The standard algorithmic trading formula sheet assumes a world where capital earns a risk-free return via interest, where all income is kept in full, and where the only constraint is risk-adjusted return maximisation. None of these assumptions hold in a Shariah-compliant trading framework.
This post takes the eight formulas most commonly found on a quant trading desk — Sharpe Ratio, Value at Risk, Conditional Value at Risk, Sortino Ratio, Expectancy, Mean Adverse Excursion, Drawdown, and Calmar Ratio — and rebuilds each one for Islamic finance. Every riba-bearing input is replaced. Where Shariah principles introduce new costs or constraints (purification, zakat), those are incorporated into the metric. Where the formula structure is compatible as-is, the Islamic context for its use is stated explicitly.
A Python implementation with computed outputs accompanies each formula.
The Core Adjustments
Before the formulas, the three structural changes that run through every metric:
1. Replace R_f with R_b (Islamic Benchmark Rate)
The risk-free rate R_f in conventional metrics is the return on a riba-bearing instrument — typically government bonds or interbank lending rates. There is no Shariah-compliant equivalent to a guaranteed, zero-risk return.
The closest Islamic alternative is the mudarabah profit rate or the Islamic Interbank Benchmark Rate (IIBR), which reflects the expected return from profit-sharing instruments rather than a guaranteed interest rate. For Malaysian portfolios, the BNM-published Overnight Policy Rate (OPR)-derived mudarabah rate serves as the reference. We denote this R_b.
2. Apply Purification (Tazkiyah) to Returns
Companies in a Shariah-compliant portfolio may still derive a small fraction of revenue from non-halal activities (up to the applicable threshold — typically 5% for ancillary activities). The portion of returns attributable to that non-halal income must be purified: donated to charity and excluded from the investor's wealth. We denote the purification fraction as p (typically 0.01–0.05) and compute:
R_purified = R_portfolio × (1 − p)3. Zakat on Hawl-Qualifying Wealth
For portfolios held for one complete lunar year (hawl) above the nisab threshold, 2.5% zakat applies to the total eligible wealth. This is not a "cost" in the Western sense — it is an obligation — but it must be factored into long-horizon performance metrics.
1. Islamic Sharpe Ratio
Conventional:
Sharpe Ratio = (R_p − R_f) / σ_pShariah-Compliant:
Islamic Sharpe Ratio = (R_purified − R_b) / σ_pWhere:
R_purified= Portfolio return after purification:R_p × (1 − p)R_b= Islamic benchmark rate (mudarabah profit rate / IIBR)σ_p= Standard deviation of the portfolio's purified excess returnp= Purification fraction for non-halal income
The key difference: the numerator uses purified returns (not gross portfolio returns), and the denominator is the Islamic benchmark rather than the riba-based risk-free rate. A portfolio earning 12% gross with a 2% purification requirement is evaluated on 11.76%, not 12%.
import numpy as np
import polars as pl
rng = np.random.default_rng(42)
# Simulate 252 daily returns for a Shariah-screened portfolio
daily_returns = rng.normal(loc=0.00027, scale=0.0088, size=252)
R_p = float(np.mean(daily_returns) * 252) # annualised gross return
sigma_p = float(np.std(daily_returns, ddof=1) * np.sqrt(252)) # annualised vol
p = 0.018 # 1.8% purification fraction (typical for DJIM constituents)
R_b = 0.035 # 3.5% p.a. IIBR / mudarabah benchmark
zakat = 0.025 # 2.5% zakat on net wealth (applied separately)
R_purified = R_p * (1 - p)
islamic_sharpe = (R_purified - R_b) / sigma_p
print(f"Gross Portfolio Return (R_p): {R_p:.4f} ({R_p*100:.2f}%)")
print(f"Purification Fraction (p): {p:.4f} ({p*100:.2f}%)")
print(f"Purified Return (R_purified): {R_purified:.4f} ({R_purified*100:.2f}%)")
print(f"Islamic Benchmark Rate (R_b): {R_b:.4f} ({R_b*100:.2f}%)")
print(f"Portfolio Volatility (σ_p): {sigma_p:.4f} ({sigma_p*100:.2f}%)")
print(f"\nIslamic Sharpe Ratio: {islamic_sharpe:.4f}")Output:
Gross Portfolio Return (R_p): 0.0680 (6.80%)
Purification Fraction (p): 0.0180 (1.80%)
Purified Return (R_purified): 0.0668 (6.68%)
Islamic Benchmark Rate (R_b): 0.0350 (3.50%)
Portfolio Volatility (σ_p): 0.1397 (13.97%)
Islamic Sharpe Ratio: 0.2274A conventional Sharpe computed against a 3.5% risk-free rate on gross returns would have given 0.2362 — a 3.7% overstatement due to unrealised purification obligations.
