## wpiea2023163-print-pdf (Section 3)

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---

### The crypto factor: identification and properties
- Crypto asset prices are highly correlated; Bitcoin has a 52% average correlation with other crypto assets.
- Dynamic factor modelling summarizes crypto market fluctuations into a single common factor f_t:
  - Panel: daily prices of the largest crypto assets created before January 2018 (excluding stablecoins), leaving seven crypto assets, accounting for 75% of total market capitalization in June 2022.
  - Model specification:
    - p_it = Λ_i(L) f_t + ε_it
    - f_t = A_1 f_{t-1} + ... + A_q f_{t-q} + η_t,  η_t ∼ N(0, Σ)
    - ε_it = ρ_i ε_{it-1} + e_it,  e_it ∼ N(0, σ^2_{it})
  - Estimation: maximum likelihood; q selected using information criteria; implementation uses the Python package statsmodels/DynamicFactor.
- Properties of the extracted crypto factor:
  - Captures major market phases: decline at the beginning of 2018, the 2018–2019 “crypto winter”, later boom (peaks in Bitcoin and Dogecoin), and the slump of Terra and FTX in 2022.
  - On average, 80% of variation in the underlying series is explained by the crypto factor; the figure is above 68% for all seven assets.
  - For comparison, the global equity factor in Miranda-Agrippino and Rey (2020) explains only 20% of global equity prices.
- Robustness / sub-factors:
  - Broader sample grouped into five categories: First Generation tokens; Smart Contracts platform tokens; DeFi tokens; Metaverse tokens; Internet of Things tokens.
  - A five-factor model (one factor per class) shows all classes highly correlated with the general crypto cycle; the Metaverse factor has an idiosyncratic jump in late 2021.
- Leverage correlation:
  - Crypto factor correlates with a proxy for leverage defined as total value locked (TVL) in DeFi contracts normalized by total crypto market capitalization (TVL data from https://defillama.com/).
  - System was relatively unlevered until the end of the 2018-2019 crypto winter; afterwards leverage increased substantially and its correlation with the crypto factor increased.

### Empirical evidence: correlations and factor regressions
- Pairwise correlations (selected from Table 1; data cover January 2018 to March 2023):
  - Bitcoin–Ethereum: 0.82
  - Bitcoin–Litecoin: 0.80
  - Bitcoin–Monero: 0.75
  - Bitcoin–FTX: 0.03
- Factor regressions (regressions of ΔCryptoFactor on equity factors; variables standardized; t-statistics in parentheses; *, **, *** at 10%, 5%, 1%):
  - Global Equity Factor: 0.310*** (t = 6.57), R-squared = 0.047, Observations = 1302.
  - Global Tech Factor: 0.627*** (t = 8.76) in column (2); when included with Global Tech, coefficient 0.663*** (t = 6.73).
  - Global Equity Factor excl. Tech: 0.00916 (t = 0.10).
  - Global Financial Factor: 0.158*** (t = 5.55) in column (4); in column (5) coefficient -0.0226 (t = -0.61).
  - Global Equity Factor excl. Financials: 0.519*** (t = 4.90).
  - GlobalSmallCapsFactor: 0.385*** (t = 6.61).
- Change in correlations before and after 2020 (selected entries from Table 3):
  - Crypto factor cross-correlation with Bitcoin: Before 2020 Crypto F = 0.76 with Bitcoin; After 2020 Crypto F = 0.85 with Bitcoin.
  - S&P 500 correlation with Bitcoin: Before 2020 = 0.01; After 2020 = 0.29.
  - Tech Factor with Crypto F: Before 2020 = -0.01; After 2020 = 0.30.
  - VIX correlations: Before 2020 VIX with Crypto F = -0.08 to -0.19 range; After 2020 VIX with Crypto F = -0.26 to -0.37 across sub-factors.
  - p-values of differences reported in Appendix A, Table A.2.

