PATH/ADVANCED/quant-portfolio-construction/6 STOPS · 3 hr

Quant Portfolio Construction

From Statistics to Markowitz to Factor Investing

Heavy-math version of Investment Management 101. Same tools, but you're building optimizers, sizing factor bets, and stress-testing the Markowitz framework against real-world frictions (estimation error, regime change, transaction costs).

Quants, systematic-fund analysts, applied-stats grad students.

By the end you will…

Build a covariance matrix and run Markowitz with shrinkage estimators.

Construct multi-asset efficient frontiers under realistic constraints.

Apply CAPM and SML diagnostics to rank a 30-stock universe.

Stress-test risk decomposition under regime change.

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01 / THE JOURNEY

The journey

6 STOPS · 3 hr · ADVANCED

STOP 1 · 25 MIN · Risk & Return

● You are here

Risk & Return Statistics

Compute E(r), σ², σ, Sharpe, skew, kurtosis from probabilistic or historical data.

Build the inputs: μ vector and σ vector for every asset in the universe.

Open lab →

Worked example

10y annualized μ and σ for a 5-asset universe.

Asset	μ (ann.)	σ (ann.)	Sharpe
SPY	12.5%	16.2%	0.53
QQQ	16.8%	21.5%	0.60
IWM	9.1%	20.8%	0.25
AGG	2.4%	5.8%	0.07
GLD	5.5%	16.0%	0.09

STOP 2 · 30 MIN · Risk & Return

Correlation Matrix

Visualize how assets co-move; quantify diversification benefit.

Estimate the covariance matrix — and shrink it toward identity.

Open lab →

Worked example

Sample 5×5 correlation matrix (10y daily).

	SPY	QQQ	IWM	AGG	GLD
SPY	1.00	0.92	0.85	−0.18	0.05
QQQ	0.92	1.00	0.78	−0.15	0.07
IWM	0.85	0.78	1.00	−0.20	0.04
AGG	−0.18	−0.15	−0.20	1.00	0.18
GLD	0.05	0.07	0.04	0.18	1.00

STOP 3 · 20 MIN · Portfolio Construction

Two-Asset Portfolio

Sliders for weights and correlation; see how σ shifts.

Sanity-check the optimizer: 2-asset case has a closed-form solution.

Open lab →

Worked example

Minimum-variance weights for SPY/AGG with ρ = −0.2

w*₁ = (σ²₂ − ρσ₁σ₂) / (σ²₁ + σ²₂ − 2ρσ₁σ₂)

= (0.01 − (−0.2)(0.20)(0.10)) / (0.04 + 0.01 + 0.008)

w*₁ = 0.21 (MVP weight on stock leg)

STOP 4 · 45 MIN · Portfolio Construction

Efficient Frontier

Multi-asset Markowitz mean-variance optimization with tangency portfolio.

Markowitz quadratic programming with constraints + tangency portfolio.

Open lab →

Worked example

5-asset efficient frontier with 100 random Monte Carlo points.

STOP 5 · 30 MIN · Asset Pricing Models

CAPM Calculator

Live β regression, expected return via Capital Asset Pricing Model.

Run β regressions on the universe and validate vs realized α.

Open lab →

Worked example

Asset βs vs market — full universe scatter.

STOP 6 · 30 MIN · Asset Pricing Models

Security Market Line

Plot the SML; find under and overpriced assets via Jensen's α.

Rank the universe by Jensen's α and look for cross-sectional bets.

Open lab →

Worked example

Top 5 alpha names from a 30-stock screen.

Ticker	β	Realized r	CAPM E(r)	α
NVDA	1.55	+28.5%	+13.3%	+15.2%
META	1.32	+18.1%	+11.9%	+6.2%
AAPL	1.20	+13.8%	+11.2%	+2.6%
JPM	1.05	+11.5%	+10.3%	+1.2%
KO	0.65	+ 8.1%	+ 7.9%	+0.2%

02 / CASE STUDIES

Apply what you learned

2 SCENARIOS

Case Study

When Markowitz breaks: estimation error eats Sharpe alive

Take a 25-asset universe. Run the optimizer with a sample-mean μ and sample-covariance Σ on the first 5 years; allocate per the tangency portfolio. Hold for next 5 years. Realized Sharpe usually disappoints by 30-50% versus in-sample. Why? Sample means have huge standard errors (μ_hat for stocks has σ ≈ 5-10% with 5y of data). Mitigations: (1) Black-Litterman with priors, (2) Ledoit-Wolf shrinkage on Σ, (3) max-weight constraints, (4) resampled efficiency. The Efficient Frontier Lab's constraint controls let you compare unconstrained vs constrained-and-shrunk versions side by side.

Try the Efficient Frontier →Try the Correlation Matrix →Try the Risk Decomposition →

Case Study

Building a low-correlation 12-asset portfolio: the multi-strat blueprint

Goal: maximize Sharpe with bounded leverage. Universe = SPY + QQQ + IWM + EFA + EEM + AGG + TLT + GLD + SLV + USO + BTC + ETH. Run the correlation matrix — most pairs in {0.4, 0.9}, but BTC-AGG ≈ 0.05, GLD-EEM ≈ 0.20. Tangency portfolio puts ~30% on BTC, ~15% gold, ~15% TLT, ~10% SPY, rest in EM and small bonds. Implementation note: BTC σ is 60%+, so your tangency weight will be tiny when you include realistic volatility constraints — but the diversification benefit is real. The Efficient Frontier Lab handles up to 15 assets simultaneously.

Try the Efficient Frontier →Try the Correlation Matrix →Try the Risk & Return Statistics →

Free for any classroom or self-study use. Each Lab works standalone too — this path is one suggested ordering of the foundations.

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Quant Portfolio Construction

From Statistics to Markowitz to Factor Investing

Quants, systematic-fund analysts, applied-stats grad students.

By the end you will…

Build a covariance matrix and run Markowitz with shrinkage estimators.

Construct multi-asset efficient frontiers under realistic constraints.

Apply CAPM and SML diagnostics to rank a 30-stock universe.

Stress-test risk decomposition under regime change.

Asset

μ (ann.)

σ (ann.)

Sharpe

SPY

12.5%

16.2%

0.53

QQQ

16.8%

21.5%

0.60

IWM

9.1%

20.8%

0.25

AGG

2.4%

5.8%

0.07

GLD

5.5%

16.0%

0.09

SPY

QQQ

IWM

AGG

GLD

SPY

1.00

0.92

0.85

−0.18

0.05

QQQ

0.92

1.00

0.78

−0.15

0.07

IWM

0.85

0.78

1.00

−0.20

0.04

AGG

−0.18

−0.15

−0.20

1.00

0.18

GLD

0.05

0.07

0.04

0.18

1.00

Ticker

Realized r

CAPM E(r)

NVDA

1.55

+28.5%

+13.3%

+15.2%