LUCIDA & FALCON: Building a Strong Cryptocurrency Asset Portfolio with Multi-Factor Strategies
The ultimate goal of mining and calculating "factors" is to accurately calculate the expected return on assets.Written by: LUCIDA & FALCON
Introduction
In June last year, I conceived a simple idea of using a multi-factor model to select cryptocurrencies. Click for details.
A year later, we have begun developing multi-factor strategies for the cryptocurrency market and have written a series of articles titled "Building a Strong Cryptocurrency Investment Portfolio with Multi-Factor Strategies."
The general framework of this series is as follows (with the possibility of minor adjustments):
Theoretical Foundation of Multi-Factor Models
Single Factor Construction
- Factor Data Preprocessing
- Data Screening
- Outlier Handling: Extremes, Errors, Missing Values
- Standardization
- Neutralization: Industry, Market, Market Capitalization
- Factor Effectiveness Assessment
- Information Ratio IC, Returns, Sharpe Ratio, Turnover Rate
- Major Factor Synthesis
- Factor Collinearity Analysis
- Orthogonal Elimination of Factor Collinearity
- Classical Weighting Methods → Composite Factors
- Equal Weighting, Rolling IC Weighting, IC_IR Weighting
- Testing Composite Factors: Returns, Group Returns, Factor Value Weighted Returns, Composite Factor IC, Group Turnover Rate
- Other Weighting Methods (when factors and returns have a non-linear relationship): Machine Learning, Reinforcement Learning (not considered due to the uniqueness of the cryptocurrency industry)
- Risk Portfolio Optimization
Below is the main content of the first article "Theoretical Foundation."
1. What is a "Factor"
A "factor" refers to the "indicator" in technical analysis and the "feature" in artificial intelligence machine learning, which determines the reasons for the rise and fall of cryptocurrency returns.
Our team identifies common types of factors in the cryptocurrency field: fundamental factors, on-chain factors, volume-price factors, derivative factors, alternative factors, and macro factors.
The ultimate goal of extracting and calculating "factors" is to accurately compute the expected returns of assets.
2. Calculation of "Factors"
(1) Derivation of Multi-Factor Models
Origin: Single Factor Model ------ CAPM
Factor research can be traced back to the 1960s, with the introduction of the Capital Asset Pricing Model (CAPM), which quantifies how risk affects a company's capital cost and thus influences expected returns. According to 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）「formula2」
E(Ri) is the mathematical expectation, Ri is the asset's return, Rf is the risk-free rate, Rm is the market portfolio return, and βi=Cov(Ri,Rm)/Var(Rm) reflects the sensitivity of asset returns to market returns, also known as the asset's exposure to market risk.
Supplementary Understanding:
In financial markets, the "risk" and "return" discussed are essentially the same thing.
From a statistical perspective, a more detailed understanding of βi
CAPM can be viewed as a bivariate regression model without an intercept Yi=β1+β2⋅X(β1=0), using ordinary least squares (OLS) to estimate the model parameters, where β1=β2=Σ(X−μX)(Y−μY)/Σ(X−μX)²=Cov(X,Y)/Var(X).
β1 measures the extent to which changes in the explanatory variable (market return) lead to average changes in the explained variable (return of asset i). In finance, this change is interpreted as the "sensitivity" or "exposure" of Y to X.
β>1 amplifies market fluctuations
β=1 is identical to market fluctuations
0<β<1 fluctuates in the same direction as the market but is less volatile
β≤0 fluctuates in the opposite direction of the market
From the perspective of finance regarding risk and return, a more detailed understanding of βi
Portfolios have two types of risk: systematic risk (i.e., market risk, non-diversifiable risk) and unsystematic risk (diversifiable risk). βi is systematic risk, which cannot be diversified away regardless of how the asset portfolio is constructed. The αi mentioned below represents unsystematic risk, which can be hedged through the construction of different strategies.
The CAPM model is the simplest linear factor model, indicating that the excess return of an asset is determined solely by the expected excess return of the market portfolio (market factor) and the asset's exposure to market risk. This model lays the theoretical foundation for subsequent research on numerous linear multi-factor pricing models.
Development: Multi-Factor Model ------ APT
Building on CAPM, it was discovered that the returns of different assets are influenced by multiple factors, leading to the emergence of Arbitrage Pricing Theory (APT), which constructs a linear multi-factor model:
E(Ri)=βi⋅λ「formula3」
Here, $E(Ri)$ represents the expected return of asset i, and λ represents the expected return of the factor (i.e., factor premium). Formula (2) replaces E(Ri)−Rf in the CAPM model to represent expected returns, using a market-neutral investment portfolio constructed through long-short hedging, where Rf is offset, making the overall expected return of the asset the difference between the expected returns of the long and short positions, thus using E(Ri) is more general.
