### Book 1- QM- Module 5 PORTFOLIO MATHEMATICS
**📅 Date:** ➤ ⌈ [[2025-11-06-Thu〚Expected Return of a Portfolio,Variance, Co-Variance〛]]⌋
**💭 What:**
➤ 联合概率： 一个事情的发生会对另一个事件的发生产生影响
➤ 两个return
- expected return of stock (用这个当我们做Expected return)
- historical return
**👀 Snap:**
➤ 行业中第十一年的收入是在前十年中加起来的两倍
⇩ 🅻🅸🅽🅺🆂 ⇩
**🏷️ Tags**: #💰/Economy
**🗂 Menu**:
➤⌈[[✢ M O C ➣ 11 ⌈N O V - 2 0 2 5⌉ ✢|2025 - N O V - MOC]]⌋
➤⌈[[✢ L O G ➢ 11 ⌈N O V - 2 0 2 5⌉ ✢|2025 - N O V - LOG]] ⌋ #👾/Private
➤ ⌈[[💰060 - 04 + ρ Correlation Coefficient & Hedging Risk]]⌋
---
**Focus:** Expected Return, Variance, Covariance, and Portfolio Risk–Return Structure
---
## 🧠 1. Conceptual Overview
Portfolio mathematics underpins **modern portfolio theory (MPT)** — understanding how assets combine to form portfolios with optimised **risk–return trade-offs**.
- **Expected return** = weighted average of asset returns.
- **Risk (variance / standard deviation)** depends not only on individual asset risk but also on **how they move together (covariance / correlation)**.
- **Diversification** reduces total risk when assets are **not perfectly correlated**.
---
## 💡 2. Expected Return of a Portfolio
$
E(R_p) = \sum_{i=1}^{n} w_i E(R_i)
$
![[Screenshot 2025-11-06 at 17.15.52.png]]
Where:
- $w_i$ = weight of asset *i* in the portfolio
- $E(R_i)$ = expected return of asset *i*
- $\sum w_i = 1$
> **Insight:** Expected return is **linear in weights**, so changes in allocation affect returns predictably, but not risk predictably.
---
## ⚙️ 3. Variance and Covariance （方差和协方差）
>[!info]
>### Variance：
>- 1个变量的值于他的平均值的变化程度
>- Measures how much a variables values deviate from it's mean (expected value)
> ![[Screenshot 2025-11-06 at 17.39.31.png]]
>### Covariance:
>- 两个变量一起波动的测量指标
>- How 2 assists moving together
>- ![[Screenshot 2025-11-06 at 17.23.30.png]]
Individual asset variance:
$
Var(R_i) = E[(R_i - E(R_i))^2]
$
Covariance between assets:
$
Cov(R_i, R_j) = E[(R_i - E(R_i))(R_j - E(R_j))]
$
Correlation:
$
\rho_{ij} = \frac{Cov(R_i, R_j)}{\sigma_i \sigma_j}
$
Range: −1 ≤ ρ ≤ +1
- ρ = +1 → perfect positive correlation (no diversification)
- ρ = 0 → uncorrelated (some diversification)
- ρ = −1 → perfect negative correlation (maximum diversification)
---
## 📈 4. Portfolio Variance (Two-Asset Case)
$
\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B Cov_{AB}
$
or using correlation:
$
\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho_{AB} \sigma_A \sigma_B
$
> **Key:** Portfolio risk depends on the *interaction* between assets, not just their individual risks.
If $ρ < 1$, total variance < weighted average variance → **diversification benefit**.
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![[Screenshot 2025-11-06 at 17.39.31 1.png]]
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![[Screenshot 2025-11-06 at 18.07.40.png]]
![[IMG_9842.jpeg]]