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

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?

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!!!

Convexity (compound percentage of percentages) can be additional source of PnL, which is outside of $F_0 (p)$ analysis

• 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