What is the Bitcoin Misery
What is the Bitcoin Misery Index?
The Bitcoin Misery Index (BMI) is an indicator designed to quantify how “painful” market conditions have been for holders of Bitcoin over different horizons. There is no single universally mandated formula; providers and researchers may compute BMI variants that combine recent returns, realized or implied volatility, and/or multi-horizon performance metrics. Higher values signal worse recent outcomes for holders (more volatility, larger negative realized returns, or both); lower values indicate calmer, more rewarding conditions.
Common forms and formulas (assumptions noted)
Below are two commonly used conceptual approaches. Before using any BMI, confirm the exact definition used by your data provider.
1) Volatility-sum BMI (volatility-based)
– Idea: add short-term and long-term volatility to capture both immediate and persistent market stress.
– Formula (example): BMI_vol = σ_30d_annualized + σ_365d_annualized
– σ_30d_annualized = standard deviation of daily log returns over the past 30 days, annualized = sd_daily * sqrt(365)
– σ_365d_annualized = standard deviation over the past 365 days, annualized
– Assumption: realized (historical) volatility is used. Providers may use implied volatility instead.
2) Return+Volatility BMI (return penalized by volatility)
– Idea: combine a negative of recent annualized return with a measure of volatility so that poor returns and high volatility both raise “misery.”
– Formula (example): BMI_ret = -R_365_annualized + k * σ_30d_annualized
– R_365_annualized = total return over past 365 days converted to annualized percentage
– k = scaling constant (for consistency of units; often k = 1)
– Assumption: sign convention makes higher BMI worse (negative returns increase BMI).
Worked numeric example (volatility-sum variant)
Step 1 — compute daily log returns for last 365 days from price series P_t:
r_t = ln(P_t / P_{t-1})
Step 2 — compute standard deviations:
– sd_30 = standard deviation of r_t over most recent 30 days
– sd_365 = standard deviation of r_t over most recent 365 days
Step 3 — annualize:
– σ_30d_annualized = sd_30 * sqrt(365)
– σ_365d_annualized = sd_365 * sqrt(365)
Step 4 — compute BMI:
– BMI_vol = σ_30d_annualized + σ_365d_annualized
Numeric example (hypothetical):
– sd_30 = 0.025 (2.5% daily)
– sd_365 = 0.015 (1.5% daily)
– σ_30d_annualized = 0.025 * sqrt(365) ≈ 0.025 * 19.105 ≈ 0.4776 → 47.8% annualized
– σ_365d_annualized = 0.015 * 19.105 ≈ 0.2866 → 28.7% annualized
– BMI_vol ≈ 47.8% + 28.7% = 76.5 (units: percentage points)
Interpretation: A BMI of 76.5 (in this formulation) indicates a high combined short- and long-term volatility environment. Compare current BMI to its historical distribution to judge whether it is unusually high or low.
Practical checklist: compute your own BMI
– Obtain daily Bitcoin price series (cleaned, consistent timezone).
– Decide BMI formula and record assumptions (realized vs. implied vol, annualization factor, weighting).
– Compute daily log returns; remove or document outliers.
– Calculate rolling standard deviations and/or cumulative returns for chosen windows.
– Annualize volatility consistently (commonly sqrt(252) for trading days or sqrt(365) for daily crypto).
– Combine components per your chosen formula.
– Plot BMI over time and compare current reading to historical percentiles (e.g., 10th, 50th, 90th).
How traders and researchers use BMI
– Risk gauge: high BMI may trigger risk-off positioning or hedging (in theory).
– Sentiment/contrarian signal: extreme high BMI values have sometimes coincided with local market bottoms, but this is probabilistic, not deterministic.
– Portfolio review: use BMI alongside other indicators (volume, on-chain metrics, macro data) to form a holistic view.
Limitations and caveats
– Definition sensitivity: BMI readings depend heavily on chosen windows, whether volatility is realized or implied, and any weighting constants. Always inspect the provider’s specification.
– No predictive guarantee: a high BMI reflects past or current pain, not assured future returns. It is a descriptive metric that may not reliably predict turning points.
– Data quality: noisy price data, differing exchange consolidation, and outliers can distort computed volatilities.
– Annualization choice: crypto trades 24/7, so some analysts use sqrt(365); traditional finance often uses sqrt(252). This
This sentence continues: This choice changes the numeric scale of BMI and can materially affect cross-sample comparisons; always report which convention you used (sqrt(365) versus sqrt(252) or another rule) and be consistent across backtests.
Practical implementation — step‑by‑step
1. Choose your inputs and conventions (document them).
– Price series: spot BTC/USD from a specific exchange or consolidated index (state the source and UTC time aggregation).
– Return frequency: typically daily returns (r_t = P_t / P_{t-1} − 1).
– Volatility type: realized (historical) or implied (options). Realized uses past returns; implied uses option prices.
– Window lengths: e.g., 30‑day rolling for short term, 90 or 180 days for medium/long term.
– Annualization: pick sqrt(365) for crypto or sqrt(252) for a finance convention and stick with it.
