Lecture 1 Heavy-tailed distributions

Question 1

Where is risk all about?

Answer 1

probabilities

Question 2

Where is tail risk about?

Answer 2

Small probabilities and Sometimes, about the probability of events we have never seen before!

Question 3

What are three methods to calculate Value-at-Risk (“VaR”)?

Answer 3

• Method 1: Normal distribution (basic)
• Method 2: Historical simulation (basic)
• Method 3: Power law tail (heavy tails)

Question 4

Definition of “Value-at-Risk” or “VaR”:

Answer 4

Definition:
The maximum loss over an t-days period with a x% confidence level.

Question 5

Formula Normal Distribution

Answer 5

𝝁 + 𝒛 ∗ 𝝈 = VaR
𝝁 = Average daily return
𝒛 = Z- score
𝝈 = Standard deviation

Question 6

What are the Z-values of these probabilities? 5% (0.95), 1% (0.99), 0.1% (0.9999)

Answer 6

5% -1.64
1% -2.33
0.1% -3.09

Question 7

What is the problem of the normal distribution?

Answer 7

tail of normal
distribution is too thin
(exponential-type shape)

Question 8

How to use the Historical simulation?

Answer 8

Rank n historical returns from low to high
Take the n*(100%-x%)th worst observation

Question 9

What are the problems with the historical sample?

Answer 9

Historical sample of returns doesn’t reflect future risk
It can’t measure the VaR of very small probabilities

Question 10

Why is the power law tail a better way than the normal distribution?

Answer 10

Also called a “fat tail” or a “heavy tail” – the power law tail ultimately results in more
probability mass for extreme outcomes than the normal distribution…
(so “heavy” or “fat” tails)

Question 11

Formula Power Law Tail

Answer 11

𝑉𝑎𝑅 = (𝐶/𝑝)^1/𝛼
𝐶 = “Scale parameter”
𝛼 = “Tail index” (usually 2.0≤α≤5)
𝑝 = probability

Question 12

What are the four remarks of the power law tail?

Answer 12

1. Models are estimated based on a limited amount of random data, so
   the number you calculate is never precisely the VaR!
   * In general: the further in the tail, the larger the estimation uncertainty…
2. Backward-looking risk models: historical returns do not reflect future risk
3. How much historical data to use?
   ➢ Did the risk characteristics of the underlying asset change?
   ❖ If yes: choose shorter estimation horizon…
   ➢ Are you interested in extremely small probability events?
   ❖ If yes: choose longer estimation horizon…
4. Volatility clustering
   * Periods of high and low volatility
   * Can result in several VaR exceptions in a short period of time