CFA L1  /  Quant  /  M5. Portfolio Mathematics
Module 5 · Quantitative Methods

Portfolio Mathematics

EN: Expected return and variance of a portfolio, the role of correlation, and Roy's safety-first criterion.
VN: Lợi suất kỳ vọng và phương sai danh mục, vai trò của tương quan, và tiêu chí Roy.

In this module
  1. Expected Portfolio Return
  2. Portfolio Variance — Two Assets
  3. Portfolio Variance — N Assets
  4. Covariance from Correlation
  5. Roy's Safety-First Ratio
  6. Shortfall Risk (Normal)

1. Expected Portfolio Return Core

About: Portfolio expected return is the weighted average of asset expected returns. Linear in weights — simple even with many assets.Tóm tắt: Lợi suất kỳ vọng danh mục = trung bình có trọng số của lợi suất kỳ vọng từng tài sản. Tuyến tính theo trọng số.
\[ E(R_p) = \sum_{i=1}^{n} w_i \cdot E(R_i),\qquad \sum w_i = 1 \]

Components / Thành phần

  • \(w_i\) Weight of asset i (sums to 1).
  • \(E(R_i)\) Expected return of asset i.
Practice problem

60% in equities (E(R) = 10%), 40% in bonds (E(R) = 4%). What is the expected portfolio return?

Show solution
\(E(R_p) = 0.60(10) + 0.40(4)\)
= 6 + 1.6 = 7.6%

2. Portfolio Variance — Two Assets Core

About: Portfolio variance is NOT a simple weighted average — it depends on covariance/correlation. Diversification benefit grows as ρ falls. Foundation of modern portfolio theory.Tóm tắt: Phương sai danh mục KHÔNG phải trung bình có trọng số — phụ thuộc covariance. Lợi ích đa dạng hóa tăng khi ρ giảm. Nền tảng MPT.
\[ \sigma_p^{2} = w_1^{2}\sigma_1^{2} + w_2^{2}\sigma_2^{2} + 2\,w_1\,w_2\,\rho_{1,2}\,\sigma_1\,\sigma_2 \]

Components / Thành phần

  • \(\sigma_i\) Standard deviation of asset i.
  • \(\rho_{1,2}\) Correlation between asset 1 and 2.

Diversification: Lower \(\rho\) → lower portfolio variance. With \(\rho = 1\) no diversification benefit; with \(\rho = -1\) full hedge possible.

Practice problem

w₁ = 0.5, σ₁ = 20%; w₂ = 0.5, σ₂ = 10%; ρ = 0.3. Compute portfolio standard deviation.

Show solution
\(\sigma_p^{2} = 0.25(400) + 0.25(100) + 2(0.5)(0.5)(0.3)(20)(10)\)
\(= 100 + 25 + 30 = 155\)
σₚ ≈ 12.45%

3. Portfolio Variance — N Assets Core

About: With N assets, variance has N variance terms + N(N−1) covariance terms. As N grows, covariances dominate — portfolio risk converges to average covariance.Tóm tắt: Với N tài sản, phương sai gồm N variance + N(N−1) covariance. Khi N tăng, covariance chi phối — rủi ro hội tụ về trung bình covariance.
\[ \sigma_p^{2} = \sum_{i=1}^{n}\sum_{j=1}^{n} w_i\,w_j\,\text{Cov}(R_i, R_j) \]

Note: \(\text{Cov}(R_i, R_i) = \sigma_i^{2}\), so the diagonal contributes the variance terms.

Practice problem

3 equal-weight assets, σ each 20%, all pairwise correlations 0.4. Compute portfolio σ.

Show solution
w = 1/3, σ² each = 0.04
Variance terms: 3 × (1/9)(0.04) = 0.01333
Covariance terms: 6 × (1/9)(0.4)(0.04) = 0.01067
σp² = 0.024 → σp = √0.024
σp ≈ 15.49%

4. Covariance ↔ Correlation Core

About: Convert between covariance and correlation. Correlation is unit-free and bounded [-1, 1] — far more intuitive for analysis.Tóm tắt: Quy đổi giữa covariance và correlation. Correlation không có đơn vị, bị chặn [-1, 1] — trực quan hơn.
\[ \text{Cov}(X, Y) = \rho_{X,Y} \cdot \sigma_X \cdot \sigma_Y \]
Practice problem

ρXY = −0.3, σX = 12%, σY = 18%. Compute Cov(X, Y).

Show solution
Cov = −0.3 × 0.12 × 0.18
= −0.00648

5. Roy's Safety-First Ratio (SFR) Core

About: Roy's safety-first criterion picks the portfolio that minimizes the probability of returns falling below a minimum threshold R_L. Portfolio with highest SFR is preferred.Tóm tắt: Tiêu chí an toàn của Roy chọn danh mục giảm thiểu xác suất rơi dưới ngưỡng R_L. Danh mục có SFR cao nhất là tốt nhất.

EN: Choose the portfolio with the highest SFR — analogous to Sharpe but uses a "minimum acceptable return" threshold instead of \(R_f\).
VN: Chọn danh mục có SFR cao nhất — giống Sharpe nhưng so với mức tối thiểu chấp nhận được.

\[ SFR = \frac{E(R_p) - R_L}{\sigma_p} \]

Components / Thành phần

  • \(R_L\) Threshold / minimum acceptable return.
  • \(\sigma_p\) Portfolio standard deviation.
Practice problem

Portfolio A: E(R) = 10%, σ = 12%. Portfolio B: E(R) = 8%, σ = 7%. Threshold R_L = 3%. Which is safer by Roy's criterion?

Show solution
SFR(A) = (10 - 3)/12 ≈ 0.583
SFR(B) = (8 - 3)/7 ≈ 0.714
Choose B — higher SFR → lower probability of falling below R_L

6. Shortfall Risk under Normality Core

About: Under normal returns, shortfall probability = N(−SFR). Lets you quantify the chance of falling below a threshold using Z-table lookup.Tóm tắt: Khi lợi suất phân phối chuẩn, xác suất shortfall = N(−SFR). Dễ tính bằng bảng Z.

EN: If returns are normally distributed, the probability of falling below \(R_L\) is \(N(-SFR)\).
VN: Nếu lợi suất phân phối chuẩn, xác suất rơi dưới \(R_L\) bằng \(N(-SFR)\).

\[ P(R_p < R_L) = N(-SFR) \]
Practice problem

SFR = 1.65. What is the shortfall probability assuming normal returns?

Show solution
\(P = N(-1.65)\)
≈ 5%