This document introduces the Forecasting Extreme Events Risk (FEER) Index, which is described as a downside risk measure that is coherent, sensitive to high moments of a distribution, imposes a complete order on gambles based on stochastic dominance, and is dynamically calibrated to implied market probability. The FEER Index aims to forecast extreme events by serving as a measure of tail risk that incorporates information from options markets to assess rare but severe risks.

1. GAL ZAHAVI, AMITAY KAUFFMAN
Financial Engineering and Risk Management Lab
TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY
THE WILLIAM DAVIDSON FACULTY OF INDUSTRIAL ENGINEERING & MANAGEMENT

2. 0
50
100
150
200
29/01/1993
S&P500
S&P500
Forecasting Extreme Events Risk - FEER Index
Efficient Financial Markets

3. 0
50
100
150
200
250
18/11/201007/01/2011
26/02/2011
17/04/2011
06/06/2011
26/07/2011
14/09/201103/11/201123/12/201111/02/2012
confidence 95%
SPY
confidence 99%
Forecasting Extreme Events Risk - FEER Index

4. If we avoid extreme losses: daily return
The total return:
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
Frequency
confidence 99%
Heavy Tail
Forecasting Extreme Events Risk - FEER Index

5. Sequence of
World Events
Associated
Gamble
Gamble
Riskiness
Investment
Rule
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6. Oprimization of Conditional Value at Risk
RT Rockafellar, S Uryasev- Journal of risk, 2000
Conditional Value at Risk VaR:
Artzner et al. (1998)
Forecasting Extreme Events Risk - FEER Index

7. Distribution Moments Sensitive
– Risk Measures
After Taylor expansion:
The Measure is sensitive to all moments of the distribution.
Is the root of the polynomial
Forecasting Extreme events Risk - FEER Index
The Measure impose a complete order on all the gambles
Personal Wealth

8. Forecasting Extreme Events Risk - FEER Index
No need to invers an integral equation to compute

9. VaR CVaR
Down Side - Risk Measures
Forecasting Extreme Events Risk - FEER Index
& Sensitive to High Moments
CVaR FEER - Index
Down Side
Sensitive to
Tail Risk
Sensitive to High Moments

10. CVaR - Real World Probability P
VaR - Risk Neutral Probability Q
Forecasting Extreme Events Risk - FEER Index
𝐶𝑉𝑎𝑅𝛼 =
−∞
𝑉𝑎𝑅
𝑉𝑎𝑅 − 𝑟 𝑝𝑡 𝑟 𝑑𝑟

11. -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Frequency
confidence 99%
Forecasting Extreme Events Risk - FEER Index

12. Forecasting Extreme Events Risk - FEER Index

13. Down Side - Risk Measures
Down Side +Tail Sensitive
Associated Gambles
FEER - Index
Underline Extreme
events
r
Forecasting Extreme Events Risk - FEER Index

14.  Down side risk measure.
 Coherent .
 Sensitive to high moments.
 Stochastic dominance.
 Generates complete order on all gambles.
 Dynamically calibrated to implied market
probability.

15. FEER – Index
The End To Our Fear

Forecasting Extreme Events Risk - FEER Index