定量方法（Quantitative Methods）— 描述性统计模块
一、什么是偏度？
偏度（Skewness） 是衡量数据分布 不对称程度 的统计量。
- 如果数据分布完全对称（如正态分布），偏度 = 0
- 如果数据分布"歪"向一边，偏度 ≠ 0
💡 直观理解：你拿一张纸画一个钟形曲线，然后把峰值往左或往右"推"——推得越远，偏度绝对值越大。
二、三种分布形态
1. 对称分布（Symmetric Distribution）
- 偏度 = 0
- 均值 = 中位数 = 众数
- 例如：标准正态分布
/\
/ \
/ \
/ \
——+————————+——
均=中=众
2. 正偏态 / 右偏（Positively Skewed / Right-Skewed）
- 偏度 > 0
- 分布的右尾更长（右侧有更多极端大值）
- 均值 > 中位数 > 众数
/\
/ \
/ \
\
\_____（长右尾）
——+——+——+——————
众 中 均
实际场景： - 股票收益率（大部分日子小幅波动，偶尔暴涨） - 房价（大部分在均价附近，少数豪宅拉高尾部） - 个人收入（少数高收入者拉长右尾）
3. 负偏态 / 左偏（Negatively Skewed / Left-Skewed）
- 偏度 < 0
- 分布的左尾更长（左侧有更多极端小值）
- 均值 < 中位数 < 众数
/\
/ \
/ \
_____/ \
——+——+——+——————
均 中 众
实际场景： - 保险理赔金额（大部分是小额理赔，少数大额拉长右尾…等一下！）
⚠️ 常见误区：保险理赔金额其实是 正偏态！因为大部分理赔金额小，少数大额理赔拉长右尾，所以均值被拉高。注意区分"大部分值集中在哪里"和"尾部在哪里"。
真正的负偏态案例： - 考试最高分附近的分布（大部分高分考生集中在90+，少数低分拉长左尾） - 产品的寿命数据（大部分产品寿命长，少数早期失效）
三、均值、中位数、众数的位置关系
这是 CFA 一级的 必考知识点：
| 分布类型 | 偏度 | 三者关系 |
|---|---|---|
| 对称分布 | = 0 | 均值 = 中位数 = 众数 |
| 正偏态 | > 0 | 均值 > 中位数 > 众数 |
| 负偏态 | < 0 | 均值 < 中位数 < 众数 |
🧠 记忆技巧：想象你在拖一条尾巴—— - 右尾（正偏）→ 均值被拖向右边 → 均值 > 中位数 - 左尾（负偏）→ 均值被拖向左边 → 均值 < 中位数
均值最容易被极端值影响，所以它永远在靠近尾巴的方向上最远。
四、偏度系数的计算
样本偏度公式（CFA 一级了解即可，不要求手算）：
$$S_K = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\frac{X_i - \bar{X}}{s}\right)^3$$
其中： - n = 样本量 - X̄ = 样本均值 - s = 样本标准差
解读标准：
| 偏度系数 | 含义 |
|---|---|
| ≈ 0 | 大致对称 |
| > 0.5 | 明显正偏 |
| < −0.5 | 明显负偏 |
| > 1 或 < −1 | 高度偏斜 |
五、偏度对投资分析的实战意义
1. 收益率分布判断
正常市场： 收益率近似对称（或轻微负偏） 极端情况： 如果某资产的收益率分布呈现高度正偏—— - 大部分时间小幅亏损 - 偶尔一次暴涨 - 适合有耐心的逆向投资者
2. 风险管理
- 正偏资产：中位数收益低于均值收益 → 只看均值会高估"典型"表现
- 负偏资产：均值低于中位数 → 尾部风险更大，需要关注下行保护
3. 策略选择（实战案例）
案例 1：可转债套利 - 可转债的收益分布天然正偏：大部分时间收利息，股价暴涨时可转股获利 - 中位数回报温和，但均值被少数大赚的年份拉高
案例 2：卖空波动率策略 - 典型负偏分布：大部分时间赚小额权利金，黑天鹅事件一次巨亏 - 2020 年 3 月 COVID 期间，卖空 VIX 的策略遭遇严重负偏事件
📊 实战口诀：投资中最怕的不是"偶尔赚不到钱"，而是"经常小赚、偶尔巨亏"——这就是负偏态的典型特征。
六、常见易错点总结
| 易错点 | 正确理解 |
|---|---|
| "右偏意味着大部分数据在右边" | ❌ 右偏 = 右尾长 = 极端值在右边，大部分数据在左边 |
| "均值永远在中间" | ❌ 均值被极端值拉走，偏态分布中均值不在中间 |
| 偏度与方差的区别 | 方差衡量 离散程度（散不散），偏度衡量 对称性（歪不歪） |
七、测试题
题目 1
某基金经理分析历史日收益率数据，发现均值 = 0.05%，中位数 = 0.08%，众数 = 0.10%。该收益率的分布最可能是：
A. 对称分布 B. 正偏态（右偏） C. 负偏态（左偏） D. 无法判断
题目 2
以下哪种资产的历史收益分布最可能呈现 负偏态？
A. 深度虚值看涨期权 B. 投资级公司债券 C. 卖出看跌期权的策略 D. 成长型科技股
题目 3
当数据分布是正偏态时，以下说法正确的是：
A. 分布的左尾比右尾更长 B. 中位数大于均值 C. 均值被右尾的极端值拉高 D. 偏度系数小于 0
题目 4（应用题）
某分析师统计了 100 只中小盘股过去一年的回报率。他发现：10% 分位数 = −35%，中位数 = +5%，90% 分位数 = +120%，均值 = +18%。
该回报分布的特征是：
A. 负偏，均值被左尾压低 B. 正偏，中位数是更可靠的典型回报指标 C. 对称，均值可以代表典型回报 D. 负偏，投资风险偏小
八、答案与解析
答案 1：C — 负偏态（左偏）
均值（0.05%）< 中位数（0.08%）< 众数（0.10%），符合负偏态特征。均值被左尾的极端低值压低。
答案 2：C — 卖出看跌期权的策略
卖出看跌期权：大部分时间收权利金（小赚），但市场大跌时面临大额赔付（偶尔巨亏）。这是典型的负偏态分布。A 选项（深度虚值看涨期权）大部分时间归零、偶尔暴涨，是正偏态。
答案 3：C — 均值被右尾的极端值拉高
