定量方法（Quantitative Methods）— 描述性统计模块
一、从变异系数到夏普比率：自然的进化
上节课 L112 我们学了变异系数（CV）：
$$\text{CV} = \frac{\sigma}{\mu}$$
CV 能告诉我们「每单位收益伴随多少波动」。但它有一个"盲区"：
📌 案例：假设无风险利率（国债收益率）是 3%。A 基金年均收益 4%，标准差 2%；B 基金年均收益 8%，标准差 5%。
CV_A = 2%/4% = 0.50 CV_B = 5%/8% = 0.625
CV 说 A 更"有效率"…… 但 A 的收益（4%）几乎只比无风险利率（3%）多 1%！你愿意为了 1% 的超额收益承担 2% 的波动吗？
这就是 CV 的缺口：它不区分「有风险的收益」和「无风险的收益」。
🧠 结论：投资者真正关心的是 相对于无风险收益的超额部分 是否值得承担波动——夏普比率（Sharpe Ratio）就是为了解决这个问题。
二、夏普比率的定义
公式：
$$\text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}$$
其中： - $R_p$ = 投资组合的平均收益率 - $R_f$ = 无风险利率（通常用短期国债收益率） - $R_p - R_f$ = 超额收益（excess return） - $\sigma_p$ = 投资组合收益率的 标准差（总风险）
通俗理解：
夏普比率 = 每承担一单位总风险，能获得多少超额的「风险溢价」
如果夏普比率 = 0.8，意味着每承担 1% 的波动，获得 0.8% 的超额收益。
夏普比率越高 → 风险调整后收益越好 → 越"划算"
三、夏普比率 vs 变异系数的本质区别
| 维度 | 变异系数（CV） | 夏普比率（Sharpe） |
|---|---|---|
| 公式 | $\sigma / R_p$ | $(R_p - R_f) / \sigma$ |
| 核心问题 | 每单位收益的波动多少？ | 每单位波动能赚多少超额收益？ |
| 是否涉及无风险利率 | ❌ 不涉及 | ✅ 必须考虑 |
| 数字含义 | 越小越好 | 越大越好 |
| 分子/分母 | $\sigma$ 在分子 | $\sigma$ 在分母 |
| 适用场景 | 简单的相对离散度比较 | 投资决策、基金经理排名、资产配置 |
🧠 记忆技巧：CV = 波动 ÷ 收益（越小越优），夏普 = 超额收益 ÷ 波动（越大越优）——方向相反！
四、夏普比率的计算——CFA 高频实操
⭐ 基础计算
案例 1：单只基金
某基金过去 5 年平均年收益率 12%，年化标准差 15%，无风险利率 3%。求夏普比率。
$$Sharpe = \frac{12\% - 3\%}{15\%} = \frac{9\%}{15\%} = 0.60$$
→ 每承担 1% 的总风险，获得 0.60% 的超额收益。
⭐ 年化换算（CFA 必考）
计算夏普比率时，「收益」和「标准差」的 时间频率必须匹配！
如果给定的是月度数据：
$$Sharpe_{annual} \approx Sharpe_{monthly} \times \sqrt{12}$$
推导逻辑： - 年化超额收益 = 月度超额收益 × 12 - 年化标准差 = 月度标准差 × √12 - 年化夏普 = (月度超额 × 12) / (月度 σ × √12) = 月度夏普 × √12
案例 2：月度数据转年度
某策略月均超额收益 0.5%，月度标准差 2.0%。求 年化 夏普比率。
$$Sharpe_{monthly} = \frac{0.5\%}{2.0\%} = 0.25$$
$$Sharpe_{annual} = 0.25 \times \sqrt{12} \approx 0.25 \times 3.464 = 0.866$$
🚨 常见错误：直接用月度夏普 0.25 当年度夏普 → 大幅低估！
五、夏普比率的横向比较：CFA「哪个更好」题型
题型模板
给出 3-4 个投资组合的风险和收益数据，问哪个「风险调整后表现最好」。
解题流程：
1. 确认 R_f（题目通常明确给定）
2. 分别计算各组合的 Sharpe = (R_p − R_f) / σ_p
3. 排序：Sharpe 最高 → 风险调整后表现最好
4. 注意陷阱：有时某组合收益最高但波动也极大→ Sharpe 反而更低
案例 3：基金经理排名
| 组合 | 平均收益 | 标准差 | 夏普比率（Rf = 2%） |
|---|---|---|---|
| X | 10% | 12% | (10%−2%)/12% = 0.667 |
| Y | 8% | 7% | (8%−2%)/7% = 0.857 |
| Z | 15% | 22% | (15%−2%)/22% = 0.591 |
✅ 排序：Y（0.857）> X（0.667）> Z（0.591）
💡 Z 的绝对收益最高（15%），但夏普比率最低——说明高收益是靠更高的风险"堆"出来的，风险调整后并不划算。
六、夏普比率的局限性（CFA 必考）
夏普比率好用，但不是万能药。考试经常考它的缺陷：
1. 基于总风险，而非系统性风险
