📖 峰度（Kurtosis）

Quantitative Methods — Descriptive Statistics Module

1. What Is Kurtosis?

Kurtosis is a statistical measure that describes the thickness of the tails of a distribution — i.e., how likely extreme values are to occur.

💡 Intuitive understanding: Kurtosis is not about how "peaked" the mountain is — it's about how fat the tails are. If the tails of a normal distribution are like chopsticks, the tails of a leptokurtic distribution are like baseball bats — thicker and heavier, meaning more extreme values.

Core CFA Level 1 definition:

Kurtosis measures the combined weight of the tails relative to the rest of the distribution.

2. Three Types: Benchmarked Against the Normal Distribution

The kurtosis of a normal distribution is 3 (derived mathematically).

Using this as the benchmark, distributions fall into three categories:

2.1 Mesokurtic

• Kurtosis = 3 / Excess Kurtosis = 0
• This is the normal distribution itself
• Tail thickness is "just right"

       /\
      /  \
     /    \
    /      \
——+————————+——
  normal tails

2.2 Leptokurtic (Fat-Tailed)

• Kurtosis > 3 / Excess Kurtosis > 0
• Tails are fatter than the normal distribution (extreme values occur more frequently)
• The peak is also taller and sharper than normal (more probability mass near the mean AND in the tails, less in the intermediate regions)

      /\
     /  \          ← sharper peak
    /    \
   /      \_____
——+————————+——
          ↑ fatter tails

🧠 Key insight: Leptokurtic = "more data near the mean" + "more data in the extreme tails" + "less data in the transition zones." In other words, it is more extreme than the normal — calmer when calm, crazier when crazy.

Real-world examples: - Daily stock returns (most days see small moves, but extreme crashes/rallies occur more often than the normal distribution predicts) - Credit default events (quiet most of the time, catastrophic when they occur) - Cryptocurrency price volatility

2.3 Platykurtic (Thin-Tailed)

• Kurtosis < 3 / Excess Kurtosis < 0
• Tails are thinner than the normal distribution (extreme values occur less frequently)
• The peak is also flatter

     _____
    /     \       ← flatter peak
   /       \
  /         \
——+————————+——
       ↑ thinner tails

Real-world examples: - Uniform distribution - Returns of heavily regulated utility stocks - Exchange rates within a central-bank-managed band

3. Excess Kurtosis

In the CFA exam, what you most often use is not kurtosis itself, but excess kurtosis:

$$\text{Excess Kurtosis} = \text{Kurtosis} - 3$$

🧠 Memory aid: - Lepto- = thin / slender (Greek) → sounds like the peak is sharp and thin → sharp peak + fat tails - Platy- = broad / flat (Greek) → sounds like a "plateau" → flat peak + thin tails

4. Kurtosis Formula (Awareness Only)

Sample Kurtosis:

$$K = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^{n} \left(\frac{X_i - \bar{X}}{s}\right)^4$$

Excess Kurtosis:

$$\text{Excess Kurtosis} = K - 3$$

⚠️ CFA Level 1 does not require manual calculation of kurtosis, but you must understand its meaning and applications. Note especially: the formula uses the 4th power, while skewness uses the 3rd power.

5. Skewness vs. Kurtosis: At-a-Glance Comparison

This is a high-frequency distinction in CFA Level 1:

📊 Practical takeaway: - Skewness tells you "which direction you fall" (direction) - Kurtosis tells you "how hard you hit the ground" (extremity)

An asset can be simultaneously negatively skewed + fat-tailed (e.g., a short put option strategy: occasional huge losses, and those losses occur more frequently than the normal distribution predicts).

6. Real-World Finance: Why Kurtosis Matters

Case 1: The Failure of VaR (Value at Risk)

During the 2008 financial crisis, risk models at major banks (built on the normal distribution assumption) severely underestimated tail risk because:

• Real-world market excess kurtosis ≈ 3–7 (far > 0) ← fat tails!
• Normal distribution prediction: probability of a daily drop of 5% ≈ once every 7,000 years
• Reality: October 2008 saw multiple such days in a single month

🧠 Conclusion: Ignoring kurtosis → underestimating extreme risk → getting destroyed by market "black swans."