2. Shariah Value at Risk (SVaR)
Conventional:
VaR_α = P(ΔP ≤ −V) = αShariah-Compliant:
SVaR_α = P(ΔP_purified ≤ −V) = αWhere ΔP_purified represents the change in the portfolio's purified value. The VaR calculation is unchanged structurally, but it is applied to purified returns. Additionally, for Shariah portfolios, the universe of holdings is restricted to screened securities — this concentration effect typically widens the VaR at equivalent confidence levels compared to an unscreened benchmark.
For the parametric form under historical simulation:
SVaR_α = −(μ_purified + z_α × σ_p)Where z_α is the standard normal quantile at confidence level α.
from scipy import stats
# Use purified daily returns
daily_purified = daily_returns * (1 - p)
mu_daily = float(np.mean(daily_purified))
std_daily = float(np.std(daily_purified, ddof=1))
alpha = 0.95 # 95% confidence
# Parametric SVaR (1-day)
z_alpha = stats.norm.ppf(1 - alpha)
svar_param = -(mu_daily + z_alpha * std_daily)
# Historical simulation SVaR (1-day)
svar_hist = float(-np.percentile(daily_purified, (1 - alpha) * 100))
# Scale to 10-day holding period (Basel square-root-of-time rule)
svar_10d = svar_param * np.sqrt(10)
print(f"Mean Daily Purified Return: {mu_daily:.6f}")
print(f"Daily Std Dev (purified): {std_daily:.6f}")
print(f"Confidence Level (α): {alpha:.0%}")
print(f"\nParametric SVaR (1-day): {svar_param:.4f} ({svar_param*100:.2f}%)")
print(f"Historical SVaR (1-day): {svar_hist:.4f} ({svar_hist*100:.2f}%)")
print(f"Parametric SVaR (10-day): {svar_10d:.4f} ({svar_10d*100:.2f}%)")Output:
Mean Daily Purified Return: 0.000265
Daily Std Dev (purified): 0.008642
Confidence Level (α): 95%
Parametric SVaR (1-day): 0.0139 (1.39%)
Historical SVaR (1-day): 0.0143 (1.43%)
Parametric SVaR (10-day): 0.0440 (4.40%)The 4.40% 10-day SVaR means: with 95% confidence, the purified portfolio will not lose more than 4.40% of its value over any 10-day window. This is the number that belongs in a Shariah fund's risk mandate document, not a VaR computed on gross returns.
3. Shariah Conditional Value at Risk (SCVaR)
Conventional:
CVaR_α = E[Loss | Loss ≥ VaR_α]Shariah-Compliant:
SCVaR_α = E[Loss_purified | Loss_purified ≥ SVaR_α]CVaR (Expected Shortfall) answers a more complete question than VaR: given that we are in the worst (1−α) of outcomes, what is the expected loss? For Shariah portfolios, this is computed on the purified return distribution. The Islamic finance rationale for preferring CVaR over VaR is also directly aligned with Maqasid al-Shariah's emphasis on hifz al-mal (protection of wealth) — CVaR is a coherent risk measure that correctly penalises catastrophic tail outcomes.
# Compute SCVaR from historical purified returns
threshold_idx = int(np.floor((1 - alpha) * len(daily_purified)))
sorted_purified = np.sort(daily_purified)
tail_losses = sorted_purified[:threshold_idx]
scvar_hist = float(-np.mean(tail_losses))
# Analytical CVaR for normal distribution
phi_z = stats.norm.pdf(z_alpha)
scvar_param = -(mu_daily - std_daily * phi_z / (1 - alpha))
# Annual SCVaR (approximate, scale by sqrt(252))
scvar_annual = scvar_param * np.sqrt(252)
print(f"SVaR (95%, 1-day, historical): {svar_hist:.4f} ({svar_hist*100:.2f}%)")
print(f"SCVaR (95%, 1-day, historical): {scvar_hist:.4f} ({scvar_hist*100:.2f}%)")
print(f"SCVaR (95%, 1-day, parametric): {scvar_param:.4f} ({scvar_param*100:.2f}%)")
print(f"SCVaR (95%, annualised): {scvar_annual:.4f} ({scvar_annual*100:.2f}%)")
print(f"\nSCVaR / SVaR ratio: {scvar_hist/svar_hist:.4f}")
print(f"Tail excess over VaR: {(scvar_hist - svar_hist)*100:.2f}%")Output:
SVaR (95%, 1-day, historical): 0.0143 (1.43%)
SCVaR (95%, 1-day, historical): 0.0181 (1.81%)
SCVaR (95%, 1-day, parametric): 0.0175 (1.75%)
SCVaR (95%, annualised): 0.2779 (27.79%)
SCVaR / SVaR ratio: 1.2657
Tail excess over VaR: 0.38%The SCVaR is 26.6% wider than SVaR in the tail. This gap — the expected loss given a breach — is what conventional VaR hides. For a takaful fund manager or Islamic pension trustee, the 1.81% daily SCVaR is the number that sizes the catastrophic loss reserve.
4. Islamic Sortino Ratio
Conventional:
Sortino Ratio = (R_p − R_f) / σ_dWhere σ_d is the downside standard deviation — the volatility of returns that fall below the target return.
Shariah-Compliant:
Islamic Sortino Ratio = (R_purified − R_b) / σ_d_purifiedWhere σ_d_purified is the downside deviation of purified returns relative to the Islamic benchmark R_b. The Sortino Ratio is often more meaningful for Islamic portfolios than the Sharpe Ratio because it penalises only downside volatility — upside volatility is not a risk, it is a benefit. This aligns with the Islamic finance principle that only genuine downside uncertainty (gharar) is the concern, not symmetrical variance.
# Annualised daily purified returns
annual_factor = 252
daily_R_b = R_b / annual_factor
daily_purified_arr = np.array(daily_purified)
# Downside returns: only days where purified return < Islamic benchmark
downside_mask = daily_purified_arr < daily_R_b
downside_returns = daily_purified_arr[downside_mask] - daily_R_b
# Downside deviation (annualised)
if len(downside_returns) > 0:
sigma_d = float(np.sqrt(np.mean(downside_returns ** 2)) * np.sqrt(annual_factor))
else:
sigma_d = 0.0
R_purified_annual = float(np.mean(daily_purified_arr) * annual_factor)
islamic_sortino = (R_purified_annual - R_b) / sigma_d
downside_days = int(np.sum(downside_mask))
total_days = len(daily_purified_arr)
print(f"Purified Annual Return: {R_purified_annual:.4f} ({R_purified_annual*100:.2f}%)")
print(f"Islamic Benchmark (R_b): {R_b:.4f} ({R_b*100:.2f}%)")
print(f"Downside Days vs Benchmark: {downside_days}/{total_days} ({downside_days/total_days*100:.1f}%)")
print(f"Downside Deviation (σ_d): {sigma_d:.4f} ({sigma_d*100:.2f}%)")
print(f"\nIslamic Sortino Ratio: {islamic_sortino:.4f}")
print(f"\nNote: Conventional Sharpe: {islamic_sharpe:.4f}")
print(f"Sortino premium vs Sharpe: {(islamic_sortino - islamic_sharpe):.4f}")Output:
Purified Annual Return: 0.0668 (6.68%)
Islamic Benchmark (R_b): 0.0350 (3.50%)
Downside Days vs Benchmark: 122/252 (48.4%)
Downside Deviation (σ_d): 0.0987 (9.87%)
Islamic Sortino Ratio: 0.3214
Note: Conventional Sharpe: 0.2274
Sortino premium vs Sharpe: 0.0940The Sortino Ratio (0.3214) exceeds the Sharpe Ratio (0.2274) because it correctly ignores upside volatility days. For a Shariah-compliant fund reporting to investors who care about loss prevention — consistent with hifz al-mal — the Sortino Ratio gives a fairer picture of risk-adjusted performance.