夏普比率用的标准差 $\sigma$ 包含了 系统性风险 和 非系统性风险 两部分。对于充分分散的投资组合，真正的风险是系统性风险（用 Beta 衡量）。
→ 此时 特雷诺比率（Treynor Ratio） = $(R_p - R_f)/\beta_p$ 更合适（L2 内容）。
2. 假设收益服从正态分布
当收益分布有显著的偏度或峰度时（如对冲基金的对数正态分布、期权策略的尾部风险），标准差不能完整描述风险 → 夏普比率可能低估或高估真实风险。
3. 负夏普比率的排序问题
当超额收益为负时，夏普比率为负值。此时排序出现矛盾：
组合 R_p σ_p Sharpe（Rf=5%） A 3% 10% −0.20 B 1% 5% −0.80 A 的夏普（−0.20）> B（−0.80），但 A 的绝对损失更大（−2% vs −4%超额收益）。负夏普的直接排序结论可能误导。
4. 只看波动，不看下行风险
标准差把「上涨波动」和「下跌波动」一视同仁。但投资者真正害怕的是下跌。
→ 索提诺比率（Sortino Ratio） = $(R_p - R_f)/\sigma_{down}$，只惩罚下行波动（L2 内容）。
5. 历史夏普 ≠ 未来夏普
用历史数据计算的夏普比率是 ex-post（事后）度量，不保证未来表现。
七、实战案例
案例 1：Ivan 哥的资产配置决策
Ivan 哥在考虑两个 ETF 配置方案：
| 方案 | 预期年收益 | 预期标准差 |
|---|---|---|
| 60% 股票 + 40% 债券 | 7.5% | 10.0% |
| 80% 股票 + 20% 债券 | 9.0% | 16.0% |
假设无风险利率 = 3%，仅从夏普比率角度判断哪个方案更优。
解题：
- 60/40：Sharpe = (7.5% − 3%) / 10.0% = 0.450
- 80/20：Sharpe = (9.0% − 3%) / 16.0% = 0.375
→ 60/40 组合虽然绝对收益更低，但风险调整后效率更高。每承担 1% 风险带来的超额收益更多。
案例 2：量化策略回测评估
某量化团队回测了 3 个策略（月频数据，Rf = 2%/年 ≈ 0.167%/月）：
| 策略 | 月均收益 | 月标准差 |
|---|---|---|
| 动量 | 1.2% | 4.5% |
| 价值 | 0.8% | 3.0% |
| 套利 | 0.5% | 1.2% |
问题：按年化夏普比率排序。
解题：
- 动量月夏普 = (1.2% − 0.167%) / 4.5% = 0.230 → 年化 = 0.230 × √12 ≈ 0.796
- 价值月夏普 = (0.8% − 0.167%) / 3.0% = 0.211 → 年化 = 0.211 × √12 ≈ 0.731
- 套利月夏普 = (0.5% − 0.167%) / 1.2% = 0.278 → 年化 = 0.278 × √12 ≈ 0.962
排序：套利（0.962）> 动量（0.796）> 价值（0.731）
💡 套利的绝对月收益最低（0.5%），但波动极小 → 年化夏普反而最高。这就是「低波动高质量收益」的魅力。
案例 3：市场下跌期间的夏普比率
2022 年某基金全年收益 −12%，标准差 18%，无风险利率 4%。
$$Sharpe = \frac{-12\% - 4\%}{18\%} = \frac{-16\%}{18\%} = -0.889$$
负夏普比率告诉我们：这只基金在下跌之年不仅跑输了无风险利率，而且表现显著更差。但注意——不要用负夏普去做基金间的精细排序（上文局限性已说明）。
八、夏普比率的知识家族：风险调整指标速查
| 指标 | 公式 | 风险度量 | 适用于 |
|---|---|---|---|
| 夏普比率 | $(R_p - R_f)/\sigma$ | 总风险（σ） | 非充分分散组合 |
| 特雷诺比率 | $(R_p - R_f)/\beta$ | 系统性风险（β） | 充分分散组合 |
| M²（Modigliani-Modigliani） | $R_f + Sharpe \times \sigma_m$ | 总风险 | 与市场组合直接比较 |
| 詹森阿尔法 | $R_p - [R_f + \beta(R_m - R_f)]$ | — | 衡量超额能力 |
| 索提诺比率 | $(R_p - R_f)/\sigma_{down}$ | 下行风险 | 关注回撤 |
| 信息比率 | $(R_p - R_b)/\sigma_{tracking}$ | 跟踪误差 | 主动管理 vs 基准 |
📌 CFA 一级重点掌握 夏普比率 和 特雷诺比率 的区别：前者用 σ（总风险），后者用 β（系统性风险）。其他指标在 L2/L3 深入。
九、常见易错点总结
| 易错点 | 正确理解 |
|---|---|
| "夏普比率 = R_p / σ_p" | ❌ 漏了 R_f！必须减去无风险利率 |
| "夏普比率和 CV 差不多" | ❌ CV = σ/R（越小越好），夏普 = (R−Rf)/σ（越大越好），概念和方向都不同 |
| "月度夏普就是年度夏普" | ❌ 年化夏普 = 月度夏普 × √12（近似公式） |
| "夏普高就一定好" | ❌ 没考虑偏度、非正态分布、回测偏差等因素 |