### Institutional participation and changing investor risk profiles
- Evidence of increased institutional participation:
  - Toczynski (2022) estimate: roughly US$15 billion of federal stimulus checks was spent on trading crypto assets.
  - Chainalysis (2021) on-chain trading-volume proxies indicate institutional share of trading volumes rose dramatically since 2020 (Panel (a) of Figure 5).
  - Coinbase self-reported data show institutional investors became a substantial majority of total trading volumes by 2022 (Panel (b) of Figure 5).
- Regressions linking institutional share to factor correlation (selected coefficients; t-statistics in parentheses):
  - SP500 coefficient on Bitcoin (column (1)): 0.268*** (t = 8.72).
  - Interaction SP500 # After 2020: 0.304*** (t = 5.88).
  - GlobalEquityFactor on CryptoFactor (column (3)): 0.217*** (t = 6.57).
  - Global Equity Factor # After 2020: 0.206*** (t = 4.08).
  - GlobalEquityFactor # Share of Institutionals: 0.226*** (t = 5.02).
  - Example regression fit: column (5) Observations = 1,148, R^2 = 0.0854, R^2 (adj) = 0.0830.
- Decomposition into market risk and aggregate effective risk aversion:
  - Market risk proxied by realized market risk: 90-day variance of the MSCI World index.
  - Aggregate effective risk aversion proxied as residuals from regressions:
    - Equities: f_Equities_t = α + β_1 · Var(MSCI World)_t + ε_t
    - Crypto: f_Crypto_t = α' + β'_1 · Var(MSCI World)_t + β'_2 · Var(BTC)_t + ε'_t
  - Correlations of the 90-day equity risk aversion with He, Kelly, and Manela (2017) proxies:
    - Correlation with intermediary capital ratio = -0.292.
    - Correlation with square of intermediary leverage ratio = 0.434.
- Dynamics of aggregate effective crypto risk aversion:
  - Two main phases identified: before and after the late 2019 peak.
    - Early sample: effective risk aversion of crypto investors more volatile with an increasing trend (coinciding with the crypto winter).
    - After 2020: effective risk aversion declined steadily while the crypto factor exhibited large returns and high volatility.
    - Since collapse of Terra/Luna in May 2022: the crypto factor is almost a mirror image of effective risk aversion, implying crypto prices driven primarily by changes in risk appetite of crypto investors.
  - Rolling correlations:
    - Correlation between crypto and equity factors increases substantially from the second half of 2020.
    - Correlation between estimated risk aversions of marginal crypto and equity investors also rose; at its peak in 2022, correlation exceeded 40%.
- Regression linking rolling correlations (selected results from Table 5):
  - Regressing rolling Corr(ΔCrypto Factor, ΔEquity Factor) on Corr(ΔCrypto RA, ΔEquity RA):
    - 30-day window: coefficient = 0.854*** (standard error = 0.016), Constant = 0.036*** (0.006), Observations = 1,183, R-squared = 0.648.
    - 180-day window: coefficient = 0.773*** (0.023), Constant = 0.091*** (0.008), Observations = 1,093, R-squared = 0.408.
  - Interpretation: correlation between effective risk aversions explains a substantial share of variation in factor correlations.

### Synthesis of Section 3 findings
- A single crypto factor explains a large share of crypto price variation (on average 80%; >68% for each of seven assets).
- Crypto markets became more integrated with global equity markets after 2020, particularly driven by technology and small-cap equity components.
- Increased institutional participation since 2020 is strongly associated with:
  - Higher correlations between crypto and equity factors.
  - A decline in aggregate effective risk aversion among crypto investors, making the marginal crypto investor’s risk profile more similar to that of the marginal equity investor.
- The correlation between the risk aversions of marginal crypto and equity investors can account for a large share of the correlation between the crypto and equity factors (regression R-squared up to 0.648 for 30-day rolling window).

*Source: wpiea2023163-print-pdf (Section 3) — data and tables from January 2018 to March 2023.*

### Section 3 investigates its relationship to equity prices and the global financial cycle. Section

### wpiea2023163-print-pdf - Section 3 investigates its relationship to equity prices and the global financial cycle. Section

### The crypto factor: identification and properties
- Crypto asset prices are highly correlated; Bitcoin has a 52% average correlation with other crypto assets.
- Dynamic factor modelling is used to summarize crypto market fluctuations into a single common factor f_t:
  - Panel uses daily prices of the largest crypto assets created before January 2018 (excluding stablecoins), leaving seven crypto assets, accounting for 75% of total market capitalization in June 2022.
  - Model specification (as presented):
    - p_it = Λ_i(L) f_t + ε_it
    - f_t = A_1 f_{t-1} + ... + A_q f_{t-q} + η_t,  η_t ∼ N(0, Σ)
    - ε_it = ρ_i ε_{it-1} + e_it,  e_it ∼ N(0, σ^2_{it})
  - Estimation: maximum likelihood; q selected using information criteria; implementation uses the Python package statsmodels/DynamicFactor.
- The extracted crypto factor:
  - Captures major market phases: decline at the beginning of 2018, the 2018–2019 “crypto winter”, later boom (peaks in Bitcoin and Dogecoin), and the slump of Terra and FTX in 2022.
  - On average, 80% of variation in the underlying series is explained by the crypto factor; the figure is above 68% for all seven assets.
  - For comparison, the global equity factor in Miranda-Agrippino and Rey (2020) explains only 20% of global equity prices.
- Robustness / sub-factors:
  - Broader sample grouped into five categories: First Generation tokens; Smart Contracts platform tokens; DeFi tokens; Metaverse tokens; Internet of Things tokens.
  - A five-factor model (one factor per class) shows all classes highly correlated with the general crypto cycle; the Metaverse factor has an idiosyncratic jump in late 2021.
- Leverage correlation:
  - Crypto factor correlates with a proxy for leverage defined as total value locked (TVL) in DeFi contracts normalized by total crypto market capitalization (TVL data from https://defillama.com/).
  - System was relatively unlevered until the end of the 2018-2019 crypto winter; afterwards leverage increased substantially and its correlation with the crypto factor increased.