– Combination rule: decide how to combine vol and pain (sum, weighted sum, product, etc.). No universal standard; choose and justify.
2. Compute rolling realized volatility (if using realized vol).
– For daily returns over an N‑day window:
– sigma_real = stdev(r_{t−N+1} … r_t)
– annualized_sigma = sigma_real * sqrt(annual_factor)
– annual_factor = 365 for 24/7 crypto, or 252 if you prefer trading days.
– Note: use sample standard deviation (N−1) or population (N) consistently and document.
3. Compute a pain metric (examples).
– Trailing cumulative return over the window: R = (P_t / P_{t−N}) − 1.
– Or annualize the window return with linear scaling: R_ann ≈ R * (annual_factor / N) (approximate).
– Or use maximum drawdown within the rolling window for a more severe “pain” measure.
4. Combine into a BMI variant (explicit formula—choose one).
– Example (simple, transparent): BMI = annualized_sigma + max(0, −R_ann)
– This adds annualized volatility (percent) to annualized negative returns (percent), treating positive returns as zero “pain.”
– Alternative (weighted sum): BMI = w_vol * annualized_sigma + w_loss * max(0, −R_ann)
– Alternative (multiplicative): BMI = annualized_sigma * (1 + k * max(0, −
R_ann)).
Explanation and units
– Make units consistent. Compute annualized_sigma and R_ann as decimals (e.g., 0.50 = 50%). If you prefer percent points, convert all pieces to percent before combining.
– Interpretation: higher BMI → more “pain” (more volatility and/or larger recent losses). Different formula families emphasize volatility, realized losses, or both.
Worked numeric example (step‑by‑step)
Assumptions:
– Window N = 90 days.
– Annualization factor = 365 (Bitcoin trades every day).
– Daily arithmetic return series has sample standard deviation σ_daily = 0.04 (4%).
– Price change over the 90‑day window: R = P_t / P_{t−90} − 1 = −0.20 (−20%).
Compute components:
1. annualized_sigma = σ_daily * sqrt(365) = 0.04 * 19.105 = 0.7642 (76.42%).
2. R_ann ≈ R * (365 / 90) = −0.20 * 4.0556 = −0.8111 (−81.11%).
3. Negative annualized loss = max(0, −R_ann) = 0.8111.
Now three BMI variants:
A. Additive (treat positive returns as zero pain)
BMI_add = annualized_sigma + max(0, −R_ann)
= 0.7642 + 0.8111 = 1.5753 (convert to percent: 157.53%)
B. Weighted additive (choose weights w_vol = 0.6, w_loss = 0.4)
BMI_w = 0.6*annualized_sigma + 0.4*max(0, −R_ann)
= 0.6*0.7642 + 0.4*0.8111 = 0.7829 (78.29%)
C. Multiplicative (k = 1)
BMI_mul = annualized_sigma * (1 + k * max(0, −R_ann))
= 0.7642 * (1 + 0.8111) = 1.3843 (138.43%)
Notes on these numbers:
– Which scale you present (decimal vs percent) matters for interpretability. Many practitioners prefer percent points (multiply decimals by 100).
– The multiplicative form amplifies volatility in proportion to how large losses were; additive form treats effects linearly.
Implementation checklist
1. Data:
– Source reliable BTC price series (CoinGecko, CoinMarketCap, exchange aggregated, Yahoo Finance).
– Use consistent timestamping (UTC) and handle missing days (
) — choose one policy and apply it consistently. Options:
– Resample to UTC midnight and forward-fill small gaps (1–2 days); this preserves continuity for 24/7 markets.
– Drop missing timestamps but compute returns only where consecutive days exist; then annualize using the effective count of observations.
– If long gaps exist (exchanges offline, data holes), exclude that period or source an alternative feed.
2. Compute returns and moments
– Decide return type: log returns r_t = ln(P_t / P_{t−1}) are preferred for volatility aggregation; simple returns R_t = (P_t/P_{t−1}) − 1 are acceptable but behave differently when compounding is relevant.
– Daily mean: mu_daily = mean(r_t) or mean(R_t) depending on choice.
– Daily volatility: sigma_daily = stdev(r_t) (population or sample? use sample stdev (N−1) for small samples).
– Annualize:
– Volatility: sigma_ann = sigma_daily * sqrt(365) (crypto trades daily). If you use trading-day conventions, use sqrt(252).
– Returns (geometric): R_ann = exp(mu_daily * 365) − 1. (This converts average daily log return into an annual compounded return.)
– Loss term: L = max(0, −R_ann). Use this nonnegative scalar in whatever BMI variant you pick.
3. Choose BMI formulation and scaling
– Use one consistent variant across time to compare readings: e.g., equal-weight additive, weighted additive, or multiplicative. Match units (percent vs decimals) and label accordingly.
– If you present percent points, multiply decimals by 100 and state that explicitly.
– Consider a cap for extreme inputs to avoid runaway values (for example cap L at 2.0 or sigma_ann at 3.0) — document any caps.
4. Windowing, smoothing, and update frequency
– Window length: common choices are 30, 90, or 365 days. Short windows react quickly but are noisy; long windows are smoother but lag.
