Portfolio Theory And Optimization

30 essential concepts in portfolio theory and optimization

What You'll Learn

Master Modern Portfolio Theory with 30 comprehensive flashcards covering MPT, CAPM, efficient frontier, alpha/beta, diversification, correlation, portfolio variance, and mean-variance optimization. Essential for finance and quant roles.

Key Topics

  • Modern Portfolio Theory (MPT) framework and principles
  • CAPM model and alpha/beta calculations
  • Efficient frontier and mean-variance optimization
  • Portfolio variance and correlation mathematics
  • Systematic vs unsystematic risk and diversification
  • Sharpe ratio and risk-adjusted returns

Looking for more finance resources? Visit the Explore page to browse related decks or use the Create Your Own Deck flow to customize this set.

How to study this deck

Start with a quick skim of the questions, then launch study mode to flip cards until you can answer each prompt without hesitation. Revisit tricky cards using shuffle or reverse order, and schedule a follow-up review within 48 hours to reinforce retention.

Preview: Portfolio Theory And Optimization

Question

What is Modern Portfolio Theory (MPT)?

Answer

Framework developed by Harry Markowitz for portfolio construction: Key principles: 1. Investors are risk-averse 2. Return alone is insufficient - must consider risk 3. Diversification reduces risk 4. Optimal portfolio balances risk-return tradeoff Goal: Maximize return for given risk level OR minimize risk for given return level

Question

What is portfolio diversification and why does it reduce risk?

Answer

Diversification: Holding multiple assets instead of one Why it reduces risk: - Assets don't move perfectly together - Losses in one offset by gains in another - Reduces unsystematic (idiosyncratic) risk - Only systematic (market) risk remains Key insight: Portfolio risk < Sum of individual risks "Don't put all eggs in one basket"

Question

What is the Efficient Frontier?

Answer

Set of optimal portfolios that offer: - Highest expected return for given risk level, OR - Lowest risk for given expected return level Properties: - Curved line in risk-return space - Portfolios below frontier are suboptimal - Portfolios above frontier are unattainable - Rational investors only choose portfolios on frontier Found through mean-variance optimization

Question

What is the risk-return tradeoff?

Answer

Fundamental principle: Higher expected returns require accepting higher risk Relationship: - Low risk → Low return (Treasury bills) - High risk → High return potential (stocks, crypto) - No free lunch: Can't get high return without risk Investor choice depends on risk tolerance: - Risk-averse: Prefer safer, lower-return assets - Risk-seeking: Accept volatility for higher returns

Question

What is systematic risk vs unsystematic risk?

Answer

Systematic Risk (Market Risk): - Affects entire market - Cannot be diversified away - Examples: Recession, interest rates, inflation - Measured by Beta (β) Unsystematic Risk (Idiosyncratic Risk): - Specific to individual asset/company - CAN be diversified away - Examples: CEO resignation, product failure - Eliminated with ~20-30 stocks Diversification only reduces unsystematic risk

Question

What is Beta (β) and what does it measure?

Answer

β measures systematic risk - sensitivity to market movements: β = Cov(stock, market) / Var(market) Interpretation: - β = 1: Moves with market (average risk) - β > 1: More volatile than market (high risk) - β < 1: Less volatile than market (low risk) - β = 0: Uncorrelated with market - β < 0: Moves opposite to market (rare) Example: β = 1.5 means stock moves 1.5× market movement

Question

What is Alpha (α) and what does it represent?

Answer

α measures excess return beyond what Beta predicts: α = Actual Return - Expected Return from CAPM Interpretation: - α > 0: Outperforming (skill/luck) - α = 0: Performing as expected - α < 0: Underperforming Represents: - Manager skill - Security selection ability - Return not explained by market exposure Goal of active management: Generate positive alpha

Question

What is CAPM (Capital Asset Pricing Model)?

Answer

Model relating expected return to systematic risk: E[R_stock] = R_f + β × (E[R_market] - R_f) OR rearranged: R_stock = α + β × R_market + ε Components: - R_f: Risk-free rate - β: Systematic risk - (E[R_market] - R_f): Market risk premium - α: Excess return (should be 0 in theory) - ε: Random error Core principle: Return = Risk-free rate + Risk premium

Question

How do you calculate portfolio Beta?

Answer

Portfolio Beta = Weighted average of individual Betas: β_portfolio = Σ(w_i × β_i) where: - w_i = weight of asset i in portfolio - β_i = Beta of asset i Example: - 60% in stock with β=1.2 - 40% in stock with β=0.8 - Portfolio β = 0.6(1.2) + 0.4(0.8) = 1.04

Question

How do you calculate portfolio expected return?

