Fractal Optimization of Market Neutral Portfolio
📝 Abstract
A fractal approach to the long-short portfolio optimization is proposed. The algorithmic system based on the composition of market-neutral spreads into a single entity was considered. The core of the optimization scheme is a fractal walk model of returns, optimizing a risk aversion according to the investment horizon. The covariance matrix of spread returns has been used for the optimization and modified according to the Hurst stability analysis. Out-of-sample performance data has been represented for the space of exchange traded funds in five period time period of observation. The considered portfolio system has turned out to be statistically more stable than a passive investment into benchmark with higher risk adjusted cumulative return over the observed period.
💡 Analysis
A fractal approach to the long-short portfolio optimization is proposed. The algorithmic system based on the composition of market-neutral spreads into a single entity was considered. The core of the optimization scheme is a fractal walk model of returns, optimizing a risk aversion according to the investment horizon. The covariance matrix of spread returns has been used for the optimization and modified according to the Hurst stability analysis. Out-of-sample performance data has been represented for the space of exchange traded funds in five period time period of observation. The considered portfolio system has turned out to be statistically more stable than a passive investment into benchmark with higher risk adjusted cumulative return over the observed period.
📄 Content
1
Fractal Optimization as Generator of Market Neutral Long-Short Portfolio
Sergey Kamenshchikov, PhD, IFCM Group/ Moscow Exchange
Ilia Drozdov, CFA, QB Capital
Abstract
A fractal approach to the long-short portfolio optimization is proposed. The algorithmic system based on the composition of market-neutral spreads into a single entity was considered. The core of the optimization scheme is a fractal walk model of returns, optimizing a risk aversion according to the investment horizon. The covariance matrix of spread returns has been used for the optimization and modified according to the Hurst stability analysis. Out-of-sample performance data has been represented for the space of exchange traded funds in five period time period of observation. The considered portfolio system has turned out to be statistically more stable than a passive investment into benchmark with higher risk adjusted cumulative return over the observed period.
Introduction
According to the research of Malkiel [1] only 14% of long-term equity funds represent an average return of 2-4% above S&P500 benchmark in ten years time frame. This statistics correspond to pre- ETF era of 1990-2001. However the typical Sharpe ratio of S&P500 reaches 1.5-2 levels only for the 5-10 years horizon. In combination with bond funds it makes S&P500 index a comfortable instrument for pension programs, but inefficient for middle-term investment of 1-5 years horizon. The local diversification of Long-Only funds doesn’t efficiently provide a systematic risk aversion. The Global Asset Allocation models simplified a diversification at the beginning. However while there are several major drivers of the global Market like the US or Asia this model still lacks a market-neutrality in long term strategies. Another approach to market neutral investment is a portfolio of hedge funds which apply short term long-short arbitrage models with high beta neutrality. Unfortunately hedge fund models still preserve properties of “black boxes” and are not comfortable for the transparent investing. In the current research we prove that a long-short model is suitable for a long term investment and may provide stable trends. This model is based on market- neutral pair spreads which use relative competitive advantages of assets. Diversification of spreads allows eliminating both systematic and non systematic portfolio risks. We introduce a fractal model of volatility to account for nonlinear risks such as volatility clustering. This approach suggests a new step outside the standard statistics. In following sections we provide description of market neutralization of spreads, their composition into the single entity and portfolio optimization.
Market neutrality
Let’s represent daily returns of assets i and j in the linear form: ) ( ) ( ) ( 0 m i m i i i r t r r t r
)
(
)
(
)
(
0
m
j
m
j
j
j
r
t
r
r
t
r
(1)
Here
0
ir and
0
jr are constant drift terms,
mr is a mutual market return. Residuals express a random
component in case of a perfect regression model. Otherwise residuals may be represented as
nonlinear functions of market returns. If we consider long-term investments, returns are to be
normalized in relation to the investment entry point at the beginning of the holding period:
0
)
(
)
(
)
(
i
i
i
i
p
t
t
p
t
p
t
r
0 ) ( ) ( ) ( j j j j p t t p t p t r (2) The market term expresses the mutual market of these two assets which has to be defined in a quantitative way. Consequently iand j are constant factors that show a relation of each asset to the market linear motion.
2
Definition of betas may be expressed through the increments of returns:
) ( ) ( ) ( m i m i i r t r t r
) ( ) ( ) ( m j m j j r t r t r (3) Here new residuals are weakly nonlinear terms , , ( ) i j m i j m r r . Hedge factors / ij i j may be defined by the relation with weakly nonlinear residual (4): ( ) / ( , ) ( ) / i i i m ij ij ij m ij j j j m r t r r r t r (4) Here ( , ) m ij r is a weakly nonlinear term. Long/Short position (spread) of assets i and j correspondingly tends to the perfect market neutral state while two conditions are satisfied: ij j i w w / 1 / , 0 . Here , i j w w are relative weights of assets i and j correspondingly. Second condition is equivalent to m ij r . The return of this spread may be represented as the superposition of constant term and weakly nonlinear term:
0
0
( )
( )
(
,
)
(
,
)
ij
i
ij
j
i
ij j
ij
m
ij
ij
m
ij
r t
r t
r
r
r
const
r
(5)
A fundamental sense of a stable spread return is a competitive advant
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