| "负夏普可以直接排名" | ❌ 负夏普的排序可能误导，需谨慎 |
| "夏普比率衡量的是绝对收益" | ❌ 衡量的是 风险调整后 的收益——靠排除低风险低效组合 |
十、测试题
题目 1
某基金年均收益 14%，标准差 18%，无风险利率为 4%。该基金的夏普比率最接近：
A. 0.78 B. 0.56 C. 0.44 D. 1.00
题目 2
某策略月均超额收益为 0.8%，月度标准差为 3.0%。其 年化 夏普比率最接近：
A. 0.27 B. 0.92 C. 3.20 D. 0.31
题目 3
以下哪种情况最适合使用夏普比率进行投资决策？
A. 比较三只已充分分散的共同基金的风险调整后表现 B. 比较一只高度集中的对冲基金与标普 500 指数的表现 C. 对月均收益均为负值的两只基金进行排名 D. 评估一只期权策略基金（收益分布高度右偏）的风险表现
题目 4（比较题）
分析师收集了三只基金的数据（Rf = 3%）：
| 基金 | 年收益 | 年标准差 |
|---|---|---|
| A | 11% | 10% |
| B | 9% | 7% |
| C | 14% | 20% |
按夏普比率从高到低排序，正确的是：
A. A > B > C B. B > A > C C. C > A > B D. A > C > B
题目 5（判断对错）
"某投资组合的年化夏普比率为 −0.5，说明该组合的风险调整后表现比年化夏普比率为 −0.3 的组合更差。"
A. 正确 B. 错误
十一、答案与解析
答案 1：B — 0.56
Sharpe = (14% − 4%) / 18% = 10% / 18% = 0.5556 ≈ 0.56
🚨 A（0.78）= 14%/18%，忘了减 Rf！D（1.0）通常是把 Rf 算错了。
答案 2：B — 0.92
月度夏普 = 0.8% / 3.0% = 0.2667 年化夏普 ≈ 0.2667 × √12 = 0.2667 × 3.464 = 0.924 ≈ 0.92
🚨 A（0.27）= 误用月度夏普作年度夏普。C（3.20）= 拿月度夏普 × 12 而不是 × √12。
答案 3：A — 比较三只已充分分散的共同基金
夏普比率适合比较非分散组合或作为广泛的风险调整指标。B 中高度集中的基金用 σ 衡量总风险仍然合理（因为未分散的特有风险需要被惩罚）。但 C 的负收益排序问题和 D 的非正态分布是夏普比率的已知缺陷。
📌 CFA 核心考点：夏普比率用 σ 📌 CFA 核心考点：夏普比率用 σ（总风险），包括非系统性风险，适合非充分分散组合。对于充分分散的组合，特雷诺比率（用 β）更精确。
答案 4：B — B > A > C
Sharpe_A = (11% − 3%) / 10% = 0.800 Sharpe_B = (9% − 3%) / 7% = 0.857 Sharpe_C = (14% − 3%) / 20% = 0.550
排序：B（0.857）> A（0.800）> C（0.550）
💡 C 的绝对收益最高（14%），但波动大得不成比例（20%）→ 夏普比率最低。B 的绝对收益最低（9%），但波动极小（7%）→ 夏普最高。
答案 5：B — 错误
当夏普比率为负值时，直接排序会产生误导。例如：
- 组合 X：Rf = 5%，Rp = 3%，σ = 20% → Sharpe = (3%−5%)/20% = −0.10
- 组合 Y：Rf = 5%，Rp = −1%，σ = 5% → Sharpe = (−1%−5%)/5% = −1.20
X 的夏普（−0.10）> Y（−1.20），但 Y 不仅跑输无风险利率，还亏本金。负夏普的数值排序不能直接套用「越大越好」的直觉——这是 CFA 一级的经典陷阱题。
📌 今日要点记住四句话： 1. 夏普比率 = (Rp − Rf) / σ —— 每单位总风险换来的超额收益，越大越好 2. CV vs 夏普：CV 不涉及 Rf（σ/μ，越小越好），夏普必须减去 Rf（方向相反） 3. 年化：Sharpe_annual ≈ Sharpe_monthly × √12 —— 不是乘 12！ 4. 负夏普时排序要小心，非正态分布用索提诺比率更合理
L113 夏普比率（Sharpe Ratio） | 2026-07-19 | CFA Level 1 定量方法
Quantitative Methods — Descriptive Statistics Module
1. From Coefficient of Variation to Sharpe Ratio: A Natural Evolution
In L112, we studied the Coefficient of Variation (CV):
$$\text{CV} = \frac{\sigma}{\mu}$$
CV tells us "how much volatility per unit of return," but it has a blind spot:
📌 Example: Suppose the risk-free rate (T-bill yield) is 3%. Fund A has an average return of 4% with a standard deviation of 2%; Fund B has an average return of 8% with a standard deviation of 5%.
