Estimated study time: 45 minutes
Content:
Modern Portfolio Theory (MPT), developed by Harry Markowitz in the 1950s, provides the mathematical framework for constructing portfolios that maximize expected return for a given level of risk, or equivalently, minimize risk for a given expected return. The key insight is that what matters for a portfolio is not the risk of each individual asset in isolation, but its contribution to overall portfolio risk — and this contribution depends on how assets co-move (their correlations). Adding an asset with low or negative correlation to an existing portfolio reduces portfolio variance even if the asset itself has high volatility. This is the formal mathematical foundation for diversification: the benefit of combining assets that don't move perfectly together.
Portfolio expected return is simply the weighted average of individual asset expected returns: E(Rp) = Σ(wi × E(Ri)). Portfolio variance, however, is not simply the weighted average of individual variances — the covariances (correlations) between assets matter: Var(Rp) = Σi Σj (wi × wj × Cov(Ri, Rj)). For a two-asset portfolio: Var(Rp) = w1^2 × σ1^2 + w2^2 × σ2^2 + 2 × w1 × w2 × σ1 × σ2 × ρ12, where ρ12 is the correlation between assets 1 and 2. When ρ12 = 1.0 (perfect positive correlation), no diversification benefit exists. When ρ12 < 1.0, portfolio variance is less than the weighted average of individual variances. When ρ12 = –1.0 (perfect negative correlation), a completely risk-free portfolio can be constructed.
The minimum variance frontier consists of all portfolios that minimize portfolio variance for each level of expected return. The efficient frontier is the upper portion of the minimum variance frontier — the set of portfolios offering the highest expected return for each level of variance. Rational, risk-averse investors should choose portfolios on the efficient frontier. The global minimum variance portfolio (GMVP) is the portfolio with the lowest possible variance. Adding a risk-free asset creates the Capital Market Line (CML): the set of optimal portfolios combining the risk-free asset with the tangency portfolio (the specific risky portfolio on the efficient frontier that has the highest Sharpe ratio). All investors will hold the same tangency portfolio of risky assets (a key implication of the CAPM) and differ only in how much they borrow or lend at the risk-free rate.
Diversification eliminates unsystematic (idiosyncratic, company-specific) risk — the risk that can be eliminated by holding a broad portfolio. Systematic risk (market risk, non-diversifiable risk) cannot be eliminated through diversification — it reflects co-movement with broad economic factors. As the number of assets in a portfolio increases, unsystematic risk declines rapidly toward zero, while systematic risk remains. Research suggests that most of the diversification benefit is achieved with 20-30 securities in a domestic portfolio; additional securities offer diminishing incremental diversification. This distinction between diversifiable and non-diversifiable risk is central to the CAPM — investors should only be compensated for bearing systematic risk, not unsystematic risk (which they can eliminate for free through diversification).
Key Terms:
Quiz Questions:
Q1. Asset A has an expected return of 10% and standard deviation of 15%. Asset B has an expected return of 6% and standard deviation of 8%. The correlation between A and B is 0.2. A portfolio holds 60% in A and 40% in B. What is the portfolio expected return?
A) 8.4% B) 7.5% C) 8.0% D) 9.2%
Answer: A — E(Rp) = 0.60 × 10% + 0.40 × 6% = 6.0% + 2.4% = 8.4%. Portfolio expected return is always the weighted average of individual returns, regardless of correlation. Correlation only affects portfolio variance (risk), not expected return. This is a key point: diversification reduces risk without reducing expected return — it is a "free lunch."
---
Q2. Using the same assets from Q1 (A: σ = 15%, B: σ = 8%, ρ = 0.2, w_A = 60%, w_B = 40%), what is the portfolio standard deviation?
A) 10.4% B) 11.2% C) 9.8% D) 12.5%
Answer: C — Var(Rp) = (0.6)^2 × (0.15)^2 + (0.4)^2 × (0.08)^2 + 2 × 0.6 × 0.4 × 0.2 × 0.15 × 0.08 = 0.36 × 0.0225 + 0.16 × 0.0064 + 0.048 × 0.0024 × 0.2... Let me compute properly: = 0.0081 + 0.001024 + 2 × 0.6 × 0.4 × 0.2 × 0.15 × 0.08 = 0.0081 + 0.001024 + 0.001152 = 0.010276. σp = √0.010276 ≈ 10.14% ≈ 10.1%. The weighted average of individual standard deviations would be 0.6 × 15% + 0.4 × 8% = 12.2% — far higher than 10.1%, demonstrating the diversification benefit from ρ = 0.2 < 1.0.
---
Q3. As the number of securities in a well-diversified portfolio increases from 5 to 50, which statement BEST describes the change in portfolio risk?
A) Total portfolio risk increases as more securities are added B) Systematic risk is eliminated, leaving only unsystematic risk C) Unsystematic risk is diversified away; systematic risk remains and cannot be eliminated D) Both systematic and unsystematic risk are eliminated through diversification
Answer: C — As a portfolio becomes more diversified, company-specific (unsystematic) risk is progressively eliminated. With 50+ stocks, a well-constructed portfolio's risk is almost entirely systematic (market) risk. Systematic risk — the exposure to market-wide factors like recessions, interest rate changes, and inflation — cannot be eliminated through diversification because all stocks are affected. This is why the CAPM rewards beta (systematic risk) but not standard deviation (total risk).
---
Q4. The Capital Market Line (CML) connects the risk-free rate to the tangency portfolio. A portfolio on the CML above the tangency portfolio is characterized by:
A) Holding only the risk-free asset and zero risky assets B) Borrowing at the risk-free rate to invest more than 100% in the tangency portfolio (leveraged portfolio) C) Holding the tangency portfolio with some cash D) Holding a combination of multiple risky asset classes beyond the tangency portfolio
Answer: B — Portfolios on the CML above the tangency portfolio are achieved by borrowing at the risk-free rate and investing the borrowed funds plus investor capital entirely in the tangency portfolio. This leverages the tangency portfolio's return and risk. For example, investing 150% in the tangency portfolio (borrowing 50%) lies above the tangency point on the CML, offering higher expected return with proportionally higher risk.
---
Q5. A portfolio manager has $10 million in a domestic equity portfolio and is considering adding an international equity fund with a correlation of 0.3 to the domestic portfolio. The expected return of the international fund is lower than the domestic portfolio, but the manager adds it anyway. This decision is BEST justified by:
A) The lower expected return of the international fund ensures capital preservation B) The low correlation reduces overall portfolio variance, potentially improving the risk-return tradeoff despite the lower return C) International equities are always necessary for regulatory diversification compliance D) The lower return of international equities guarantees loss of portfolio value
Answer: B — Adding an asset with low correlation (0.3) can improve the portfolio's risk-return tradeoff even if that asset has a lower expected return, because the diversification benefit (variance reduction) partially or fully compensates for the lower return contribution. The mean-variance analysis might show that the new portfolio (with the international fund) lies above the original portfolio on the efficient frontier — offering either the same return with less risk or more return for the same risk. This is the fundamental insight of MPT.
---