# Power-law Portfolios

Jan Rosenzweig

###### Abstract

Portfolio optimization methods suffer from a catalogue of known problems,
mainly due to the facts that pair correlations of asset returns are unstable,
and that extremal risk measures such as maximum drawdown are difficult to
predict due to the non-Gaussianity of portfolio returns.
In order to look at optimal portfolios for arbitrary risk penalty functions,
we construct portfolio shapes where the penalty is proportional to a moment of
the returns of arbitrary order $p>2$.
The resulting component weight in the portfolio scales sub-linearly with its
return, with the power-law $w\propto\mu^{1/(p-1)}$. This leads to
significantly improved diversification when compared to Kelly portfolios, due
to the dilution of the winner-takes-all effect.
In the limit of penalty order $p\rightarrow\infty$, we recover the simple
trading heuristic whereby assets are allocated a fixed positive weight when
their return exceeds the hurdle rate, and zero otherwise. Infinite order
power-law portfolios thus fall into the class of perfectly diversified
portfolios.

Key words: optimal portfolios, fat-tailed risk, ICA

Key takeaways:

  * •

power-law portfolios address a number of know problems of covariance-based
portfolios

  * •

weights of components scale sub-linearly with their performance, reducing
portfolio concentration and the winner-takes-all problem

  * •

portfolio diversification improves with the increasing order of penalty,
ultimately leading to a perfectly diversified portfolio as the penalty order
tends to infinity.

##  1 Introduction

Textbook portfolio construction starts with the assumption that all assets are
Gaussian, with perfectly known returns, variances and correlations. It then
proceeds to apply the Kelly criterion to construct the Markowitz mean-variance
portfolio through it mass-adoption variant, the Capital Asset Pricing Model.
The resulting portfolio is strongly dependent on the returns, variances and
covariances. If the components were indeed Gaussian and the parameters were
determined to a sufficient degree of accuracy, it would be maximising the
Sharpe ratio for the given asset universe [8, 9, 10, 11, 13, 16].

In the real world, the components are not Gaussian, and their returns,
variances and correlations are, at best, rough estimates. This leads to a
number of well-known problems in portfolio construction, and a series of
various partial solutions to those problems [2, 14, 15].

At the core of our approach is the simple notion that, while Gaussian random
variables are completely determined by their return and covariance, non-
Gaussian random variables carry non-trivial information at all moments.

Similarly to [12], we therefore proceed to look into portfolio construction
when the risk penalty is attributed some arbitrary high order moment, rather
than just the covariance.

A number of studies have looked at various parametric non-Gaussian random
variables and various risk-based penalty functions [1, 5, 4, 6].

We, on the other hand, make no specific assumptions about either the form of
the underlying random variables, nor the specific risk measure involved.
Rather, we seek to understand how different moments of the joint distribution
influence the resulting portfolio construction.

We recover a simple scaling law relating the component weight to its return.
We also look at what happens when the order of the penalty movement goes to
inifinity, and in this limit we recover the well nown trading heuristic [6]

> _Don’t trade if the signal is too small. If it is large enough, buy a fixed
> size._

The paper is set out as follows. Second section addresses the power-law
portfolio construction in general. Third section dealis with the infinite
order limit. Fourth section looks at what happens if the underlying variables
were actually Gaussian. Fifth section optimizes a portfolio of S&P 500 stocks
over the period of 12 years using power-law portfolios for values of $p$
varying over orders of magnitude. The final section discusses the results.

##  2 Power-law weights

Let $S^{(1)}_{t}..S^{(N)}_{t}$ denote prices of $N$ assets at time $t$,
forming a vector of asset prices ${\bf S}_{t}$, and $d{\bf S}_{t}$ is the
vector of its increments.

We denote by

${\bf m}_{t}=E\left(d{\bf S}\right)$

the mean return of the joint distribution of $d{\bf S}_{t}$, and by ${\bf
r}_{t}$ the vector of funding rates.

The portfolio optimization problem that we are interested in is the selection
of a normalized weights vector ${\bf w}$ such that

${\bf w}.({\bf m}_{t}-{\bf r}_{t})-\lambda\ E\left|{\bf w}.(d{\bf S}_{t}-{\bf
m}_{t})\right|^{p}\rightarrow\max$ (1)

for some choice of $\lambda,p>0$. In other words, we are looking for portfolio
weights that maximize the return of the portfolio, while penalizing for its
absolute moment of order $p$.