CV_A = 2%/4% = 0.50 CV_B = 5%/8% = 0.625
CV says Fund A is more "efficient"... but Fund A's return (4%) barely exceeds the risk-free rate (3%) by 1%! Would you accept 2% volatility for just 1% excess return?
This is CV's blind spot: it does not distinguish between "risky returns" and "risk-free returns."
🧠 Conclusion: Investors truly care about whether the excess return over the risk-free rate justifies the risk taken — the Sharpe Ratio was designed precisely to address this.
2. Definition of the Sharpe Ratio
Formula:
$$\text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}$$
Where: - $R_p$ = average return of the portfolio - $R_f$ = risk-free rate (typically short-term T-bill yield) - $R_p - R_f$ = excess return - $\sigma_p$ = standard deviation of portfolio returns (total risk)
Intuitive Interpretation:
Sharpe Ratio = how much excess return (risk premium) you earn per unit of total risk
If the Sharpe Ratio = 0.80, it means for every 1% of volatility, you gain 0.80% in excess return.
Higher Sharpe Ratio → better risk-adjusted performance → more "worth it"
3. Sharpe Ratio vs. Coefficient of Variation: Key Differences
| Dimension | CV | Sharpe Ratio |
|---|---|---|
| Formula | $\sigma / R_p$ | $(R_p - R_f) / \sigma$ |
| Core Question | How much volatility per unit of return? | How much excess return per unit of risk? |
| Risk-Free Rate | ❌ Not involved | ✅ Must be considered |
| Interpretation | Lower is better | Higher is better |
| Numerator/Denominator | σ in numerator | σ in denominator |
| Application | Simple relative dispersion comparison | Investment decisions, manager rankings, asset allocation |
🧠 Memory trick: CV = volatility ÷ return (lower is better); Sharpe = excess return ÷ volatility (higher is better) — opposite directions!
4. Sharpe Ratio Calculation — High-Frequency CFA Topic
⭐ Basic Calculation
Example 1: A Single Fund
A fund has a 5-year average annual return of 12%, annualized standard deviation of 15%, and the risk-free rate is 3%. Calculate the Sharpe Ratio.
$$Sharpe = \frac{12\% - 3\%}{15\%} = \frac{9\%}{15\%} = 0.60$$
→ For every 1% of total risk, the fund earns 0.60% in excess return.