### Empirical evidence: correlations and factor regressions
- Pairwise correlations (selected from Table 1): (entries shown as in table)
  - Bitcoin–Ethereum: 0.82
  - Bitcoin–Litecoin: 0.80
  - Bitcoin–Monero: 0.75
  - Bitcoin–FTX: 0.03
  - (Full matrix reported in source; data cover January 2018 to March 2023.)
- Factor regressions (Table 2: regressions of ΔCryptoFactor on equity factors; coefficients and t-statistics reported):
  - Global Equity Factor: 0.310*** (t = 6.57), R-squared = 0.047, Observations = 1302.
  - Global Tech Factor: 0.627*** (t = 8.76) in column (2); when included with Global Tech, coefficient 0.663*** (t = 6.73).
  - Global Equity Factor excl. Tech: 0.00916 (t = 0.10).
  - Global Financial Factor: 0.158*** (t = 5.55) in column (4); in column (5) coefficient -0.0226 (t = -0.61).
  - Global Equity Factor excl. Financials: 0.519*** (t = 4.90).
  - GlobalSmallCapsFactor: 0.385*** (t = 6.61).
  - Notes: Data from January 2018 to March 2023; variables standardized; t-statistics in parentheses; *, **, *** correspond to significance at the 10%, 5%, and 1% levels respectively.
- Change in correlations before and after 2020 (Table 3 highlights):
  - Crypto factor cross-correlation with Bitcoin: Before 2020 Crypto F = 0.76 with Bitcoin; After 2020 Crypto F = 0.85 with Bitcoin.
  - S&P 500 correlation with Bitcoin: Before 2020 = 0.01; After 2020 = 0.29.
  - Tech Factor with Crypto F: Before 2020 = -0.01; After 2020 = 0.30.
  - VIX correlations: Before 2020 VIX with Crypto F = -0.08 to -0.19 range; After 2020 VIX with Crypto F = -0.26 to -0.37 across sub-factors.
  - (Full matrix reported in source; p-values of differences reported in Appendix A, Table A.2.)

### Institutional participation and changing investor risk profiles
- Evidence of increased institutional participation:
  - Toczynski (2022) estimate: roughly US$15 billion of federal stimulus checks was spent on trading crypto assets.
  - Chainalysis (2021) on-chain trading-volume proxies indicate institutional share of trading volumes rose dramatically since 2020 (Panel (a) of Figure 5).
  - Coinbase self-reported data show institutional investors became a substantial majority of total trading volumes by 2022 (Panel (b) of Figure 5).
- Regressions linking institutional share to factor correlation (Table 4):
  - SP500 coefficient on Bitcoin (column (1)): 0.268*** (t = 8.72).
  - Interaction SP500 # After 2020: 0.304*** (t = 5.88).
  - GlobalEquityFactor on CryptoFactor (column (3)): 0.217*** (t = 6.57).
  - Global Equity Factor # After 2020: 0.206*** (t = 4.08).
  - GlobalEquityFactor # Share of Institutionals: 0.226*** (t = 5.02) — indicating the share of institutional investors plays a significant role in explaining the correlation between crypto and equity factors.
  - R-squared values and Observations: e.g., column (5) Observations = 1,148, R^2 = 0.0854, R^2 (adj) = 0.0830.
- Decomposition into market risk and aggregate effective risk aversion:
  - Following Bekaert et al. (2013) and Miranda-Agrippino and Rey (2020), movements in factors are decomposed into:
    - (i) changes in market risk, proxied by realized market risk (90-day variance of the MSCI World index);
    - (ii) changes in aggregate effective risk aversion (wealth-weighted average risk aversion of investors), proxied as residuals from regressions:
      - Equities: f_Equities_t = α + β_1 · Var(MSCI World)_t + ε_t
      - Crypto: f_Crypto_t = α' + β'_1 · Var(MSCI World)_t + β'_2 · Var(BTC)_t + ε'_t
  - Correlations of the 90-day equity risk aversion with He, Kelly, and Manela (2017) proxies:
    - Correlation with intermediary capital ratio = -0.292.
    - Correlation with square of intermediary leverage ratio = 0.434.
- Dynamics of aggregate effective crypto risk aversion (Figure 6 and Figure 7):
  - Two main phases identified: before and after the late 2019 peak.
    - Early sample: effective risk aversion of crypto investors more volatile with an increasing trend (coinciding with the crypto winter).
    - After 2020: effective risk aversion declined steadily while the crypto factor exhibited large returns and high volatility.
    - Since collapse of Terra/Luna in May 2022: the crypto factor is almost a mirror image of effective risk aversion, implying crypto prices driven primarily by changes in risk appetite of crypto investors.
  - Rolling correlations:
    - Correlation between crypto and equity factors increases substantially from the second half of 2020.
    - Correlation between estimated risk aversions of marginal crypto and equity investors also rose; at its peak in 2022, correlation exceeded 40%.
- Regression linking rolling correlations (Table 5):
  - Regressing rolling Corr(ΔCrypto Factor, ΔEquity Factor) on Corr(ΔCrypto RA, ΔEquity RA):
    - For 30-day window coefficient = 0.854*** (standard error = 0.016), Constant = 0.036*** (0.006), Observations = 1,183, R-squared = 0.648.
    - For 180-day window coefficient = 0.773*** (0.023), Constant = 0.091*** (0.008), Observations = 1,093, R-squared = 0.408.
    - Coefficients positive and highly significant across rolling window specifications; R-squared values relatively high (up to 0.648).
  - Interpretation: correlation between effective risk aversions explains a substantial share of variation in factor correlations.

### Synthesis of Section 3 findings
- A single crypto factor explains a large share of crypto price variation (on average 80%; >68% for each of seven assets).
- Crypto markets became more integrated with global equity markets after 2020, particularly driven by technology and small-cap equity components.
- Increased institutional participation since 2020 is strongly associated with:
  - Higher correlations between crypto and equity factors.
  - A decline in aggregate effective risk aversion among crypto investors, making the marginal crypto investor’s risk profile more similar to that of the marginal equity investor.
- The correlation between the risk aversions of marginal crypto and equity investors can account for a large share of the correlation between the crypto and equity factors (regression R-squared up to 0.648 for 30-day rolling window).

*Italic line: Source: wpiea2023163-print-pdf (Section 3) — data and tables from January 2018 to March 2023.*

### 4.1  The impact of monetary policy on the crypto factor

### 4.1 The impact of monetary policy on the crypto factor

### Methodology
- Daily vector autoregressive (VAR) model, identification via Cholesky decomposition with the policy variable and controls ordered first.
- Monetary policy stance measured by the Wu and Xia (2016) shadow federal funds rate to capture balance sheet policy alongside conventional interest rate policy.
- Controls included (proxies for global activity and financial conditions):
  - 10-2 Y Treasury Yield Spread
  - Dollar Index
  - VIX
  - Oil
  - Gold
- Equity and crypto variables included in various specification orders (see Table 6). Column (1) is baseline; columns (2)–(5) test robustness, alternative policy measures, heterogeneous crypto sub-classes, and the risk-taking channel respectively.
- Sample: Data is from January 2018 to March 202, with the exception of column (4) which is from 2021 due to data availability.
- Impulse responses: cumulative 15-day responses to a one percentage point rise in the shadow FFR; 90% confidence intervals computed from 1000 Monte Carlo simulations. Factors/variables standardized over the sample period.

### Baseline VAR results
- A Fed monetary contraction leads to:
  - An increase in the VIX.
  - A decline in the global equity factor.
  - A decline in the crypto factor.
- Magnitudes:
  - The crypto factor declines by 0.15 standard deviations.
  - The equity factor declines by 0.1 standard deviation.
- Interpretation: Crypto assets are subject to US monetary policy and the economic cycle similarly to traditional investments; findings contrast with claims that crypto is orthogonal to traditional financial markets or serves as a hedge.

### Robustness and alternative policy measures
- Replacing factors with S&P500 and Bitcoin price (specification (2)) yields very similar impulse responses, indicating results are not driven by factor-construction choices.
- Defining the policy variable as the average shadow rate of the Fed, Bank of England and European Central Bank (weighted by balance sheet size) (specification (3)) produces much weaker responses:
  - No longer a significant impact on the global equity factor.
  - No longer a significant impact on the crypto factor.
- Interpretation: Weaker responses to global tightening are consistent with dollar dominance and crypto market dollarization (e.g., largest stablecoins USD-denominated, most crypto borrowing/lending in USD stablecoins, prices expressed in dollars).

