Mean-Variance Mechanics

Now we compute. With the covariance matrix understood, this lesson turns the optimization problem into actual weights. Two portfolios sit at the heart of everything: the minimum-variance portfolio — the calmest possible mix, which needs no return forecast at all — and the maximum-Sharpe (tangency) portfolio — the best risk-adjusted bet, the point the capital market line touches the frontier. We’ll see where their closed-form solutions come from (the Lagrangian, $\Sigma^{-1}$), and grind through a full two-asset example by hand so the formulas stop being symbols and start being numbers. This is the mechanical core of the course; the later lessons are all about why these clean formulas misbehave on real data.

The minimum-variance portfolio

Analogy. Forget about chasing returns for a moment. Imagine you just want the smoothest ride — the portfolio that wobbles least, whatever the assets happen to earn. That’s the minimum-variance portfolio: the leftmost tip of the efficient frontier, the single point where risk bottoms out.

The problem. Minimize portfolio variance subject only to the budget constraint: $\min_w\ w^\top \Sigma\, w \quad \text{s.t.}\quad \mathbf{1}^\top w = 1.$ Notice μ never appears. To solve it, form the Lagrangian (variance plus a multiplier $\lambda$ times the constraint): $\mathcal{L} = w^\top \Sigma\, w - \lambda(\mathbf{1}^\top w - 1).$ Set the derivative to zero ($2\Sigma w = \lambda \mathbf{1}$), solve for $w$, and impose the constraint. The result is the closed-form minimum-variance weights: $w_{\min} = \frac{\Sigma^{-1}\mathbf{1}}{\mathbf{1}^\top \Sigma^{-1}\mathbf{1}}.$ In words: apply $\Sigma^{-1}$ to a vector of ones, then rescale so the weights sum to one.

Worked example (two assets). Use $\sigma_1 = 20\%$, $\sigma_2 = 10\%$, $\rho = 0.3$, so $\Sigma = \begin{pmatrix} 0.04 & 0.006 \\ 0.006 & 0.01 \end{pmatrix}.$ For two assets there’s a tidy shortcut formula: $w_1 = \frac{\sigma_2^2 - \sigma_1\sigma_2\rho}{\sigma_1^2 + \sigma_2^2 - 2\sigma_1\sigma_2\rho}.$ Plug in: numerator $= 0.01 - 0.006 = 0.004$; denominator $= 0.04 + 0.01 - 2(0.006) = 0.05 - 0.012 = 0.038$. So $w_1 = \frac{0.004}{0.038} \approx 0.105, \qquad w_2 = 1 - 0.105 = 0.895.$ The min-variance portfolio holds about 10.5% in the volatile asset and 89.5% in the calm one — sensible, since the low-vol asset deserves most of the weight. Its variance: $w^\top\Sigma w = 0.105^2(0.04) + 0.895^2(0.01) + 2(0.105)(0.895)(0.006) \approx 0.000441 + 0.008012 + 0.001127 \approx 0.00958$, so volatility $\approx \sqrt{0.00958} \approx 9.79\%$ — below even the calmer asset’s 10%, thanks to diversification.

The maximum-Sharpe (tangency) portfolio

Analogy. Now bring reward back in. Of all the portfolios you could build, which gives the most return per unit of risk? Draw a line from the risk-free rate and pivot it upward until it just kisses the frontier — that touch point is the tangency portfolio, and the steepest such line is the capital market line. Its slope is the highest Sharpe ratio attainable.

The problem. Maximize the Sharpe ratio $\max_w\ \frac{w^\top\mu - r_f}{\sqrt{w^\top\Sigma w}} \quad \text{s.t.}\quad \mathbf{1}^\top w = 1.$ Working through the optimization (define excess returns $\mu^e = \mu - r_f\mathbf{1}$), the tangency weights have the clean closed form $w_{\tan} = \frac{\Sigma^{-1}\mu^e}{\mathbf{1}^\top\Sigma^{-1}\mu^e}.$ Compare it to min-var: the only change is that the ones-vector is replaced by the excess-return vector $\mu^e$. Min-var asks “what’s calmest?”; tangency asks “what’s calmest per unit of expected reward?” — so it tilts toward high-excess-return, low-risk assets.

Worked example (two assets). Keep $\Sigma = \begin{pmatrix}0.04 & 0.006\\ 0.006 & 0.01\end{pmatrix}$, let $\mu = (8\%, 4\%)$ and $r_f = 2\%$, so excess returns $\mu^e = (6\%, 2\%) = (0.06, 0.02)$.

