定量方法（Quantitative Methods）— 描述性统计模块
一、背景：标准差不够用的时候
学完方差和标准差后，我们有了度量数据离散程度的工具。但思考一个问题：
📌 案例：A 基金年均收益率 10%，标准差 5%；B 基金年均收益率 50%，标准差 20%。
哪只基金风险更大？
如果只看标准差，B（20%） > A（5%）→ B 更"波动"，但这公平吗？
B 基金的收益率是 A 的 5 倍，标准差大一些很正常。这就引出一个核心概念：
🧠 结论：标准差是 绝对离散度（absolute dispersion），有时需要用 相对离散度（relative dispersion） 来做公平比较。
变异系数（Coefficient of Variation, CV）就是专门解决这一问题的工具。
二、变异系数的定义
公式：
$$\text{CV（总体）} = \frac{\sigma}{\mu}$$
$$\text{CV（样本）} = \frac{s}{\bar{x}}$$
即：标准差 ÷ 均值
CFA 一级通常用样本公式：
$$\text{CV} = \frac{s}{\bar{X}}$$
解读：
- CV 衡量的是 每单位均值的风险/波动——"每赚 1% 的收益，伴随多少波动？"
- CV 是一个 无量纲（unit-free） 数字——没有单位，可以跨资产、跨策略、跨时间比较
- CV 通常以小数或百分比表示（如 CV = 0.35 → 每单位均值对应 35% 的波动）
三、回到开篇案例
| 指标 | A 基金 | B 基金 |
|---|---|---|
| 均值收益率 | 10% | 50% |
| 标准差 | 5% | 20% |
| CV | 5/10 = 0.50 | 20/50 = 0.40 |
A 的 CV（0.50）> B 的 CV（0.40）→ 每单位收益所承受的波动，A 反而比 B 更大！
🧠 结论：B 基金的绝对波动更大，但考虑到它的高收益，波动是"划算"的。A 基金虽然标准差小，但「波动/收益」比值更差。
四、CV 的核心应用场景（CFA 高频考点）
⭐ 场景一：不同均值的比较
两只股票预期收益不同，直接比标准差不公平 → 用 CV 做 "波动-收益效率比"。
⭐ 场景二：不同单位的比较
| 指标 | 美国市场 | 欧洲市场 |
|---|---|---|
| 均值 | $120 | €80 |
| 标准差 | $30 | €18 |
直接用标准差比？$30 vs €18 —— 单位不同，没法比！CV 消除了单位差异：
- 美国：$30 / $120 = 0.25
- 欧洲：€18 / €80 = 0.225
→ 欧洲市场每单位价格对应的波动更低（CV 更小）。
⭐ 场景三：投资组合构建中的风险调整
在比较不同策略/基金经理的业绩时，CV 是简单的风险-收益效率筛选工具：
CV 越低 → 每单位收益承担的风险越少 → 越"划算"
五、CV 的局限性（🚨 重要）
CFA 考试既考 CV 怎么用，也考它什么时候不能用：
| 局限性 | 说明 |
|---|---|
| 均值接近 0 时无意义 | 当 $\bar{X} \to 0$，CV $\to \infty$，失去比较意义 |
| 均值负值时产生误导 | 均值 = −2%，标准差 = 4% → CV = −2.0，负数 CV 无法正常比较 |
| 只用了均值和标准差两个参数 | 忽略了偏度、峰度等信息，分布形状可能完全不同但 CV 相同 |
| 区间尺度（interval scale）数据用 CV 要谨慎 | 如摄氏温度：0°C 和 10°C 标准差的 CV，因为 0°C 是任意零点，比值无意义 |
| 比值尺度（ratio scale）数据才完全适用 CV | 如价格、收益率、收入——"零"代表真正的"无"，比值有意义 |
🧠 简单的判断标准：CV 适用于 比率尺度数据 + 均值 > 0 的场景。
六、CFA 的 "哪个更好" 题型模板
题型 1：给定两组数据，问谁相对波动更大
🔴 CFA 经典问题："Based on the coefficient of variation, which asset exhibits greater relative variability?"