正偏态 = 右尾更长，极端大值把均值拉高。A 错误（右尾更长），B 错误（中位数 < 均值），D 错误（偏度系数 > 0）。
答案 4：B — 正偏，中位数是更可靠的典型回报指标
均值（+18%）远高于中位数（+5%），且 90% 分位数高达 +120%，说明右尾极长，是正偏态。此时中位数（+5%）比均值（+18%）更能反映"大多数股票的典型回报"。
📌 今日要点记住三句话： 1. 偏度衡量分布的 不对称性（歪不歪），方差衡量 离散程度（散不散） 2. 正偏 = 右尾长 = 均值 > 中位数 > 众数 3. 投资中识别偏态 → 防范"小赚多次+一次巨亏"的负偏陷阱
L109 偏度（Skewness） | 2026-07-15 | CFA Level 1 定量方法
Quantitative Methods — Descriptive Statistics Module
1. What is Skewness?
Skewness is a statistical measure of the degree of asymmetry in a data distribution.
- A perfectly symmetric distribution (e.g., normal distribution) has skewness = 0
- If the distribution is "tilted" to one side, skewness ≠ 0
💡 Intuitive understanding: Imagine drawing a bell curve on paper, then "pushing" the peak to the left or right — the further you push, the larger the absolute value of skewness.
2. Three Distribution Shapes
1. Symmetric Distribution
- Skewness = 0
- Mean = Median = Mode
- Example: Standard normal distribution
/\
/ \
/ \
/ \
——+————————+——
M=Md=Mo
2. Positively Skewed / Right-Skewed
- Skewness > 0
- Longer right tail (more extreme large values on the right)
- Mean > Median > Mode
/\
/ \
/ \
\
\_____（long right tail）
——+——+——+——————
Mo Md M
Real-world scenarios: - Stock returns (mostly small daily moves, occasional surges) - Housing prices (most near the average, a few luxury homes pull the tail) - Personal income (a few high earners stretch the right tail)
3. Negatively Skewed / Left-Skewed
- Skewness < 0
- Longer left tail (more extreme small values on the left)
- Mean < Median < Mode
/\
/ \
/ \
_____/ \
——+——+——+——————
M Md Mo
Real-world scenarios: - Exam scores near the top (most students score 90+, a few low scores stretch the left tail) - Product lifespan data (most products last long, a few fail early)
⚠️ Common misconception: Insurance claim amounts are actually positively skewed! Most claims are small, and a few large claims stretch the right tail, pulling the mean upward. Focus on "where most values cluster" vs. "where the tail is."
3. Relationship Between Mean, Median, and Mode
This is a must-know for CFA Level 1:
| Distribution Type | Skewness | Relationship |
|---|---|---|
| Symmetric | = 0 | Mean = Median = Mode |
| Positively Skewed | > 0 | Mean > Median > Mode |
| Negatively Skewed | < 0 | Mean < Median < Mode |
🧠 Memory tip: Picture yourself pulling a tail — - Right tail (positive skew) → the mean is dragged to the right → Mean > Median - Left tail (negative skew) → the mean is dragged to the left → Mean < Median
The mean is the most sensitive to extreme values, so it always sits furthest in the direction of the tail.
