Building Your A-Share Quantitative System from Scratch (Episode 50): Modern Portfolio Theory (MPT) — Mean-Variance Optimization and the Efficient Frontier (with Code and Charts)

【Answer 1】 Term One-Sentence Definition MPT Modern Portfolio Theory. Core idea: Don’t evaluate individual assets in isolation—assess them collectively based on their combined return and risk. Sharpe Ratio Excess return per unit of volatility (i.e., [return − risk-free rate] ÷ volatility). A value of 0.62 means “0.62% excess return per 1% of volatility”—higher is better. Efficient Frontier The upper boundary of the scatter plot formed by plotting all possible portfolios in (risk, return) space—the set of portfolios offering the highest return for a given level of risk. Feasible Set The full scatter plot itself—all portfolios whose weights sum to 1. The efficient frontier equals the upper-left boundary of the feasible set. GMVP Global Minimum Variance Portfolio—the portfolio with the lowest possible volatility among all feasible portfolios, requiring no return forecasts. Tangency Portfolio The portfolio at the point where a line drawn from the risk-free rate is tangent to the efficient frontier. Mathematically equivalent to the maximum-Sharpe-ratio portfolio. CML Capital Market Line—the straight line connecting the risk-free rate point to the tangency portfolio. Two-Fund Separation Theorem Any portfolio on the efficient frontier can be expressed as a weighted combination of the GMVP and the tangency portfolio. Short Selling / Negative Weights Weights 1 Borrowing money to amplify position size. A weight of 3.43 means using ¥1 of capital to control ¥3.43 worth of exposure. Adjusted-for-Splits Pricing / Ex-Dividend & Ex-Right Adjustment Dividends and bonus issues cause artificial price drops on ex-date—not real price declines. Adjusted-for-splits pricing retroactively removes these gaps from historical prices to ensure continuous, calculable returns. In-Sample / Out-of-Sample In-sample = the historical period used to estimate parameters and compute portfolio weights; out-of-sample = entirely new data unseen by the model. In-sample performance is largely meaningless. Estimation Error The difference between estimated parameters (μ̂, Σ̂) derived from finite samples and their true values (μ*, Σ*). This error does not cancel out randomly—it systematically biases results upward. Noise Random fluctuations unrelated to underlying patterns. Much of what appears “optimal” in-sample is merely noise mistaken for signal. Overfitting Memorizing noise as if it were genuine structure. Symptom: excellent in-sample performance, catastrophic out-of-sample failure. Shrinkage Pulling estimates slightly toward a more conservative, simpler target—trading some accuracy for greater stability. Singular / Rank-Deficient / Positive-Definite Positive-definite = matrix is invertible and yields strictly positive portfolio variance—essential for valid optimization; singular / rank-deficient = non-invertible matrix, causing Σ⁻¹ to blow up. Risk Contribution The share of total portfolio volatility attributable to a given asset. Risk parity = equalizing each asset’s risk contribution. Monte Carlo Using random numbers to simulate outcomes extensively. The 1,000 gray dots in Figure 7 of Section 7 were generated by randomly sampling 1,000 sets of portfolio weights to visualize the feasible region. pp Percentage point—a unit of absolute difference between two percentages (e.g., 4.18 pp = 15.63% − 11.45%), not a relative change of 4.18%. 7.05e-09 notation Scientific notation meaning 0.00000000705. In this article, values below e−6 are effectively interpreted as “identical.” Baseline A benchmark. Here, it refers to the equally-weighted portfolio—any smarter strategy must first outperform this. Walk-Forward Rolling out-of-sample testing: optimize weights on an initial window, validate on the immediately following window, then shift the entire window forward by one step and repeat. Regime Market state (bull/bear/choppy). Correlations shift across regimes—assets tend to move together during crises. Frobenius Distance A metric measuring how similar two matrices are: square root of the sum of squared element-wise differences—smaller values indicate greater similarity. MLE Maximum Likelihood Estimation—the standard statistical method for parameter estimation. In this article’s code, “MLE distance” measures how far the unshrunk sample covariance matrix deviates from the true covariance matrix. Sortino / Calmar / MaxDD Three risk-adjusted performance metrics: Sortino modifies Sharpe by penalizing only downside volatility; Calmar Ratio = annualized return ÷ maximum drawdown; MaxDD = the largest peak-to-trough decline.

[Python Quantitative Data Engineering Practical #02] Can’t go beyond candlestick charts? Let’s implement momentum, volatility, and money flow factors in practice!

【Answer 2】 def build_features(df): """Input: cleaned single-stock OHLCV DataFrame (with columns o, h, l, c, v, t) Output: feature matrix as a DataFrame""" df = df.sort_values("t").reset_index(drop=True) for col in ["o", "h", "l", "c", "v"]: df[col] = pd.to_numeric(df[col], errors="coerce") # Momentum df["ret_5"] = df["c"].pct_change(5) * 100 df["ret_10"] = df["c"].pct_change(10) * 100 df["ret_20"] = df["c"].pct_change(20) * 100 # Volatility df["tr"] = np.maximum(df["h"] - df["l"], np.maximum((df["h"] - df["c"].shift(1)).abs(), (df["l"] - df["c"].shift(1)).abs())) df["atr_14"] = df["tr"].rolling(14).mean() df["vol_20"] = df["c"].pct_change().rolling(20).std() * 100 # Money Flow df["obv"] = 0 for i in range(1, len(df)): if df.loc[i, "c"] > df.loc[i-1, "c"]: df.loc[i, "obv"] = df.loc[i-1, "obv"] + df.loc[i, "v"] elif df.loc[i, "c"] < df.loc[i-1, "c"]: df.loc[i, "obv"] = df.loc[i-1, "obv"] - df.loc[i, "v"]

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