– Rolling vs expanding: a rolling window keeps a fixed lookback; expanding uses all history up to now. Rolling windows are generally preferred for a “current misery” metric.
– Smoothing: exponential moving average (EMA) on sigma_daily and mu_daily can stabilize readings. Choose an alpha consistent with your effective memory (e.g., alpha = 2/(N+1)).
– Update cadence: compute BMI daily (end-of-day) for consistency with daily data.
5. Visualization and communication
– Plot BMI alongside its components (annualized volatility and negative annualized return) on the same axis or stacked with consistent units.
– Annotate large market events for context (sharp spikes, halving dates, exchange issues).
– Provide a short legend describing formula, window, and data source on every chart.
6. Sensitivity and robustness checks
– Test multiple return definitions (log vs simple), several window lengths, and different annualization conventions (365 vs 252). Report how sensitive your BMI is to these choices.
– Run a bootstrap or rolling-sample analysis to estimate typical sampling variability (confidence intervals) for the BMI reading.
– Check for structural breaks: if BTC market microstructure changed (e.g., derivatives, ETFs), earlier vol/return behavior may not be comparable.
7. Backtesting and interpretation (educational)
– Use historical BMI to compute summary statistics: mean, median, quantiles, and frequency of extreme readings.
– If you investigate predictive relationships (e.g., whether high BMI precedes outperformance or underperformance), treat this as a research exercise: use out-of-sample testing, avoid look-ahead bias, and report p-values and effect sizes — do not assume causation.
– Consider plotting subsequent-period returns or drawdowns binned by prior BMI decile; this illustrates correlations but is not an investment rule.
8. Implementation pseudocode (high level)
1) Load daily BTC price series, timestamped UTC.
2) Impute or drop missing days per your policy.
3) Compute daily log returns r_t.
4) For the chosen rolling window N:
– sigma_daily = rolling_std(r_t, N)
– mu_daily = rolling_mean(r_t, N)
– sigma_ann = sigma_daily * sqrt(365)
– R_ann = exp(mu_daily * 365) − 1
– loss = max(0, −R_ann)
– BMI = combine(sigma_ann, loss) using your formula
5) Store and plot BMI with component series.
Worked numeric mini-example (rounded)
– Suppose over a 90-day window
— Suppose over a 90-day window you observe the following (rounded for clarity):
Inputs (observed from the 90-day sample)
– sigma_daily (sample standard deviation of daily log returns) = 0.030 (3.0% per day)
– mu_daily (sample mean of daily log returns) = −0.0005 (−0.05% per day)
Step-by-step calculations
1) Annualize volatility
– sigma_ann = sigma_daily * sqrt(365)
– sigma_ann = 0.030 * sqrt(365) ≈ 0.030 * 19.105 ≈ 0.573
– As a percentage: sigma_ann ≈ 57.3%
2) Annualize the mean log-return (to get compound annual growth)
– mu_ann = mu_daily * 365 = −0.0005 * 365 = −0.1825
3) Convert mu_ann to an annualized simple return (R_ann)
– R_ann = exp(mu_ann) − 1 = exp(−0.1825) − 1 ≈ 0.8332 − 1 = −0.1668
– As a percentage: R_ann ≈ −16.7%
4) Loss component (only counts negative annualized returns)
– loss = max(0, −R_ann) = max(0, 0.1668) = 0.1668
– As a percentage: loss ≈ 16.7%
5) Bitcoin Misery Index (BMI)
– Using the simple additive formula: BMI = sigma_ann + loss
– BMI = 0.573 + 0.1668 = 0.7398
– As a percentage: BMI ≈ 73.98% → round to ≈ 74.0%
Interpretation (for this example)
– The sample suggests high realized risk (≈57% annualized volatility) plus a negative annualized return (≈−16.7%). The BMI of ≈74% aggregates these into a single “misery” number: higher = worse from a combined risk-and-loss viewpoint.
– This is not a forecast; it is a descriptive summary of the chosen past window and calculation choices.
Quick sensitivity notes
– Window length (N): shorter N → noisier sigma_daily and mu_daily; longer N → smoother but slower to react.
– Return type: using log returns as above makes annualization additive for mu; using simple returns changes formulas.
– Annualization factor: using sqrt(252) vs sqrt(365) matters for sigma_ann; choose consistent trading-calendar assumptions and state them.
Practical checklist before using BMI in analysis or a backtest
– Decide and document: return type (log vs simple), trading days convention (252 vs 365), and rolling-window length.
– Handle missing days: drop vs impute; test impact on results.
– Unit tests: verify that a zero-volatility flat series gives BMI = 0 when returns are non-negative.
– Edge cases: check behavior when returns are extremely negative (loss cap?) or when sigma_daily = 0.
– Backtest hygiene: avoid look-ahead bias, use out-of-sample periods, and respect transaction costs and slippage if turning decisions into trades.
– Statistical checks: compute correlations between prior BMI deciles and subsequent returns/drawdowns; report p-values and effect sizes; visualize distributions.
Suggested diagnostic plots
– Time series: