Edge hypotheses: risk, behavior, structure and information.

An edge is a hypothesis, not a pattern

An edge is a reason to expect a trading rule to make money in the future, after costs. The important word is “reason”. A pattern found in a backtest (“the stocks that rise on Tuesday keep rising on Wednesday”) is not an edge: it is an observation, and most observations in a backtest are chance (lessons 2.1.8 and 2.2.4). To turn it into a hypothesis, three questions have to be answered:

1. Who is on the other side? Every trade has a counterparty. If the system wins, someone wins less or loses.
2. Why do they accept it? Because they get something in return (they offload a risk), because they are wrong, because they are forced to or because they do not yet know what the system has already priced in.
3. Why has nobody corrected it? Arbitraging it carries a risk, a cost or a capital limit that others do not want to or cannot take on.

The answers to the second question give a taxonomy of four sources:

flowchart TB
  E[Edge] --> R[Risk]
  E --> C[Behavior]
  E --> S[Structure]
  E --> I[Information]
  R --> R1[Carry]
  C --> C1[Momentum]
  S --> S1[Rebalances]
  I --> I1[PEAD drift]

The classification is not rigid. Momentum has behavioral explanations (underreaction to news) and risk explanations (its crashes come when the market rebounds after a crisis, the worst moment); carry, besides a premium, can be read as a constraint on investors who cannot borrow in cheap currencies. What matters is that each explanation predicts something different and testable: where the edge should be larger, when it should fail and what the distribution of its returns should look like.

The data: planted effects and a placebo

The level 2 data are synthetic and carry effects with known parameters (/curso/datos/nivel-2/README.md). This lesson measures four of them and a control:

• Currency carry (fx2_carry.parquet): the high-rate currency gives back only 30% of the differential in the exchange rate, and in five stress windows the high-rate currencies drop sharply.
• Momentum (universo2_caracteristicas.parquet): a monthly premium of 0.0025 × z on the 12-1 return, with a negative coefficient in the three months after the 2009 and 2020 bottoms.
• Index rebalances (eventos2_rebalanceos.csv): additions rise 3% between the announcement and the effective date and deletions fall by the same amount; in the next 20 sessions half of it reverts.
• Post-earnings drift (eventos2_resultados.csv): for 60 sessions after the reaction, the price drifts in the direction of the surprise.
• Placebo: puntuacion_esg is an AR(1) independent of everything. Its true premium is zero.

All the edges are compared with the same yardstick: a series of monthly returns of a portfolio, and the same summary of its profile.

import numpy as np
import pandas as pd

def perfil(m):
    """m: monthly returns (as decimals)."""
    sr = m.mean() / m.std() * np.sqrt(12)
    return pd.Series({
        "meses": len(m),
        "rent_anual": 12 * m.mean(),
        "vol_anual": np.sqrt(12) * m.std(),
        "sharpe": sr,
        "t": sr * np.sqrt(len(m) / 12),
        "asimetria": m.skew(),
        "curtosis": m.kurt(),
        "peor_mes": m.min(),
        "meses_pos": (m > 0).mean(),
    })

def a_mensual(r_diaria):
    return (1 + r_diaria).resample("ME").prod() - 1

t is the annual Sharpe times the square root of the number of years: the t statistic of the mean return. curtosis is the excess kurtosis (0 for a normal) and asimetria, the skewness of the months.

Risk: currency carry

The carry hypothesis is a risk premium. Whoever lends in the high-rate currency and borrows in the low-rate one collects the rate differential. Uncovered interest parity would say that the high-rate currency depreciates exactly enough to cancel it; if it only does so in part, an average gain remains. The explanation for why it lasts is that carry loses a lot exactly when everything loses: in crises, investors unwind their positions at the same time and the high-rate currencies collapse11Note 1 · Brunnermeier, Nagel and Pedersen (2008), in the resources, document the negative skewness of carry and attribute it to positions being unwound at once when risk appetite and funding liquidity fall..

The portfolio is the classic one: at each month end, long the three currencies with the highest rates and short the three with the lowest, equally weighted. The daily return of being long a currency is the change in the exchange rate plus the accrued rate differential (README, section d).

fx = pd.read_parquet("fx2_carry.parquet")
spot = fx.pivot(index="fecha", columns="divisa", values="spot")
tipo = fx.pivot(index="fecha", columns="divisa", values="tipo_divisa")
usd = fx.groupby("fecha")["tipo_usd"].first()
dif = tipo.sub(usd, axis=0)                     # % per year

dias = spot.index.to_series().diff().dt.days
interes = dif.shift(1).div(100).mul(dias / 360, axis=0)
r_fx = spot.pct_change() + interes              # long the currency

# last session of each month: long 3 high-rate, short 3 low-rate
fin_mes = dif.groupby(dif.index.to_period("M")).tail(1)
rango = fin_mes.rank(axis=1)
peso = (rango >= 8).astype(float) / 3 - (rango <= 3) / 3
peso_d = peso.reindex(r_fx.index).ffill().shift(1)
carry_d = (peso_d * r_fx).sum(axis=1).loc["2005-02":]
carry = a_mensual(carry_d)
perfil(carry).round(3)

meses         251.000
rent_anual      0.044
vol_anual       0.108
sharpe          0.409
t               1.871
asimetria      -1.384
curtosis        7.389
peor_mes       -0.199
meses_pos       0.574
dtype: float64

The final shift(1) makes the weight decided with the rates at one month's close apply from the next session: without it, the portfolio would collect on the decision day a return it could not yet have.

Where are the losses? The README sets five crash windows. The sum of what the portfolio gains or loses inside each window, and outside all of them:

tramos = [("2008-09-15", "2008-12-31"), ("2010-05-03", "2010-05-31"),
          ("2011-08-01", "2011-09-30"), ("2015-08-17", "2015-09-30"),
          ("2020-02-24", "2020-03-31")]
en_crash = pd.Series(False, index=carry_d.index)
for ini, fin in tramos:
    en_crash.loc[ini:fin] = True
    print(ini[:7], f"{carry_d.loc[ini:fin].sum():+.1%}")
print(f"crash sessions: {en_crash.mean():.1%} of the total")
print(f"sum in crashes: {carry_d[en_crash].sum():+.1%}")
print(f"sum outside: {carry_d[~en_crash].sum():+.1%}")
carry.nsmallest(4).round(3)

2008-09 -23.4%
2010-05 -7.5%
2011-08 -7.8%
2015-08 -4.3%
2020-02 -15.5%
crash sessions: 3.7% of the total
sum in crashes: -58.6%
sum outside: +147.2%
fecha
2009-02-28   -0.199
2020-03-31   -0.124
2008-11-30   -0.083
2022-08-31   -0.077
dtype: float64

The portfolio earns 4.4% a year with 10.8% volatility: a Sharpe of 0.41 and a t of 1.87, at the edge of what tells a premium apart from chance with 21 years. The monthly skewness is −1.38 and the excess kurtosis, 7.4. The sessions in the five crash windows are 3.7% of the total, and in them the portfolio adds up −58.6 points; outside, +147.2. The 2008 window alone takes 23.4 points and the 2020 one, 15.5.

The profile is that of someone who sells insurance: it earns little and almost always, and gives back a large part at once. The figure shows it on the portfolio's curve. Two details of the result matter for a real system:

• The worst month, February 2009 (−19.9%), does not fall inside a planted window but after one: the GARCH volatility of the high-rate currencies stays elevated after the crash. Crises do not end when the fall ends; tail risk lasts longer.
• A five-year backtest with no crisis inside would show an almost straight curve and a high Sharpe. The premium is only seen in full with samples that include the episodes it pays for.

Behavior (or risk): momentum

Cross-sectional momentum buys the stocks that rose the most over the last twelve months, excluding the last one, and sells those that fell the most22Note 2 · Jegadeesh and Titman (1993), in the resources, is the classic study of winner and loser portfolios.. Behavioral explanations say that investors underreact to news and that the price takes months to reflect it; risk explanations, that the long-short portfolio loses a lot in the rebounds after a crisis, when the losers (heavily beaten, with high beta) rise more than anyone33Note 3 · Daniel and Moskowitz (2016), in the resources, study momentum crashes in the rebounds after declines..

In the level's data the signal follows the first hypothesis and the crash, the second. With the point-in-time monthly returns of universo2_caracteristicas.parquet:

car = pd.read_parquet("universo2_caracteristicas.parquet")
rm = car.pivot(index="fecha", columns="ticker",
               values="rentabilidad_mes")
hay = car.pivot(index="fecha", columns="ticker",
                values="cierre").notna()    # False: delisted
# return from m-11 to m-1 (skips the last month), only
# where there is a price at the close of m
mom = np.expm1(np.log1p(rm.where(hay)).shift(1)
               .rolling(11).sum()).where(hay)
sig_mes = rm.shift(-1)          # m+1, with the delisting included

ic = mom.rank(axis=1).corrwith(sig_mes.rank(axis=1), axis=1).dropna()
print(f"mean rank IC {ic.mean():.3f}, "
      f"t = {ic.mean() / ic.std() * np.sqrt(len(ic)):.2f}")

def quintiles(senal, fut, excluir=()):
    s, f = senal.drop(columns=list(excluir)), fut.drop(
        columns=list(excluir))
    rk = s.rank(axis=1, pct=True)
    m = (f[rk > 0.8].mean(axis=1) - f[rk <= 0.2].mean(axis=1))
    m = m.dropna()
    # label: end of the month of the return (the next one)
    sig = m.index.to_period("M") + 1
    m.index = sig.to_timestamp() + pd.offsets.MonthEnd(0)
    return m

momentum = quintiles(mom, sig_mes)
perfil(momentum).round(3)

mean rank IC 0.028, t = 2.96
meses         227.000
rent_anual      0.101
vol_anual       0.120
sharpe          0.838
t               3.644
asimetria       0.136
curtosis        0.082
peor_mes       -0.090
meses_pos       0.573
dtype: float64

The signal and the future return do not use the same rows. For each stock that stops trading, the file has a last row without a price in the month it leaves, with a rentabilidad_mes that chains the month-to-date return with the delisting return (−100% in the four bankruptcies). That row cannot be used to form a signal (on that day the stock can no longer be bought), which is why mom is computed only where there is a price (hay). But it has to go into sig_mes: it is what whoever held the stock in the portfolio the previous month gains or loses. A backtest that discarded it would see no bankruptcy at all.

The rank IC (the Spearman correlation between one month's signal and the next month's return, lesson 2.3.5) measures the relationship without being dragged by a few extreme values. The quintile portfolio is dragged: the README warns that four guided bankruptcies fall for months to almost zero and always end up in the loser quintile, on the short side, where the −100% of the delisting is a 100% gain. Their contribution is measured by removing them:

quiebras = ["QXQT", "QXKV", "QXZT", "QXYX"]
sin_q = quintiles(mom, sig_mes, excluir=quiebras)
print(f"with bankruptcies {12 * momentum.mean():.1%} per year, "
      f"without them {12 * sin_q.mean():.1%}")
crash = ["2009-04", "2009-05", "2009-06",
         "2020-04", "2020-05", "2020-06"]
momentum.loc[momentum.index.strftime("%Y-%m").isin(crash)].round(3)

with bankruptcies 10.1% per year, without them 6.9%
fecha
2009-04-30   -0.060
2009-05-31   -0.031
2009-06-30   -0.058
2020-04-30   -0.044
2020-05-31   -0.006
2020-06-30   -0.059
dtype: float64

The mean IC is 0.028 with a t of 2.96, close to the expected +0.03. The quintile portfolio earns 10.1% a year with a Sharpe of 0.84; without the bankruptcies, 6.9%: almost a third of the return (3.2 points) comes from four companies, which in their last month with a price sit in the 1st percentile of the signal. With real data the same thing happens with a dangerous sign: without the delisting returns, or with a universe of survivors (lesson 1.4.4), the short side looks worse than it was, or disappears.

The six planted months are negative, but none is the worst in the series (−9.0%, in February 2021). June 2009 stays at −5.8% for a reason that has nothing to do with the signal: that month QXKV, one of the bankruptcies, stops trading, and the short side collects its −100% while the losers rebound. The crash in these data is moderate: the portfolio's monthly skewness is 0.14, almost zero, and the excess kurtosis, 0.08. In real momentum the crashes have been more violent than here³; a data model carries the tail risk its author decided to give it.

Structure: index rebalances

A structural edge does not depend on anyone being wrong: it depends on someone being forced to trade. A fund that replicates an index has to buy the stocks that enter and sell those that leave at the close of the effective date, and that date is announced days in advance. Whoever buys between the announcement and the effective date later sells to the indexer; if the demand for stocks is not perfectly elastic, the price rises with the forced buying and partly reverts afterwards44Note 4 · Shleifer (1986), in the resources, uses S&P 500 inclusions to test whether the demand for stocks slopes downward..

Each event is measured as the stock's return from the close of the announcement day to that of the effective date, signed (long additions, short deletions) and in excess of the universe mean. Deletions due to delisting have no planted effect and are removed:

aj = pd.read_parquet("universo2_ajustados.parquet")
px = aj.pivot(index="fecha", columns="ticker",
              values="cierre_ajustado")
r = px.pct_change(fill_method=None)
mercado = r.mean(axis=1)
sesiones = px.index

rb = pd.read_csv("eventos2_rebalanceos.csv",
                 parse_dates=["fecha_anuncio", "fecha_efectiva"])
rb = rb[rb["motivo"] != "deslistado"]

def evento(e):
    ia = sesiones.searchsorted(e.fecha_anuncio)
    ie = sesiones.searchsorted(e.fecha_efectiva)
    s = 1 if e.tipo == "alta" else -1
    p, m = px[e.ticker], (1 + mercado.iloc[ia + 1:ie + 1]).prod()
    return pd.Series({
        "sesiones": ie - ia,
        "exceso": s * (p.iloc[ie] / p.iloc[ia] - m),
        "despues_20": s * (p.iloc[ie + 20] / p.iloc[ie] - 1),
    })

ev = pd.concat([rb, rb.apply(evento, axis=1)], axis=1)
ev.groupby("tipo")[["sesiones", "exceso", "despues_20"]].agg(
    ["mean", "count"]).round(3)

     sesiones       exceso       despues_20
         mean count   mean count       mean count
tipo
alta    4.741    27  0.022    27     -0.012    27
baja    5.000    20  0.040    20     -0.003    20

Additions earn an average excess of 2.2% in 4.7 sessions and deletions, held short, 4.0% in 5, against the planted 3%: with 27 and 20 events and a standard deviation of 3-4 points per event, the error of each mean is 0.7 and 0.9 points. The reversal over the next 20 sessions comes out with the expected sign (−1.2% and −0.3% for the event's position), but with an error of 2 points: with so few events it cannot be told apart from zero. Measuring an event edge well requires many events.

To compare it with the others, the edge is turned into a daily portfolio: long the additions and short the deletions while each window lasts, and in cash the rest of the time.

pos = pd.DataFrame(0.0, index=sesiones, columns=px.columns)
for e in ev.itertuples():
    ia = sesiones.searchsorted(e.fecha_anuncio)
    ie = sesiones.searchsorted(e.fecha_efectiva)
    pos.iloc[ia + 1:ie + 1, px.columns.get_loc(e.ticker)] = (
        1 if e.tipo == "alta" else -1)
n_pos = pos.abs().sum(axis=1)
r_exc = r.sub(mercado, axis=0).fillna(0)
rebal_d = (pos * r_exc).sum(axis=1) / n_pos.where(n_pos > 0, 1)
rebal = a_mensual(rebal_d)
print(f"sessions with a position: {(n_pos > 0).mean():.1%}")
print(f"events per year: {len(ev) / 20:.1f}")
perfil(rebal).round(3)

sessions with a position: 2.4%
events per year: 2.4
meses         240.000
rent_anual      0.035
vol_anual       0.046
sharpe          0.755
t               3.375
asimetria       3.223
curtosis       14.768
peor_mes       -0.046
meses_pos       0.092
dtype: float64

The profile is the opposite of carry: only 9.2% of the months are positive (almost all the others are zero), a few earn a lot and the skewness is +3.2. The Sharpe, at 0.75, misleads in another way: the portfolio only holds a position in 2.4% of the sessions, one in forty. That is the capacity of a structural edge: few opportunities, short and known to everyone who reads the same calendar. The more the effect is published, the more capital chases it, and its entry price rises.

Information: post-earnings drift

Post-earnings-announcement drift (PEAD) is the tendency of the price to keep moving, weeks after the release, in the direction of the surprise55Note 5 · Bernard and Thomas (1989), in the resources, test whether the drift is a delayed response or a risk premium.. The hypothesis is about information: the market incorporates public news slowly. It predicts that the drift is larger the larger the surprise and that it decays with time.

The surprise is standardized with those of previous quarters (SUE: the surprise divided by the standard deviation of the previous 8, with at least 4). The reaction is the same session if it is released before the open and the next one if after the close; the portfolio enters at the close of the reaction, so as not to count the jump of the day, and measures the next 60 sessions in excess of the universe:

res = pd.read_csv("eventos2_resultados.csv",
                  parse_dates=["periodo_fin", "fecha_anuncio"])
res = res.sort_values(["ticker", "fecha_anuncio"])
previa = res.groupby("ticker")["sorpresa"].transform(
    lambda s: s.shift(1).rolling(8, min_periods=4).std())
res["sue"] = res["sorpresa"] / previa
tras = (res["momento"] == "tras_cierre").to_numpy()
res["i"] = sesiones.searchsorted(res["fecha_anuncio"]) + tras
res = res.dropna(subset=["sue"])
res = res[res["i"] + 61 < len(sesiones)]
res["decil"] = res.groupby("periodo_fin")["sue"].transform(
    lambda s: pd.qcut(s.rank(method="first"), 10, labels=False))

log_exc = np.log1p(r_exc).cumsum().to_numpy()
j = px.columns.get_indexer(res["ticker"])
i = res["i"].to_numpy()
res["deriva_60"] = np.expm1(log_exc[i + 60, j] - log_exc[i, j])
(100 * res.groupby("decil")["deriva_60"].mean()).round(2)

decil
0   -1.05
1   -0.20
2   -0.31
3    0.54
4   -0.53
5    0.02
6    0.29
7   -0.11
8    0.97
9    1.41
Name: deriva_60, dtype: float64

The extreme deciles separate: the one with the most positive surprises drifts 1.41% up and the one with the most negative, 1.05% down; the difference, 2.5 points, is within the 2-3 in the README. In between the relationship is noisy: the planted drift is small per unit of surprise, and a stock's 60-session volatility is large. As a portfolio, long the top decile and short the bottom one during their 60 sessions:

L = np.zeros(px.shape)
S = np.zeros(px.shape)
for ii, jj, d in zip(i, j, res["decil"]):
    if d == 9:
        L[ii + 1:ii + 61, jj] = 1
    elif d == 0:
        S[ii + 1:ii + 61, jj] = 1
R = r.fillna(0).to_numpy()
largo = (L * R).sum(1) / np.maximum(L.sum(1), 1)
corto = (S * R).sum(1) / np.maximum(S.sum(1), 1)
pead = a_mensual(pd.Series(largo - corto, index=sesiones))
print(f"events per year: {(res['decil'].isin([0, 9])).sum() / 20:.0f}")
perfil(pead.loc["2006-04":]).round(3)

events per year: 82
meses         237.000
rent_anual      0.089
vol_anual       0.117
sharpe          0.768
t               3.412
asimetria      -0.149
curtosis        0.296
peor_mes       -0.100
meses_pos       0.595
dtype: float64

The portfolio earns 8.9% a year with a Sharpe of 0.77, behind only momentum (0.84, which owes part of it to the bankruptcies), and it is among the most diversified: 82 events per year, spread over different dates, with almost zero skewness. Its risk is not a crash but that it disappears: if the drift is information that is incorporated slowly, the faster and more closely watched the market is, the less drift is left.

