Portfolio Analysis in pandas: Correlation and Covariance Matrices, Weighted Portfolio Volatility and a Rebalancing Simulation

What you will learn

• How to align several price series into one returns DataFrame and compute .cov() and .corr() correctly annualised.
• The matrix formula for portfolio volatility, w.T @ Σ @ w, and how to read each asset's contribution to risk.
• How to simulate a buy-and-hold portfolio that drifts versus one rebalanced on a calendar.
• Which numerical traps (misaligned dates, mixed currencies, look-ahead in the covariance window) produce wrong answers.

Explanation

Correlation and portfolio heat (Risk & Trade Management, lesson 6), combining two systems (Systematic Trading, lesson 10) and rebalancing rules (ETFs, lesson 10) were taught elsewhere; this lesson implements them for any number of assets in code.

From prices to a returns matrix

Load each instrument's adjusted close (lesson 4) into one DataFrame, one column per ticker: prices = pd.concat(series_dict, axis=1). Holidays that differ between markets create NaN rows; decide explicitly between prices.dropna() (common days only) and prices.ffill() (carry the last price forward). Then rets = prices.pct_change().dropna(). Convert everything to one currency first; a US ETF held by a euro investor must be translated with the EUR/USD series, or its volatility and correlations describe the wrong thing.

Covariance, correlation and annualisation

rets.cov() is the sample covariance matrix of daily returns; multiply by 252 to annualise (variance scales with time). rets.corr() is unit-free and needs no scaling. The diagonal holds each asset's variance, so np.sqrt(np.diag(cov_a)) gives annualised volatilities. The relation Σ_ij = ρ_ij × σ_i × σ_j lets you build a matrix from volatilities and correlations to test a hypothetical scenario such as "correlations rise to 0.8 in a crisis".

Portfolio volatility and risk contributions

• w = np.array([0.5, 0.3, 0.2]) # weights, sum to 1
• cov_a = rets.cov().values * 252
• port_var = w @ cov_a @ w
• port_vol = np.sqrt(port_var)
• marginal = cov_a @ w # d(var)/d(w)
• risk_contrib = w * marginal / port_var # shares that sum to 1

@ is matrix multiplication. risk_contrib tells you what fraction of the portfolio's variance each asset is responsible for; it is usually far from the capital weight, and that gap is the starting point of risk budgeting (Portfolio Construction, lesson 1).

A rebalancing simulation

Buy-and-hold lets weights drift: winners grow into a larger share. Calendar rebalancing resets to targets at fixed dates. In pandas, resample returns to the rebalancing frequency and loop over the periods:

• period_rets = (1 + rets).resample('ME').prod() - 1 # monthly returns
• target = np.array([0.6, 0.4]); hold = 10_000 * target; path = []
• for _, r in period_rets.iterrows():
• hold = hold * (1 + r.values) # grow each sleeve
• path.append(hold.sum())
• hold = hold.sum() * target # rebalance to targets

For the drifting version, delete the last line. Use 'QE' for quarterly and 'YE' for yearly (pandas 2.2 and later; older versions use 'M', 'Q', 'A'). Turnover of a rebalance is abs(new_hold - old_hold).sum(); multiply by a cost rate such as 0.1 % and subtract. In a German taxable account rebalancing also realises gains; check the tax effect with a Steuerberater.

Worked example

Hypothetical example with invented inputs; fees, spreads and taxes excluded.

Part 1: three-asset volatility. Annualised volatilities 20 %, 10 %, 15 %; correlations ρ₁₂ = 0.2, ρ₁₃ = 0.6, ρ₂₃ = 0.1; weights 50 / 30 / 20 %.

1. Covariances: Σ₁₂ = 0.2 × 0.20 × 0.10 = 0.0040; Σ₁₃ = 0.6 × 0.20 × 0.15 = 0.0180; Σ₂₃ = 0.1 × 0.10 × 0.15 = 0.0015. Diagonal: 0.0400, 0.0100, 0.0225.
2. Portfolio variance = Σ wᵢ wⱼ Σᵢⱼ = 0.25 × 0.04 + 0.09 × 0.01 + 0.04 × 0.0225 + 2 × (0.5 × 0.3 × 0.004 + 0.5 × 0.2 × 0.018 + 0.3 × 0.2 × 0.0015) = 0.0100 + 0.0009 + 0.0009 + 2 × (0.0006 + 0.0018 + 0.00009) = 0.01678.
3. Portfolio volatility = √0.01678 = 12.95 %, against a weighted average of volatilities of 16.0 %; the difference is the diversification benefit.
4. Risk contributions: asset 1 carries 73.9 % of the variance with 50 % of the capital; asset 2 carries 9.5 % with 30 %; asset 3 carries 16.6 % with 20 %.

Part 2: drift versus rebalancing. €10,000 split 60 / 40 between asset A and asset B. Annual returns: A +20 %, −10 %, +15 %; B +2 %, +4 %, +1 %.

1. Drift: A: 6,000 × 1.20 × 0.90 × 1.15 = €7,452; B: 4,000 × 1.02 × 1.04 × 1.01 = €4,285.63. Total €11,737.63; A's weight has drifted to 63.5 %.
2. Yearly rebalancing: year 1 total = 7,200 + 4,080 = €11,280 → reset to 6,768 / 4,512; year 2 = 6,091.20 + 4,692.48 = €10,783.68 → reset; year 3 = 7,440.74 + 4,356.61 = €11,797.35.
3. Rebalancing ended €59.72 higher because it sold A after +20 % and bought back after −10 %. With a steadily trending A, drift would have won; rebalancing controls risk, it does not guarantee return.

Common mistake

Practice activity

On paper: two assets with annualised volatilities 18 % and 6 %, correlation 0.3, weights 70 / 30 %. (a) Build the 2 × 2 covariance matrix. (b) Compute portfolio variance and volatility. (c) Compute each asset's risk contribution as a share of variance. (d) Recompute the volatility with the correlation at 0.9 and state in one sentence why this matters.

Key takeaways

• Align dates and currencies before computing returns; decide explicitly how to treat missing days.
• Annualise covariance by multiplying by the periods per year; correlation needs no scaling.
• Portfolio volatility is √(wᵀ Σ w); risk contributions w × (Σw) / variance show where the risk really sits.
• Simulate rebalancing by resampling returns and resetting holdings; estimate covariance only from data available at the time.

Next step

Lesson 12 automates the daily routine: a scheduled script that fetches data, computes signals and writes a report, with API keys and rate limits handled properly.