Crypto-specific strategies: momentum, funding arbitrage and basis.

A market that never closes and moves as a block

The crypto market has three features that change how a strategy is designed (lessons 1.2.6 and 1.3.5). It never closes: the daily “session” is a convention (here, the close at 00:00 UTC) and weekends count. It has derivatives of its own: the perpetual, a future with no expiry anchored to spot with periodic funding payments, and dated futures whose price departs from spot by the basis. And almost all coins rise and fall at the same time.

flowchart TB
  C[Crypto market] --> F[Common factor]
  C --> M[Cross-sect. momentum]
  C --> D[Derivatives]
  D --> P[Perpetual: funding]
  D --> T[Future: basis]
  P --> K[Cash-and-carry]
  T --> K

import numpy as np
import pandas as pd

cd = pd.read_parquet("cripto2_diario.parquet")
P = cd.pivot(index="fecha", columns="moneda", values="cierre")
r = P.pct_change(fill_method=None)
resumen = pd.DataFrame({
    "desde": P.apply(lambda s: s.first_valid_index()).dt.date,
    "hasta": P.apply(lambda s: s.last_valid_index()).dt.date,
    "vol_anual_pct": 100 * np.log1p(r).std() * np.sqrt(365),
})
print(resumen.round(0).sort_values("vol_anual_pct").to_string())
print("worst day of KJT and KLU:",
      (100 * r[["KJT", "KLU"]].min()).round(1).to_dict())

             desde       hasta  vol_anual_pct
moneda
KAL     2018-01-01  2025-12-31           67.0
KMV     2019-07-01  2025-12-31           73.0
KCX     2018-01-01  2025-12-31           74.0
KGN     2018-01-01  2025-12-31           75.0
KES     2018-01-01  2025-12-31           78.0
KFL     2018-01-01  2025-12-31           81.0
KNX     2020-09-15  2025-12-31           85.0
KRM     2018-01-01  2025-12-31           98.0
KHX     2018-01-01  2025-12-31           99.0
KBR     2018-01-01  2025-12-31          103.0
KIO     2018-01-01  2025-12-31          103.0
KLU     2018-01-01  2022-11-30          108.0
KDL     2018-01-01  2025-12-31          109.0
KOB     2021-03-01  2025-12-31          111.0
KJT     2018-01-01  2022-05-27          138.0
worst day of KJT and KLU: {'KJT': -61.7, 'KLU': -45.5}

Twelve coins trade for all eight years; KMV, KNX and KOB start later, and KJT and KLU stop trading in 2022 after losing 99% and 90% in a few days (their worst day, −61.7% and −45.5%). Annual volatility ranges from 67% for KAL, the largest and most liquid, to 138% for KJT: between three and seven times that of a stock index. A universe built today from the coins that exist would have neither KJT nor KLU.

The common factor

The crypto market can be summarized in an equal-weighted index of the coins trading each day. A regression of each coin on that index gives its beta and its $R^2$, the fraction of its daily variance explained by the common factor:

mercado = r.mean(axis=1)
filas = {}
for mon in r:
    d = pd.concat([r[mon], mercado], axis=1).dropna()
    b, a = np.polyfit(d.iloc[:, 1], d.iloc[:, 0], 1)
    filas[mon] = {"beta": b, "r2": d.corr().iloc[0, 1] ** 2}
fc = pd.DataFrame(filas).T.sort_values("r2")

corr = r.corr().to_numpy()
media_corr = corr[np.triu_indices_from(corr, 1)].mean()
completas = r.dropna()
valores = np.linalg.eigvalsh(np.corrcoef(completas.T.to_numpy()))
print(f"mean correlation: {media_corr:.2f}; first component: "
      f"{valores[-1] / valores.sum():.0%} of the variance")
fc.round(2).T

mean correlation: 0.58; first component: 70% of the variance
       KJT   KBR   KHX   KOB   KIO   KMV   KES   KFL   KRM   KCX   KLU   KDL   KGN   KNX   KAL
beta  0.91  0.94  0.91  1.31  1.08  0.86  0.85  0.91  1.12  0.86  1.06  1.37  0.94  1.15  0.89
r2    0.40  0.40  0.42  0.54  0.54  0.59  0.59  0.62  0.63  0.67  0.71  0.75  0.75  0.77  0.84

The average pairwise correlation is 0.58 and the first principal component explains 70% of the variance of the fifteen. The $R^2$ against the index ranges from 0.40 for KJT and KBR to 0.84 for KAL: for the large coin, more than four fifths of what happens each day is the whole market; for those with more idiosyncratic volatility, less than half. Betas are between 0.85 and 1.37. For a strategy, the consequence is direct: a long-only portfolio is, above all, a bet on the common factor, and the cross-sectional signal can only be isolated with a long-short portfolio.

Weekly cross-sectional momentum

The planted hypothesis is a short-term momentum: the coins that have risen most over the last four weeks keep doing slightly better than the rest the following week. Liu, Tsyvinski and Wu (2022) find in real data that a three-factor model (market, size and momentum) explains the cross-section of crypto returns; Liu and Tsyvinski (2021) document a strong time series momentum.

The signal's timing matters. The signal is computed with Sunday's close (28-day return up to that close) and the position is taken at Monday's close, the next session:

flowchart LR
  D[Sunday close] --> S[28-day signal]
  S --> L[Monday order]
  L --> H[7 days]

from scipy import stats

lunes = [i for i, f in enumerate(P.index)
         if f.dayofweek == 0 and 29 <= i < len(P) - 8]
logp = np.log(P)
ic = {"at Sunday close": [], "at Monday close": []}
for i in lunes:
    senal = logp.iloc[i - 1] - logp.iloc[i - 29]
    for clave, ent in [("at Sunday close", i - 1),
                       ("at Monday close", i)]:
        fut = logp.iloc[ent + 7] - logp.iloc[ent]
        ok = senal.notna() & fut.notna()
        ic[clave].append(stats.spearmanr(senal[ok], fut[ok])[0])
ic = pd.DataFrame(ic)
pd.DataFrame({"ic_medio": ic.mean(),
              "t": ic.mean() / ic.std() * np.sqrt(len(ic)),
              "semanas_pos_pct": 100 * (ic > 0).mean()}).round(3)

                 ic_medio      t  semanas_pos_pct
at Sunday close     0.068  4.820           59.223
at Monday close     0.069  4.842           60.922

The weekly rank IC is +0.068 executing at Sunday's close (the signal's) and +0.069 at Monday's, with a t of 4.8 in both cases: the planted signal (expected IC ≈ +0.05) is clearly recovered, and it is positive in 59% to 61% of weeks. That next-day execution does not make it worse is a property of this particular signal: the planted return is spread evenly across the whole week, and the Monday-to-Monday window also captures almost all of it. A signal whose effect were concentrated in the first few hours would not survive it.

The portfolio buys the five coins with the highest signal and sells the five with the lowest, equally weighted, enters at Monday's close and is rebuilt every week. The cost is an assumption of the lesson: 10 basis points per side on traded notional, between commission and slippage, in the order of what a retail client taking liquidity on a large exchange with liquid coins pays. Short positions are assumed to be in perpetuals, so they would also pay or receive funding, which is ignored here.

COSTE = 10 / 1e4
pesos = pd.DataFrame(0.0, index=P.index, columns=P.columns)
for i in lunes:
    senal = (logp.iloc[i - 1] - logp.iloc[i - 29]).dropna()
    orden = senal.rank()
    w = (pd.Series(0.0, index=P.columns)
         .add((orden > len(orden) - 5) / 5, fill_value=0)
         .sub((orden <= 5) / 5, fill_value=0))
    pesos.iloc[i + 1:i + 8] = w.to_numpy()      # from Tuesday
pesos = pesos.where(P.notna(), 0.0)

giro = pesos.diff().abs().sum(axis=1)
ls = (pesos * r.fillna(0)).sum(axis=1) - COSTE * giro
ls = ls[ls.index >= P.index[lunes[0] + 1]]
ew = mercado.reindex(ls.index)

def metricas(x):
    eq = (1 + x).cumprod()
    return pd.Series({
        "rent_anual_pct": 100 * x.mean() * 365,
        "vol_anual_pct": 100 * x.std() * np.sqrt(365),
        "sharpe": x.mean() / x.std() * np.sqrt(365),
        "max_dd_pct": 100 * (eq / eq.cummax() - 1).min(),
        "beta": np.polyfit(ew, x, 1)[0]})

coste_anual = 100 * COSTE * giro[ls.index].mean() * 365
print(f"annual commission cost: {coste_anual:.2f} %")
pd.DataFrame({"long_short_neto": metricas(ls),
              "indice_equiponderado": metricas(ew)}).round(2)

annual commission cost: 5.80 %
                long_short_neto  indice_equiponderado
rent_anual_pct            46.89                 30.46
vol_anual_pct             39.94                 70.79
sharpe                     1.17                  0.43
max_dd_pct               -41.45                -84.27
beta                      -0.05                  1.00

Net of commissions, the portfolio returns 46.9% a year with a volatility of 39.9%: a Sharpe of 1.17 and a beta of −0.05 against the index, which is almost zero. Commissions, with high weekly turnover, take 5.8 points a year. The equal-weighted index returns 30.5% with a Sharpe of 0.43 and a maximum drawdown of 84%: that is the common factor's premium in this data, not the signal's.