There is a closely a related problem,

${\bf w}.({\bf m}_{t}-{\bf r}_{t})-\lambda\ \left|E\left({\bf w}.(d{\bf
S}_{t}-{\bf m}_{t})\right)^{p}\right|\rightarrow\max$ (2)

where the penalty term is the absolute value of the signed central moment, as
opposed to the absolute central moment.

We are primarly motivated by the cases where $p$ is an even integer, so that
both formulations are exactly the same. We can, however, expand the results to
other positive values of $p$.

Following [12], we solve it using the Independent Component Analysis (ICA)
[7]. Given the independent components $IC^{(1)},...IC^{(N)}$ with respective
means $\mu^{(i)}$, funding rates $r^{(i)}$, $p$th absolute central moments
$M_{p}^{(i)}$ and $p$th signed central moments $m_{p}^{(i)}$, we look for the
solution in the form

$\Pi\propto\sum_{i}w^{(i)}IC^{(i)}.$ (3)

Substituting (3) into (1), we can always find a local maximum, which is under
reasonable conditions also the global maximum of (1) (see Appendix); it has
the asymptotic expansion

$w^{(i)}\propto\left\\{\begin{array}[]{ll}0&\mu^{(i)}-r^{(i)}\leq 0;\\\
\left(\frac{\mu^{(i)}-r^{(i)}}{M_{p}^{(i)}}\right)^{1/(p-1)}+C_{0}^{(i)}+C_{1}^{(i)}\left(\mu^{(i)}-r^{(i)}\right)^{-1/(p-1)}+...&\texttt{otherwise.}\end{array}\right.$
(4)

for the formulation (1), and

$w^{(i)}\propto\left\\{\begin{array}[]{ll}0&\mu^{(i)}-r^{(i)}\leq 0;\\\
\left(\frac{\mu^{(i)}-r^{(i)}}{|m_{p}^{(i)}|}\right)^{1/(p-1)}+C_{0}^{(i)}+C_{1}^{(i)}\left(\mu^{(i)}-r^{(i)}\right)^{-1/(p-1)}+...&\texttt{otherwise.}\end{array}\right.$
(5)

for the formulation (2).

In particular, the leading order term in (4) is separable, i.e. $w^{(i)}$
depends only on the properties of $IC^{(i)}$ at leading order, and not on any
of the $IC^{(j)}$ for $j\neq i$. Separability does not extend to higher order
coefficients $C_{m}^{(i)}$, which generally do depend on $IC^{(j)}$ for $j\neq
i$.

In practice, the leading order term is sufficient in most cases, leading to
the approximate component weight formulas

$w^{(i)}\propto\left(\frac{\mu^{(i)}-r^{(i)}}{M_{p}^{(i)}}\right)^{1/(p-1)}$
(6)
$w^{(i)}\propto\left(\frac{\mu^{(i)}-r^{(i)}}{|m_{p}^{(i)}|}\right)^{1/(p-1)}$
(7)

for problems (1) and (2), respectively.

The case of $p=2$ is well known; equation (6) then becomes the well known
Kelly criterion for maximising the Sharpe ratio of the portfolio [8, 9, 10,
11, 13]. Each component is normalised by its volatility, and then weighted in
proportion to its Sharpe ratio.

The case of $p=4$ has also been studied before; $M_{4}$ is the kurtosis of the
component, and the exponent is $1/3$; we therefore recover the Fat-tailed
ratio of Rosenzweig [12], which maximizes the ratio of portfolio return to its
kurtosis.

Intuitively, as discussed above, we are motivated by the cases where $p$ is an
even integer, so that the absolute central moment and the signed central
moment are exactly the same. The weights formula (6), however, obviously works
for any non-integer $p\neq 1$, as long as the moment in the penalty function
is the absolute central moment. If $p\leq 1$, the penalty grows slower than
the gain, and (1) is optimized by setting all weights to zero.