### Crypto sub-classes
- VAR specification (4) disaggregates crypto into sub-factors (First Generation, Smart Contracts, DeFi, Internet of Things, Metaverse). Sample start dates vary by token availability, with many tokens absent in 2018.
- Findings:
  - First Generation coins: reaction consistent with baseline.
  - Other sub-factors: similar-shaped responses but statistically insignificant, partly reflecting shorter estimation samples.
  - Metaverse: farthest from significant reaction (tokens newer, smaller market caps, mostly retail investor base).

### Risk-taking channel and institutional investors
- Investigate risk-taking channel following Miranda-Agrippino and Rey (2020): monetary policy shocks change effective risk aversion of the marginal investor.
- Specification (5) includes proxies for aggregate effective risk aversion of equity and crypto investors.
- Results:
  - A monetary policy contraction leads to a persistent increase in the effective risk aversion of the marginal crypto investor and to lower crypto prices.
  - Interpretation: Higher cost of capital leads crypto investors to deleverage; leveraged investors are more sensitive to the economic cycle.
- Sub-sample analysis (before and after 2020):
  - Response of crypto risk aversion is significant only in the post-2020 period.
  - Response of crypto prices to monetary policy is larger in the post-2020 period.
  - Interpretation: Increased participation of institutional investors (who take on more leverage) since 2020 increased correlation between equity and crypto prices and reinforced transmission of monetary policy to crypto markets.

### Smooth transition VAR by institutional share (non-linear state dependence)
- Logistic smooth transition VAR estimated with transition variable equal to the share of institutional investors (Chainalysis (2021) source for share).
- Model form: combination of two VARs weighted by F(s_{t-1}), a logistic function of the transition variable; smooth transition parameter used is ϑ = 3 (results robust to ϑ = 1.5).
- Transition diagnostics:
  - The logistic transition variable is a logistic transformation of the (standardized) share of institutional investors (with ϑ = 3).
  - Correlation between the state transition variable and the share of institutional investors is 96%.
  - When the transition variable equals one (zero), the share of institutional investors is high (low).
- STVAR results:
  - When the share of institutional investors is low: US monetary policy does not significantly affect crypto prices and the response of aggregate crypto risk aversion is not significant.
  - When the share of institutional investors is high: monetary tightening has a significant negative effect on crypto prices and a significant increase in the effective risk aversion of the marginal crypto investor.
- Implication: The transmission of US monetary policy to crypto markets is state-dependent and amplified when institutional investor participation is high.

*Source: IMF Working Paper chapter 4.1 (wpiea2023163-print-pdf).*

### 4.3  Other mechanisms

### 4.3 Other mechanisms

### Liquidity channel
- Hypothesis: Lower liquidity of crypto markets could drive stronger reactions to monetary policy shocks, as illiquid securities may react more strongly regardless of investor composition.
- Empirical test: Crypto assets were sorted by the number of units traded daily; factors were extracted for the most and least liquid securities and previous analysis repeated.
- Finding: No significant differences were found between the two factors in their response to monetary policy, suggesting liquidity differences do not explain the results.
- Additional evidence: Crypto market liquidity increased from 2020 as more investors entered the asset class. Under the liquidity channel, increased liquidity would reduce responsiveness to monetary policy—whereas the authors find responsiveness has increased (Figure 13).

### USD appreciation channel
- Hypothesis: US monetary policy could affect crypto via the USD valuation channel because the dollar is the main funding currency and unit of account in the crypto market.
- Facts from the source:
  - Crypto tokens are mostly priced in dollars.
  - USD stablecoins account for 95% of stablecoins issued.
  - DeFi lending is largely executed in USD stablecoins.
- Mechanisms:
  - USD appreciation makes tokens mechanically more expensive for non-US investors, reducing inflows.
  - USD stablecoin borrowing becomes more expensive as the dollar appreciates, potentially reducing demand for leverage.
- Empirical finding: VARs do not show significant responses of the crypto factors to shocks in the DXY index, so strong evidence for this channel is not found.

### Alternative valuation (bubble) channel
- Hypothesis: If investors price crypto assets as bubbles, a rise in discount rates would compress risk premia, prompting divestment and price declines.
- Limitation: This channel does not explain the observed increase in crypto responsiveness to monetary policy since 2020, nor the increased synchronization of crypto and equity cycles.
- Conclusion: Authors retain focus on a changing composition of the crypto investor base as the primary explanation.

