Portfolio Frontier Mathematics

Lorenzo Naranjo

Fall 2026

Introduction

The Tradeoff

• In modern portfolio theory, the tradeoff between risk and return determines investment decisions.
• It is customary to use the variance of investment returns as a proxy for risk, whereas expected return quantifies its reward.
• Even though the variance might not be the best statistic to quantify the risk embedded in a risk-averse utility function, it is the proper statistical moment to represent linear pricing functionals without arbitrage opportunities (e.g. Cochrane 2009).
  • Mean-variance analysis is always right.
  • Might not be the best way to map utility functions.
  • Investors might prefer inefficient portfolios that have other characteristics such as skewness, kurtosis, different exposures to other sources of consumption growth risk, etc.

Diversification

• Markowitz (1952) was the first to put into solid foundations the analysis of the investment problem.
• He showed that by combining risky assets, it is possible to diversify away part of the risk contributed by each asset.
• The remaining portfolio risk cannot be diversified, i.e., it is systematic.
• Therefore, for a given level of target expected return, it is then possible to find a combination of the risky assets that would achieve a minimum level of variance.

The Investment Opportunity Set

• The resulting mean-variance analysis is a cornerstone of modern finance.
• Without short-selling constraints, a region inside an hyperbola in (\mu, \sigma) space describes the investment opportunity set available to investors.
• All the portfolios that achieve the minimum level of risk for a given expected rate of return determine the hyperbola, which is itself known as a minimum variance frontier (MVF).
• The analysis follows Chapter 3 in Huang and Litzenberger (1988), although some derivations are original.

Risky Assets

Portfolios of Risky Assets

• The investment opportunity set of the economy is spanned by n-risky assets with returns \{ r_{1}, r_{2}, \ldots, r_{n}\}.
• For the moment there is no risk-free asset to invest, which is equivalent to say that no combination of the risky assets can synthesize a portfolio with zero variance.
• Investors can form portfolios by buying or selling the risky assets in proportions \{w_{1}, w_{2}, \ldots, w_{n}\}.
• These portfolio weights must satisfy \sum_{i = 1}^{n} w_{i} = 1 to guarantee that all the funds are invested in the risky assets.
  • The sign and magnitude of each weight is unrestricted.
  • A positive weight means that the investor is buying the asset whereas a negative weight means that the investor is selling short the asset.

Notation

• With this notation, the return r of a portfolio is r = \sum_{i = 1}^{n} w_{i} r_{i}.
• Throughout these notebooks we use matrix notation to simplify mathematical expressions.
  • We use lowercase boldface symbols to denote vectors and uppercase boldface symbols for matrices.
  • Given a matrix \mathbf{A} we write its transpose by \mathbf{A}'.

Notation (cont’d)

• We denote by \mathbf{r} = (r_{1}, r_{2}, \ldots, r_{n})' the vector of risky returns.
• Similarly, we can write the vector of portfolio weights as \mathbf{w} = (w_{1}, w_{2}, \ldots, w_{n})'.

Portfolio Returns

• Therefore, the return of a portfolio can be written more compactly as r = \sum_{i = 1}^{n} w_{i} r_{i} = \mathbf{w}' \mathbf{r}. \tag{1}
• If we denote by \pmb{\iota} = \begin{pmatrix} 1 & 1 & \cdots & 1 \end{pmatrix}' a conformal vector of ones, we can write more compactly the restriction that the sum of the weights equals one \mathbf{w}' \pmb{\iota} = \sum_{i = 1}^{n} w_{i} = 1.

Expected Returns

• Portfolio optimization involves minimizing the portfolio variance subject to an expected return constraint.
• We write the vector of expected returns of the risky assets as \mathbf{e} = (\operatorname{E}(r_{1}), \operatorname{E}(r_{2}), \ldots, \operatorname{E}(r_{n}))', where \operatorname{E}(\cdot) denotes the expectation operator.

