A Market Index for Cryptocurrencies

Introduction

Main purpose in this notebook:

  • Apply what we just learned about model fitting and interpretation in a real finance setting.
  • Build a crypto market factor (MKT) and use regressions to separate common market exposure (beta) from coin-specific risk (R^2 gaps).
  • Compare crypto-market returns with stock-market returns (SPY).

Core model: one-factor return decomposition for each coin i. r_i = \alpha_i + \beta_i r_{\text{MKT}} + e_i.

Interpretation:

  • r_{\text{MKT}}: common crypto market shock (systematic component).
  • \beta_i: sensitivity of coin i to that common shock.
  • \alpha_i: average return component not explained by the factor.
  • e_i: idiosyncratic component (coin-specific variation).

Why this model is useful:

  • It decomposes returns into common risk vs coin-specific risk.
  • It gives a compact way to compare coins with two summary statistics:
    • Beta for sensitivity.
    • R^2 for fraction of variance explained by the common factor.

Important distinction for interpretation:

  • High beta does not necessarily imply high R^2.
  • A coin can react strongly to market moves on average (high beta), yet still have large idiosyncratic volatility (low R^2).

Coins in the index:

Ticker Coin
BTC Bitcoin
ETH Ethereum
BCH Bitcoin Cash
LTC Litecoin
XRP Ripple
DOGE Dogecoin
XLM Stellar 
import numpy as np
import pandas as pd
import yfinance as yf
import matplotlib.pyplot as plt
import seaborn as sns
import statsmodels.formula.api as smf
import warnings

warnings.simplefilter(action='ignore', category=FutureWarning)

coins = ['BTC-USD','ETH-USD','LTC-USD','XRP-USD','DOGE-USD','BCH-USD','XLM-USD']

df = (
    yf.download(coins, start='2015-01-01', progress=False)['Close']
      .resample('ME')
      .last()
      .pct_change()
      .dropna()
)
df.columns = [c.replace('-USD','') for c in df.columns]

Sample period used in the code:

  • Monthly returns from January 2015 onward (subject to data availability for each asset).

Building a Crypto Market Index

Factor model used for each coin i: r_i = \alpha_i + \beta_i r_{\text{MKT}} + e_i.

MKT is a fixed-weight crypto index (weights based on market-cap shares).

Modeling choice:

  • We use fixed weights for transparency and stability of interpretation.
  • With time-varying weights, measured beta can mix true exposure changes with index-reweighting effects.
w_raw = pd.Series({
    'BTC': 57.5,
    'ETH': 11.6,
    'XRP': 3.8,
    'DOGE': 0.7,
    'BCH': 0.4,
    'LTC': 0.2,
    'XLM': 0.2
})

w = w_raw / w_raw.sum()
weights = pd.DataFrame(np.tile(w.values, (len(df.index), 1)), index=df.index, columns=w.index)
df['MKT'] = (weights * df[w.index]).sum(axis=1)

Analyzing the Relationship Between Cryptocurrencies and the Crypto Market Index

First, we benchmark crypto market volatility against equities:

Key results (in this sample):

  • The crypto market index is much more volatile than SPY.

BTC vs MKT

beta_MKT p_value(beta=0) R_squared
0 0.8612 0.0 0.9023

BTC interpretation:

  • BTC has strong positive exposure to the crypto market factor.
  • The fit is tight (high R^2), consistent with BTC’s large index weight.

BCH vs MKT

beta_MKT p_value(beta=0) R_squared
0 1.1018 0.0 0.4748

BCH interpretation:

  • BCH still has meaningful market exposure.
  • The relationship is noisier than BTC (lower R^2), so more variation is coin-specific.

All Coins on MKT

Coin Beta R-squared
7 MKT 1.00 1.000
1 BTC 0.86 0.902
3 ETH 1.08 0.721
4 LTC 1.06 0.690
0 BCH 1.10 0.475
5 XLM 2.07 0.433
6 XRP 2.59 0.368
2 DOGE 2.13 0.229

Cross-coin interpretation:

  • High beta with low R^2 means strong average market sensitivity but substantial idiosyncratic risk.
  • DOGE, XLM, and XRP show relatively high betas but lower R^2, indicating strong average market sensitivity alongside substantial coin-specific risk.
  • Sanity check: the MKT row should show beta =1 and R^2=1 when regressing MKT on itself.

Statistical reading guide:

  • Testing \beta=0: asks whether market exposure is statistically different from zero.
  • Confidence intervals: show plausible ranges for true exposure.
  • Economic significance (beta size) and statistical significance (p-value) are related but distinct.

Crypto Market Index vs. Stock Market

Scatter relationship:

beta_SPY p_value(beta_SPY=0) p_value(beta_SPY=1) R_squared
0 1.8102 0.0001 0.0682 0.1489

Key results (in this sample):

  • SPY loading in MKT ~ SPY is positive and statistically significant (reject \beta_{SPY}=0).
  • Testing \beta_{SPY}=1: at the 5% level we fail to reject equality to 1, though the result is borderline at the 10% level.

Economic interpretation:

  • A positive, significant SPY loading indicates directional co-movement between crypto and equities.
  • But R^2 is still crucial: co-movement can be significant while a large share of crypto variation remains unexplained by equities.

Takeaways

  • This notebook applies the same regression logic from previous notes to a realistic asset-pricing use case.
  • Beta measures average market sensitivity; R^2 measures how much variation the market factor explains.
  • Crypto and equities co-move, but a large share of crypto risk remains crypto-specific.
  • Scope note: this notebook is an in-sample modeling exercise (fit and interpretation), not a trading backtest.