# Tail Risk Premia versus Pure Alpha


## Core empirical observations

https://www.trendfollowing.com/cfm-short.pdf

In a model-independent analysis of various possible strategies, JP Bouchaud et al found (see also a long paper) that

• The risk of (almost) any strategy is the tail risk
• TSmom (Time-series momentum) is a special strategy with a genuine premium which is not based on the tail risk

## [long paper for details](https://arxiv.org/pdf/1409.7720.pdf)

## my summary

- issues

a) PnL plots **do not consider path dependence and autocorrelation**
but still one drwas such plots for TSmom as well

**Non-linear allocation function redistributes information about autocorrelation into a good strategy** (pos+ skewness, large Sharpe, etc)
Need pos+ convexity of allocation/leverage as a function of autocorrelation

do we want to draw $F_0 (p)$ function always for the strategies? 
Also, how to scan quickly through autocorrelation on all timescales?
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b) **PnL is about right leverage at right times**.
**Tail-risk is about the area and the shape** of $F_0 (p)$ function. The area can be even negative!!!
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**Convexity (compound percentage of percentages) can be additional source of PnL**, which is outside of $F_0 (p)$ analysis⏎
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- observations

a) focuses on isolating tail risk/skewness - new definition related to the classic one
https://en.wikipedia.org/wiki/Skewness
b) hump structure of $F_0 (p)$ is due to **dominance of pos+ mid-size returns and dominance of neg- large-size returns**
c) SR (Sharpe ratio) is negatively dependent on the volatility


- main points

Model-independent analysis means statistical analysis

Start by ranking returns by absolute amplitude. This allows to plot three plots

- math 

a) usual time-ordered plot of SP500
b) ranked PnL plot $F(p)$ where $p$ is the rank of the absolute return from 0 to 1. The cumulative PnL function $F (p)= \int_0^p dy y (P(y) - P(-y))$ where we split positive-returns $P(y)$ and negative-returns  $P(-y)$ parts of the distribution function.

$F(p)=U(p)- D(p)$ whith $U(p)$ and $D(p)$ are up and down returns. The positive returns are $U(p) = \int_0^x P(y) dy$ over $P(y)$ positive returns distribution function
c) **symmetrised ranked PnL** $F_s (p)= F(p) - F_0 (p)$
$F_0 (p) = \int^p_0 y (P(y) - P(-y)) dy$ where for each return is the mean substracted.
$P(y)$ is the probability distribution over positive returns AFTER the substraction of the mean!!!!

Importantly $F_0 (p=1)=0$ because the mean was substracted!
By definition $F_s (p) = F (p) - F_0 (p) = \int^p_0 dy (y P(y) - y PminusM (y)) - ....  $

## other research

https://research-center.amundi.com/files/nuxeo/dl/d1fddc0d-a0c5-43db-9754-2782180b6b3a

negatively skewed
https://www.trendfollowing.com/whitepaper/skewed.pdf

# Parents

* TSeries Momentum - technical papers