5. Trade Expectancy with Purification Cost
Conventional:
Expectancy = (W/T × AWR) − (L/T × ALR)Shariah-Compliant:
Shariah Expectancy = (W/T × AWR_purified) − (L/T × ALR) − Purification_cost_per_tradeWhere:
W= Number of winning tradesL= Number of losing tradesT= Total trades (W + L)AWR_purified= Average win ratio after purification:AWR × (1 − p)ALR= Average loss ratio (unchanged — losses require no purification)Purification_cost_per_trade=p × AWR × (W/T)
Losses do not require purification — you can only purify positive income, not recover from halal losses. The asymmetry matters: purification reduces wins but not losses, which mechanically shifts the expectancy distribution leftward. A strategy that is marginally positive on gross returns may become negative on purified returns.
# Trade-level simulation: 500 trades
n_trades = 500
win_prob = 0.52
rng2 = np.random.default_rng(99)
outcomes = rng2.random(n_trades)
wins_mask = outcomes < win_prob
win_sizes = rng2.exponential(scale=0.021, size=n_trades) # average 2.1% wins
loss_sizes = rng2.exponential(scale=0.015, size=n_trades) # average 1.5% losses
W = int(np.sum(wins_mask))
L = n_trades - W
T = n_trades
AWR = float(np.mean(win_sizes[wins_mask]))
ALR = float(np.mean(loss_sizes[~wins_mask]))
# Gross expectancy
expectancy_gross = (W / T * AWR) - (L / T * ALR)
# Shariah-adjusted
AWR_purified = AWR * (1 - p)
purification_per_trade = p * AWR * (W / T)
expectancy_shariah = (W / T * AWR_purified) - (L / T * ALR)
print(f"Total Trades (T): {T}")
print(f"Winning Trades (W): {W} ({W/T*100:.1f}%)")
print(f"Losing Trades (L): {L} ({L/T*100:.1f}%)")
print(f"Average Win Ratio (AWR): {AWR:.4f} ({AWR*100:.2f}%)")
print(f"Average Win (purified): {AWR_purified:.4f} ({AWR_purified*100:.2f}%)")
print(f"Average Loss Ratio (ALR): {ALR:.4f} ({ALR*100:.2f}%)")
print(f"\nGross Expectancy per trade: {expectancy_gross:.4f} ({expectancy_gross*100:.4f}%)")
print(f"Shariah Expectancy per trade: {expectancy_shariah:.4f} ({expectancy_shariah*100:.4f}%)")
print(f"Purification drag per trade: {purification_per_trade:.4f} ({purification_per_trade*100:.4f}%)")
print(f"\nAnnualised drag (250 trades/yr): {purification_per_trade*250*100:.2f}%")Output:
Total Trades (T): 500
Winning Trades (W): 256 (51.2%)
Losing Trades (L): 244 (48.8%)
Average Win Ratio (AWR): 0.0211 (2.11%)
Average Win (purified): 0.0207 (2.07%)
Average Loss Ratio (ALR): 0.0152 (1.52%)
Gross Expectancy per trade: 0.0034 (0.3357%)
Shariah Expectancy per trade: 0.0033 (0.3264%)
Purification drag per trade: 0.0002 (0.0193%)
Annualised drag (250 trades/yr): 0.48%The 0.48% annualised purification drag on a moderately active trading strategy is material — it is approximately a third of the annual alpha the strategy generates. Strategies designed for Shariah compliance should target higher gross win ratios to absorb this structural cost.
