Methodology · 統計方法

主頁所有 risk-adjusted 指標 / forward expectation / Monte Carlo projection 嘅完整公式、sample window 同 benchmark 對齊細節。意圖 — 任何熟識 quant 嘅讀者可以從公式重現顯示嘅數字。

Sample Definition

Period

2019-01 → 2026-05 · 89 monthly returns · 7.3 years

Granularity

Monthly returns derived from daily MTM equity (entry-month attribution)

SPY benchmark

Same period · monthly close · total return basis (dividend reinvestment baked into ETF NAV)

Risk-free rate

4.0% annualized · applied uniformly to both Metis 同 SPY

Risk-Adjusted Ratios

Sharpe Ratio

Sharpe = (annualized return − RFR) / (monthly std × √12)
= (1.524 − 0.04) / (0.1304 × √12)
= 3.29

SPY same-period: (0.173 − 0.04) / (0.0485 × √12) = 0.79

Note: Higher granularity (daily) typically yields lower Sharpe due to autocorrelation discounting. Monthly granularity used here for comparability with industry-standard reporting.

Sortino Ratio

Sortino = (annualized return − RFR) / (downside std × √12)
Downside std = √(Σ neg_r² / N_total)
= 15.07

⚠ Reading caveat: Sortino 異常高源於 only 1 negative month over the sample. Downside deviation denom 細到極端 → ratio 不穩定. Interpret directionally as「downside contained」唔係「15× alpha」. SPY same-period Sortino 1.29.

Calmar Ratio

Calmar = post-2019 CAGR / Max Drawdown
= 1.343 / 0.268 = 5.01

SPY same-period: 0.157 / 0.248 = 0.63. Numerator advantage (8.6× CAGR) dominant — denom DD comparable (Metis 26.8% vs SPY 24.8%).

Forward 3-Month Expected Return (FCER)

主頁 FCER 顯示嘅係 post-2019 89-month sample 嘅 3-month rolling compound returns distribution (87 windows). 唔係 Monte Carlo. 純 historical roll.

• Mean +26.16% · Median +17.28% · 95.4% positive (83 / 87 windows positive)
• Sample window 包含 2020 COVID crisis + 2025-26 cycle peak — distribution skewed right
• Forward expectation 應 frame 為「assuming regime persists」, 非 forecasted

Compounding Logic · 1mo / 3mo / 12mo Aggregation

主頁 Section 06 顯示 6 個 horizon-aggregated stats (1mo / 3mo / 12mo · 各有 median 同 mean)。呢度展示佢哋之間嘅 mathematical relationships。

Mean compounds approximately

Expected: (1 + monthly_mean)^N − 1

1mo: +8.02% ← base
3mo: (1.0802)³ − 1 = +26.00% ≈ shown +26.16% diff +0.16pp ✓
12mo: (1.0802)¹² − 1 = +152.4% vs shown +147.0% diff −5.4pp

Mean 跟 compounding 行 within a few percentage points. Long-horizon 嘅 5pp gap = variance drag — 當 monthly returns 唔均勻 (std 13.04%), compound geometric mean < arithmetic mean (Jensen's inequality)。

Median ≠ compound

Naive: (1 + monthly_median)^N − 1 ← 不正確

1mo: +6.57% ← base
3mo: (1.0657)³ − 1 = +21.08% vs shown +17.28% diff −3.8pp
12mo: (1.0657)¹² − 1 = +115.5% vs shown +135.2% diff +19.7pp

Median 係 order statistic, 唔可以 compound — 直接係從 87 (3mo) / 78 (12mo) rolling windows 嘅實際 distribution 入面攞 median。所以同 naive compound 唔同。

3mo median 比 naive compound 低 (variance drag dominates), 但 12mo median 反而高過 naive — 因為 long horizon 入面有大機會包含至少一個 fat-tail event (COVID +76% / tariff +46%), 拉高 distribution 嘅 middle。即係 positive skew kick-in。

Mean − Median Gap = Skew Detector

Gap 隨 horizon 變大 = 少數巨型 winner 拉高 mean 但 median 仍 representative of typical experience。

• 1mo gap: +1.45pp (modest right skew)
• 3mo gap: +8.88pp (skew emerging)
• 12mo gap: +11.8pp (skew kicked in fully)

呢個 pattern 係 asymmetric payoff mechanism signature — 唔係 luck, 係 system 結構特性 (capped downside via SL/TIME exit, uncapped upside via TIME extension + V-shape capture). 詳 Section 04 嘅 「Risk-Adjusted Anatomy」 同 Section 04 8-trade COVID case study.

How to read the 6 numbers together

1. 揀 horizon — 1mo / 3mo / 12mo · 對應你關心嘅體驗時間
2. Mean = average outcome (跨 many parallel scenarios) · 基本 compound 邏輯
3. Median = typical outcome · 直接讀 distribution, 唔好 mental compound
4. Gap (Mean − Median) = skew strength · positive 大 = upside fat-tail dominant
5. Cross-horizon sanity check: Mean 應接近 (1+r)^N−1 · Median 唔應該 — 反映 distribution shape evolution

Forward Risk Projection (Monte Carlo)

12-month bootstrap, 50,000 runs. 每 run 隨機 resample 12 個 monthly returns from post-2019 89-month pool (independent monthly draws, no autocorrelation modeling). Per-run track terminal NLV + intra-year max drawdown (running peak). Percentiles taken across all runs.

• Caveat 1: Independent bootstrap 假設 monthly returns iid → likely overstate diversification benefit. Real markets cluster volatility (COVID 2020-03 + 2020-04 negatively correlated → bootstrap 不能重現).
• Caveat 2: 7-year sample 含 2 個 era-defining events (COVID 2020 + tariff 2025-04) — outsized influence on MC outcomes.
• Caveat 3:「Mechanism breakdown」5% scenario 為 judgmental, 唔係 derived from data.

SPY Benchmark Computation

Monthly return = (close_t − close_{t-1}) / close_{t-1}
Rolling 12mo = ∏(1 + r_i) − 1, i=t-11..t
Sharpe / Sortino / Calmar — same formula 同 RFR as Metis

SPY ETF used as proxy for SPX total return.

Era Caveat

所有 risk-adjusted 數字反映 post-2019 era. Pre-2019 era CAGR 顯著較低 (詳 Section 03「Per-Trade ROI · 3 Periods」). 機制 alpha 跨 era 穩定但 signal frequency 同 absolute return 因 market microstructure 演化而擴大 — 詳細討論 Section 03.

僅供方法論參考。Sample window / formula 假設可進一步要求討論。