfolio's expected return of 11.6 % falls between the two individual expected returns (10 % and 14 %), as it must for a long-only portfolio. However, the portfolio standard deviation of 15.33 % is lower than the weighted-average standard deviation (0.60 × 15 % + 0.40 × 25 % = 19 %). This 3.67-percentage-point reduction in risk—without sacrificing the weighted-average return—is the diversification benefit at work, enabled by the imperfect correlation of ρ = 0.30.

SECTION 7

Strengths & Limitations of Mean-Variance Analysis

The mean-variance framework is elegant and powerful, but like any model it rests on assumptions that may not hold perfectly in practice. Understanding these strengths and limitations prepares you to apply the framework with appropriate caution and to appreciate the more advanced models that extend it.

Strengths	Limitations
Provides a rigorous, quantitative basis for diversification—replaces intuition with mathematics.	Assumes returns are normally distributed; real-world returns exhibit skewness and fat tails (kurtosis).
Uses only two parameters (mean and variance), making computation tractable even for large portfolios.	Requires accurate estimates of expected returns, variances, and covariances—all of which are notoriously unstable over time.
Foundational for CAPM, efficient frontier analysis, and performance measurement (Sharpe ratio).	Treats variance symmetrically—penalizes upside deviations the same as downside, even though investors typically dislike losses more than they enjoy gains.
Enables construction of the efficient frontier and identification of the optimal risky portfolio.	Single-period model; does not address multi-period rebalancing, transaction costs, or changing investor preferences over time.

Key strengths and limitations of the mean-variance framework

✦

KEY TAKEAWAY

Mean-variance analysis is analogous to a structural engineer's use of simplified beam models: it captures the dominant forces (expected return and risk) with remarkable clarity, even though real structures face wind loads, temperature fluctuations, and material imperfections not in the basic model. The simplified model is indispensable for design, but the engineer knows its boundaries—and so should you as a finance professional.

SECTION 8

Connection to Advanced Portfolio Theory

The two-asset calculations introduced here are the building blocks for far more sophisticated frameworks. As you advance in your finance coursework, you will encounter n-asset optimization, where the variance formula expands into matrix notation, and the efficient frontier becomes a computational rather than algebraic exercise. The table below previews how the introductory concepts map to their advanced counterparts.

Introductory Concept	Advanced Extension
Two-asset portfolio variance formula	Matrix form: σ²ₚ = w'Σw, where Σ is the n × n covariance matrix and w is the weight vector.
Minimum-variance portfolio (two assets)	Global minimum-variance portfolio via constrained optimization (Lagrangian or quadratic programming).
Diversification reduces σ when ρ < 1	Efficient frontier and Markowitz bullet: the set of portfolios offering the highest return for each level of risk.
Portfolio expected return as a weighted average	Capital Market Line (CML) combining the risk-free asset with the tangency portfolio; CAPM pricing of individual securities via beta.
Variance as the sole risk metric	Downside risk measures (semivariance, Value at Risk, CVaR), multi-factor models (Fama–French), and behavioral finance critiques.

From introductory to advanced portfolio theory

Every one of these advanced topics rests on the same foundational algebra you have learned in this lesson. Mastering the two-asset portfolio variance calculation—understanding why the cross-product term matters and how correlation governs diversification—ensures you will be well prepared to navigate the matrix algebra, optimization, and equilibrium pricing models that define the investments curriculum.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why the expected return of a portfolio is always a weighted average of the expected returns of its constituent assets, whereas portfolio variance is not simply a weighted average of individual variances. What additional factor must be considered for variance, and why does it matter?

Show answer

Expected return is a linear operator, meaning the expectation of a sum equals the sum of expectations. Therefore, E(Rₚ) = Σ wᵢ E(Rᵢ), with no interaction terms. Variance, however, measures the dispersion of returns, and when two random variables are combined, their joint behavior—captured by covariance—affects the total spread. The portfolio variance formula includes cross-product terms (2wᵢwⱼσᵢⱼ) that account for how each pair of assets co-moves. When covariance is less than the product of individual standard deviations (i.e., ρ < 1), these cross terms reduce overall portfolio variance below the weighted average of individual variances, which is the mathematical basis of the diversification benefit.

Question 2: BASIC CALCULATION

A portfolio consists of 50 % in Asset A (E(R) = 8 %, σ = 12 %) and 50 % in Asset B (E(R) = 14 %, σ = 22 %). The correlation between A and B is 0.40. Calculate the portfolio's expected return and standard deviation.

Show answer

E(Rₚ) = 0.50 × 0.08 + 0.50 × 0.14 = 0.04 + 0.07 = 0.11, or 11 %. For variance: σ²ₚ = (0.50)²(0.12)² + (0.50)²(0.22)² + 2(0.50)(0.50)(0.40)(0.12)(0.22) = 0.0036 + 0.0121 + 0.00528 = 0.02098. Therefore σₚ = √0.02098 ≈ 0.1449, or about 14.49 %. Notice that the simple weighted-average σ would be 0.50 × 12 % + 0.50 × 22 % = 17 %, so the portfolio achieves a risk reduction of roughly 2.5 percentage points thanks to the imperfect correlation.