解题流程：
1. 分别计算 CV_A = σ_A / μ_A
2. 分别计算 CV_B = σ_B / μ_B
3. 比较：CV 越大 → 相对波动越大
4. 排除只看标准差的干扰选项
题型 2：情景分析——是否应该用 CV？
题目可能给出一组均值很小的数据（甚至为负），问：
"The mean return is −0.5% and the standard deviation is 3%. Is the coefficient of variation meaningful?"
答：否。均值负数时 CV 为负，无法做有意义的比较。
七、实战案例
案例 1：基金经理筛选
Ivan 哥正在评估两位基金经理过去三年的表现：
| 指标 | 经理 X | 经理 Y |
|---|---|---|
| 年均超额收益 | 8.5% | 14.0% |
| 超额收益标准差 | 7.2% | 13.0% |
问题：仅从相对离散度角度看，哪位经理的收益更"稳定划算"？
解题：
- 经理 X：CV = 7.2% / 8.5% = 0.847
- 经理 Y：CV = 13.0% / 14.0% = 0.929
→ 经理 X 的 CV 更低（0.847 < 0.929），每单位超额收益伴随的波动更少。
💡 虽然经理 Y 绝对收益更高，但相对离散度也更大。后续可用 夏普比率（Sharpe Ratio）结合无风险利率做更全面的判断——这恰好是 L113 的内容！
案例 2：不同市场的波动效率比较
分析师收集了以下数据：
| 市场 | 平均日收益 | 日收益标准差 |
|---|---|---|
| 新兴市场 ETF | 0.08% | 1.50% |
| 发达市场 ETF | 0.04% | 0.60% |
问题：哪个市场的收益波动效率更高（每单位收益承受的波动更少）？
解题：
- 新兴市场 CV = 1.50% / 0.08% = 18.75
- 发达市场 CV = 0.60% / 0.04% = 15.00
→ 发达市场 CV 更低，每单位收益的波动更少，效率更高。
💡 虽然新兴市场绝对收益更高，但要获得那点额外收益，你得忍受不成比例的额外波动——这正是 CV 揭示的信息。
八、CV 与其他离散度指标的关系总结
| 指标 | 类型 | 用途 | 局限 |
|---|---|---|---|
| 极差（Range） | 绝对离散度 | 快速看全距 | 极端值敏感 |
| 方差（Variance） | 绝对离散度 | 理论推导 | 单位平方，不直观 |
| 标准差（SD） | 绝对离散度 | 最常见的离散度量 | 不同均值不能直接比 |
| 变异系数（CV） | 相对离散度 | 跨均值/跨单位比较 | 均值 ≤ 0 时失效 |
九、常见易错点总结
| 易错点 | 正确理解 |
|---|---|
| "CV 大说明收益好" | ❌ CV 是「波动/收益」——越大越差，越小越有效率 |
| "标准差小就一定更好" | ❌ 均值不同时，需要 CV 来做相对比较 |
| "任何数据都可以用 CV" | ❌ 均值 ≤ 0 时 CV 无意义；区间尺度数据（如温度）慎用 |
| "CV 可以用来代替夏普比率" | ❌ CV 不涉及无风险利率，只衡量相对波动，不衡量风险调整后收益 |
| "CV = σ² / μ" | ❌ 分子是标准差，不是方差！这是最常见的计算错误 |
十、测试题
题目 1
某分析师收集了一只股票的历史数据：
- 平均月收益率 = 1.2%
- 月收益率标准差 = 3.6%
该股票的变异系数（CV）最接近：
A. 0.33 B. 0.30 C. 3.00 D. 0.03
题目 2
以下哪种情况 不适合 使用变异系数做比较？
A. 比较标普 500 和日经 225 的相对波动率 B. 比较黄金（均值 0.5%）和比特币（均值 4.0%）的风险效率 C. 比较两只月均收益率均为负值的对冲基金的表现 D. 比较不同面值股票（$50 vs $800）的价格波动
题目 3
资产 P 的均值为 6%，标准差为 15%。资产 Q 的均值为 2%，标准差为 4%。关于两者的相对离散度，以下说法正确的是：
A. P 的相对离散度高于 Q B. Q 的相对离散度高于 P C. 两者相对离散度相同 D. 信息不足，无法比较
题目 4（应用题）
某量化团队在测试两种择时策略：
| 策略 | 月均超额收益 | 超额收益标准差 |
|---|---|---|
| 策略 Alpha | 0.45% | 1.80% |
| 策略 Beta | 0.30% | 0.90% |
如果团队希望选择「每单位超额收益波动最小」的策略，应选择：
A. 策略 Alpha，因为 CV 更低 B. 策略 Beta，因为 CV 更低 C. 策略 Alpha，因为绝对收益更高 D. 策略 Beta，因为绝对标准差更小
十一、答案与解析
答案 1：C — 3.00
CV = s / X̄ = 3.6% / 1.2% = 3.0
🚨 易错：A（0.33）是 X̄ / s 的结果，颠倒了分子分母！注意 CV = 标准差 ÷ 均值，不是均值 ÷ 标准差。
答案 2：C — 比较两只月均收益率均为负值的对冲基金的表现
当均值为负时，CV 为负值，丢失了比较意义。例如均值 −1%、标准差 5% → CV = −5；均值 −2%、标准差 5% → CV = −2.5。负数之间的排序不再反映"相对波动"的直观含义。
答案 3：A — P 的相对离散度高于 Q
CV_P = 15% / 6% = 2.50 CV_Q = 4% / 2% = 2.00 2.50 > 2.00 → P 每单位收益承担的波动更大。
答案 4：B — 策略 Beta，因为 CV 更低
CV_Alpha = 1.80% / 0.45% = 4.0 CV_Beta = 0.90% / 0.30% = 3.0
Beta 的 CV 更低（3.0 < 4.0）→ 每单位超额收益的波动更少。
D 看到「标准差小」就选了，但忽略了均值也小——这正是 CV 要纠正的思维陷阱！
📌 今日要点记住三句话： 1. CV = 标准差 ÷ 均值，衡量"每单位均值的波动量"——越小越有效率 2. CV 的杀手应用：比较 不同均值 或 不同单位 的数据集时，标准差不够用 3. 硬伤记牢：均值 ≤ 0 时 CV 失效，CV 不涉及无风险利率（后续 L113 夏普比率会加入）
L112 变异系数（CV） | 2026-07-18 | CFA Level 1 定量方法
Quantitative Methods — Descriptive Statistics Module
1. Background: When Standard Deviation Falls Short
After learning variance and standard deviation, we have tools to measure dispersion. But consider this question:
📌 Case Study: Fund A has an average annual return of 10% with a standard deviation of 5%. Fund B has an average annual return of 50% with a standard deviation of 20%.
Which fund is riskier?
If we only look at standard deviation, Fund B (20%) > Fund A (5%) → Fund B appears "more volatile." But is that fair?
Fund B's return is 5× that of Fund A, so a larger standard deviation is expected. This leads to a key insight:
🧠 Conclusion: Standard deviation measures absolute dispersion. For fair comparison, we often need relative dispersion.
The Coefficient of Variation (CV) is the tool designed for exactly this problem.
2. Definition of Coefficient of Variation
Formula:
$$\text{CV (population)} = \frac{\sigma}{\mu}$$
$$\text{CV (sample)} = \frac{s}{\bar{x}}$$
In simple terms: Standard Deviation ÷ Mean
The CFA Level I exam typically uses the sample formula:
$$\text{CV} = \frac{s}{\bar{X}}$$
Interpretation:
- CV measures risk/volatility per unit of mean — "How much fluctuation accompanies each 1% of return?"
- CV is unit-free (dimensionless) — no units, enabling cross-asset, cross-strategy, and cross-period comparisons
- CV is typically expressed as a decimal or percentage (e.g., CV = 0.35 → 35% fluctuation per unit of mean)
3. Returning to the Opening Case
| Metric | Fund A | Fund B |
|---|---|---|
| Mean Return | 10% | 50% |
| Standard Deviation | 5% | 20% |
| CV | 5/10 = 0.50 | 20/50 = 0.40 |
Fund A's CV (0.50) > Fund B's CV (0.40) → Per unit of return, Fund A actually carries MORE volatility!
🧠 Conclusion: Fund B has greater absolute volatility, but given its high returns, the volatility is "worth it." Fund A, despite having a lower standard deviation, has a worse "fluctuation-to-return" ratio.