4. Computing the Skewness Coefficient
Sample skewness formula (CFA Level 1 — understand conceptually, not required to calculate by hand):
$$S_K = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\frac{X_i - \bar{X}}{s}\right)^3$$
Where: - n = sample size - X̄ = sample mean - s = sample standard deviation
Interpretation guidelines:
| Skewness Coefficient | Interpretation |
|---|---|
| ≈ 0 | Roughly symmetric |
| > 0.5 | Notably positively skewed |
| < −0.5 | Notably negatively skewed |
| > 1 or < −1 | Highly skewed |
5. Practical Implications for Investment Analysis
1. Assessing Return Distributions
Normal markets: Returns are approximately symmetric (or slightly negatively skewed) Extreme case: If an asset's return distribution is highly positively skewed — - Small losses most of the time - Occasional massive gains - Suitable for patient, contrarian investors
2. Risk Management
- Positively skewed assets: Median return is lower than mean return → relying solely on the mean overstates "typical" performance
- Negatively skewed assets: Mean is lower than median → greater tail risk, need to focus on downside protection
3. Strategy Selection (Practical Cases)
Case 1: Convertible Bond Arbitrage - Convertible bond returns are naturally positively skewed: earn coupon income most of the time, profit from conversion when the stock surges - Median returns are modest, but the mean is pulled up by a few highly profitable years
Case 2: Short Volatility Strategies - Classic negatively skewed distribution: earn small premiums most of the time, suffer a single catastrophic loss during a black swan event - During the March 2020 COVID crash, short VIX strategies experienced severe negative skew events
📊 Practical takeaway: The scariest scenario in investing isn't "occasionally missing out on gains" — it's "frequent small gains + occasional massive loss." That is the hallmark of negative skewness.
6. Common Pitfalls Summary
| Pitfall | Correct Understanding |
|---|---|
| "Right-skewed means most data is on the right" | ❌ Right-skewed = long right tail = extreme values on the right, most data on the left |
| "The mean is always in the middle" | ❌ The mean is pulled by extreme values; in skewed distributions, the mean is not in the middle |
| Skewness vs. Variance | Variance measures dispersion (how spread out), skewness measures symmetry (how tilted) |
7. Practice Questions
Question 1
A fund manager analyzes historical daily return data and finds that Mean = 0.05%, Median = 0.08%, and Mode = 0.10%. The return distribution is most likely:
A. Symmetric B. Positively skewed (right-skewed) C. Negatively skewed (left-skewed) D. Cannot be determined
Question 2
Which of the following assets is most likely to have a negatively skewed historical return distribution?
A. Deep out-of-the-money call options B. Investment-grade corporate bonds C. A strategy of writing (selling) put options D. Growth technology stocks
Question 3
When a data distribution is positively skewed, which of the following statements is correct?
A. The left tail is longer than the right tail B. The median is greater than the mean C. The mean is pulled higher by extreme values in the right tail D. The skewness coefficient is less than 0
Question 4 (Application)
An analyst compiles the past year's returns for 100 small/mid-cap stocks. The findings: 10th percentile = −35%, Median = +5%, 90th percentile = +120%, Mean = +18%.
The return distribution is best characterized as:
A. Negatively skewed, with the mean depressed by the left tail B. Positively skewed, with the median being a more reliable measure of typical return C. Symmetric, with the mean representing typical return D. Negatively skewed, with low investment risk
8. Answers and Explanations
Answer 1: C — Negatively skewed (left-skewed)
Mean (0.05%) < Median (0.08%) < Mode (0.10%), consistent with negative skewness. The mean is pulled lower by extreme low values in the left tail.
Answer 2: C — A strategy of writing (selling) put options
Selling put options: collect premiums most of the time (small gains), but face large payouts during market crashes (occasional massive losses). This is a classic negatively skewed distribution. Option A (deep OTM call options) mostly expires worthless with occasional huge gains — positively skewed.
Answer 3: C — The mean is pulled higher by extreme values in the right tail
Positive skew = longer right tail, extreme large values pull the mean upward. A is incorrect (right tail is longer), B is incorrect (Median < Mean), D is incorrect (skewness coefficient > 0).
Answer 4: B — Positively skewed, with the median being a more reliable measure of typical return
Mean (+18%) is far higher than Median (+5%), and the 90th percentile reaches +120%, indicating a very long right tail — positively skewed. In this case, the median (+5%) better reflects "the typical return for most stocks" than the mean (+18%).
📌 Three key takeaways for today: 1. Skewness measures asymmetry (how tilted), while variance measures dispersion (how spread out) 2. Positive skew = long right tail = Mean > Median > Mode 3. In investing, identify skewness → guard against the negative skew trap of "many small wins + one catastrophic loss"
L109 Skewness | 2026-07-15 | CFA Level 1 Quantitative Methods