5.1.2 The Efficient Frontier

Basic Prerequisites: Calculus & KKT Optimization and Linear Algebra.

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1. Intuition & Practical Objective

The efficient frontier is the set of portfolios with minimum variance for their expected return - the “no-waste” portfolios. Every other feasible portfolio is dominated: it has higher risk, or lower return, or both, than some frontier point. The practical objective of this page is the Merton (1972) closed form: the entire frontier as a single parabola in mean–variance space, plus the closed-form weights of every frontier portfolio.

The punchline is geometric. In $(\sigma ,\mu )$ space the frontier is a smooth concave curve whose optimal branch is exactly the efficient (north-east) half; in $({\sigma }^2,\mu )$ space it is a parabola. Everything on it can be written explicitly with just four scalars derived from ${\Sigma }^{-1}$: the Merton constants $A,B,C,D$. That closed form is what you need to code a frontier that traces any target return, and it is the mathematical spine of the whole topic-folder.

Why formula-first matters. The frontier is a quadratic program whose Lagrange solution is linear in ${\Sigma }^{-1}$. Seeing the closed form is what lets a practitioner understand why it’s fragile (estimation error in $\Sigma$ and $\mu$ enters through ${\Sigma }^{-1}$) and how individual risk preferences slot in (each investor just picks a point on the same curve).

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2. Mathematical Ground Truth & Derivations

The program (Markowitz 1952). Minimize variance at a target expected return $R^{\ast }$, with full budget normalization, allowing short sales (Merton’s setup - all weights unconstrained):

$$\underset{w}{\min }\frac{1}{2}{w}^T\Sigma w\text{s.t.}{w}^T\mu =R^{\ast },w^{T}1=1.$$

Lagrangian: $\frac{1}{2}{w}^T\Sigma w-{\lambda }_1(w^T\mu -R^{\ast })-{\lambda }_2(w^{T}1-1)$. First-order condition ${\nabla }_w\mathcal{L}=0$:

$$\Sigma w={\lambda }_1\mu +{\lambda }_{2}1⟹w={\Sigma }^{-1}({\lambda }_1\mu +{\lambda }_{2}1).$$

The Merton "$A,B,C,D$" trick. Multiply by constraints to solve for the multipliers. Define the four scalars

$$A=1^T{\Sigma }^{-1}\mu ,B={\mu }^T{\Sigma }^{-1}\mu ,C=1^T{\Sigma }^{-1}1,D=BC-A^2>0.$$

(All quadratic forms of the positive-definite ${\Sigma }^{-1}$; $D>0$ by Cauchy–Schwarz applied to $1$ and $\mu$.) The multiplier system collapses to $\left[\begin{array}{cc}C& A\\ A& B\end{array}\right]\left[\begin{array}{c}{\lambda }_2\\ {\lambda }_1\end{array}\right]=\left[\begin{array}{c}1\\ R^{\ast }\end{array}\right]$, giving the frontier portfolio

$$\boxed{w^{\text{f}}(R^{\ast })={\Sigma }^{-1}(\frac{B-A{R}^{\ast }}{D}1+\frac{C{R}^{\ast }-A}{D}\mu )}$$

The frontier equation (parabola). Sandwiching by $w^T\Sigma w$ gives

$${\sigma }^2(R^{\ast })=w^T\Sigma w=\frac{C{R}^{\ast 2}-2A{R}^{\ast }+B}{D}.$$

$R^{\ast }$ and ${\sigma }^2$ trace a parabola ($\frac{d^2{\sigma }^2}{d{R}^{\ast 2}}=\frac{2C}{D}>0$); the efficient branch is the half with $R^{\ast }\ge A/C$ (Merton eq. 12–18). The asymptote slopes in $(\sigma ,\mu )$ space are $\pm \sqrt{D/C}$.

The minimum-variance portfolio (lowest return and lowest variance - the tip of the parabola):

$$\boxed{{\mu }_{\text{mv}}=\frac{A}{C},{\sigma }_{\text{mv}}^2=\frac{1}{C},w_{\text{mv}}=\frac{{\Sigma }^{-1}1}{1^T{\Sigma }^{-1}1}}$$

Two-fund/separation in the all-risky case. Write Merton’s decomposition $w={\Sigma }^{-1}1(\frac{B-A{R}^{\ast }}{D})+{\Sigma }^{-1}\mu (\frac{C{R}^{\ast }-A}{D})$ as $w=R^{\ast }g+h$, i.e. any frontier portfolio is a linear combination of two fixed “mutual funds” $g={\Sigma }^{-1}\mu$-direction and $h$-direction (Merton §III). Equivalently, $w=w_{\text{mv}}+\lambda (w_{\text{tan}}-w_{\text{mv}})$. With Tobin’s (1958) addition of a riskless asset, the two funds may be taken to be cash and one fully-risky efficient portfolio - the result used for the CAPM (next page).

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3. Computational Implementation - tracing the frontier

Stdlib only (Gauss–Jordan inverse + dot products). Computes the frontier weights for three target returns, verifies them against the analytic parabola, and confirms two-fund separation to machine precision.

(The true tangency two-fund combination is carried out with the Sharpe-maximal portfolio on the next page; the section below verifies the all-risky “two-fund” identity using the minimum-variance portfolio and an arbitrary second frontier portfolio.)

The direct weight computation (top) and the analytic parabola (right column) agree to five decimals, and any two frontier funds re-generate every other frontier portfolio to machine precision - two-fund separation, verified numerically.

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4. Failure Modes & First-Principles Breakdowns

1. The parabola assumes free shorting. The closed form allows negative weights ($w_1$ goes from $+0.655$ to $+0.053$ as $R^{\ast }$ rises, but any frontier portfolio can short assets). A long-only investor cannot reach the full parabola - the feasible set is a restricted, piecewise curve (04 · Constraints).
2. The min-variance point is the only input-free point. Moving along the frontier is controlled by $R^{\ast }$, but the shape of the curve is $\Sigma$‘s business. Errors in $\Sigma$ warp the whole frontier, and errors in $\mu$ shift the efficient branch - see 05 · Failure Modes.
3. $D>0$ is what keeps the parabola convex and the program well-posed. It fails exactly when $1$ and $\mu$ are collinear under ${\Sigma }^{-1}$ (e.g. every asset risks identically) - then no unique frontier exists, the numerical symptom being a near-singular 2×2 multiplier system.
4. Frontier ≠ return forecast. Tracing the frontier requires an input $\mu$; the frontier answers “for this return belief, the least risk,” not “what return will happen.” Computer said “frontier,” forecast said otherwise - the optimizer inherits the forecast’s errors.

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5. References

• Merton, Robert C.: An Analytic Derivation of the Efficient Portfolio Frontier, JFQA 7(4):1851–1872 (1972)
• Markowitz, Harry: Portfolio Selection, Journal of Finance 7(1):77–91 (1952)
• Tobin, James: Liquidity Preference as Behavior Toward Risk, RES 25(2):65–86 (1958)
• Elton, Gruber, Brown & Goetzmann, Modern Portfolio Theory and Investment Analysis

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Tangency & CAPM · 04 · Min-Variance & Constraints
• Base: Calculus & KKT Optimization · Linear Algebra