With that in mind, are restricting ourselves to $p\geq 2$, while allowing $p$
to be non-integer. It is, however, important to note that, if $p$ is not an
even integer, further technical conditions are needed to ensure that the
asymptotic expansion (4) approximates the global maximum of (1).

The most interesting feature of (6) is the sub-linear scaling of the component
weight with its performance. In Sharpe-maximizing Kelly portfolios, the weight
is proportional to the performance. A component with twice the performace of
another will generally receive twice its weight.

Formula (6) shows, however, that that is a special case. For any choice of
$p>2$, the scaling is sub-linear, and the component with twice the performance
will receive less than twice the weight. This, in turn, directly addresses the
winner-takes-all problem inherent in Kelly portfolios, whereby the portfolio
is dominated by a small number of highly performing components, which in turn
negates the benefits of diversification.

Using a higher order penalty for portfolio construction as per (1) directly
addresses this issue, by effectively limiting the over-weighting of higly
performing components.

##  3 Infinite Order Penalty

The most interesting result of the previous section concerns the sub-linear
dependence of each component weight on its return. This becomes more prominent
as the penalty order $p$ becomes larger.

A graphical representation of the dependence of the weight of a component on
its return is shown in Figure 1.

As seen from Figure 1, the weight/return diagram has a simple limiting
behaviour for large values of $p$; it approximates a step function with the
step set at the hurdle rate $r$.

This is immediately obvious by examining the functional form of equation (6);
as $p$ tends to inifinity, the exponent $1/(p-1)$ tends to zero and
$\mu^{1/(p-1)}$ tends to 1.

While this observation is mathematically trivial, it has deep implications for
portfolio management.

There is a well known trading heuristic, as cited by Giller [6]:

> _Don’t trade if the signal is too small. If it is large enough, buy a fixed
> size._

The reason is that a portfolio composed of equally weighted independent
components is a perfectly diversified portfolio. It has a variance that
decreases as $1/N$, and excess kurtosis that decreases as either $1/N$ or
$1/N^{2}$, depending on whether the components are only orthogonal, or
independent to higher orders [12].

The limit of our equation (6) as $p\rightarrow\infty$ directly recovers the
step function implied by the heuristic. Noting that ICs are by construction
normalized to the same volatility [7], we can formulate it in words as:

> _Don’t trade if the expected return is smaller than the hurdle rate. If it
> is larger, buy a fixed volatility._

If hurdle rate $r=0$, the power law portfolio for $p\rightarrow\infty$ is a
perfectly diversified portfolio. Otherwise, the power-law portfolio will not
include the components that return less than the hurdle rate; it will
therefore not have full $N$ components, and its variance decays as
$1/N^{\prime}$ for some smaller number of components $N^{\prime}\leq N$.

The portfolios which still retain non-trivial dependence on the return are
generally not perfectly diversified, due to the fact that their components are
not equally weighted in volatility. This is the case for all finite values of
$p$.

On the other hand, power-law portfolios come closer to being perfectly
diversified as $p$ increases. We thus get a useful further rule-of-thumb for
interpreting the order $p$. For small $p$, the portfolios are very dependent
on their returns, at the expense of diversification. As $p$ increases, the
portfolios give up return in exchange for the benefit of diversification. In
the limit of $p\rightarrow\infty$, the portfolios become perfectly
diversified, and the return is only used to determine whether the holding is
long, short or zero.

##  4 What if they are Gaussian?

While our primary motivation is to study portfolios with non-normal returns,
it is still worthwhile to examine the case where each $IC^{(i)}$ is Gaussian
with the mean $\mu^{(i)}$ and standard deviation $\sigma^{(i)}$. In that case,
dropping the superscripts for a moment, we have a simple formula for the
absolute central moments,

$M_{p}=\frac{1}{\sqrt{\pi}}\ 2^{p/2}\
\Gamma\left(\frac{p+1}{2}\right)\sigma^{p}$ (8)

where $\Gamma()$ denotes the Gamma function and $\sigma$ is the volatility of
the component.

