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
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)) - .... $
## 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?
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