Covariance Matrix

• The covariance matrix of returns is denoted by \mathbf{V} and defined as \mathbf{V} = \begin{pmatrix} \sigma_{11} & \sigma_{12} & \cdots & \sigma_{1n} \\ \sigma_{21} & \sigma_{22} & \cdots & \sigma_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{n1} & \sigma_{n2} & \cdots & \sigma_{nn} \\ \end{pmatrix}, where \mathbf{V}_{ij} = \sigma_{ij} = \operatorname{Cov}(r_{i}, r_{j}).
• The covariance matrix is symmetric since \operatorname{Cov}(r_{i}, r_{j}) = \operatorname{Cov}(r_{j}, r_{i}).
• It is also positive semidefinite since for any \mathbf{y} \in \mathbb{R}^{n} we have that \begin{aligned} 0 \leq \operatorname{E}\left((\mathbf{r} - \mathbf{e})' \mathbf{y}\right)^{2} & = \operatorname{E}\left( \mathbf{y}' (\mathbf{r} - \mathbf{e})(\mathbf{r} - \mathbf{e})' \mathbf{y} \right) \\ & = \mathbf{y}' \operatorname{E}\left( (\mathbf{r} - \mathbf{e})(\mathbf{r} - \mathbf{e})' \right) \mathbf{y} \\ & = \mathbf{y}' \mathbf{V} \mathbf{y}. \\ \end{aligned}

Definition 1 (The Economy) We consider an economy spanned by n-risky assets with returns \mathbf{r} such that \mathbf{V}^{-1} exists. The returns r of any portfolio can be expressed as r = \mathbf{w}' \mathbf{r}.

Assumptions

• We will assume that \mathbf{V} is invertible to guarantee that the n-risky basis assets are linearly independent, i.e. that no combination of them generates a risk-free asset.
  • It is not possible to combine the risky assets to build a portfolio that has zero variance.
• Since the covariance matrix is symmetric, all its eigenvalues are real numbers.
  • Therefore, being positive semidefinite implies that all its eigenvalues are non-negative.
• Assuming that \mathbf{V} is invertible implies that all eigenvalues are different from zero.
  • All the eigenvalues of the covariance matrix are positive, implying that \mathbf{V} is positive definite, i.e. \mathbf{y}' \mathbf{V} \mathbf{y} > 0 for any \mathbf{y} \neq \mathbf{0}.

Property 1 (Portfolio Statistics) For portfolios of risky assets we have the following relations \begin{aligned} \operatorname{E}(r_{p}) & = \mathbf{w}_{p}' \mathbf{e}, \\ \sigma^{2}(r_{p}) & = \mathbf{w}_{p}' \mathbf{V} \mathbf{w}_{p}, \\ \operatorname{Cov}(r_{p}, r_{q}) & = \mathbf{w}_{q}' \mathbf{V} \mathbf{w}_{p}. \end{aligned}

The Minimum Variance Frontier

• We want to solve the following problem: \begin{aligned} \min_{\mathbf{w}} \quad & \dfrac{1}{2} \mathbf{w}' \mathbf{V} \mathbf{w} \\ \textrm{s.t.} \quad & \begin{aligned} \mathbf{w}' \mathbf{e} & = \mu \\ \mathbf{w}' \pmb{\iota} & = 1 \end{aligned} \end{aligned}
• For this we form the Lagrangian \mathcal{L} = \dfrac{1}{2} \mathbf{w}' \mathbf{V} \mathbf{w} + \lambda_{1} (\mu - \mathbf{w}' \mathbf{e}) + \lambda_{2} (1 - \mathbf{w}' \pmb{\iota}).