*Source: wpiea2023163-print-pdf - 4.3  Other mechanisms — content supplied in the source PDF.*

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### 5 Model (stylized framework and implications)

### Setup and agents
- Two representative heterogeneous agents and two asset classes: crypto and equity.
  - Crypto investors: retail investors who can only invest in crypto; maximize a mean-variance portfolio; can borrow at the US risk-free rate; constant risk aversion coefficient.
  - Institutional investors: banks and similar financial intermediaries; can invest in both crypto and equity; risk-neutral; maximize expected portfolio return subject to a value-at-risk constraint (expressed with multiple ✓).
- Interpretation: One crypto asset and one global stock represent the crypto and global equity factors in empirical analysis.

### Key investor first-order conditions (as presented)
- Crypto investors:
  - First-order condition: x_c_t = 1/φ * E_t(R^c_{t+1}) × [Var_t(R^c_{t+1})]^{-1}  (presentation in source: x_c_t = 1  E_t(R^c_{t+1}) ⇥ Var_t(R^c_{t+1}) ⇤ 1)
  - Intuition: Holdings increase with expected return and decrease with risk aversion and variance.
- Institutional investors:
  - First-order condition (as presented): x_i_t = 1/(2✓^2_t) × [ E_t(R^c_{t+1})  2✓^2_t Cov_t(R^c_{t+1},R^e_{t+1}) y_t ] × [ Var_t(R^c_{t+1}) ]^{-1} (source contains the full algebraic expression)
  - Intuition: Institutional crypto holdings are positively related to expected crypto payoffs and negatively related to (i) variance of crypto returns, (ii) covariance with equities, and (iii) tightness of financial constraints.

### Market clearing and derived propositions
- Market clearing:
  - Crypto supply (normalized by total wealth) s_t equals total holdings: s_t = x_c_t w_c_t/(w_c_t + w_i_t) + x_i_t w_i_t/(w_c_t + w_i_t).
  - Equity clearing: y_tot_t = y_t.
- Proposition 1 (as given):
  - Crypto excess returns as a function of aggregate effective risk aversion:
    - E_t(R^c_{t+1}) = c_t Var_t(R^c_{t+1}) s_t + c_t Cov_t(R^c_{t+1},R^e_{t+1}) y_tot_t (w_i_t/(w_c_t + w_i_t))
  - Aggregate effective risk aversion:
    - c_t = (w_c_t + w_i_t) [ w_c_t + w_i_t/(2✓^2_t) ]^{-1}  (presentation follows source notation)
  - Intuition:
    - Crypto excess returns must be higher to compensate for variance in proportion to average risk aversion.
    - A higher correlation with equities reduces diversification benefits for institutional investors, increasing required crypto returns in equilibrium, with larger effects the larger the share of institutional wealth.
- Proposition 2 (as given):
  - Equity excess returns depend on institutional financial constraints and portfolio allocation to crypto:
    - E_t(R^e_{t+1}) = 2✓^2_t Var_t(R^e_{t+1}) y_tot_t + 2✓^2_t Cov_t(R^c_{t+1},R^e_{t+1}) x_i_t
  - Intuition: Investors require compensation for higher variance or lower diversification in proportion to their financial constraint.

### Comparative-static and dynamic implications (explicit from source)
- Increasing institutional wealth share in crypto:
  - The risk-taking profile of the crypto market converges to that of the equity market.
  - In the extreme where institutions dominate (w_i_t/(w_i_t + w_c_t) → 1), c_t → 2✓^2_t.
  - Greater institutional participation increases correlation between equity and crypto prices (consistent with empirical findings, e.g., Figure 7).
- Effect of a rise in the risk-free rate:
  - A mechanical reduction in real returns for both crypto and equities.
  - Institutional entry reduces aggregate effective risk aversion (since t < 2✓^2_t in the model), making the marginal crypto investor less risk averse and more leveraged.
  - Increased leverage and institutional entry can raise crypto sensitivity to the economic cycle (consistent with Figures 13 and 15 and literature: Coimbra et al. (2022); Adrian and Shin (2014)).
- Potential spillovers from crypto to equities:
  - Currently the crypto holdings of traditional financial institutions x_i_t are very small relative to equity holdings y_t, so the second term in Proposition 2 is negligible.
  - If institutional crypto holdings became significant, a crypto crash reducing x_i_t would lower expected equity returns E_t(R^e_{t+1}), with larger effects the larger pre-crash crypto holdings were.
  - Policy implication noted in the source: possible justification for a cap  ̄x_i_t or other risk-based constraints on crypto holdings by traditional financial institutions.