4. Core Applications of CV (High-Frequency CFA Topics)
⭐ Scenario 1: Comparing Different Means
Two stocks have different expected returns. Direct standard deviation comparison is unfair → use CV as a "volatility-return efficiency ratio."
⭐ Scenario 2: Comparing Different Units
| Metric | US Market | European Market |
|---|---|---|
| Mean Price | $120 | €80 |
| Std Dev | $30 | €18 |
Can we directly compare $30 vs €18? No — different units! CV eliminates the unit difference:
- US: $30 / $120 = 0.25
- Europe: €18 / €80 = 0.225
→ European market has lower volatility per unit of price (smaller CV).
⭐ Scenario 3: Risk-Adjusted Portfolio Construction
When comparing performance across strategies or fund managers, CV serves as a simple risk-return efficiency screening tool:
Lower CV → Less risk per unit of return → More "efficient"
5. Limitations of CV (🚨 Important)
The CFA exam tests both how to use CV AND when NOT to use it:
| Limitation | Explanation |
|---|---|
| Mean near zero → meaningless | As $\bar{X} \to 0$, CV $\to \infty$, losing comparative value |
| Negative mean → misleading | Mean = −2%, SD = 4% → CV = −2.0. Negative CVs cannot be meaningfully compared |
| Only uses mean and SD | Ignores skewness, kurtosis, etc. Distributions with the same CV can have completely different shapes |
| Interval scale data requires caution | e.g., temperature in Celsius: 0°C is an arbitrary zero, so CV ratios are meaningless |
| Ratio scale data is fully suitable | e.g., prices, returns, income — where "zero" truly means "none" and ratios are meaningful |
🧠 Simple rule of thumb: CV is appropriate for ratio scale data with a mean > 0.
6. CFA "Which Is Better" Question Template
Question Type 1: Given two datasets, determine which has greater relative variability
🔴 Classic CFA question: "Based on the coefficient of variation, which asset exhibits greater relative variability?"
Solution flow:
1. Compute CV_A = σ_A / μ_A
2. Compute CV_B = σ_B / μ_B
3. Compare: Larger CV → greater relative variability
4. Eliminate distractors that only look at standard deviation
Question Type 2: Scenario analysis — should CV be used?
A question may present data with a very small (or negative) mean and ask:
"The mean return is −0.5% and the standard deviation is 3%. Is the coefficient of variation meaningful?"
Answer: No. When the mean is negative, CV is negative and cannot provide a meaningful comparison.
7. Practical Case Studies
Case 1: Fund Manager Selection
Ivan is evaluating two fund managers' performance over the past three years:
| Metric | Manager X | Manager Y |
|---|---|---|
| Avg. Annual Excess Return | 8.5% | 14.0% |
| Excess Return Std Dev | 7.2% | 13.0% |
Question: From a relative dispersion perspective, which manager's returns are more "stable and efficient"?
Solution:
- Manager X: CV = 7.2% / 8.5% = 0.847
- Manager Y: CV = 13.0% / 14.0% = 0.929
→ Manager X has a lower CV (0.847 < 0.929), meaning less volatility per unit of excess return.
💡 Although Manager Y has higher absolute returns, relative dispersion is also greater. For a more comprehensive assessment incorporating the risk-free rate, use the Sharpe Ratio — which happens to be the topic of L113!
Case 2: Comparing Volatility Efficiency Across Markets
An analyst collected the following data:
| Market | Avg. Daily Return | Daily Return Std Dev |
|---|---|---|
| Emerging Market ETF | 0.08% | 1.50% |
| Developed Market ETF | 0.04% | 0.60% |
Question: Which market has higher return-volatility efficiency (less volatility per unit of return)?
Solution:
- Emerging Market CV = 1.50% / 0.08% = 18.75
- Developed Market CV = 0.60% / 0.04% = 15.00
→ The developed market has a lower CV, meaning less volatility per unit of return — higher efficiency.
💡 While the emerging market offers higher absolute returns, you must endure a disproportionate amount of extra volatility to capture that incremental return. This is precisely the insight CV reveals.