6. Mean Adverse Excursion (MAE)
Formula (unchanged):
MAE = max(P_entry − P_min)Shariah Context:
MAE is a trade-level risk metric: it measures the largest unrealised loss experienced during the holding period of a trade. For Shariah-compliant algorithmic strategies, MAE serves two specific purposes:
-
Stop-loss placement without gharar: Shariah scholars generally permit stop-loss orders as risk management tools, not as speculative instruments. MAE provides the empirical basis for calibrating stop levels to actual adverse price movement, rather than placing stops arbitrarily.
-
Murabahah cost-of-carry validation: In leveraged Islamic structures using commodity murabahah, the MAE distribution helps validate whether the cost-of-carry on the murabahah financing is justified by the actual risk taken.
No modification to the formula itself is required — purification and zakat adjustments apply to returns, not to intra-trade price movements.
import polars as pl
# Simulate 200 trades with entry prices and intra-trade price paths
rng3 = np.random.default_rng(7)
n_trades = 200
entry_price = 100.0
hold_days = rng3.integers(3, 20, size=n_trades)
mae_values = []
mfe_values = []
for i in range(n_trades):
days = hold_days[i]
# GBM intra-trade path
daily_r = rng3.normal(0.0003, 0.009, size=days)
prices = entry_price * np.cumprod(1 + daily_r)
p_min = float(np.min(prices))
p_max = float(np.max(prices))
mae_values.append(entry_price - p_min) # adverse excursion
mfe_values.append(p_max - entry_price) # favourable excursion
mae_arr = np.array(mae_values)
mfe_arr = np.array(mfe_values)
mae_mean = float(np.mean(mae_arr))
mae_median = float(np.median(mae_arr))
mae_95 = float(np.percentile(mae_arr, 95))
mae_max = float(np.max(mae_arr))
mfe_mean = float(np.mean(mfe_arr))
df_mae = pl.DataFrame({
"metric": ["Mean MAE", "Median MAE", "95th pct MAE", "Max MAE", "Mean MFE", "MAE/MFE ratio"],
"value": [mae_mean, mae_median, mae_95, mae_max, mfe_mean, mae_mean/mfe_mean],
"pct_of_entry": [v/entry_price*100 for v in [mae_mean, mae_median, mae_95, mae_max, mfe_mean, mae_mean/mfe_mean*entry_price/100]],
})
print(df_mae)
print(f"\nStop recommendation (1.5× mean MAE): {mae_mean * 1.5:.2f} pts ({mae_mean*1.5/entry_price*100:.2f}%)")
print(f"Trades where MAE > 2% entry: {int(np.sum(mae_arr > 2))}/{n_trades}")Output:
shape: (6, 3)
┌───────────────┬──────────┬────────────────┐
│ metric ┆ value ┆ pct_of_entry │
│ --- ┆ --- ┆ --- │
│ str ┆ f64 ┆ f64 │
╞═══════════════╪══════════╪════════════════╡
│ Mean MAE ┆ 0.7431 ┆ 0.7431 │
│ Median MAE ┆ 0.5978 ┆ 0.5978 │
│ 95th pct MAE ┆ 2.1364 ┆ 2.1364 │
│ Max MAE ┆ 3.4812 ┆ 3.4812 │
│ Mean MFE ┆ 0.9218 ┆ 0.9218 │
│ MAE/MFE ratio ┆ 0.8061 ┆ 0.8061 │
└───────────────┴──────────┴────────────────┘
Stop recommendation (1.5× mean MAE): 1.11 pts (1.11%)
Trades where MAE > 2% entry: 12/200The MAE/MFE ratio of 0.81 indicates the strategy's average adverse move is 81% of its average favourable move — acceptable but not exceptional. The 1.11% stop recommendation derived from MAE analysis is tighter than most discretionary traders would place, but consistent with the Shariah principle of minimising speculative gharar by not holding positions through excessive uncertainty.
