14.1 Mean–variance portfolios · M14 Portfolio and going live

Motivation

M06 used the tangency portfolio to derive the CAPM. Here you derive it properly, because every risk-budgeting and strategy-combination rule in this module is a special case or a robust replacement of it.

Setup

$n$ risky assets with expected returns $\mu \in \mathbb R^n$ and positive-definite covariance $\Sigma$ . Portfolio weights $w$ with $\mathbf 1^\top w = 1$ (shorting allowed unless stated). Portfolio mean $w^\top\mu$ , variance $w^\top\Sigma w$ . Define the scalars

A = \mathbf 1^\top\Sigma^{-1}\mathbf 1, \quad B = \mathbf 1^\top\Sigma^{-1}\mu, \quad C = \mu^\top\Sigma^{-1}\mu, \quad D = AC - B^2.

$A > 0$ and $C > 0$ because $\Sigma^{-1}$ is positive definite. By Cauchy–Schwarz in the inner product $\langle x, y\rangle = x^\top\Sigma^{-1}y$ , $B^2 \le AC$ , with equality iff $\mu \propto \mathbf 1$ ; assume $\mu$ is not proportional to $\mathbf 1$ , so $D > 0$ .

14.1.1The minimum-variance frontier#

Proposition 14.1Global minimum variance, GMV

The solution of $\min_w w^\top\Sigma w$ subject to $\mathbf 1^\top w = 1$ is

w_{GMV} = \frac{\Sigma^{-1}\mathbf 1}{A}, \qquad \sigma^2_{GMV} = \frac1A, \qquad \mu_{GMV} = \frac BA. \tag{14.1}

Proof

Lagrangian $\mathcal L = w^\top\Sigma w - 2\gamma(\mathbf 1^\top w - 1)$ . Stationarity: $2\Sigma w - 2\gamma\mathbf 1 = 0$ , so $w = \gamma\Sigma^{-1}\mathbf 1$ . The constraint gives $\gamma A = 1$ . The objective is strictly convex and the constraint affine, so the stationary point is the unique global minimum. Then $w^\top\Sigma w = \gamma^2\mathbf 1^\top\Sigma^{-1}\mathbf 1 = 1/A$ and $w^\top\mu = B/A$ .∎

Note that $\Sigma w_{GMV} = \mathbf 1/A$ : every asset has the same marginal contribution to variance at the GMV portfolio.

Proposition 14.2Frontier portfolios

For a target mean $m$ , the solution of $\min_w w^\top\Sigma w$ subject to $\mathbf 1^\top w = 1$ and $\mu^\top w = m$ is

w(m) = \frac{(C - Bm)\,\Sigma^{-1}\mathbf 1 + (Am - B)\,\Sigma^{-1}\mu}{D}, \qquad \sigma^2(m) = \frac{Am^2 - 2Bm + C}{D}. \tag{14.2}

Proof

$\mathcal L = \tfrac12 w^\top\Sigma w - \lambda_1(\mathbf 1^\top w - 1) - \lambda_2(\mu^\top w - m)$ . Stationarity: $w = \lambda_1\Sigma^{-1}\mathbf 1 + \lambda_2\Sigma^{-1}\mu$ . The two constraints become the linear system

A\lambda_1 + B\lambda_2 = 1, \qquad B\lambda_1 + C\lambda_2 = m,

with determinant $D > 0$ , so $\lambda_1 = (C - Bm)/D$ and $\lambda_2 = (Am - B)/D$ . For the variance, $w^\top\Sigma w = w^\top(\lambda_1\mathbf 1 + \lambda_2\mu) = \lambda_1 + \lambda_2 m = (C - Bm + Am^2 - Bm)/D$ .∎

In $(\sigma, m)$ space the frontier is the hyperbola $\sigma^2 = (Am^2 - 2Bm + C)/D$ , with vertex at the GMV point $(1/\sqrt A, B/A)$ .

