Phase 10 - Lesson 10.5

Applications: Portfolio Optimization and Regularization

Convex optimization at work: minimum-variance portfolios and ridge/lasso estimators.

⏱ 55 min● Intermediate🔗 Prereqs: 10.2, 10.3

Builds on: 10.2–10.3 gave optimality and KKT; here they solve real quant problems.
Leads to: Phase 11 (stat learning) and 14 (math finance) build on these formulations.

Learning Objectives

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Formulate the minimum-variance portfolio as a convex quadratic program and solve it via KKT.
Derive the two-asset closed-form minimum-variance weights.
Formulate mean-variance (Markowitz) optimization and interpret the risk-aversion trade-off.
Express ridge and lasso regression as convex problems and explain their effect.
Implement and verify the closed-form two-asset minimum-variance solution.

Key Vocabulary

Covariance matrix $\Sigma$: PSD matrix of asset return covariances; $w^\top\Sigma w$ is portfolio variance.

Minimum-variance portfolio: Weights minimizing $w^\top\Sigma w$ subject to $\mathbf1^\top w=1$ (fully invested).

Mean-variance / Markowitz: Trade off return against variance: $\max\ \mu^\top w-\tfrac{\gamma}{2}w^\top\Sigma w$.

Risk aversion $\gamma$: The weight on variance; larger $\gamma$ gives more conservative portfolios.

Ridge regression: Least squares with an $\ell_2$ penalty $\lambda\|\beta\|_2^2$; shrinks coefficients, convex, closed form.

Lasso: Least squares with an $\ell_1$ penalty $\lambda\|\beta\|_1$; convex, induces sparsity (variable selection).

Intuition & Motivation

Intuition

Two of the most-used tools in quant finance are the same kind of object: a convex quadratic minimized under simple constraints or penalties. The minimum-variance portfolio asks ‘how do I hold $1 across assets to wobble least?’ - a quadratic in the weights. Ridge and lasso ask ‘how do I fit coefficients without overfitting?’ - a quadratic plus a convex penalty. Because all are convex, the solutions are global, stable, and (for min-variance and ridge) closed-form.

The minimum-variance portfolio

Let $\Sigma\succ0$ be the return covariance of $n$ assets and $w$ the portfolio weights. Fully invested means $\mathbf1^\top w=1$. Portfolio variance is $w^\top\Sigma w$. The problem

\[\min_w\ \tfrac12\,w^\top\Sigma w\quad\text{s.t.}\quad \mathbf 1^\top w=1\] (10.9)

is a convex QP with one equality constraint - exactly the KKT worked example of 10.3 with $Q=\Sigma,\ a=\mathbf1$. Its solution is

\[w^\star=\frac{\Sigma^{-1}\mathbf 1}{\mathbf 1^\top\Sigma^{-1}\mathbf 1},\qquad \sigma_{\min}^2=\frac{1}{\mathbf 1^\top\Sigma^{-1}\mathbf 1}.\] (10.10)

Worked Example - Two-asset closed form

Let assets have variances $\sigma_1^2,\sigma_2^2$ and correlation $\rho$, so $\Sigma=\begin{pmatrix}\sigma_1^2&\rho\sigma_1\sigma_2\\ \rho\sigma_1\sigma_2&\sigma_2^2\end{pmatrix}$.

Write $w=(w_1,1-w_1)$ and minimize $v(w_1)=w_1^2\sigma_1^2+(1-w_1)^2\sigma_2^2+2w_1(1-w_1)\rho\sigma_1\sigma_2$.

Set $v'(w_1)=0$ and solve: $w_1^\star=\dfrac{\sigma_2^2-\rho\sigma_1\sigma_2}{\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2}$, $w_2^\star=1-w_1^\star$.

Sanity checks: if $\sigma_1=\sigma_2$ then $w_1^\star=1/2$ (symmetry); if $\rho=0$, weights tilt inversely to variance ($w_1\propto1/\sigma_1^2$).

Mean–variance (Markowitz)

Investors want return too. Maximize expected return penalized by risk:

\[\max_w\ \mu^\top w-\tfrac{\gamma}{2}\,w^\top\Sigma w\quad\text{s.t.}\quad\mathbf 1^\top w=1,\] (10.11)

a concave (hence convex-minimization) problem. Its KKT solution $w^\star=\tfrac1\gamma\Sigma^{-1}(\mu-\nu\mathbf1)$ traces the efficient frontier as $\gamma$ varies: large $\gamma$ (very risk-averse) approaches the minimum-variance portfolio (10.10); small $\gamma$ chases return.