8. CV vs. Other Dispersion Measures: Summary
| Measure | Type | Purpose | Limitation |
|---|---|---|---|
| Range | Absolute dispersion | Quick spread overview | Sensitive to outliers |
| Variance | Absolute dispersion | Theoretical derivations | Squared units, unintuitive |
| Standard Deviation | Absolute dispersion | Most common dispersion measure | Cannot compare different means |
| Coefficient of Variation (CV) | Relative dispersion | Cross-mean / cross-unit comparison | Fails when mean ≤ 0 |
9. Common Pitfalls
| Pitfall | Correct Understanding |
|---|---|
| "Higher CV means better returns" | ❌ CV = volatility/return — higher is WORSE, lower is more efficient |
| "Lower SD is always better" | ❌ When means differ, use CV for relative comparison |
| "CV works for any dataset" | ❌ CV is meaningless when mean ≤ 0; use caution with interval scale data (e.g., temperature) |
| "CV can replace the Sharpe Ratio" | ❌ CV does not incorporate the risk-free rate; it measures relative volatility, not risk-adjusted return |
| "CV = σ² / μ" | ❌ The numerator is standard deviation, NOT variance! This is the most common calculation error |
10. Practice Questions
Question 1
An analyst collected the following data for a stock:
- Average monthly return = 1.2%
- Monthly return standard deviation = 3.6%
The coefficient of variation (CV) for this stock is closest to:
A. 0.33 B. 0.30 C. 3.00 D. 0.03
Question 2
In which of the following situations is the coefficient of variation NOT appropriate for comparison?
A. Comparing the relative volatility of the S&P 500 and Nikkei 225 B. Comparing the risk efficiency of gold (mean 0.5%) and Bitcoin (mean 4.0%) C. Comparing the performance of two hedge funds both with negative average monthly returns D. Comparing price volatility of stocks with different face values ($50 vs $800)
Question 3
Asset P has a mean of 6% and a standard deviation of 15%. Asset Q has a mean of 2% and a standard deviation of 4%. Regarding their relative dispersion, which of the following is correct?
A. P has higher relative dispersion than Q B. Q has higher relative dispersion than P C. Both have equal relative dispersion D. Insufficient information to compare
Question 4 (Application)
A quantitative team is testing two market-timing strategies:
| Strategy | Avg. Monthly Excess Return | Excess Return Std Dev |
|---|---|---|
| Strategy Alpha | 0.45% | 1.80% |
| Strategy Beta | 0.30% | 0.90% |
If the team wants to select the strategy with the "lowest volatility per unit of excess return," they should choose:
A. Strategy Alpha, because its CV is lower B. Strategy Beta, because its CV is lower C. Strategy Alpha, because absolute returns are higher D. Strategy Beta, because absolute standard deviation is lower
11. Answers and Explanations
Answer 1: C — 3.00
CV = s / X̄ = 3.6% / 1.2% = 3.0
🚨 Common mistake: A (0.33) results from X̄ / s — reversing numerator and denominator! Remember: CV = Standard Deviation ÷ Mean, not the other way around.
Answer 2: C — Comparing two hedge funds both with negative average monthly returns
When the mean is negative, CV becomes negative and loses comparative meaning. For example: mean = −1%, SD = 5% → CV = −5; mean = −2%, SD = 5% → CV = −2.5. The ordering among negative CVs no longer reflects the intuitive concept of "relative variability."
Answer 3: A — P has higher relative dispersion than Q
CV_P = 15% / 6% = 2.50 CV_Q = 4% / 2% = 2.00 2.50 > 2.00 → P carries more volatility per unit of return.
Answer 4: B — Strategy Beta, because its CV is lower
CV_Alpha = 1.80% / 0.45% = 4.0 CV_Beta = 0.90% / 0.30% = 3.0
Beta's CV is lower (3.0 < 4.0) → less volatility per unit of excess return.
Option D focuses on the "smaller standard deviation" but ignores the smaller mean — exactly the cognitive trap that CV is designed to overcome!
📌 Three key takeaways for today: 1. CV = Standard Deviation ÷ Mean — measures "volatility per unit of mean" — the smaller, the more efficient 2. CV's killer application: comparing datasets with different means or different units when standard deviation alone is insufficient 3. Remember the hard limits: CV fails when mean ≤ 0; CV does not incorporate the risk-free rate (coming up in L113 — the Sharpe Ratio!)
L112 Coefficient of Variation | 2026-07-18 | CFA Level 1 Quantitative Methods