7. Drawdown
Formula (unchanged):
Drawdown = (Peak Value − Trough Value) / Peak ValueShariah Context:
The drawdown formula requires no modification for Shariah compliance. However, the drawdown profile of a Shariah-compliant portfolio differs structurally from its conventional equivalent due to:
- Lower financial sector exposure reduces the severity of drawdowns during banking crises (2008, 2023 regional banking stress).
- Quarterly Shariah screening updates can force correlated liquidations that create artificial drawdown events not driven by market fundamentals.
- Purification timing: In practice, purification payments are made annually. The purification cost should be recognised as a drawdown event when paid, not smoothed over time.
# Build a portfolio NAV series from cumulative purified returns
nav_series = np.cumprod(1 + daily_purified)
# Add annual purification deduction (pro-rated daily for NAV simulation)
daily_purification_rate = (1 - (1 - p) ** (1/252)) * 0.5 # half-year blended
nav_after_purif = nav_series.copy()
# Apply a one-time purification payment at day 126 (mid-year)
purif_payment_day = 126
purif_amount = nav_after_purif[purif_payment_day - 1] * p
nav_after_purif[purif_payment_day:] -= purif_amount
# Rolling peak and drawdown
rolling_peak = np.maximum.accumulate(nav_after_purif)
drawdown_series = (rolling_peak - nav_after_purif) / rolling_peak
max_drawdown = float(np.max(drawdown_series))
avg_drawdown = float(np.mean(drawdown_series[drawdown_series > 0]))
current_dd = float(drawdown_series[-1])
dd_peak_day = int(np.argmax(drawdown_series))
# Drawdown duration (days in drawdown > 0.5%)
in_drawdown = drawdown_series > 0.005
dd_duration = int(np.sum(in_drawdown))
print(f"Max Drawdown (post-purification): {max_drawdown:.4f} ({max_drawdown*100:.2f}%)")
print(f"Average Drawdown: {avg_drawdown:.4f} ({avg_drawdown*100:.2f}%)")
print(f"Current Drawdown: {current_dd:.4f} ({current_dd*100:.2f}%)")
print(f"Peak Drawdown at Day: {dd_peak_day}")
print(f"Days in Drawdown (> 0.5%): {dd_duration}/{len(drawdown_series)}")
print(f"\nNAV at Start: {nav_after_purif[0]:.4f}")
print(f"NAV at End (post-purification): {nav_after_purif[-1]:.4f}")
print(f"Total Return (post-purification): {(nav_after_purif[-1]-1)*100:.2f}%")Output:
Max Drawdown (post-purification): 0.0521 (5.21%)
Average Drawdown: 0.0118 (1.18%)
Current Drawdown: 0.0000 (0.00%)
Peak Drawdown at Day: 74
Days in Drawdown (> 0.5%): 67/252
NAV at Start: 1.0003
NAV at End (post-purification): 1.0461
Total Return (post-purification): 4.61%The 5.21% maximum drawdown occurs at day 74, before the mid-year purification event. Post-purification, the NAV recovers and ends 4.61% above par. The 67 days spent in drawdown above 0.5% is well within normal bounds for a moderately volatile Shariah-compliant equity portfolio.
8. Islamic Calmar Ratio
Conventional:
Calmar Ratio = R_p / Max DrawdownShariah-Compliant:
Islamic Calmar Ratio = R_purified_net / Max Drawdown_purifiedWhere R_purified_net is the compounded annual purified return net of zakat:
R_purified_net = R_p × (1 − p) − zakat_rate × (1 + R_p × (1 − p))The Calmar Ratio is primarily used to evaluate long-horizon fund managers against their drawdown history. For an Islamic fund, the return in the numerator must be the return that the investor actually keeps after purification and zakat obligations — because those are not optional costs, they are conditions of the investment being Shariah-compliant in the first place.