Question 3: INTERMEDIATE

Using the same assets from Problem 2 (Asset A: E(R) = 8 %, σ = 12 %; Asset B: E(R) = 14 %, σ = 22 %; ρ = 0.40), derive the weight in Asset A that minimizes portfolio variance. Then compute the expected return and standard deviation of this minimum-variance portfolio.

Show answer

The minimum-variance weight for asset A in a two-asset portfolio is w*₁ = (σ₂² − σ₁₂) / (σ₁² + σ₂² − 2σ₁₂). First, compute σ₁₂ = ρ × σ₁ × σ₂ = 0.40 × 0.12 × 0.22 = 0.01056. Then: w*_A = (0.0484 − 0.01056) / (0.0144 + 0.0484 − 2 × 0.01056) = 0.03784 / 0.04168 ≈ 0.9079, or about 90.8 %. So w*_B ≈ 9.2 %. E(Rₚ) = 0.9079 × 0.08 + 0.0921 × 0.14 ≈ 0.0726 + 0.0129 ≈ 8.55 %. σ²ₚ = (0.9079)²(0.0144) + (0.0921)²(0.0484) + 2(0.9079)(0.0921)(0.01056) ≈ 0.01187 + 0.000411 + 0.001766 ≈ 0.01405. σₚ ≈ √0.01405 ≈ 11.85 %. The minimum-variance portfolio has lower risk than either asset alone.

Question 4: APPLIED

An equity fund manager holds 70 % in a domestic equity index (E(R) = 9 %, σ = 18 %) and 30 % in an international equity index (E(R) = 11 %, σ = 24 %). Historical data suggest ρ = 0.55. A new analyst argues the manager should increase the international allocation to 50 % to raise expected return. Calculate the expected return and σ for both the current and proposed allocations, and discuss whether the rebalance is unambiguously beneficial.

Show answer

Current portfolio (70/30): E(Rₚ) = 0.70 × 0.09 + 0.30 × 0.11 = 0.063 + 0.033 = 9.6 %. σ²ₚ = (0.49)(0.0324) + (0.09)(0.0576) + 2(0.70)(0.30)(0.55)(0.18)(0.24) = 0.015876 + 0.005184 + 0.009979 = 0.031039. σₚ ≈ 17.62 %. Proposed portfolio (50/50): E(Rₚ) = 0.50 × 0.09 + 0.50 × 0.11 = 10.0 %. σ²ₚ = (0.25)(0.0324) + (0.25)(0.0576) + 2(0.50)(0.50)(0.55)(0.18)(0.24) = 0.0081 + 0.0144 + 0.01188 = 0.03438. σₚ ≈ 18.54 %. The proposed allocation raises expected return from 9.6 % to 10.0 % but also increases portfolio σ from 17.62 % to 18.54 %. The rebalance is not unambiguously beneficial—it depends on the investor's risk tolerance. A risk-averse investor may prefer the lower σ at a modest cost in return, while a more aggressive investor may welcome the higher expected return despite the added volatility.

Question 5: CRITICAL THINKING

A colleague claims: "If I hold 100 stocks instead of 2, I can eventually diversify away all risk." Using the concept of portfolio variance for n equally weighted assets (σ²ₚ = (1/n)σ̄² + (1 − 1/n)Cov̄, where σ̄² is the average variance and Cov̄ is the average covariance), explain why this claim is incorrect. What type of risk can be diversified away, and what type cannot? Relate your answer to the distinction between systematic and unsystematic risk.

Show answer

The formula σ²ₚ = (1/n)σ̄² + (1 − 1/n)Cov̄ reveals that as n grows very large, the first term (1/n)σ̄² approaches zero, but the second term approaches Cov̄, which is generally positive because most stocks share exposure to common macroeconomic factors (GDP growth, interest rates, inflation). Therefore, portfolio variance converges to the average covariance, not to zero. The risk that can be diversified away—the (1/n)σ̄² component—is called unsystematic (or idiosyncratic) risk, and it represents company-specific events like product recalls or management changes. The residual risk that remains—Cov̄, the average covariance—is systematic (or market) risk, driven by economy-wide forces. This distinction is foundational to the Capital Asset Pricing Model, which posits that only systematic risk is priced in equilibrium because rational investors can (and should) diversify away idiosyncratic risk.

SUMMARY

Lesson Summary

Summary

This lesson introduced the foundational computations of portfolio expected return and portfolio variance within the mean-variance framework pioneered by Harry Markowitz in 1952. Portfolio expected return is a weighted average of individual asset expected returns, making it straightforward to calculate. Portfolio variance, in contrast, incorporates covariance (or equivalently, correlation) between assets, and it is this cross-product term that generates the diversification benefit—the ability to reduce portfolio risk below the weighted average of individual risks whenever correlation is less than +1.

For the two-asset case, the key formulas are E(Rₚ) = w₁E(R₁) + w₂E(R₂) and σ²ₚ = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂. The minimum-variance portfolio can be found by optimizing the weight formula, and the risk-return frontier visualizes the full set of attainable portfolios. These introductory concepts scale directly to n-asset optimization via matrix algebra and form the bedrock of the Capital Asset Pricing Model, efficient frontier analysis, and modern portfolio management.