The Sharpe of 1.17 is not a promise for real crypto: it is what a planted signal with an IC of 0.07 gives across fifteen synthetic coins. What is useful is to compare it with what was planted. With an average difference of 1.96 between the normal rank scores of the two legs, the generator promises about 70 points a year before costs, and the portfolio earns about 53 (46.9 plus 5.8 of commissions). With a volatility of 40% over eight years, the standard error of the mean annual return is about 14 points: the difference is little more than one standard error, consistent with chance. And the maximum drawdown of 41%, in a market-neutral portfolio, calls for a look at the worst days:

peores = ls.nsmallest(3)
print((100 * peores).round(1).to_string())
d = peores.index[0]
aporte = (pesos.loc[d] * r.fillna(0).loc[d]).nsmallest(2)
print("worst day, by coin (% of capital):",
      (100 * aporte).round(1).to_dict())
m = aporte.index[0]
print(f"{m} that day: {100 * r.loc[d, m]:+.0f} %, "
      f"weight {pesos.loc[d, m]:.2f}")
eq = (1 + ls).cumprod()
dd = eq / eq.cummax() - 1
print("maximum drawdown:", eq[:dd.idxmin()].idxmax().date(), "to",
      dd.idxmin().date(), f"({100 * dd.min():.1f} %)")

fecha
2022-05-03   -16.5
2023-08-22   -10.6
2022-06-21   -10.0
worst day, by coin (% of capital): {'KOB': -15.6, 'KES': -1.2}
KOB that day: +78 %, weight -0.20
maximum drawdown: 2021-03-08 to 2022-06-26 (-41.4 %)

The worst day is May 3, 2022: KOB, which the portfolio held short with 20% of capital, rises 78% and the short leg loses 15.6% of capital in one session; the rest of the positions barely count. The next two worst days, of −10.6% and −10.0%, are of the same magnitude, and the maximum drawdown, 41.4%, runs from March 8, 2021 to June 26, 2022. The generator clips idiosyncratic innovations at ±5 standard deviations, and even so, with the volatility of these coins, there are days of up to +84% according to the data README. It is the risk specific to the short leg: a short position in a small, volatile asset can lose in one day what the signal earns in months. Per-coin weight limits, stops on the short leg and inverse-volatility weighting are the usual answer, and all of them change the result.

The figure shows the portfolio's equity curve against the index's, on a log scale. The portfolio rises strongly in 2018 and 2019, flattens in 2020 and 2021, falls in the first half of 2022 and rises again from 2023. The index, in contrast, loses more than half in 2018 and in 2022 and gains in the bullish stretches: two almost independent curves, as the beta says.

Perpetual funding

A perpetual does not expire. To keep its price from drifting away from spot, every 8 hours longs pay shorts a percentage of notional, the funding, if the perpetual trades above the index, and the other way round if it trades below (lesson 1.2.6). When the market is euphoric and many want to be long with leverage, funding rises. cripto2_funding.parquet has the rate for each period of KAL, KBR and KCX:

fu = pd.read_parquet("cripto2_funding.parquet")
fu["fecha"] = fu["ts"].dt.tz_convert(None).dt.normalize()
diario = fu.groupby(["fecha", "moneda"])["tasa_funding"].sum().unstack()
anual = 100 * diario * 365              # % per year, not compounded

tabla = pd.DataFrame({
    "media_pct": anual.mean(),
    "dias_negativo_pct": 100 * (diario < 0).mean(),
    "peor_30d_pct": anual.rolling(30).mean().min(),
    "mejor_30d_pct": anual.rolling(30).mean().max(),
})
r7 = np.log(P[anual.columns]).diff(7)
tabla["corr_con_r7"] = [anual[c].corr(r7[c]) for c in anual]
print(tabla.round(2))
print((anual.groupby(anual.index.year).mean()).round(1).T)

        media_pct  dias_negativo_pct  peor_30d_pct  mejor_30d_pct  corr_con_r7
moneda
KAL         11.52              35.63        -36.30          52.86         0.84
KBR         11.40              42.09        -36.93          63.70         0.89
KCX         14.04              34.60        -37.91          63.72         0.89
fecha   2018  2019  2020  2021  2022  2023  2024  2025
moneda
KAL      1.4  15.2  13.1  11.3   2.6  20.3  18.9   9.4
KBR      5.9   9.0   9.4  14.3  14.5  12.3   9.8  16.0
KCX      5.8  20.8  15.5  13.8   9.0  16.4  12.4  18.7

Average funding, annualized without compounding, is 11.4% to 14.0%: on average, longs pay shorts somewhat more than the 0.01% base rate per period. But it is not stable. Between 35% and 42% of days it is negative; the worst 30-day average is −36% to −38% annualized, and the best, +53% to +64%. The correlation with the previous 7 days' return is 0.84 to 0.89: it rises after rallies, which is the planted formula ($0.0001 + 0.0006 \tanh(r_7 / 0.15)$ per period, plus noise). By year, KAL's ranges from 1.4% in 2018 to 20.3% in 2023.

Cash-and-carry: collecting funding without betting on price

The cash-and-carry buys the coin on spot and sells the same amount in the perpetual. The two legs offset each other: if the price rises, spot gains and the perpetual loses by the same amount. What is left is the funding the short leg collects when it is positive (and pays when it is negative), minus the costs of entering and exiting both legs.

flowchart TB
  K[Capital] --> S[Long spot]
  K --> M[Short margin]
  M --> P[Short perpetual]
  P --> F[Collects funding]
  S --> N[Price neutral]
  P --> N

f_kal = diario["KAL"]
cobrado = f_kal.groupby(f_kal.index.year).sum() * 100
print("funding collected by the short, % of notional per year:")
print(cobrado.round(1).to_string())

# capital: 1 in spot + margin of the short perpetual
# (1/3 of notional: 3x leverage)
rend_capital = f_kal.sum() / (1 + 1 / 3) / (len(f_kal) / 365)
print(f"annual return on capital: {100 * rend_capital:.1f} %"
      " (before costs)")
subida_30d = (P["KAL"] / P["KAL"].rolling(30).min() - 1).max()
print(f"largest 30-day rise in KAL: {100 * subida_30d:.0f} %")

funding collected by the short, % of notional per year:
fecha
2018     1.4
2019    15.2
2020    13.1
2021    11.3
2022     2.6
2023    20.3
2024    18.9
2025     9.4
annual return on capital: 8.6 % (before costs)
largest 30-day rise in KAL: 116 %

The short in the KAL perpetual collects between 1.4% (2018) and 20.3% (2023) of notional each year, and no year is negative. But the capital required is not just the spot notional: the short perpetual needs margin. With 3x leverage on that leg (a third of notional), the return on capital drops to 8.6% a year, before costs. And the last line is the risk: KAL rose as much as 116% in 30 days. With margin of a third of notional, a rise like that eats it several times over; it has to be topped up in time, by selling part of the spot or with cash, or the exchange liquidates the short leg.

The basis of dated futures

The quarterly future expires on a fixed date and converges to spot at expiry. Its annualized difference from spot is the basis: $b = \ln(F/S) \times 365 / \text{days}$. Buying spot and selling the future locks in that basis at entry: at expiry both legs are worth the same and the result is the initial difference, whatever happens to the price. In the data, the basis is formed from funding: about 0.8 times the annualized 7-day funding plus 2%.

ba = pd.read_parquet("cripto2_basis.parquet")
f7 = (diario.rolling(7).mean() * 365).stack().rename("funding_7d")
f7.index.names = ["fecha", "moneda"]
ba = ba.join(f7, on=["fecha", "moneda"]).dropna()

for mon, g in ba.groupby("moneda"):
    b, a = np.polyfit(g["funding_7d"], g["basis_anual"], 1)
    print(f"{mon}: basis = {a:.3f} + {b:.2f} × funding_7d,"
          f" corr {g['basis_anual'].corr(g['funding_7d']):.2f}")

# one cash-and-carry per contract: opened 80-90 days before expiry
entrada = ba[ba["dias_al_vencimiento"].between(80, 90)]
entrada = entrada.groupby(["moneda", "contrato"]).first()
fijado = 100 * entrada["basis_anual"]
print("\nbasis locked in at entry (% per year) by coin:")
print(fijado.groupby("moneda").describe()[
    ["count", "mean", "min", "max"]].round(1))

KAL: basis = 0.025 + 0.80 × funding_7d, corr 0.97
KBR: basis = 0.012 + 0.80 × funding_7d, corr 0.99
KCX: basis = 0.012 + 0.81 × funding_7d, corr 0.97

basis locked in at entry (% per year) by coin:
        count  mean   min   max
moneda
KAL      33.0  17.1 -25.9  58.6
KBR      33.0   8.1 -49.0  74.2
KCX      33.0  15.2 -30.0  84.4

The regression recovers the planted relationship: a slope of 0.80 to 0.81 on 7-day funding and a correlation of 0.97 to 0.99. The intercept comes out between 1.2% and 2.5% (truth 2%). For the cash-and-carry with a future, what matters is the second table: opening 80-90 days before expiry, the average locked-in basis is 17.1% a year in KAL, 8.1% in KBR and 15.2% in KCX. The minimum is negative in all three: in panic quarters the future trades below spot, and the cash-and-carry would have to be done in reverse (sell spot and buy the future), which requires borrowing the coin. Unlike the perpetual, the basis is locked in at entry: if spot and the future are bought and sold at those prices, the result at expiry is that basis, whatever funding does in the meantime.

Costs, commissions and what the backtest does not see

• Commissions per side. Exchange fees depend on the client's volume and on whether it provides or takes liquidity; a weekly long-short pays four sides per coin and week. The lesson's 10 basis point assumption takes several points a year.
• Weekends and hours. There is no official close: the “closing” price is whatever the vendor chooses (lesson 1.3.5). Two sources with different cutoff times give different signals.
• Collapses. KJT and KLU lose 99% and 90% in a few days and stop trading. A universe built from today's coins erases them (lesson 1.4.4); a point-in-time one includes them, and a portfolio long in them pays for it.
• Liquidations and leverage. Crypto derivatives trade with margin and automatic liquidation; a wick lasting a few minutes can close a position that in daily data was never in danger.
• Counterparty. The money is on the exchange. Diversifying across exchanges and periodically withdrawing profits are part of risk management, not an operational detail.

Exercises

1. Repeat the weekly rank IC (execution at Monday's close) with signal windows of 7, 14, 28 and 56 days. Which gives the best IC? What does the result tell you about searching for the optimal window?

def ic_ventana(n):
    v = []
    for i in lunes:
        if i - 1 - n < 0:
            continue
        senal = logp.iloc[i - 1] - logp.iloc[i - 1 - n]
        fut = logp.iloc[i + 7] - logp.iloc[i]
        ok = senal.notna() & fut.notna()
        v.append(stats.spearmanr(senal[ok], fut[ok])[0])
    v = np.array(v)
    return v.mean(), v.mean() / v.std() * np.sqrt(len(v))

pd.DataFrame({n: ic_ventana(n) for n in [7, 14, 28, 56]},
             index=["ic_medio", "t"]).T.round(3)

    ic_medio      t
7      0.034  2.414
14     0.057  3.893
28     0.069  4.847
56     0.069  4.597

2. A conditional cash-and-carry only holds the position when the average funding of the last 7 days, annualized, exceeds 5%, and pays 10 basis points per side on each leg on entry and exit. Compute its annual result in KAL versus holding it all the time.

f7_kal = diario["KAL"].rolling(7).mean() * 365
dentro = (f7_kal > 0.05).shift(1, fill_value=False)   # decided yesterday
cambios = dentro.astype(int).diff().abs().fillna(0)
coste = cambios * 2 * 2 * 10 / 1e4      # 2 legs × entry or exit
anos = len(f_kal) / 365
siempre = f_kal.sum() / anos
condicional = ((f_kal * dentro).sum() - coste.sum()) / anos
print(f"always in:   {100 * siempre:.1f} % of notional per year")
print(f"conditional: {100 * condicional:.1f} % per year "
      f"({int(cambios.sum())} entries and exits, "
      f"{100 * dentro.mean():.0f} % of the time in)")

always in:   11.5 % of notional per year
conditional: 4.7 % per year (183 entries and exits, 60 % of the time in)

3. The backtest of the KAL cash-and-carry collects net funding every year. List what would have to happen on a real exchange for it to lose money, even if funding were the one in the data.