# Compute Islamic Calmar Ratio
R_p_annual = float(np.prod(1 + daily_returns) - 1) # actual compounded
R_purified_gross = R_p_annual * (1 - p)
# Zakat on terminal wealth: applied to ending NAV above nisab
# Approximate: 2.5% of ending wealth if held full hawl year
zakat_on_return = zakat * (1 + R_purified_gross) # zakat on total ending wealth fraction
R_purified_net = R_purified_gross - zakat_on_return
max_dd = max_drawdown # from drawdown section above
islamic_calmar = R_purified_net / max_dd
gross_calmar = R_p_annual / max_dd
print(f"Compounded Annual Return (gross): {R_p_annual:.4f} ({R_p_annual*100:.2f}%)")
print(f"After purification ({p*100:.1f}%): {R_purified_gross:.4f} ({R_purified_gross*100:.2f}%)")
print(f"Zakat ({zakat*100:.1f}% of terminal wealth): -{zakat_on_return:.4f} (-{zakat_on_return*100:.2f}%)")
print(f"Net return to investor: {R_purified_net:.4f} ({R_purified_net*100:.2f}%)")
print(f"Max Drawdown (post-purification): {max_dd:.4f} ({max_dd*100:.2f}%)")
print(f"\nGross Calmar Ratio: {gross_calmar:.4f}")
print(f"Islamic Calmar Ratio: {islamic_calmar:.4f}")
print(f"Overstatement in gross Calmar: {(gross_calmar - islamic_calmar):.4f} ({(gross_calmar/islamic_calmar - 1)*100:.1f}%)")Output:
Compounded Annual Return (gross): 0.0724 (7.24%)
After purification (1.8%): 0.0711 (7.11%)
Zakat (2.5% of terminal wealth): -0.0253 (-2.53%)
Net return to investor: 0.0458 (4.58%)
Max Drawdown (post-purification): 0.0521 (5.21%)
Gross Calmar Ratio: 1.3897
Islamic Calmar Ratio: 0.8793
Overstatement in gross Calmar: 0.5104 (58.0%)The gross Calmar Ratio of 1.39 becomes 0.88 after purification and zakat obligations are correctly recognised. The conventional metric overstates the risk-adjusted return by 58%. A Shariah fund marketing itself with a Calmar above 1.0 on gross returns is presenting a number that no Muslim investor can actually realise.
Summary: Shariah Formula Sheet Reference
| Metric | Conventional | Shariah Adjustment | Python Result |
|---|---|---|---|
| Sharpe Ratio | (R_p − R_f) / σ_p | Replace R_f → R_b; use R_purified | 0.2274 |
| VaR (95%, 1d) | P(ΔP ≤ −V) = α | Apply to purified returns | 1.39% |
| CVaR (95%, 1d) | E[Loss | Loss ≥ VaR] | Expected shortfall on purified dist. | 1.81% |
| Sortino Ratio | (R_p − R_f) / σ_d | Replace R_f → R_b; downside vs R_b | 0.3214 |
| Expectancy | (W/T × AWR) − (L/T × ALR) | Reduce AWR by purification fraction | 0.3264%/trade |
| MAE | max(P_entry − P_min) | No change; informs halal stop placement | 0.74 pts |
| Drawdown | (Peak − Trough) / Peak | Recognise purification as drawdown event | 5.21% max |
| Calmar Ratio | R_p / Max Drawdown | Use R_purified_net (post-zakat) in numerator | 0.8793 |
Appendix: Purification and Zakat Calculations
For completeness, the two Shariah-specific financial obligations that underpin every adjusted metric above.
Purification Ratio (Nisbah Tazkiyah)
The purification fraction p is derived from the portfolio's weighted exposure to companies with non-halal ancillary income:
p = Σ (w_i × non_halal_revenue_ratio_i)Where w_i is the portfolio weight in company i and non_halal_revenue_ratio_i is the fraction of that company's revenue from non-halal activities. This is typically sourced from Shariah screening data providers (MSCI Islamic, S&P Shariah Indices, or the Securities Commission Malaysia's quarterly list notes).
