Module 1 · Portfolio Management

Portfolio Risk and Return — Part I

EN: Diversification, efficient frontier, capital allocation line.
VN: Đa dạng hóa, đường biên hiệu quả, capital allocation line.

1. Portfolio Return & Risk (2 assets) Core

\[ E(R_p) = w_1 E(R_1) + w_2 E(R_2) \] \[ \sigma_p^{2} = w_1^{2}\sigma_1^{2} + w_2^{2}\sigma_2^{2} + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2 \]

See Quant M5 for detailed components.

2. Diversification Ratio Core

About: ρ = +1 no diversification. ρ = 0 significant diversification. ρ = −1 perfect hedge possible. Real-world correlations rise during crises (when you need diversification most).Tóm tắt: ρ = 1 không đa dạng. ρ = 0 có lợi. ρ = −1 hedge hoàn hảo. Khủng hoảng correlations tăng.

Effect of correlation

ρ = +1 Linear combination — no diversification benefit.
ρ = 0 Significant diversification.
ρ = −1 Perfect hedge possible — riskless portfolio achievable.

3. Efficient Frontier Core

About: Set of portfolios with maximum E(R) for each level of σ. Below the frontier = inferior portfolios. MVP at leftmost.Tóm tắt: Tập danh mục có E(R) cao nhất cho mỗi mức σ. Dưới frontier = inferior. MVP ở trái cùng.

Upper boundary of mean-variance feasible set. CAL/CML tangent to frontier at the optimal risky portfolio.

Set of portfolios with the maximum expected return for a given standard deviation. The minimum-variance portfolio sits at the leftmost point. Investors choose along the frontier based on risk preference.

4. Capital Allocation Line (CAL) Core

About: Combinations of risk-free asset + risky tangency portfolio. Slope = Sharpe ratio of tangency. CML is the special case where tangency = market portfolio.Tóm tắt: Kết hợp risk-free + tangency. Slope = Sharpe của tangency. CML khi tangency = market.

\[ E(R_p) = R_f + \frac{E(R_T) - R_f}{\sigma_T} \cdot \sigma_p \]

Components

T Tangency portfolio (touches efficient frontier).
Slope = Sharpe ratio of tangency portfolio.
Capital Market Line (CML) Special case where tangency portfolio = market portfolio.

Practice problem

Tangency portfolio E(R) = 12%, σ = 18%. Rf = 3%. Build a portfolio with σp = 9% on the CAL.

Show solution

Slope = (12 − 3)/18 = 0.5

E(Rp) = 3 + 0.5(9)

= 7.5% (half exposure to tangency)

Practice problem Practice

Practice problem

Two assets: σ₁ = 20%, σ₂ = 15%. Equal weights. Correlations of 1.0, 0.0, −1.0. Compare portfolio σ in each case.

Show solution

σp² = 0.25(400) + 0.25(225) + 2(0.5)(0.5)ρ(20)(15)

= 156.25 + 150ρ

ρ = 1.0: σp² = 306.25 → σp = 17.5%

ρ = 0: σp² = 156.25 → σp ≈ 12.50%

ρ = −1.0: σp² = 6.25 → σp = 2.5%

ρ=1.0: 17.50%; ρ=0: 12.50%; ρ=−1.0: 2.50% — diversification grows as ρ falls.