Incorporating (8) into (6) and returning the superscripts, we get

$w^{(i)}\propto\frac{1}{\sigma^{(i)}}\left(\frac{\mu^{(i)}-r^{(i)}}{\sigma^{(i)}}\right)^{1/(p-1)}2^{-p/2(p-1)}\Gamma\left(\frac{p+1}{2}\right)^{-1/(p-1)}.$
(9)

In other words, each weight is, after its corresponding component is
normalized to unit volatility (term $1/\sigma^{(i)}$), proportional to the
Sharpe ratio of the component, raised to the power of $1/(p-1)$.

The constant term involving the Gamma function is just a proportionality
constant which will in practice be over-riden by normalization. We can,
however, still simplify it further to better understand the infinite order
limit from the previous section.

Using the Striling’s formula

$\Gamma(z)=\sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^{z}\left(1+O\left(\frac{1}{z}\right)\right),$

we get

$w^{(i)}\propto\frac{1}{\sigma^{(i)}}\left(\frac{\mu^{(i)}-r^{(i)}}{\sigma^{(i)}}\right)^{1/(p-1)}\frac{e^{p/2(p-1)}}{p^{p/2(p-1)}2^{(p+1)/2(p-1)}}$
(10)

When $p$ is sufficiently large, $p/(p-1)\approx 1$ and this further simplifies
to

$w^{(i)}\propto\frac{1}{\sigma^{(i)}}\left(\frac{\mu^{(i)}-r^{(i)}}{\sigma^{(i)}}\right)^{1/(p-1)}\sqrt{\frac{e}{2}}\frac{1}{\sqrt{p}}.$
(11)

The weight includes normalization to unit volatility (term $1/\sigma^{(i)}$),
and then allocation proportional to the Sharpe ratio to the power of
$1/(p-1)$. The amplitude of the weight decays as $1/\sqrt{p}$, which is in
practice over-riden by the normalization of the weights.

Again, we can express the allocation as a heuristic in words for any given
$p$:

> _Don’t trade if the expected return is smaller than the hurdle rate. If it
> is larger, normalize to unit volatility and buy a size in proportion to the
> Sharpe ratio to the power of $1/(p-1)$._

In the limit of $p$ going to infinity, this remains as in the previous
section:

> _Don’t trade if the expected return is smaller than the hurdle rate. If it
> is larger, buy a fixed volatility._

##  5 S&P 500 stocks

We looked at the same data set as in [12], namely S&P 500 stocks over a period
of 12 years, from the 1st January 2007 until the 31st December 2018. To
counteract the effects of stocks drifting in and out of the index over such a
long time frame, we have divided the time frame into four buckets, each
lasting three calendar years; from 1st January 2007 until 31st December 2009,
from 1st January 2010 until 31st December 2012, from 1st January 2013 until
31st December 2015 and from 1st January 2016 until 31st December 2018. The
basket for each bucket was selected as consisting of the index constituents on
the last business day prior to the start of the bucket, and these stocks were
followed until the end of the bucket. Any stock that was de-listed before the
end of a bucket in which it appeared was deemed to have returned $0\%$ from
its last trading day until the end of the bucket. There were no adjustments
for stocks entering or leaving the index over the duration of any of the
buckets.

We have extracted the first ten ICs and constructed the resulting power-law
portfolios corresponding to $p=2,4,100$ and $\infty$. The performance of the
resulting portfolios is shown in Figure 2.

As expected from the theoretical analysis, the Kelly portfolio for $p=2$ is
the most aggressive in each bucket, having the highest weighting by return.
The portfolios become incresingly less aggressive for increasing $p$. Perhaps
counter-intuitively, $p=\infty$ is not always the least aggressive portfolio.
In the two earliest buckets, 2007-2009 and 2010-2012, the portfolio for
$p=100$ is less aggressive than the portfolio for $p=\infty$.

This is not as surprising as it seems. The buckets 2007-2009 and 2010-2012
include the global financial crisis when stock returns were highly erratic,
which was reflected in the high order return moments. The $p=\infty$ case is
agnostic of return moments, and it relies purely on diversification.

The portfolio statistics are shown in Table 1. The immediately obvious feature
is that, in each bucket, the $p=2$ portfolio has the highest Sharpe Ratio, and
the $p=4$ portfolio has the highest Fat-tailed Ratio. This is entirely
unsurprising in light of the theoretical results above, since the $p=2$
portfolio by construction maximizes the Sharpe ratio, and the $p=4$ portfolio
by construction maximizes the Fat-tailed Ratio. We did not show the $p=100$
power-law ratio, but, by construction, it is maximized by the $p=100$
portfolio. And for any other choice of $p$, the specific $p$-portfolio
maximizes the $p$th power-law ratio. Note that there is no simple $p=\infty$
ratio to compare.