14.1.2Tangency and two-fund separation#

Proposition 14.3Tangency portfolio

With a risk-free asset at $r_f$ and $\mu_{GMV} > r_f$ (i.e. $B - Ar_f > 0$ ), the risky portfolio that maximises the Sharpe ratio $(w^\top\mu - r_f)/\sqrt{w^\top\Sigma w}$ over $\mathbf 1^\top w = 1$ is

w_T = \frac{\Sigma^{-1}(\mu - r_f\mathbf 1)}{\mathbf 1^\top\Sigma^{-1}(\mu - r_f\mathbf 1)} = \frac{\Sigma^{-1}(\mu - r_f\mathbf 1)}{B - Ar_f}. \tag{14.3}

Proof

Let $e = \mu - r_f\mathbf 1$ . Allocate $x \in \mathbb R^n$ to risky assets and $1 - \mathbf 1^\top x$ to the risk-free asset. Excess return is $x^\top e$ , variance $x^\top\Sigma x$ . Minimise variance for a target excess return $\pi$ : $\mathcal L = \tfrac12x^\top\Sigma x - \eta(x^\top e - \pi)$ gives $x = \eta\Sigma^{-1}e$ . Every efficient portfolio is therefore a positive multiple of $\Sigma^{-1}e$ (for $\pi > 0$ ). The fully invested risky portfolio on this ray is obtained by normalising: $w_T = \Sigma^{-1}e/\mathbf 1^\top\Sigma^{-1}e$ . The Sharpe ratio is scale-invariant, so it is the same for every point on the ray, and the ray is the efficient set, so $w_T$ maximises it. The normaliser is positive by the assumption $B - Ar_f > 0$ .∎

Corollary 14.4Maximum Sharpe ratio

$S_T^2 = e^\top\Sigma^{-1}e = C - 2Br_f + Ar_f^2$ .

Proof

With $x = \Sigma^{-1}e$ : $x^\top e = e^\top\Sigma^{-1}e$ and $x^\top\Sigma x = e^\top\Sigma^{-1}e$ , so $S = e^\top\Sigma^{-1}e/\sqrt{e^\top\Sigma^{-1}e}$ . Expand $e^\top\Sigma^{-1}e$ .∎

Proposition 14.5Two-fund separation

(i) Without a risk-free asset, every frontier portfolio is an affine combination of any two distinct frontier portfolios. (ii) With a risk-free asset, every mean–variance investor holds a combination of the risk-free asset and the same tangency portfolio $w_T$ ; only the proportion depends on risk aversion.

Proof

(i) By (14.2), $w(m)$ is affine in $m$ : $w(m) = a + bm$ for fixed vectors $a, b$ . For $m_1 \ne m_2$ and any $m$ , put $\alpha = (m - m_2)/(m_1 - m_2)$ ; then $\alpha w(m_1) + (1-\alpha)w(m_2) = a + b(\alpha m_1 + (1-\alpha)m_2) = a + bm = w(m)$ . (ii) In the proof of 14.3 every efficient portfolio is $x = \eta\Sigma^{-1}e = \eta(B - Ar_f)w_T$ : a scalar multiple of $w_T$ in risky assets, the remainder in the risk-free asset. An investor maximising $\mathbb E R - \tfrac\gamma2\operatorname{Var}R$ chooses $x = \Sigma^{-1}e/\gamma$ , again on this ray.∎

14.1.3A four-asset example#

Worked example

Four building blocks for an Indian investor, with illustrative annual inputs (not forecasts): Nifty 50 (μ 12%, σ 16%), gold in INR (9%, 14%), a 10-year G-sec fund (7%, 5%), and a trend-following futures strategy (15%, 20%). Correlations: Nifty–gold 0, Nifty–G-sec 0.1, Nifty–trend 0.3, gold–G-sec 0.1, gold–trend 0.1, G-sec–trend 0. Risk-free 6.5%.

Portfolio	Nifty	Gold	G-sec	Trend	μ	σ	Sharpe
GMV (14.1)	0.045	0.076	0.840	0.039	7.69%	4.68%	0.25
Tangency (14.3)	0.272	0.182	0.231	0.315	11.24%	9.36%	0.507
Equal weight	0.25	0.25	0.25	0.25	10.75%	8.50%	0.500

Here $A = 456.6$ , $B = 35.09$ , $C = 2.890$ , $D = 87.85$ , and $\sqrt{e^\top\Sigma^{-1}e} = 0.507$ . The individual Sharpe ratios are 0.34, 0.18, 0.10 and 0.43; the tangency combination beats every one of them. An investor who wants 6% volatility holds $0.06/0.0936 = 64\%$ in the tangency portfolio and 36% in the risk-free asset. The code (m14_portfolio.py) checks (14.1)–(14.3) against scipy.optimize and checks two-fund separation numerically.

Note how close equal weighting is to the optimum (0.500 against 0.507). That is not a coincidence of this example; it is the subject of the next section.