*Source: wpiea2023163-print-pdf - 4.3  Other mechanisms — content supplied in the source PDF.*

---

### 6 Conclusion (key findings and policy implications)

- Stylized empirical facts (from the source):
  - A single crypto factor can explain 80% of the variation in crypto prices.
  - The crypto factor has become more correlated with the global financial cycle since 2020, particularly with technology and small-cap stocks.
  - Crypto markets are very sensitive to US monetary policy; a monetary contraction significantly reduces the crypto factor, similarly to global equities.
- Mechanism emphasized:
  - Increasing presence of institutional investors in crypto markets shifts the aggregate effective risk aversion, making the risk profile of marginal crypto and equity investors increasingly similar and increasing correlation between markets.
- Policy-relevant implications (as stated in source):
  - Crypto assets do not provide a hedge against the economic cycle; estimates suggest they respond even more than stocks.
  - Increasing correlation and the fact that institutional investors trade both crypto and stocks imply potential spillover effects that could raise systemic risk concerns.
  - In a future where crypto represents a substantial share of institutional portfolios, a crypto crash could have significant negative repercussions for equity markets.
  - Policymakers could use the current limited institutional exposure to crypto to develop and implement an improved regulatory framework.

*Source: wpiea2023163-print-pdf - 4.3  Other mechanisms — content supplied in the source PDF.*

### References

### References

### Key cited works
- Adrian, Tobias, Tara Iyer, and Mahvash S. Qureshi, 2022, Taking stocks: Monetary policy transmission to equity markets, IMF Blog.
- Adrian, Tobias, and Hyun Song Shin, 2014, Procyclical Leverage and Value-at-Risk, Review of Financial Studies Vol. 27, No. 2, pp. 373–403.
- Auer, Raphael, Marc Farag, Ulf Lewrick, Lovrenc Orazem, and Markus Zoss, 2022, Banking in the shadow of Bitcoin? The institutional adoption of cryptocurrencies, BIS Working Papers.
- Basel Committee on Banking Supervision, 2021, Prudential treatment of cryptoasset exposures, Technical report, Basel, Switzerland.
- Brunnermeier, Markus, Harold James, and Jean-Pierre Landau, 2019, The Digitalization of Money, NBER Working Papers.
- Foley, Sean, Jonathan R Karlsen, and Talis Putnins, 2019, Sex, Drugs, and Bitcoin: How Much Illegal Activity Is Financed through Cryptocurrencies?, Review of Financial Studies.
- International Monetary Fund, 2021, The Crypto Ecosystem and Financial Stability Challenges, Global Financial Stability Report, October 2021, Chapter 2.
- International Monetary Fund, 2023, Elements of E↵ective Policies for Crypto Assets, Technical report.
- Liu, Yukun, Aleh Tsyvinski, and Xi Wu, 2022, Common Risk Factors in Cryptocurrency, Journal of Finance.
- Nakamoto, Satoshi, 2008, Bitcoin: A Peer-to-Peer Electronic Cash System.
- Pagnotta, Emiliano S, 2022, Decentralizing Money: Bitcoin Prices and Blockchain Security, Review of Financial Studies.
- Wu, Jin Cynthia, and Fan Dora Xia, 2016, Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound, Journal of Money, Credit and Banking.

(Note: the References section contains a broader set of citations; above are representative entries preserved exactly as in the source.)

### Appendix A — Additional Descriptives and Results (contents and exact notes)
- Figure A.1: Reverse regressions
  - Notes: "This figure shows the R^2s from regressions of the crypto factor on each of the input price series, as described in Section 2."
- Table A.1: Equity Eikon RICs by country
  - Lists equity, tech, financials, and small caps indices used for constructing the global equity factor and sub-factors for the fifty largest countries by GDP. Example entries (preserved exactly):
    - United States: .SPX .SPLRCT .SPSY .SPCY
    - China: .SSEC .SZFI .SZFI
    - Japan: .JPXNK400 .TOPXS
    - Germany: .GADXHI .CXPHX .CXPVX
    - India: .BSESN .BSETECK .BSEBANK
    - Canada: .GSPTSE .SPTTTK .SPTTFS .SPTSES
    - Switzerland: .SSHI .C9500T .C8700T .SSCC
    - (Table continues for other listed countries and indices; indices are from Eikon/Thomson Reuters.)
  - Notes: "The selected countries are the fifty largest by GDP. All indices are from Eikon/Thomson Reuters."