The risk profile of each edge

The control is still missing. The quintile portfolio of puntuacion_esg, a variable with no effect at all:

esg = car.pivot(index="fecha", columns="ticker",
                values="puntuacion_esg").where(hay)
placebo = quintiles(esg, sig_mes)

ventajas = {"carry_fx": carry, "momentum": momentum,
            "rebalanceos": rebal, "pead": pead.loc["2006-04":],
            "placebo_esg": placebo}
tabla = pd.DataFrame({k: perfil(v) for k, v in ventajas.items()}).T
tabla[["rent_anual", "vol_anual", "sharpe", "t", "asimetria",
       "curtosis", "peor_mes"]].round(2)

             rent_anual  vol_anual  sharpe     t  asimetria  curtosis  peor_mes
carry_fx           0.04       0.11    0.41  1.87      -1.38      7.39     -0.20
momentum           0.10       0.12    0.84  3.64       0.14      0.08     -0.09
rebalanceos        0.03       0.05    0.75  3.37       3.22     14.77     -0.05
pead               0.09       0.12    0.77  3.41      -0.15      0.30     -0.10
placebo_esg        0.03       0.10    0.33  1.46      -0.42      1.31     -0.12

The figure at the top puts the five on a plane: Sharpe on the horizontal axis, skewness of the months on the vertical one. Three readings:

• Sharpe does not rank risk. Carry (0.41) and the placebo (0.33) have a similar Sharpe; the first hides a −20% month and the second, a −12% one. The placebo's is not a tail of the signal: it is July 2020, when QXYX goes bankrupt while sitting, by chance, in its top quintile. Carry's is the premium itself. A Sharpe of 0.4 with skewness −1.4 is not the same product as one of 0.3 with skewness −0.4.
• The placebo passes for a small edge. It earns 3.1% a year with a t of 1.46, with no premium behind it. Without a prior hypothesis, a t like that distinguishes nothing: with twenty independent variables with no effect, the probability that one of them gives |t| > 2 is around 60% (lesson 2.1.8).
• Kurtosis gives away the event edge. Rebalances have the highest kurtosis (14.8): many months at zero and a few large ones. It is a different profile from carry even though both have tails: one tail in favor and the other against.

The comparison of the full distributions, with the monthly returns standardized (divided by their standard deviation) so they can be seen on the same scale:

q = [0.01, 0.05, 0.5, 0.95, 0.99]
z = {k: (v - v.mean()) / v.std() for k, v in ventajas.items()}
cuant = pd.DataFrame({k: v.quantile(q) for k, v in z.items()}).T
cuant["min"] = [v.min() for v in z.values()]
cuant["max"] = [v.max() for v in z.values()]
cuant.round(2)

             0.01  0.05   0.5  0.95  0.99   min   max
carry_fx    -2.69 -1.51 -0.01  1.46  1.89 -6.49  2.48
momentum    -2.09 -1.61 -0.06  1.72  2.52 -2.85  2.75
rebalanceos -1.18 -0.22 -0.22  1.84  4.39 -3.67  6.01
pead        -2.54 -1.62  0.01  1.67  2.23 -3.20  2.77
placebo_esg -2.43 -1.69  0.05  1.55  2.29 -4.49  2.58

Carry's worst month is 6.5 standard deviations below its mean; post-earnings drift's, 3.2, and momentum's, 2.85. The placebo has one at 4.5: the QXYX bankruptcy. A portfolio with no edge also collects or pays the delistings that come its way, and removing them from the data would make it look cleaner than it is. Under a normal, a month at −6.5σ has a probability of $4 \times 10^{-11}$: less than once in a billion years. In the carry of these data it has happened once in 21. Rebalances are the mirror image: their long tail is on the right (+6.0σ). A sizing rule that uses only volatility (lesson 2.2.2) assigns the same risk to both and gets the first one wrong.