Looking at the correlations in Table 2, it is noticeable that all portfolios
reproduce the same factors. Correlations are positive and high across the
board. We see correlations occasionally dipping towards 80% in a handful of
places, always between $p=2$ and one of the higher order portfolios, either
$p=100$ or $p=\infty$. Otherwise, they are comfortably above 90%, and often
above 95%.

The differences in the Sharpe ratio between different portfolios in the same
bucket are between 10 and 20%, with 20% being reached between the $p=2$ and
either $p=100$ (in 2007-2009) or $p=\infty$ (in 2016-2018). Those particular
buckets also seem to have the highest difference in the Fat-tailed ratio, this
time in favour of the higher order portfolios.

Differences in the Sharpe ratio of 10% or less can be attributed to the
perfect hindsignt that was used in our portfolio construction, and it is
unlikely that they would translate into forward-looking portfolio construction
in the real world.

The conclusion seems to be that, in low volatility environments, there is not
much to choose between the portfolios for different values of $p$. In high
volatility environments, however, we have a clear choice of whether to push
the risk out of the volatility and into the tails (for $p=2$), or out of the
tails and into the volatility (for large values of $p$). There appears to be
no choice of $p$ that would predictably and simultaneously reduce all risk
measures at the same time.

##  6 Conclusions

The method describeed here is a straightforward generalization of the Kelly
criterion to non-Gaussian portfolios, obtained by moving the risk penalty from
the second moment, variance, to an arbitrary $p$th absolute moment of the
returns, for some $p\geq 2$.

By doing so, we can significantly reduce the dependence of the portfolio
weight of a component on its return. The resulting weight scales with return
to the power of $1/(p-1)$, which is sub-linear when $p>2$. The resulting
portfolio is better diversified than a corresponding Kelly portfolio, and less
susceptible to the winner-takes-all problem in which a handful of strongly
performing components attract a lion’s share of the capital.

The diversification effect becomes stronger as $p$ increases. In the limit of
$p$ going to infinity, the weight becomes a simple 0-1 digital step function,
whereby a component is assigned either a fixed weight, if its return exceeds a
hurdle, or zero otherwise. This is the well known trading heuristic,

> _Don’t trade if the signal is too small. If it is large enough, buy a fixed
> size._

By formalizing it, we have strengthened it to

> _Don’t trade if the signal is smaller than the hurdle rate. If it is larger,
> buy a fixed volatility._

The resulting portfolios capture the same factors regardless of the chosen
value of $p$. The choice of $p$ only affects the ultimate risk profile of the
resulting portfolio.

There is no free lunch in finance, and this portfolio construction method is
not a free lunch. By penalizing for moments of any given order $p$, we succeed
in pushing the risk away from the $p$th moment, but it only moves into other
moments. The portfolio construction method we present follows a simple logic:

  * •

_If you can hedge it, hedge it._

  * •

_If you can’t hedge it, diversify it._

  * •

_If you can neither hedge nor diversify it, push it somewhere else._

The hedging arises through the use of the Independent Component Analysis,
which generates components within which individual assets hedge each other as
far as possible. The diversification arises through weights given to the
components, which generate as diversified a portfolio of independent
components as specified for the given value of $p$.

The final step, of pushing risk away from the monitored moment into other
moments is the most problematic.

On the more positive side, choosing a high value of $p$ results in risk being
pushed out of the tails and into volatility, where it is the easiest to
monitor. This is, arguably, preferable to Kelly portfolios which push risk out
of volatility and into the tails, where it is more difficult to monitor.

A further benefit of the Independent Component Analysis used here as opposed
to Principal Component Analysis is reduced dependence on pairwise asset
correlations, which is recognised as a primary weakness in Kelly portfolios.
By choosing components which are independent to all orders, as opposed to just
orthogonal, we arguably recover more stable components which are less likely
to arise due to sampling bias [12].