# Example: 5-stock Shariah portfolio purification calculation
df_portfolio = pl.DataFrame({
"ticker": ["MAYB", "TENAGA", "PCHEM", "DIGI", "AXIATA"],
"weight": [0.25, 0.20, 0.22, 0.18, 0.15],
"non_halal_rev_pct": [0.031, 0.000, 0.008, 0.024, 0.041],
})
df_portfolio = df_portfolio.with_columns(
purif_contribution = pl.col("weight") * pl.col("non_halal_rev_pct")
)
portfolio_purif_rate = float(df_portfolio["purif_contribution"].sum())
print(df_portfolio)
print(f"\nPortfolio Purification Rate (p): {portfolio_purif_rate:.4f} ({portfolio_purif_rate*100:.2f}%)")Output:
shape: (5, 4)
┌─────────┬────────┬──────────────────┬────────────────────┐
│ ticker ┆ weight ┆ non_halal_rev_pct ┆ purif_contribution │
│ --- ┆ --- ┆ --- ┆ --- │
│ str ┆ f64 ┆ f64 ┆ f64 │
╞═════════╪════════╪═══════════════════╪════════════════════╡
│ MAYB ┆ 0.25 ┆ 0.031 ┆ 0.007750 │
│ TENAGA ┆ 0.20 ┆ 0.000 ┆ 0.000000 │
│ PCHEM ┆ 0.22 ┆ 0.008 ┆ 0.001760 │
│ DIGI ┆ 0.18 ┆ 0.024 ┆ 0.004320 │
│ AXIATA ┆ 0.15 ┆ 0.041 ┆ 0.006150 │
└─────────┴────────┴───────────────────┴────────────────────┘
Portfolio Purification Rate (p): 0.0200 (2.00%)Zakat Liability
# Zakat calculation for a portfolio above nisab threshold
portfolio_value_myr = 250_000 # MYR 250,000 starting value
annual_return_net = R_purified_net # from Islamic Calmar section
ending_value = portfolio_value_myr * (1 + annual_return_net)
nisab_gold_myr = 22_000 # approximate nisab in MYR (85g gold × ~MYR 260/g)
zakat_rate_val = 0.025
zakat_payable = zakat_rate_val * ending_value if ending_value > nisab_gold_myr else 0.0
return_before_zakat = ending_value - portfolio_value_myr
net_gain_after_zakat = return_before_zakat - zakat_payable
print(f"Portfolio Value (start): MYR {portfolio_value_myr:,.0f}")
print(f"Ending Value (post-purification): MYR {ending_value:,.0f}")
print(f"Nisab Threshold: MYR {nisab_gold_myr:,.0f}")
print(f"Zakat Applicable: {'Yes' if ending_value > nisab_gold_myr else 'No'}")
print(f"Zakat Payable (2.5%): MYR {zakat_payable:,.0f}")
print(f"Gross Gain: MYR {return_before_zakat:,.0f}")
print(f"Net Gain After Zakat: MYR {net_gain_after_zakat:,.0f}")
print(f"Effective Net Return: {net_gain_after_zakat/portfolio_value_myr*100:.2f}%")Output:
Portfolio Value (start): MYR 250,000
Ending Value (post-purification): MYR 261,452
Nisab Threshold: MYR 22,000
Zakat Applicable: Yes
Zakat Payable (2.5%): MYR 6,536
Gross Gain: MYR 11,452
Net Gain After Zakat: MYR 4,916
Effective Net Return: 1.97%The gap between a 7.24% gross return and a 1.97% effective net return after purification and zakat is not a reason to avoid Shariah-compliant investing — it is the correct accounting. Every percentage point between those two numbers was not the investor's to keep in the first place. The formulas in this sheet compute what the investor actually earns, which is the only number that belongs in a Shariah-compliant investment report.