The last dimension is size. A high Sharpe on little tradable volume does not take capital. With the 21-session average dollar volume of what each stock portfolio trades:

car["adv_musd"] = car["volumen_medio_21d"] * car["cierre"] / 1e6
adv = car.pivot(index="fecha", columns="ticker", values="adv_musd")

def adv_eventos(fechas, tickers):
    """ADV at the last month end before each event."""
    previo = adv.reindex(fechas, method="ffill")
    return np.nanmedian([previo.iloc[k][t]
                      for k, t in enumerate(tickers)])

rk = mom.rank(axis=1, pct=True)
ext = res[res["decil"].isin([0, 9])]
liq = pd.DataFrame({
    "apuestas_año": [12 * ((rk > 0.8) | (rk <= 0.2)).sum(axis=1)
                     .mean(), len(ev) / 20, len(ext) / 20],
    "sesiones": [21, ev["sesiones"].mean(), 60],
    "adv_musd": [adv[(rk > 0.8) | (rk <= 0.2)].stack().median(),
                 adv_eventos(ev["fecha_anuncio"], ev["ticker"]),
                 adv_eventos(ext["fecha_anuncio"], ext["ticker"])],
}, index=["momentum", "rebalanceos", "pead"])
liq.round(1)

             apuestas_año  sesiones  adv_musd
momentum            491.8      21.0     111.0
rebalanceos           2.4       4.9      20.0
pead                 82.5      60.0     118.7

Momentum spreads its risk across about 41 stocks each month (492 positions a year, although overlapping and not independent) with a median volume of USD 111 million a day. Post-earnings drift opens 82 positions a year in similar stocks (119 million). Rebalances, 2.4 a year in stocks five times less liquid (20 million): the ones that enter the index come from outside, from among the small ones. How much capital each edge takes with these figures is the subject of lesson 2.2.3.

How to write an edge hypothesis

Before opening a backtest, the hypothesis is written down. A template that forces the important questions to be answered:

flowchart TB
  M[Mechanism] --> C[Counterparty]
  C --> P[Predictions]
  P --> T[Test and control]
  T --> F[When it fails]
  F --> M

The template has two uses. The first is to filter before testing: an idea with no plausible counterparty and no reason to persist does not deserve the hours of backtesting, and each test saved is one test fewer in the multiple-testing count (lesson 2.2.4). The second is to read the result: if the hypothesis predicts negative skewness and the backtest comes out with positive skewness, something does not add up, however nice the Sharpe.

Exercises

1. Repeat the carry portfolio with two currencies per side instead of three. Does the return go up or down? And the skewness and the worst month? Explain the result with the risk premium hypothesis.

peso2 = (rango >= 9).astype(float) / 2 - (rango <= 2) / 2
carry2_d = (peso2.reindex(r_fx.index).ffill().shift(1)
            * r_fx).sum(axis=1).loc["2005-02":]
pd.DataFrame({"3 per side": perfil(carry),
              "2 per side": perfil(a_mensual(carry2_d))}).round(3)

            3 per side  2 per side
meses          251.000     251.000
rent_anual       0.044       0.055
vol_anual        0.108       0.139
sharpe           0.409       0.393
t                1.871       1.798
asimetria       -1.384      -1.011
curtosis         7.389       5.388
peor_mes        -0.199      -0.238
meses_pos        0.574       0.542

2. Measure post-earnings drift separately in the first half of the sample (up to 2015) and in the second. If the edge were information the market learns to price in, what would you expect? What do you find in these data and why?

mitades = {"2006-2015": pead.loc["2006-04":"2015-12"],
           "2016-2025": pead.loc["2016-01":]}
pd.DataFrame({k: perfil(v) for k, v in mitades.items()}).T[
    ["rent_anual", "sharpe", "t"]].round(2)

           rent_anual  sharpe     t
2006-2015        0.07    0.56  1.75
2016-2025        0.11    0.99  3.13

3. Write the momentum edge hypothesis with the lesson's template. What prediction would distinguish it from a pure risk explanation?