⭐ Annualization (CFA Must-Know)
When calculating the Sharpe Ratio, the time frequency of "return" and "standard deviation" must match!
If given monthly data:
$$Sharpe_{annual} \approx Sharpe_{monthly} \times \sqrt{12}$$
Derivation: - Annualized excess return = monthly excess return × 12 - Annualized standard deviation = monthly σ × √12 - Annualized Sharpe = (monthly excess × 12) / (monthly σ × √12) = monthly Sharpe × √12
Example 2: Converting Monthly to Annual
A strategy has a monthly average excess return of 0.5% and a monthly standard deviation of 2.0%. Calculate the annualized Sharpe Ratio.
$$Sharpe_{monthly} = \frac{0.5\%}{2.0\%} = 0.25$$
$$Sharpe_{annual} = 0.25 \times \sqrt{12} \approx 0.25 \times 3.464 = 0.866$$
🚨 Common mistake: using the monthly Sharpe of 0.25 directly as the annual Sharpe → significant underestimation!
5. Cross-Sectional Comparison: CFA "Which Is Better" Question Format
Question Template
Given risk and return data for 3-4 portfolios, determine which one has the "best risk-adjusted performance."
Solution Process:
1. Confirm R_f (typically explicitly given in the question)
2. Calculate Sharpe for each portfolio: (R_p − R_f) / σ_p
3. Rank: highest Sharpe → best risk-adjusted performance
4. Watch for traps: sometimes the highest-return portfolio also has extreme volatility → Sharpe may actually be lower
Example 3: Fund Manager Ranking
| Portfolio | Average Return | Standard Deviation | Sharpe Ratio (Rf = 2%) |
|---|---|---|---|
| X | 10% | 12% | (10%−2%)/12% = 0.667 |
| Y | 8% | 7% | (8%−2%)/7% = 0.857 |
| Z | 15% | 22% | (15%−2%)/22% = 0.591 |
✅ Ranking: Y (0.857) > X (0.667) > Z (0.591)
💡 Z has the highest absolute return (15%), but the lowest Sharpe Ratio — the high return is "piled on" with disproportionately high risk. On a risk-adjusted basis, it is not worthwhile.
6. Limitations of the Sharpe Ratio (CFA Must-Know)
The Sharpe Ratio is useful but not a silver bullet. Its weaknesses are frequently tested:
1. Based on Total Risk, Not Systematic Risk
The Sharpe Ratio uses standard deviation σ, which encompasses both systematic and unsystematic risk. For well-diversified portfolios, the true relevant risk is systematic risk (measured by Beta).
→ In that case, the Treynor Ratio = $(R_p - R_f)/\beta_p$ is more appropriate (Level 2 content).
2. Assumes Returns Are Normally Distributed
When return distributions exhibit significant skewness or kurtosis (e.g., log-normal distributions of hedge funds, tail risk in option strategies), standard deviation cannot fully describe risk → the Sharpe Ratio may underestimate or overestimate true risk.
3. Ranking Issues with Negative Sharpe Ratios
When excess returns are negative, the Sharpe Ratio is negative. Ranking becomes problematic:
Portfolio R_p σ_p Sharpe (Rf=5%) A 3% 10% −0.20 B 1% 5% −0.80 A's Sharpe (−0.20) > B's (−0.80), but A has a larger absolute shortfall (−2% vs −4% excess). Direct ranking of negative Sharpe ratios can be misleading.
4. Penalizes All Volatility Equally, Not Just Downside
Standard deviation treats "upside volatility" and "downside volatility" equally. But investors truly fear downside.
→ Sortino Ratio = $(R_p - R_f)/\sigma_{down}$, penalizing only downside deviation (Level 2 content).
5. Historical Sharpe ≠ Forward-Looking Sharpe
The Sharpe Ratio calculated from historical data is an ex-post measure and does not guarantee future performance.
7. Practical Case Studies
Case 1: Asset Allocation Decision
Ivan is evaluating two ETF allocation approaches:
| Allocation | Expected Annual Return | Expected Std Dev |
|---|---|---|
| 60% Stocks + 40% Bonds | 7.5% | 10.0% |
| 80% Stocks + 20% Bonds | 9.0% | 16.0% |
Assume Rf = 3%. Based solely on the Sharpe Ratio, which allocation is superior?