In summary, sub-linear power law portfolios constitute a powerful portfolio
construction method which addresses some well known deficiencies of Kelly
portfolios. In the form of digital, fixed-size-or-nothing allocation of the
infinite order limit, it has already been a mainstay of real-world portfolio
construction for decades, if not centuries. We here provide a simple rationale
for its use, and we put it into a rational, objective framework.

## Acknowledgments

The author reports no conflicts of interest. The author alone is responsible
for the content and writing of the paper.

## Appendix - Power Law Formula

This appendix outlines the proof of the power-law formula (4).

First, we focus on the cases where $p$ is an even integer, $p=2k$. Then, the
absolute moment is the same as the signed moment, and we can drop the absolute
value and expand the moment calculation. Using the multinomial theorem and
independence of ICs, we get

$\begin{array}[]{rl}E\left|d\Pi\right|^{p}&=E\left(d\Pi\right)^{2k}=E\left(\sum_{i=0}^{N}w^{(i)}dIC^{(i)}\right)^{2k}=\\\
&=E\left[\sum_{k_{1}+k_{2}+...+k_{N}=2k}\prod_{i=1}^{N}\left(w^{(i)}dIC^{(i)}\right)^{k_{i}}\right]=\\\
&=\sum_{k_{1}+k_{2}+...+k_{N}=2k}\prod_{i=1}^{N}\left(w^{(i)}\right)^{k_{i}}m^{(i)}_{k_{i}}\end{array}$
(12)

It is a standard result that powers of independent random variables are
independent, and therefore all cross moments of independent random variables
are separable [7].

In particular, we can also note that $m_{1}^{(i)}=0$ for all $i$ due to
centrallity.

Then, substituting (12) into (1) and taking the first derivatives wrt each
$w^{(i)}$, we get a system of equations of the form

$\left(\mu^{(i)}-r^{(i)}\right)-P_{i}\left(w^{(i)}\right)=0$ (13)

where each $P_{i}$ has the following properties:

  * •

$P_{i}$ is a polynomial of order $2k-1$

  * •

$P_{i}(0)=0$

  * •

the leading coefficient of each $P_{i}$ is $m_{2k}^{(i)}$, which is non-
negative, and independent of $w^{(j)}$ for all $j\neq i$.

If $m_{2k}^{(i)}=0$, the weight of that component can be increased without
incurring the penalty; the maximum of (1) is then reached at the boundary,
$w^{(i)}=1,$ $w^{(j)}=0$ for $j\neq i$.

In general, if all $m_{2k}^{(i)}>0$, (13) can have up to $2k-1$ real roots for
each $i$, corresponding to up to $(2k-1)^{N}$ maxima and minima for (1). We
specifically focus on the largest zero of (13) for each $i$, which we denote
$\hat{w}^{(i)}$.

Then, if $\mu^{(i)}-r^{(i)}>0$, the following propositions hold:

  1. 1.

$\hat{w}^{(i)}\geq 0$ for all $i$;

  2. 2.

${\hat{\bf w}}=(\hat{w}^{(i)})$ is a local maximum of (1);

  3. 3.

if all coefficients of each $P_{i}$ are non-negative, then ${\hat{\bf w}}$ is
the only non-negative maximum of (1);

  4. 4.

for each $i$, there is a lower bound $b^{(i)}<0$ such that, if all
coefficients of each $P_{i}$ are greater than $b^{(i)}$, then ${\hat{\bf w}}$
is the global non-negative maximum of (1).

Briefly, 1 and 2 are direct consequences of the Descartes’ rule of signs; 3
follows from the fact that non-negativity of coefficients makes each $P_{i}$ a
monotonically increasing function on $[0,\infty)$; and 4 follows from $3$ by
continuity.

Note that 3 (and therefore 4) is trivially satisfied if the ICs are all
normally distributed, or if they folllow any symmetric distribution with
finite moments up to order $2k$.

While one can not generally guarantee that any return distribution of interest
in finance will always satisfy 4, most of them do. We can therefore quite
generally accept ${\hat{\bf w}}$ as a universal local maximum, and a nearly-
universal global maximum of (1).