Answer

Portfolio Expected Return = Weighted average of individual returns: E[R_p] = Σ(w_i × E[R_i]) where: - w_i = weight of asset i - E[R_i] = expected return of asset i - Σw_i = 1 (weights sum to 100%) Example: - 50% stock A (E[R]=10%) - 50% stock B (E[R]=6%) - Portfolio E[R] = 0.5(10%) + 0.5(6%) = 8%

Question

How do you calculate portfolio variance (risk)?

Answer

Portfolio Variance (2 assets): σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂) OR using correlation: σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂ Key insight: - Variance depends on correlations - Low correlation → lower portfolio variance - Diversification benefit from correlation < 1 Portfolio risk ≠ weighted average of individual risks

Question

What is correlation and how does it affect diversification?

Answer

Correlation (ρ): Measures how assets move together Range: -1 to +1 - ρ = +1: Perfect positive (move together) - ρ = 0: Uncorrelated (independent) - ρ = -1: Perfect negative (move opposite) Diversification benefit: - ρ < 1: Diversification reduces risk - ρ = 1: No diversification benefit - ρ < 0: Maximum diversification benefit Goal: Combine low-correlated assets

Question

What is the Sharpe Ratio in portfolio context?

Answer

Sharpe Ratio = (E[R_p] - R_f) / σ_p Measures risk-adjusted return: - Excess return per unit of total risk - Higher is better - Used to compare portfolios Key insight: - Portfolio on efficient frontier has highest Sharpe ratio for given risk - Optimal portfolio maximizes Sharpe ratio Also called reward-to-variability ratio

Question

What is the Information Ratio?

Answer

Information Ratio (IR) = E[R_p - R_b] / σ(R_p - R_b) OR IR = α / σ(ε) Measures: - Excess return over benchmark per unit of tracking error - Manager skill in generating alpha - Risk-adjusted alpha Interpretation: - Higher IR = better active management - IR > 0.5: Good - IR > 1.0: Excellent Similar to Sharpe ratio but focuses on alpha

Question

What is tracking error?

Answer

Tracking Error = Standard deviation of (Portfolio return - Benchmark return) TE = σ(R_p - R_b) Measures: - How closely portfolio follows benchmark - Active risk taken by manager - Consistency of outperformance Interpretation: - Low TE: Portfolio tracks benchmark closely - High TE: Portfolio deviates significantly Used in denominator of Information Ratio

Question

What is mean-variance optimization?

Answer

Mathematical approach to find optimal portfolio weights: Objective: Minimize: w^T Σ w (portfolio variance) Subject to: w^T μ = target return Σw_i = 1 (fully invested) w_i ≥ 0 (no short selling, optional) Inputs: - μ: Expected returns vector - Σ: Covariance matrix - Target return or risk level Output: Optimal weights for efficient frontier Foundation of Modern Portfolio Theory

Question

What are the problems with mean-variance optimization?

Answer

Major issues: 1. Garbage In, Garbage Out: - Sensitive to input estimates - Small errors in expected returns → huge weight changes 2. Estimation Error: - Need to estimate returns and covariances - Historical estimates may not hold 3. Concentrated Portfolios: - Often suggests extreme weights - Not practically implementable 4. Instability: - Optimal weights change dramatically over time - High turnover, transaction costs 5. Ignores other factors: - Transaction costs - Taxes - Liquidity

Question

What is the covariance matrix and why is it important?

Answer

Covariance Matrix (Σ): Contains all pairwise covariances For n assets, n×n matrix: - Diagonal: Variances (σ_i²) - Off-diagonal: Covariances (Cov(i,j)) Importance: - Required for portfolio variance calculation - Captures diversification benefits - Input to mean-variance optimization Challenges: - n assets → n(n+1)/2 parameters to estimate - 100 stocks → 5,050 parameters! - Estimation error compounds - Noisy estimates lead to poor optimization

Question

What is Hierarchical Risk Parity (HRP)?

Answer

Alternative portfolio construction method that: 1. Uses hierarchical clustering 2. Groups similar assets 3. Allocates based on risk contribution 4. No need for expected returns (only covariance) Advantages over mean-variance: ✓ More stable weights ✓ Less sensitive to estimation error ✓ Better out-of-sample performance ✓ No need to estimate expected returns ✓ Considers asset relationships Disadvantages: ✗ More complex ✗ Less theoretical foundation

Question

What is risk parity?