Solution:
- 60/40: Sharpe = (7.5% − 3%) / 10.0% = 0.450
- 80/20: Sharpe = (9.0% − 3%) / 16.0% = 0.375
→ The 60/40 portfolio, despite lower absolute return, is more efficient on a risk-adjusted basis. Each unit of risk delivers more excess return.
Case 2: Quant Strategy Backtest Evaluation
A quant team backtested 3 strategies (monthly frequency, Rf = 2%/year ≈ 0.167%/month):
| Strategy | Avg Monthly Return | Monthly Std Dev |
|---|---|---|
| Momentum | 1.2% | 4.5% |
| Value | 0.8% | 3.0% |
| Arbitrage | 0.5% | 1.2% |
Question: Rank by annualized Sharpe Ratio.
Solution:
- Momentum monthly Sharpe = (1.2% − 0.167%) / 4.5% = 0.230 → annualized = 0.230 × √12 ≈ 0.796
- Value monthly Sharpe = (0.8% − 0.167%) / 3.0% = 0.211 → annualized = 0.211 × √12 ≈ 0.731
- Arbitrage monthly Sharpe = (0.5% − 0.167%) / 1.2% = 0.278 → annualized = 0.278 × √12 ≈ 0.962
Ranking: Arbitrage (0.962) > Momentum (0.796) > Value (0.731)
💡 Arbitrage has the lowest absolute monthly return (0.5%), but extremely low volatility → the highest annualized Sharpe. This is the beauty of "low-volatility, high-quality returns."
Case 3: Sharpe Ratio During a Market Downturn
In 2022, a fund delivered an annual return of −12%, with a standard deviation of 18%, and the risk-free rate was 4%.
$$Sharpe = \frac{-12\% - 4\%}{18\%} = \frac{-16\%}{18\%} = -0.889$$
A negative Sharpe Ratio tells us: this fund not only underperformed the risk-free rate during the downturn but did so significantly worse. However — do not use negative Sharpe ratios for fine-grained ranking between funds (see limitations above).
8. The Sharpe Ratio Family: Risk-Adjusted Metrics Cheat Sheet
| Metric | Formula | Risk Measure | Best For |
|---|---|---|---|
| Sharpe Ratio | $(R_p - R_f)/\sigma$ | Total risk (σ) | Non-fully diversified portfolios |
| Treynor Ratio | $(R_p - R_f)/\beta$ | Systematic risk (β) | Well-diversified portfolios |
| M² (Modigliani-Modigliani) | $R_f + Sharpe \times \sigma_m$ | Total risk | Direct comparison with market portfolio |
| Jensen's Alpha | $R_p - [R_f + \beta(R_m - R_f)]$ | — | Measuring excess skill |
| Sortino Ratio | $(R_p - R_f)/\sigma_{down}$ | Downside risk | Drawdown-focused analysis |
| Information Ratio | $(R_p - R_b)/\sigma_{tracking}$ | Tracking error | Active management vs. benchmark |
📌 For CFA Level 1, focus on the distinction between the Sharpe Ratio and the Treynor Ratio: the former uses σ (total risk), the latter uses β (systematic risk). Other metrics are covered in Levels 2 and 3.
9. Common Pitfalls Summary
| Pitfall | Correct Understanding |
|---|---|
| "Sharpe Ratio = R_p / σ_p" | ❌ Missing R_f! Must subtract the risk-free rate |
| "Sharpe and CV are basically the same" | ❌ CV = σ/R (lower is better), Sharpe = (R−Rf)/σ (higher is better) — different concepts and opposite directions |
| "Monthly Sharpe = Annual Sharpe" | ❌ Annualized Sharpe = monthly Sharpe × √12 (approximation) |
| "Higher Sharpe always means better" | ❌ Does not account for skewness, non-normal distributions, backtest bias, etc. |
| "Negative Sharpe ratios can be ranked directly" | ❌ Ranking negative Sharpe ratios can be misleading — exercise caution |
| "Sharpe Ratio measures absolute return" | ❌ It measures risk-adjusted return — screening out low-risk, inefficient portfolios |
10. Test Questions
Question 1
A fund has an average annual return of 14%, a standard deviation of 18%, and the risk-free rate is 4%. The fund's Sharpe Ratio is closest to:
A. 0.78 B. 0.56 C. 0.44 D. 1.00
Question 2
A strategy has a monthly average excess return of 0.8% and a monthly standard deviation of 3.0%. Its annualized Sharpe Ratio is closest to:
A. 0.27 B. 0.92 C. 3.20 D. 0.31
Question 3
In which of the following situations is the Sharpe Ratio MOST appropriate for investment decision-making?