We can finally come back to the assumption that $\mu^{(i)}-r^{(i)}>0$ above.
ICs are generally sign-agnostic; ICA decomposition is unaffected by the
transformation $IC^{(i)}\mapsto-IC^{(i)}$; in finance terms, ICs are generally
long-short portfolios with no obvious sign. However, given the difference in
the funding cost of long and short positions, the funding rate of $-IC^{(i)}$
is generally not $-r^{(i)}$.

Therefore, if the assumption that $\mu^{(i)}-r^{(i)}>0$ can not be satisfied
by changing the sign of $IC^{(i)}$, i.e. if $IC^{(i)}$ can not cover its cost
of funding on either the long or the short side, then it can not contribute
positive return to the portfolio; therefore its best possible contribution to
the maximum of (1) is achieved by setting its weight to zero. Mathematically,
if the constant term in (13) is negative, Descartes’ rule of signs no longer
implies that (13) has a positive zero, and therefore the maximum is reached on
the boundary, $\hat{w}^{(i)}=0$.

It now remains to estimate $\hat{w}^{(i)}$, and thereby ${\hat{\bf w}}$, for
components that satisfy $\mu^{(i)}-r^{(i)}>0$. This is straightforward;
$\hat{w}^{(i)}$ has a known Puiseux series expansion [3]

$w^{(i)}=\left(\frac{\mu^{(i)}-r^{(i)}}{m_{2k}^{(i)}}\right)^{1/(2k-1)}+C_{0}^{(i)}+C_{1}^{(i)}\left(\mu^{(i)}-r^{(i)}\right)^{-1/(2k-1)}+....$
(14)

Then, noting that $m_{2k}^{(i)}=M_{2k}^{(i)}$ and that $2k=p$, we get

$w^{(i)}=\left(\frac{\mu^{(i)}-r^{(i)}}{M_{p}^{(i)}}\right)^{1/(p-1)}+C_{0}^{(i)}+C_{1}^{(i)}\left(\mu^{(i)}-r^{(i)}\right)^{-1/(p-1)}+...,$
(15)

, or

$w^{(i)}=\left(\frac{\mu^{(i)}-r^{(i)}}{|m_{p}^{(i)}|}\right)^{1/(p-1)}+C_{0}^{(i)}+C_{1}^{(i)}\left(\mu^{(i)}-r^{(i)}\right)^{-1/(p-1)}+...,$
(16)

When $p$ is not an even integer, the situation is somewhat more complicated.
We can verify that there is a local extremum of (1) satisfying (15) by
substituting (15) into (1) and setting the first derivatives wrt each
$w^{(i)}$ to zero.

However, proving global properties of (15) is more difficult. We generally do
it by approximating the moment function with a polynomial of even order, and
then using the polynomial analysis above,

One way forward relies on noting that, if $p$ is sufficiently large, we can
always write $p=2k(1+\epsilon)$ for some integer $k$ and
$\epsilon=(p-2k)/2k\ll 1$; we can then expand (1) in powers of $\epsilon$ and
revert to the polynomial analysis above, resulting in (15) plus an error term
of the order $O(\epsilon)=O(1/p)$.

This leaves us with the case when when $p$ is not an even integer and not
large. Whether or not the local maximum (15) is generally the global maximum
of (1) in this case, and under what conditions, is currently, to the best
knowledge of the author, unknown.

Formulation (2) has some simple solutions for non-even values of $p$. For
$p=3$, all cross terms in the polynomial are of the form
$m^{(j)}_{1}m^{(k)}_{2}$ and they are all equal to zero, given the vanishing
first central moments $m^{(j)}_{1}$. The formula (16) is therefore trivially
satisfied at the leading order, with $C_{0}=C_{1}=....=0$.

Higher non-even values of $p$ are not as simple. For $p=5$, for example, we
get the cross term $m^{(j)}_{2}m^{(k)}_{3}$ which might be zero, positive or
negative. This, after taking the absolute value, thus increases the number of
potential maxima, making the construction of simple analytical solutions
incresasingly difficult. Once $p$ becomes sufficiently high, however, the
asymptotics $p=2k(1+\epsilon)$ starts to work.

## References

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Figure 1: Schematic representation of the portfolio weight $w^{(i)}$ as a
function of its return $\mu^{(i)}$, for non-zero hurdle rate $r$ various
values of $p$.