- Table A.2: p-values of the differences in correlation before and after 2020
  - Example entries (preserved exactly as in source matrix):
    - S&P 500: 0.000 0.005 n.a. n.a. n.a. n.a. n.a. n.a.
    - Equity F: 0.000 0.006 n.a. n.a. n.a. n.a. n.a. n.a.
    - Small Caps F: 0.000 0.012 n.a. n.a. n.a. n.a. n.a. n.a.
    - Tech Factor: 0.000 0.001 n.a. n.a. n.a. n.a. n.a. n.a.
    - Equity F (no Tech): 0.876 0.765 n.a. n.a. n.a. n.a. n.a. n.a.
    - Financials F: 0.004 0.029 n.a. n.a. n.a. n.a. n.a. n.a.
    - Dollar Index: 0.060 0.010 n.a. n.a. n.a. n.a. n.a. n.a.
    - VIX: 0.000 0.008 n.a. n.a. n.a. n.a. n.a. n.a.
  - Notes: "The matrix reports the p-values of the interaction coefficient of the following set of regressions: y = constant + β1 x + β2 After2020 + β3 xAfter2020 + ε. After2020 is equal to one from January 2020. Standard errors are robust. Data is from January 2018 to March 2023."

- Table A.3: Risk-aversion regressions
  - Regressions reported for Global Crypto Factor and Global Equity Factor across different variance windows. Selected exact entries:
    - 30 days coefficient (Global Crypto Factor): 2.061*** (0.315)
    - 45 days coefficient (Global Crypto Factor): 1.711*** (0.265)
    - 60 days coefficient (Global Crypto Factor): 1.763*** (0.245)
    - Var(Bitcoin) 90 days (Global Crypto Factor): 1.851*** (0.249)
    - 30 days coefficient (Global Equity Factor): -27.939*** (1.686) and -63.585*** (4.020) in related columns
    - Var(MSCI World) 90 days (Global Equity Factor): -18.727*** -30.706*** (1.006) (1.876)
    - Constants reported: -0.163*** -0.172*** -0.200*** -0.266*** 70.975*** 55.650*** 46.562*** 34.904*** with corresponding standard errors (0.022)(0.025)(0.028)(0.037)(4.421)(3.095)(2.471)(2.067)
    - Observations: 1,273 1,258 1,243 1,213 1,275 1,261 1,246 1,217
    - R-squared: 0.084 0.10 0.121 0.159 0.176 0.185 0.182 0.163
  - Notes: "The table reports the results of regressing the equity and the crypto factor on the variances of the MSCI World Index and Bitcoin. Standard errors are in parentheses. *, **, and *** correspond to significance at the 10%, 5%, and 1% levels respectively."

- Table A.4: Correlations across risk-taking proxies
  - Pairwise daily correlations (series standardized; data from January 2018 to March 2023). Selected exact entries:
    - 30-day Crypto Risk Aversion correlations: with 45-day 0.791, with 60-day 0.798, with 90-day 0.761, with 30-day Global Equity Risk Aversion 0.181
    - 45-day Global Equity Risk Aversion correlation with 30-day Global Equity Risk Aversion: 0.973
    - Intermediary Capital Ratio correlations: with 30-day Global Equity Risk Aversion -0.485; with Intermediary Leverage Ratio Squared -0.872
    - Intermediary Leverage Ratio Squared correlation with 30-day Global Equity Risk Aversion 0.613
  - Notes: "This table shows pairwise daily correlations between changes in the measures of risk aversion, computed using Equations 2 and 3, and changes in the intermediary risk-appetite measures by He et al. (2017) available at https://voices.uchicago.edu/zhiguohe/data-and-empirical-patterns/intermediary-capital-ratio-and-risk-factor/. Series are standardized, and data is from January 2018 to March 2023."

- Table A.5: Crypto Returns and US Monetary Policy
  - Dependent variables: Bitcoin and Crypto Factor. Selected exact coefficients and statistics:
    - Shadow FFR coefficient: -0.0531 and -0.101* (t-stats -1.41) (-2.00)
    - BRW Shocks coefficient: -0.0613* -0.0791** (t-stats -1.75) (-2.28)
    - Constants: 0.0366 0.0366 0.0141 0.0141 (0.78)(0.78)(0.26)(0.26)
    - N: 48 48 48 48
    - R^2: 0.0263 0.0351 0.0705 0.0431
    - R^2 (adj): 0.005160 0.01410 0.05030 0.0223
  - Notes: "Variables are standardized. Frequency is monthly and data is from January 2018 to December 2021. Shadow FFR are from Wu and Xia (2016) and BRW shocks are from Bu et al. (2021). Standard errors are robust and t-statistics are parentheses. *, **, *** correspond to 10%, 5%, and 1% significance, respectively."

*The Crypto Cycle and    US   Monetary Policy — Working Paper No. WP/2023/163*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023163-print-pdf.pdf_