Construct a powerful crypto asset portfolio using multi-factor strategies #Theoretical Basics#

LUCIDA & FALCON

11-14

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Preface

In June last year, I conceived the simple idea of using a multi-factor model to select currencies.

https://mirror.xyz/lucidafund.eth/UdOfxxKgD_Xuc_KrvGvsjrWZZCwKlWPAYNx991ZgmIA

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One year later, we have begun to develop a multi-factor strategy for the crypto-asset market, and have written the overall strategy framework into a series of articles "Building a Powerful Crypto-Asset Portfolio Using Multi-Factor Strategies."

The general framework of this series is as follows (the possibility of fine-tuning is not ruled out):

1. Theoretical basis of multi-factor model
2. Single factor construction
Factor data preprocessing
Data filtering
Exception value handling: extreme values, error values, null values
standardization
Neutrality: industry, market, market capitalization
Factor validity judgment
Information ratio IC, rate of return, Sharpe ratio, turnover rate
3. Synthesis of major categories of factors
factor collinearity analysis
Orthogonal elimination factor collinearity
Classic weighting method→synthetic factor
Equal weighting, rolling IC weighting, IC_IR weighting
Test of synthetic factors: rate of return, group rate of return, factor value weighted rate of return, synthetic factor IC, group turnover rate
Other weighting methods (non-linear relationship between factors and returns): machine learning, reinforcement learning (not considered due to the particularity of the cryptocurrency industry)
4. Risk portfolio optimization

The following is the text of the first **#Theoretical Basics#**.

1. What is “factor”?

"Factors" are the "indicators" in technical analysis and the "features" of artificial intelligence and machine learning, which determine the rise and fall of cryptocurrency yields .

Our team combines the common factor types in the cryptocurrency field: fundamental factors, on-chain factors, volume and price factors, derivatives factors, alternative factors and macro factors.

The ultimate goal of mining and calculating "factors" is to accurately calculate the expected rate of return of an asset.

2. Calculation of “factor”

(1) Derivation of multi-factor model

Origin: One-factor model—CAPM

Factor research can be traced back to the 20C60s, with the advent of the Capital Asset Pricing Model (CAPM). This model quantifies how risk affects a company's cost of capital and thus the expected rate of return . According to the CAPM theory, the expected excess return of a single asset can be determined by the following univariate linear model:

$$$ E(Ri) - Rf = βi(E(Rm)-Rf) 「formula 2」 $$$

$$E(Ri)$$ is the mathematical expectation, $$Ri$$ is the return rate of the asset, $$Rf$$ is the risk-free return rate, $$Rm$$ is the return rate of the market portfolio, $$βi = Cov(Ri,Rm)/Var(Rm)$$ reflects the sensitivity of asset returns to market returns, also known as the asset's exposure to market risks.

Additional understanding:
In the financial market, the "risk" and "return" talked about are essentially the same thing.
From a statistical perspective, a more detailed understanding of $$βi$$
CAPM can be regarded as a bivariate regression model without intercept term $$ Yi = β1 + β2 · $$β1 = β2 = Σ(X-μX)(Y-μY)/ Σ(X-μX)² = Cov(X,Y)/Var(X)$$.
$$β1$$ measures the change in the explanatory variable (market return) in units, and the degree of average change in the explained variable (the return on asset i). The financial field interprets this degree of change as the "sensitivity" or "exposure" of Y to X "degree.
$$β>1$$ amplifies market fluctuations
$$β = 1$$ is exactly the same as market fluctuations
$$0<β<1$$ fluctuates in the same direction as the market, but is less volatile than the market
$$β≤ 0$$ fluctuates in the opposite direction to the market
From the perspective of financial risk and return, a more detailed understanding of $$ βi$$
There are two types of risk in an investment portfolio, systematic risk (i.e., market risk, non-offsetting risk) and unsystematic risk (i.e., offsetting risk). $$βi$$ is systemic risk , which is unique to the system and cannot be offset no matter how the portfolio is constructed. The $$αi$$ mentioned below is unsystematic risk and can be hedged by constructing different strategies.

The CAPM model is the simplest linear factor model, which points out that the excess return of an asset is determined only by the expected excess return of the market portfolio (market factor) and the asset's exposure to market risk. This model lays a theoretical foundation for subsequent research on a large number of linear multi-factor pricing models.

Development: Multifactor Model—APT

Based on CAPM, people found that the return rates of different assets are affected by multiple factors. Arbitrage Pricing Theory (APT) came out and built a linear multi-factor model:

$$$ E(Ri) = βi · λ 「formula 3」 $$$

Among them, $E(Ri) $ represents the expected return of asset $$ i$$, and $$λ$$ represents the factor expected return (i.e., factor premium). Formula (2) uses

$$E(Ri)$$ replaces $$E(Ri) - Rf$$ in the CAPM model to represent the expected return. For a capital-neutral portfolio asset constructed using long-short hedging, $$Rf$$ is offset, and the entire The expected return of an asset is the difference between the long and short expected returns, so it is more general to express it as $$E(Ri)$$.