Figure 2: Power-law portfolios for the S&P 500 components for the 2007-2009, 2010-2012, 2013-2015 and 2016-2018 buckets with $p=2,4,100$ and $\infty$. The hurdle rate $r$ is set to 0, portfolio weights are normalised to $\sum_{i}\left(w^{(i)}\right)^{2}=1$, and % returns are % of $\sqrt{\sum_{i}\left(w^{(i)}\right)^{2}}$.  Bucket | Statistic, annualized | $p=2$ | $p=4$ | $p=100$ | $p=\infty$
---|---|---|---|---|---
2007-2009 | Return | 146.16% | 135.07% | 118.47% | 132.24%
| Standard Deviation | 57.85% | 57.85% | 57.85% | 57.85%
| Kurtosis | 2.537 | 1.611 | 1.488 | 1.407
| Sharpe Ratio | 2.527 | 2.335 | 2.048 | 2.286
| Fat-tailed Ratio | 0.832 | 0.943 | 0.927 | 0.979
2010-2012 | Return | 130.86% | 126.44% | 115.53% | 118.50%
| Standard Deviation | 57.81% | 57.81% | 57.81% | 57.81%
| Kurtosis | 1.135 | 0.580 | 0.824 | 1.324
| Sharpe Ratio | 2.263 | 2.187 | 1.998 | 2.050
| Fat-tailed Ratio | 1.048 | 1.296 | 1.119 | 0.964
2013-2015 | Return | 123.95% | 121.55% | 116.44% | 108.58%
| Standard Deviation | 57.80% | 57.81% | 57.81% | 57.81%
| Kurtosis | 3.888 | 3.758 | 4.440 | 7.210
| Sharpe Ratio | 2.145 | 2.103 | 2.014 | 1.878
| Fat-tailed Ratio | 0.683 | 0.686 | 0.640 | 0.532
2016-2018 | Return | 96.08% | 89.68% | 80.54% | 80.17%
| Standard Deviation | 57.81% | 57.81% | 57.81% | 57.81%
| Kurtosis | 2.425 | 1.365 | 1.625 | 1.439
| Sharpe Ratio | 1.662 | 1.551 | 1.393 | 1.387
| Fat-tailed Ratio | 0.734 | 0.869 | 0.791 | 0.823
Table 1: Return statistics for the 2007-2009, 2010-2012, 2013-2015 and 2016-2018 buckets with $p=2,4,100$ and $\infty$. The hurdle rate $r$ is set to 0, portfolio weights are normalised to $\sum_{i}\left(w^{(i)}\right)^{2}=1$, and % returns are % of $\sqrt{\sum_{i}\left(w^{(i)}\right)^{2}}$.  Bucket |  | $p=2$ | $p=4$ | $p=100$ | $p=\infty$
---|---|---|---|---|---
2007-2009 | $p=2$ | 100.00% | 92.41% | 81.06% | 90.49%
| $p=4$ |  | 100.00% | 96.11% | 98.00%
| $p=100$ |  |  | 100.00% | 95.88%
| $p=\infty$ |  |  |  | 100.00%
2010-2012 | $p=2$ | 100.00% | 96.62% | 88.29% | 90.55%
| $p=4$ |  | 100.00% | 96.63% | 95.26%
| $p=100$ |  |  | 100.00% | 96.77%
| $p=\infty$ |  |  |  | 100.00%
2013-2015 | $p=2$ | 100.00% | 98.06% | 93.94% | 87.60%
| $p=4$ |  | 100.00% | 97.77% | 89.17%
| $p=100$ |  |  | 100.00% | 93.31%
| $p=\infty$ |  |  |  | 100.00%
2016-2018 | $p=2$ | 100.00% | 93.33% | 83.82% | 83.44%
| $p=4$ |  | 100.00% | 97.18% | 95.91%
| $p=100$ |  |  | 100.00% | 97.73%
| $p=\infty$ |  |  |  | 100.00%
Table 2: Return correlations for the 2007-2009, 2010-2012, 2013-2015 and
2016-2018 buckets with $p=2,4,100$ and $\infty$. The hurdle rate $r$ is set to
0.