First Order Conditions

The first order conditions for this problem are \begin{aligned} \dfrac{\partial \mathcal{L}}{\partial \mathbf{w}} & = \mathbf{V} \mathbf{w} - \lambda_{1} \mathbf{e} - \lambda_{2} \pmb{\iota} = 0 \\ \dfrac{\partial \mathcal{L}}{\partial \lambda_{1}} & = \mu - \mathbf{w}' \mathbf{e} = 0 \\ \dfrac{\partial \mathcal{L}}{\partial \lambda_{2}} & = 1 - \mathbf{w}' \pmb{\iota} = 0 \end{aligned} \tag{2}

Solving for the Lagrange Multipliers

• From the first FOC in (2) we get that \mathbf{w} = \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \begin{pmatrix} \lambda_{1} \\ \lambda_{2} \end{pmatrix}. \tag{3}
• Using (3) in the second an third FOC in (2) we can compute \begin{pmatrix} \mu \\ 1 \end{pmatrix} = \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix}' \mathbf{w} = \underbrace{\begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix}' \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix}}_{\pmb{\Gamma}} \begin{pmatrix} \lambda_{1} \\ \lambda_{2} \end{pmatrix}.
• Thus \begin{pmatrix} \lambda_{1} \\ \lambda_{2} \end{pmatrix} = \pmb{\Gamma}^{-1} \begin{pmatrix} \mu \\ 1 \end{pmatrix}. \tag{4}

Solving for the Portfolio Variance

• Using (4) in (3), \mathbf{w} = \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu \\ 1 \end{pmatrix}. \tag{5}
• Hence, \begin{aligned} \sigma^{2} & = \mathbf{w}' \mathbf{V} \mathbf{w} \\ & = \begin{pmatrix} \mu & 1 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix}' \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu \\ 1 \end{pmatrix} \\ & = \begin{pmatrix} \mu & 1 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu \\ 1 \end{pmatrix}. \end{aligned} \tag{6}

Writing \pmb{\Gamma} and \pmb{\Gamma}^{-1} Explicitly

• Note that we can write \pmb{\Gamma} = \begin{pmatrix} B & A \\ A & C \end{pmatrix}, where A = \pmb{\iota}' \mathbf{V}^{-1} \mathbf{e}, B = \mathbf{e}' \mathbf{V}^{-1} \mathbf{e} and C = \pmb{\iota}' \mathbf{V}^{-1} \pmb{\iota}.
• Therefore, \pmb{\Gamma}^{-1} = \frac{1}{D} \begin{pmatrix} C & -A \\ -A & B \end{pmatrix}, where D = BC - A^{2}.

Rewriting the Portfolio Variance

• We can then express (6) as \sigma^{2} = \dfrac{1}{D} (B - 2 A \mu + C \mu^{2}), which can be written as \frac{\sigma^{2}}{1/C} - \frac{(\mu - A/C)^{2}}{D/C^{2}} = 1. \tag{7}
• Equation (7) describes an hyperbola in (\mu, \sigma) space with vertex \left( 1/\sqrt{C}, A/C \right) and asymptotes \mu = A/C \pm \sigma \sqrt{D/C}.

Property 2 (The Minimum Variance Frontier) The minimum variance frontier contains all the portfolios that achieve the minimum possible variance for a given expected return. It determines the frontier of the investment opportunity set. It is an hyperbola characterized in (\mu, \sigma) space by \frac{\sigma^{2}}{1/C} - \frac{(\mu - A/C)^{2}}{D/C^{2}} = 1, where A = \pmb{\iota}' \mathbf{V}^{-1} \mathbf{e}, B = \mathbf{e}' \mathbf{V}^{-1} \mathbf{e} and C = \pmb{\iota}' \mathbf{V}^{-1} \pmb{\iota}.

Spanning the Minimum Variance Frontier

• Consider two frontier portfolios, p and q, with expected returns \mu_{p} and \mu_{q}, and a portfolio b consisting of 1 - \beta of p and \beta of q.
• The portfolio weights of p and q are given by (5) and equal to \begin{aligned} \mathbf{w}_{p} & = \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix} \\ \mathbf{w}_{q} & = \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{q} \\ 1 \end{pmatrix} \\ \end{aligned}
• Portfolio b has an expected return of \mu_{b} = (1 - \beta) \mu_{p} + \beta \mu_{q}, and its portfolio weights are given by \mathbf{w}_{b} = \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{b} \\ 1 \end{pmatrix}, which shows that b is also a frontier portfolio.