Mature: Multi-factor model – Alpha return & Beta return

Taking into account the actual pricing errors in the financial market and the APT model, from a time series perspective, the expected rate of return of a single asset is determined by the following multivariate linear model:

$$$ Rᵉit = αi + βi · λt + εit 「formula 4」 $$$

Among them, $$Rᵉit$$ represents the return of asset $$i$$ at time $t $, $$λt$$ represents the factor rate of return (i.e., factor premium) at time $$t$$, $$εit$$ Represents the random perturbation at $t $ time. $$αi$$ represents the pricing error between the actual expected return of asset $$i$$ and the expected return implied by the multi-factor model. If it deviates statistically significantly from zero, it represents the opportunity to obtain excess returns. $$βi = Cov(Ri,λ)/Var(λ)$$ represents the factor exposure or factor loading of asset $$i$$, which describes the sensitivity of asset returns to factor returns.

The multi-factor model focuses on the cross-sectional differences in the expected returns of assets. It is essentially a model about the mean, and the expected return is the average of the returns in the time series. Based on (3), the multivariate linear model of the cross-section angle can be derived:

$$$ E[Rᵉi] = αi + βi · λ 「formula 5」 $$$

Among them, $$E[Rᵉi]$$ represents the expected excess return of asset $$i$$, and $$εit$$ is averaged over time series, then $$E(εit)=0$$.

Additional understanding:
From an academic perspective, according to the market efficiency theory, an effective asset portfolio should be able to offset the risk completely to 0 , the actual return rate is equal to the expected return rate, and the expected asset return rate only depends on the market's systemic risk, that is, $ $E[Rᵉi] = βi · λ$$, there is no excess return (Abnormal Return, AR) , that is, $$AR = Ri - E(Rᵉi) = 0$$. But in the real financial world, the market is usually inefficient, and there is excess rate of return, that is, $$AR = α$$.

Assume that the investment portfolio consists of $$N$$ assets, and the factor return rate $$λ$$ corresponding to each asset $$i$$ is expanded according to different factors, and the following portfolio return rate of the multi-factor model is obtained:

$$$ Rp = ∑ᴺᵢ₌₁Wi(αi+∑ᴹⱼ₌₁βᵢⱼfᵢⱼ) $$$

Among them, $$Rp$$ is the excess return of the portfolio, $Wi $$ is the weight of each asset in the portfolio, $$βij $$ is the risk exposure of each asset on each factor, $$λ = ∑ᴹⱼ ₌₁βᵢⱼfᵢⱼ$, fᵢⱼ is the factor return corresponding to each unit factor loading of each factor for each asset.

Combined with statistical knowledge, this model implies three layers of assumptions:

The $$Beta$$ returns of each asset are uncorrelated with the $$Alpha$$ returns: $$Cov(αi,βiλ)=0$$
The idiosyncratic returns between different assets are also not correlated: $$Cov(αi,αj)=0$$
The factor must be related to the return on assets: $$Cov(Rᵉi,βiλ)≠0$$

A comprehensive explanation of $$Beta$$ returns and $$Alpha$$ returns:
Combined with the specific financial market, $$βiλ$$ is the $$Beta$$ return attributed to the overall performance of the market, and $$αi$$ is the $$Alpha$$ return specifically brought by the asset itself, that is, outperforming the market How many points. The return rate of each asset is composed of Beta return and $$Alpha$$ return. People can use the $$αi$$ value corresponding to each asset in the multi-factor model to score or assign weight to each asset, so as to Construct a portfolio and use futures to short the $$Beta$$ return to hedge risks, thereby obtaining $$Alpha$$ returns.

(2) Volatility of multi-factor model

When constructing an investment portfolio, it is necessary to strike a balance between the risk and return of the portfolio, and the above model needs to be converted into a constrained planning problem to solve. The risk of the portfolio is the volatility of the portfolio $$σ²p$$. $$σ²p$$ is derived below. Detailed analysis involving portfolio construction is explained in the "Risk Portfolio Optimization" section.

Based on the matrix expression $$Rp = W(β ∧ + α)$$ of formula (3), the volatility of the combination can be obtained:

formula 7

Among them, $$W$$ is the weight matrix of assets, $$β$$ is the weight matrix of factors, which represents the factor loading matrix $$(N of $$N$$ assets on $$K$$ risk factors ×K)$$：

formula 8

$$∧$$ represents the factor return covariance matrix $$(K×K)$$ of $$K$$ factors:

formula 9

According to Assumption 3, the idiosyncratic returns of different assets are not correlated, and the Δ matrix can be obtained as:

formula 10

About LUCIDA & FALCON

Lucida ( https://www.lucida.fund/ ) is an industry-leading quantitative hedge fund that entered the Crypto market in April 2018. It mainly trades CTA/statistical arbitrage/option volatility arbitrage and other strategies, with a current management scale of US$30 million. .

Falcon ( https://falcon.lucida.fund/ ) is a new generation of Web3 investment infrastructure. It is based on a multi-factor model and helps users "select", "buy", "manage" and "sell" crypto assets. Falcon was hatched by Lucida in June 2022.

More content can be found at https://linktr.ee/lucida_and_falcon

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Disclaimer: The content above is only the author's opinion which does not represent any position of Followin, and is not intended as, and shall not be understood or construed as, investment advice from Followin.

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