Regularization as convex optimization

The same machinery tames regression. Ordinary least squares $\min_\beta\|y-X\beta\|_2^2$ overfits when features are many or collinear. Add a convex penalty:

Ridge: $\min_\beta\|y-X\beta\|_2^2+\lambda\|\beta\|_2^2$. Strictly convex; closed form $\hat\beta=(X^\top X+\lambda I)^{-1}X^\top y$. Shrinks all coefficients, fixes ill-conditioning (note the $+\lambda I$ improves the condition number - see 10.4).
Lasso: $\min_\beta\|y-X\beta\|_2^2+\lambda\|\beta\|_1$. Convex but non-smooth; drives some coefficients exactly to zero, performing variable selection. Solved by KKT/subgradient or coordinate descent.

Both penalties are convex, so the estimators are global optima - statistically stable and reproducible. This connects convex optimization directly to statistical learning (Phase 11).

Interactive: solve a two-asset portfolio

Common Mistakes to Avoid

Forgetting the budget constraint $\mathbf1^\top w=1$; the unconstrained variance minimizer is $w=0$ (invest nothing).
Assuming minimum-variance weights are nonnegative. Without a no-short constraint, $w_i$ can be negative (a short position).
Inverting a near-singular $\Sigma$ from noisy data; regularize/shrink $\Sigma$ (echoing ridge) for stable weights.
Treating lasso like ridge - lasso is non-smooth, so plain gradient descent needs subgradients/proximal steps.

Quant Practitioner Tips

Estimation error in $\Sigma$ and $\mu$ dominates real portfolio risk; shrinkage (a convex regularizer on $\Sigma$) often beats the raw optimum.
Ridge = $\ell_2$ shrinkage (keeps all vars, improves conditioning); lasso = $\ell_1$ (sparse, selects vars). The choice is a modeling decision.
Adding a no-short constraint $w\ge0$ turns the QP into one needing KKT with inequality multipliers - still convex, solvable by projection/QP solvers.

Knowledge Check

Q1 Medium

The minimum-variance portfolio weights (fully invested) are:

$\Sigma^{-1}\mu$

$\Sigma^{-1}\mathbf1/(\mathbf1^\top\Sigma^{-1}\mathbf1)$

$\mathbf1/n$

$\Sigma\mathbf1$

Q2 Medium

Ridge regression differs from OLS by:

Using an $\ell_1$ penalty

Adding $\lambda\|\beta\|_2^2$, giving $\hat\beta=(X^\top X+\lambda I)^{-1}X^\top y$

Being non-convex

Removing the intercept

Q3 Easy

For two assets with equal variances and $\rho\lt 1$, the minimum-variance weight on asset 1 is:

$1$

$1/2$

$0$

Undefined

Practical Exercise

Two assets: $\sigma_1=0.10,\ \sigma_2=0.20,\ \rho=0.25$. (a) Compute the minimum-variance weights. (b) Compute the resulting portfolio variance. (c) Is either weight a short position?

▶ Show full solution

(a) $\text{cov}=\rho\sigma_1\sigma_2=0.25\cdot0.10\cdot0.20=0.005$. $w_1=\dfrac{\sigma_2^2-\text{cov}}{\sigma_1^2+\sigma_2^2-2\text{cov}}=\dfrac{0.04-0.005}{0.01+0.04-0.01}=\dfrac{0.035}{0.04}=0.875$, $w_2=0.125$.

(b) $\sigma^2=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\text{cov}=0.875^2(0.01)+0.125^2(0.04)+2(0.875)(0.125)(0.005)$ $=0.007656+0.000625+0.001094\approx0.009375$, so $\sigma\approx0.0968$ (below asset 1's 0.10).

(c) Both weights are positive (0.875 and 0.125), so no shorting. Diversification lowered risk below the least-risky asset alone.

After the reveal, answer for yourself: How would a negative correlation $\rho\lt 0$ change the achievable minimum variance?

Lesson Summary

Minimum-variance and Markowitz portfolios are convex quadratic programs whose KKT solutions are closed-form: $w^\star=\Sigma^{-1}\mathbf1/(\mathbf1^\top\Sigma^{-1}\mathbf1)$ for minimum variance, with the two-asset case reducing to a simple ratio. Ridge ($\ell_2$) and lasso ($\ell_1$) regularization are the same convex-optimization idea applied to regression - shrinking or selecting coefficients - yielding stable, global estimators that bridge to statistical learning.

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: Give the minimum-variance portfolio formula and the two-asset weight.
A: $w^\star=\Sigma^{-1}\mathbf1/(\mathbf1^\top\Sigma^{-1}\mathbf1)$; two-asset $w_1=(\sigma_2^2-\rho\sigma_1\sigma_2)/(\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2)$.

Q: How do ridge and lasso differ, and why are both convex?
A: Ridge adds $\lambda\|\beta\|_2^2$ (smooth, shrinks all, closed form); lasso adds $\lambda\|\beta\|_1$ (non-smooth, induces sparsity). Both add a convex penalty to a convex loss, so the problems are convex.

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

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