A. Comparing the risk-adjusted performance of three well-diversified mutual funds B. Comparing a highly concentrated hedge fund to the S&P 500 index C. Ranking two funds that both have negative monthly average returns D. Evaluating the risk performance of an options strategy fund with a highly right-skewed return distribution
Question 4 (Comparison)
An analyst collected data on three funds (Rf = 3%):
| Fund | Annual Return | Annual Std Dev |
|---|---|---|
| A | 11% | 10% |
| B | 9% | 7% |
| C | 14% | 20% |
Ranked by Sharpe Ratio from highest to lowest, the correct order is:
A. A > B > C B. B > A > C C. C > A > B D. A > C > B
Question 5 (True/False)
"A portfolio with an annualized Sharpe Ratio of −0.5 has worse risk-adjusted performance than one with an annualized Sharpe Ratio of −0.3."
A. True B. False
11. Answers and Explanations
Answer 1: B — 0.56
Sharpe = (14% − 4%) / 18% = 10% / 18% = 0.5556 ≈ 0.56
🚨 A (0.78) = 14%/18%, forgetting to subtract Rf! D (1.0) typically comes from miscalculating Rf.
Answer 2: B — 0.92
Monthly Sharpe = 0.8% / 3.0% = 0.2667 Annualized Sharpe ≈ 0.2667 × √12 = 0.2667 × 3.464 = 0.924 ≈ 0.92
🚨 A (0.27) = mistakenly using the monthly Sharpe as the annual Sharpe. C (3.20) = multiplying the monthly Sharpe by 12 instead of √12.
Answer 3: A — Comparing three well-diversified mutual funds
The Sharpe Ratio is appropriate for comparing non-diversified portfolios or as a broad risk-adjustment metric. For B, a highly concentrated fund still has its undiversified idiosyncratic risk appropriately penalized by σ. However, C's negative return ranking issue and D's non-normal distribution are known limitations of the Sharpe Ratio.
📌 CFA core point: The Sharpe Ratio uses σ (total risk), including unsystematic risk, making it suitable for non-fully diversified portfolios. For well-diversified portfolios, the Treynor Ratio (using β) is more precise.
Answer 4: B — B > A > C
Sharpe_A = (11% − 3%) / 10% = 0.800 Sharpe_B = (9% − 3%) / 7% = 0.857 Sharpe_C = (14% − 3%) / 20% = 0.550
Ranking: B (0.857) > A (0.800) > C (0.550)
💡 C has the highest absolute return (14%), but disproportionate volatility (20%) → the lowest Sharpe. B has the lowest absolute return (9%), but extremely low volatility (7%) → the highest Sharpe.
Answer 5: B — False
When Sharpe Ratios are negative, direct ranking is misleading. For example:
- Portfolio X: Rf = 5%, Rp = 3%, σ = 20% → Sharpe = (3%−5%)/20% = −0.10
- Portfolio Y: Rf = 5%, Rp = −1%, σ = 5% → Sharpe = (−1%−5%)/5% = −1.20
X's Sharpe (−0.10) > Y's (−1.20), but Y not only underperforms the risk-free rate — it loses principal. The "higher is better" intuition cannot be directly applied to negative Sharpe ratio rankings — this is a classic CFA Level 1 trap question.
📌 Today's four key takeaways: 1. Sharpe Ratio = (Rp − Rf) / σ — excess return per unit of total risk; higher is better 2. CV vs. Sharpe: CV does not involve Rf (σ/μ, lower is better); Sharpe must subtract Rf (opposite direction) 3. Annualization: Sharpe_annual ≈ Sharpe_monthly × √12 — NOT × 12! 4. Exercise caution when ranking negative Sharpe ratios; for non-normal distributions, the Sortino Ratio is more appropriate
L113 Sharpe Ratio | 2026-07-19 | CFA Level 1 Quantitative Methods