Case 2: Two Portfolio Managers

Looking only at the Sharpe ratio, you might think Fund A is better. But adding skewness and kurtosis analysis reveals that Fund A's tail risk is far greater than Fund B's.

7. Common Pitfalls

8. Test Questions

Question 1

A portfolio's historical daily return distribution shows: 95% of days have returns between −1% and +1%, but 1% of days have returns outside the −5% to +5% range. The normal distribution predicts the probability of exceeding ±5% as only 0.01%.

This return distribution is most likely:

A. Mesokurtic B. Leptokurtic (fat-tailed) C. Platykurtic (thin-tailed) D. Uniform

Question 2

Regarding excess kurtosis, which of the following is incorrect?

A. The normal distribution has an excess kurtosis of 0 B. Excess kurtosis greater than 0 means extreme values occur more frequently than predicted by the normal distribution C. A distribution with excess kurtosis equal to 3 is leptokurtic D. Excess kurtosis can be negative

Question 3

An analyst finds that an asset's return distribution has: skewness < 0, excess kurtosis > 0. Which statement is correct?

A. The asset mostly posts small losses and occasionally large gains B. The asset's return distribution is safer than the normal distribution assumption C. The asset exhibits left-tail fat-tailed characteristics; extreme loss risk is higher than under the normal distribution assumption D. The asset's mean must be greater than its median

Question 4 (Application)

An analyst uses a normal distribution model to estimate a hedge fund's daily VaR (95% confidence) at −2%. However, backtesting reveals that actual daily losses exceeding −2% occur on 12% of days (versus the expected 5%). The most likely explanation is:

A. The fund's returns are positively skewed B. The fund's actual kurtosis is below 3 C. The fund's actual returns follow a fat-tailed distribution (excess kurtosis > 0) D. The fund's standard deviation has been underestimated

9. Answers and Explanations

Answer 1: B — Leptokurtic (fat-tailed)

The actual probability of extreme values (1%) is far higher than the normal distribution prediction (0.01%), indicating tails fatter than normal → leptokurtic (excess kurtosis > 0).

Answer 2: C — "A distribution with excess kurtosis equal to 3 is leptokurtic"

Any excess kurtosis > 0 is leptokurtic; it does not need to reach 3. Excess kurtosis = 3 means kurtosis = 6, which is extremely fat-tailed. Option C claims "equal to 3 is then leptokurtic," which is misleading — the threshold is > 0, not 3.

Answer 3: C — The asset exhibits left-tail fat-tailed characteristics; extreme loss risk is higher than under the normal distribution assumption

Skewness < 0 = left-skewed (negatively skewed) → extreme values are on the left; excess kurtosis > 0 = fat tails → extreme losses occur more frequently than predicted by the normal distribution. Combined, they mean "occasional huge losses happen more often than models predict."

Answer 4: C — The fund's actual returns follow a fat-tailed distribution (excess kurtosis > 0)

Using the normal distribution (excess kurtosis = 0) produced a VaR prediction that only 5% of days would breach −2%, but the actual figure is 12%, indicating extreme losses are far more frequent than the normal distribution predicts → fat tails. While D could also be true, the most direct and likely explanation is a fat-tailed distribution — when tails are fatter than normal, VaR systematically underestimates extreme risk.

📌 Three key takeaways for today: 1. Kurtosis measures tail thickness (not peak height) — the fatter the tails, the more frequent the extreme events 2. Excess kurtosis > 0 (leptokurtic) → the norm in financial markets; the normal distribution does not apply 3. Skewness tells you the direction (which way it's lopsided), kurtosis tells you the extremity (how badly you get hit) — you need both

L110 Kurtosis | 2026-07-16 | CFA Level 1 Quantitative Methods