Answer

Portfolio construction where each asset contributes equally to total risk: Goal: Equal risk contribution from all assets Not equal weights! - Low-risk assets get higher weights - High-risk assets get lower weights Risk contribution: RC_i = w_i × (∂σ_p/∂w_i) Set RC_1 = RC_2 = ... = RC_n Popular in institutional investing

Question

What is the tangency portfolio?

Answer

Portfolio on efficient frontier with highest Sharpe ratio: - Tangent to efficient frontier from risk-free rate - Optimal risky portfolio for all investors - Combination of tangency portfolio + risk-free asset forms Capital Market Line Properties: - Maximum risk-adjusted return - Same for all investors (in theory) - Market portfolio in CAPM equilibrium Investors adjust risk by lending/borrowing at R_f

Question

What is the Capital Market Line (CML)?

Answer

Line connecting risk-free asset to tangency portfolio: E[R] = R_f + (E[R_m] - R_f)/σ_m × σ Represents: - All efficient portfolios for investors - Combination of tangency portfolio + risk-free asset - Trade-off between risk and return for efficient portfolios Slope = Sharpe ratio of market portfolio Investors choose point on CML based on risk tolerance

Question

What is the difference between CML and Security Market Line (SML)?

Answer

Capital Market Line (CML): - Risk = Total risk (standard deviation) - Only for efficient portfolios - Vertical axis: Expected return - Horizontal axis: Total risk (σ) Security Market Line (SML): - Risk = Systematic risk (Beta) - For all assets (efficient or not) - Vertical axis: Expected return - Horizontal axis: Beta (β) - Graphical representation of CAPM CML: Portfolio choice SML: Asset pricing

Question

What is the market portfolio?

Answer

Theoretical portfolio containing all risky assets: Properties: - Weighted by market capitalization - Lies on efficient frontier - Tangency portfolio in CAPM - Beta = 1 by definition In practice: - Approximated by broad market index (S&P 500, Total Stock Market) - Cannot truly hold ALL assets globally CAPM assumes all investors hold market portfolio

Question

What is portfolio rebalancing and why is it needed?

Answer

Rebalancing: Adjusting portfolio weights back to target allocation Why needed: - Asset values change → weights drift - High-return assets become overweight - Low-return assets become underweight - Risk profile changes Rebalancing strategies: 1. Calendar-based: Monthly, quarterly, annually 2. Threshold-based: When weights drift >5% 3. Opportunistic: During market extremes Trade-off: Maintaining allocation vs transaction costs

Question

What is Monte Carlo simulation for portfolio analysis?

Answer

Generate synthetic data to test portfolio strategies: Process: 1. Estimate return distribution (mean, covariance) 2. Generate random returns from distribution 3. Simulate portfolio performance 4. Repeat thousands of times 5. Analyze distribution of outcomes Uses: - Test robustness to estimation error - Stress testing - Risk assessment - Confidence intervals for returns More robust than single backtest

Question

What is the minimum variance portfolio?

Answer

Portfolio with lowest possible risk on efficient frontier: Objective: Minimize: w^T Σ w Subject to: Σw_i = 1 Properties: - Leftmost point on efficient frontier - Ignores expected returns - Only uses covariance matrix - Often better out-of-sample than mean-variance Why: - Returns hard to estimate - Variances/covariances more stable - Less estimation error

Question

What is the difference between ex-ante and ex-post analysis?

Answer

Ex-Ante (Before the fact): - Based on expectations/forecasts - Used for portfolio construction - CAPM expected returns - Optimization inputs Ex-Post (After the fact): - Based on realized outcomes - Used for performance evaluation - Historical returns - What actually happened Key insight: - Optimize ex-ante - Evaluate ex-post - Ex-post ≠ ex-ante (forecasts rarely perfect)

Question

What are constraints in portfolio optimization?

Answer

Common constraints: 1. Budget constraint: Σw_i = 1 (fully invested) 2. Long-only: w_i ≥ 0 (no short selling) 3. Position limits: w_i ≤ max_weight 4. Sector constraints: Σw_i in sector ≤ limit 5. Turnover constraint: Limit trading 6. Risk constraints: σ_p ≤ max_risk More constraints: - More realistic - Easier to implement - But: May sacrifice optimality

Question

What is portfolio turnover and why does it matter?

Answer

Turnover = Sum of |weight changes| / 2 OR Turnover = Σ|w_new - w_old| / 2 Measures trading activity: - 100% turnover = entire portfolio replaced - 0% turnover = buy and hold Why it matters: - Higher turnover → higher transaction costs - Taxes on realized gains - Market impact costs - Reduces net returns Goal: Balance optimization benefit vs costs