Maturity: Multi-Factor Model ------ Alpha Returns & Beta Returns
Considering the pricing errors that actually exist in financial markets and the APT model, from a temporal perspective, the expected return of a single asset is determined by the following multivariate linear model:
Reit=αi+βi⋅λt+εit「formula4」
Where Reit represents the return of asset i at time $t$, λt represents the factor return at time t (i.e., factor premium), and εit represents the random disturbance at time t. αi represents the pricing error between the actual expected return of asset i and the expected return implied by the multi-factor model; if statistically significantly different from zero, it indicates an opportunity for excess returns. βi=Cov(Ri,λ)/Var(λ) represents the factor exposure or factor loading of asset i, depicting the sensitivity of asset returns to factor returns.
The multi-factor model focuses on the cross-sectional differences in expected asset returns, essentially being a model about means, while expected returns are the average of returns over time. Based on (3), we can derive the multivariate linear model from a cross-sectional perspective:
E[Rei]=αi+βi⋅λ「formula5」
Where E[Rei] represents the expected excess return of asset i, and εit averages over time, thus E(εit)=0.
Supplementary Understanding:
From an academic perspective, according to the theory of market efficiency, an efficient asset portfolio should have a total risk of zero; the actual return equals the expected return, and the expected asset return depends solely on the systematic risk of the market, i.e., E[Rei]=βi⋅λ, there are no excess returns (Abnormal Return, AR), i.e., AR=Ri−E(Rei)=0. However, the real financial world is often inefficient, resulting in excess returns, i.e., AR=α.
Assuming the portfolio consists of N assets, and expanding the factor return λ corresponding to each asset i according to different factors, we obtain the following multi-factor model of portfolio returns:
Rp=∑i=1NWi(αi+∑j=1Mβijfij)
Where Rp is the excess return of the portfolio, $Wi is the weight of each asset in the portfolio, βij is the risk exposure of each asset to each factor, λ = ∑ᴹⱼ₌₁βᵢⱼfᵢⱼ$, and fᵢⱼ is the factor return corresponding to each unit factor loading of each asset.
Combining statistical knowledge, this model implies three layers of assumptions:
The Beta returns and Alpha returns of each asset are uncorrelated: Cov(αi,βiλ)=0
The characteristic returns among different assets are also uncorrelated: Cov(αi,αj)=0
Factors are definitely related to asset returns: Cov(Rei,βiλ)≠0
For a comprehensive explanation of Beta returns and Alpha returns:
In the context of specific financial markets, βiλ represents the Beta returns attributable to the overall performance of the market, while αi represents the Alpha returns brought by the specific characteristics of the asset, indicating how many points it outperforms the market. The return of each asset is composed of Beta returns and Alpha returns, and investors can use the αi values corresponding to each asset in the multi-factor model to score or weight each asset, thereby constructing a portfolio and using futures to short the Beta return portion to hedge risks, thus obtaining Alpha returns.
(2) Volatility of Multi-Factor Models
When constructing a portfolio, it is necessary to balance the risk and return of the portfolio, requiring the transformation of the above model into a constrained optimization problem for solving. The risk of the portfolio is the portfolio's volatility σ²p, and the following derives σ²p. A detailed analysis of portfolio construction will be elaborated in the "Risk Portfolio Optimization" section.
Based on the matrix expression of formula (3) Rp=W(β∧+α), we can obtain the portfolio's volatility:
Where W is the weight matrix of the assets, β is the weight matrix of the factors, representing the factor loading matrix of N assets on K risk factors (N×K):
∧ represents the covariance matrix of the factor returns for K factors (K×K):
From assumption 3, since the characteristic returns among different assets are also uncorrelated, we can derive the Δ matrix as:
About LUCIDA & FALCON
Lucida (https://www.lucida.fund/) is an industry-leading quantitative hedge fund that entered the Crypto market in April 2018, primarily trading strategies such as CTA / statistical arbitrage / options volatility arbitrage, currently managing $30 million.
Falcon (https://falcon.lucida.fund) is a new generation of Web3 investment infrastructure based on multi-factor models, helping users "select," "buy," "manage," and "sell" cryptocurrency assets. Falcon was incubated by Lucida in June 2022.
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