First compute $\Sigma^{-1}$. The determinant is $0.04\times0.01 - 0.006^2 = 0.0004 - 0.000036 = 0.000364$. So $\Sigma^{-1} = \frac{1}{0.000364}\begin{pmatrix} 0.01 & -0.006 \\ -0.006 & 0.04 \end{pmatrix} \approx \begin{pmatrix} 27.47 & -16.48 \\ -16.48 & 109.89 \end{pmatrix}.$ Now $\Sigma^{-1}\mu^e$:

• Row 1: $27.47(0.06) + (-16.48)(0.02) = 1.648 - 0.330 = 1.319$.
• Row 2: $-16.48(0.06) + 109.89(0.02) = -0.989 + 2.198 = 1.209$.

Sum $= 1.319 + 1.209 = 2.528$. Normalize: $w_{\tan} = \left(\frac{1.319}{2.528},\ \frac{1.209}{2.528}\right) \approx (0.522,\ 0.478).$ The tangency portfolio holds about 52% in the high-return asset and 48% in the low-return one — much more aggressive than min-var’s 10.5/89.5 split, because now the optimizer is paying for the first asset’s higher excess return.

The Lagrangian and the structure of the solution

Why a Lagrangian? We’re minimizing (or maximizing) subject to a constraint (weights sum to 1, maybe also hit a target return). The Lagrangian trick converts a constrained problem into an unconstrained one by adding a penalty term $-\lambda(\text{constraint})$; at the optimum, the gradient of the objective is parallel to the gradient of the constraint, which the multiplier $\lambda$ measures.

The general efficient portfolio. For a target return $r^*$ with both constraints ($\mathbf{1}^\top w = 1$ and $w^\top\mu = r^*$), the solution is a linear combination of two fixed portfolios — a beautiful result called two-fund separation: $w^*(r^*) = a + b\, r^*,$ where $a$ and $b$ are fixed vectors built from $\Sigma^{-1}$, $\mathbf{1}$, and $\mu$. As you slide $r^*$, the weights move along a straight line in weight-space, sweeping out the entire efficient frontier. Every efficient portfolio is a blend of just two — say, the minimum-variance portfolio and the tangency portfolio. That’s why those two are the fundamental objects.

The unifying thread. Min-var ($\Sigma^{-1}\mathbf{1}$), tangency ($\Sigma^{-1}\mu^e$), and the general efficient portfolio all route through $\Sigma^{-1}$. That shared dependence is the whole reason a noisy, ill-conditioned $\Sigma$ poisons every one of them — and why the fixes in later lessons (shrinking $\Sigma$, constraining $w$) help all of them at once.

A common trap: confusing the two portfolios

Pitfall — “the optimal portfolio is the one with the lowest risk.” No. The lowest-risk portfolio is the min-variance one, but it’s usually not the best risk-adjusted bet. Unless you happen to want minimum risk above all else, the tangency portfolio (highest Sharpe) is the one to combine with cash to hit your risk target. Min-var is a corner of the frontier; tangency is the pivot point of the capital market line. They coincide only in the degenerate case where all expected excess returns are equal.

Pitfall — “more assets always lowers min-var risk.” Adding assets can lower min-variance risk in theory, but with estimation error a 500-asset min-var portfolio can be worse out of sample than a 30-asset one, because you’ve injected 125,000 noisy parameters. The clean formula assumes Σ is known; reality charges you for every parameter you estimate.

Putting it together

The minimum-variance portfolio ($w = \Sigma^{-1}\mathbf{1}$, normalized) is the calmest mix and depends on Σ alone — dodging the noisy μ estimate, which is why it’s robust and empirically strong. The tangency / maximum-Sharpe portfolio ($w = \Sigma^{-1}\mu^e$, normalized) is the best risk-adjusted bet — the touch point of the capital market line — and needs both μ and Σ. Both come from a Lagrangian that builds the budget constraint into the objective, and both invoke $\Sigma^{-1}$. By two-fund separation, every efficient portfolio is a linear blend of these two, so they’re the fundamental building blocks. The shared reliance on $\Sigma^{-1}$ is exactly why a noisy covariance matrix destabilizes the whole family — the problem we diagnose next.

Next — estimation error and instability: we’ve built the clean machinery, so now we break it. We’ll see exactly why feeding estimated μ and Σ into these $\Sigma^{-1}$ formulas turns the optimizer into an “error maximizer”, why the weights swing violently from sample to sample, and why the razor-sharp “optimal” portfolio so often loses to a dumb equal-weight one out of sample.