# Sample size for the pre-registered randomised experiment. Participants build a hypothetical
# 100,000-unit portfolio from cash, a broad index fund, a 2x daily-leveraged index fund and a
# put-writing ("short volatility") fund, after seeing ONE of two disclosures about the two
# leveraged products:
#   A (usual): average annual returns over past years, in a table (labelled index back-tests)
#   B (tail):  the same averages PLUS the worst back-test drawdowns and 1,000 simulated 5-year paths
#              from a fat-tailed (GARCH) model, shown as a fan of outcomes with the 5 % worst marked
# Primary outcome: share of the portfolio put into the two leveraged products (0-100 %).
# Effect in SD units; normal approximation; two-sided alpha 0.05 (the direction is not assumed:
# earlier studies found that showing simulated outcomes can INCREASE risk taking).
import math
from statistics import NormalDist
N = NormalDist()

def n_per_arm(d, alpha=0.05, power=0.90):
    return math.ceil(2 * (N.inv_cdf(1 - alpha / 2) + N.inv_cdf(power)) ** 2 / d ** 2) + 1

for d in (0.20, 0.25, 0.30, 0.40):
    print(f"standardised effect {d:.2f}: {n_per_arm(d, power=0.8)} per arm for 80 % power, {n_per_arm(d)} for 90 %")
n = n_per_arm(0.30)
rec = math.ceil(n / 0.85)
print(f"chosen: 0.30 SD, 90 % power -> {n} analysable per arm; allowing 15 % exclusions"
      f" (failed attention or comprehension checks) recruit {rec} per arm, {2 * rec} in all")
# Randomisation is stratified by prior experience with leveraged or option products (yes / no);
# stratum enters the analysis as a covariate.
# standardised effect 0.20: 394 per arm for 80 % power, 527 for 90 %
# standardised effect 0.25: 253 per arm for 80 % power, 338 for 90 %
# standardised effect 0.30: 176 per arm for 80 % power, 235 for 90 %
# standardised effect 0.40: 100 per arm for 80 % power, 133 for 90 %
# chosen: 0.30 SD, 90 % power -> 235 analysable per arm; allowing 15 % exclusions (failed attention or comprehension checks) recruit 277 per arm, 554 in all

# Decision rule, fixed before data collection. d = arm B (tail disclosure) minus arm A (average
# returns only) in the share allocated to leveraged products, standardised, with its 95 % CI.
# Confirmatory test: is d different from zero (two-sided alpha 0.05)? The trial is powered (90 %) to
# detect d = 0.30 SD, the smallest effect of interest; how the CI sits relative to +-0.30 is reported
# descriptively, not as a second test.
# Validity check: in arm B at least 70 % must correctly answer "In the simulations, how often did the
# 2x fund lose more than half its value within 5 years?" - otherwise the disclosure was not understood.
SESOI = 0.30

def verdict(d, lo, hi, understood):
    if understood < 0.70:
        return f"effect {d:+.2f}: NOT INTERPRETABLE - only {understood:.0%} of arm B understood the disclosure"
    if hi < 0:
        v = "the tail disclosure REDUCES leveraged allocation"
    elif lo > 0:
        v = "the tail disclosure INCREASES leveraged allocation"
    else:
        v = "no difference shown"
    if lo >= -SESOI and hi <= SESOI:
        size = "CI within +-0.30: any effect is smaller than the smallest effect of interest"
    elif hi <= -SESOI or lo >= SESOI:
        size = "CI wholly beyond 0.30 in size"
    else:
        size = "CI includes effects both above and below 0.30 in size"
    return f"effect {d:+.2f} ({lo:+.2f} to {hi:+.2f}): {v}; descriptively, {size}"

for r in ((-0.48, -0.65, -0.31, 0.84), (-0.20, -0.37, -0.03, 0.81), (0.04, -0.13, 0.21, 0.88),
          (0.22, 0.05, 0.39, 0.79), (-0.40, -0.57, -0.23, 0.55)):
    print(verdict(*r))
print("Secondary outcomes (expected drawdown, confidence, choice of the cash buffer) are reported with intervals, never decisive.")
# effect -0.48 (-0.65 to -0.31): the tail disclosure REDUCES leveraged allocation; descriptively, CI wholly beyond 0.30 in size
# effect -0.20 (-0.37 to -0.03): the tail disclosure REDUCES leveraged allocation; descriptively, CI includes effects both above and below 0.30 in size
# effect +0.04 (-0.13 to +0.21): no difference shown; descriptively, CI within +-0.30: any effect is smaller than the smallest effect of interest
# effect +0.22 (+0.05 to +0.39): the tail disclosure INCREASES leveraged allocation; descriptively, CI includes effects both above and below 0.30 in size
# effect -0.40: NOT INTERPRETABLE - only 55% of arm B understood the disclosure
# Secondary outcomes (expected drawdown, confidence, choice of the cash buffer) are reported with intervals, never decisive.