Property 3 (Spanning) The investment opportunity set has dimension n and is spanned by n risky assets such that their covariance matrix is invertible. The minimum variance frontier has dimension 2 and is spanned by any two different frontier portfolios.

The Minimum Variance Portfolio

• We solve the problem: \begin{aligned} \min_{\mathbf{w}} \quad & \dfrac{1}{2} \mathbf{w}' \mathbf{V} \mathbf{w} \\ \textrm{s.t.} \quad & \mathbf{w}' \pmb{\iota} = 1 \end{aligned}
• For this we form the Lagrangian \mathcal{L} = \dfrac{1}{2} \mathbf{w}' \mathbf{V} \mathbf{w} + \lambda (1 - \mathbf{w}' \pmb{\iota}).

Solving for Portfolio Weights

• The first order conditions for this problem are \begin{aligned} \dfrac{\partial \mathcal{L}}{\partial \mathbf{w}} & = \mathbf{V} \mathbf{w} - \lambda \pmb{\iota} = 0, \\ \dfrac{\partial \mathcal{L}}{\partial \lambda} & = 1 - \mathbf{w}' \pmb{\iota} = 0. \end{aligned}
• Hence, \begin{aligned} \mathbf{w}_{mv} & = \lambda \mathbf{V}^{-1} \pmb{\iota}, \\ 1 = \mathbf{w}_{mv}' \pmb{\iota} & = \lambda \pmb{\iota}' \mathbf{V}^{-1} \pmb{\iota} = \lambda C, \\ \mathbf{w}_{mv} & = \dfrac{1}{C} \mathbf{V}^{-1} \pmb{\iota}. \end{aligned}

The Covariance of MV with Any Other Portfolio

• The covariance of the minimum variance portfolio with any other portfolio i is given by \operatorname{Cov}(r_{i}, r_{mv}) = \mathbf{w}_{i}' \mathbf{V} \mathbf{w}_{mv} = \mathbf{w}_{i}' \mathbf{V} \left( \frac{1}{C} \mathbf{V}^{-1} \pmb{\iota} \right) = \frac{1}{C}.
• Since \operatorname{V}(r_{mv}) = \operatorname{Cov}(r_{mv}, r_{mv}) = 1 / C, we have that \operatorname{Cov}(r_{i}, r_{mv}) = \operatorname{V}(r_{mv}).
• Note that the standard deviation of the minimum variance portfolio corresponds to the \sigma-coordinate of the vertex of the hyperbola described by (5).

Property 4 (Minimum Variance Portfolio) There is a portfolio that minimizes the variance among all portfolios called the minimum variance portfolio. The covariance of the minimum variance portfolio with any other portfolio, not necessarily a frontier portfolio, is always the same and equal to the variance of the minimum variance portfolio.

A Really Cool Result

• Consider an arbitrary frontier portfolio with expected return \mu_{p} and pick any asset or portfolio with expected return \mu_{i}.
• The covariance between r_{p} and r_{i} is \begin{aligned} \operatorname{Cov}(r_{i}, r_{p}) & = \mathbf{w}_{i}' \mathbf{V} \mathbf{w}_{p} \\ & = \mathbf{w}_{i}' \mathbf{V} \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix} \\ & = \begin{pmatrix} \mu_{i} & 1 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix} \\ \end{aligned}
• This shows that two assets with the same expected return will have the same covariance with a given frontier portfolio p.
  • The expected return of any asset is determined by how much it covaries with a frontier portfolio regardless of its total risk.

Residuals

• To look at this result in more detail, consider the frontier portfolio r_{p, i} with the same expected return as asset i. Define the residual \varepsilon_{i} = r_{i} - r_{p, i}. \tag{8}
• By construction, the residual has zero mean \operatorname{E}(\varepsilon_{i}) = \operatorname{E}(r_{i}) - \operatorname{E}(r_{p, i}) = 0, and if we pick any frontier portfolio p we also have that \operatorname{Cov}(r_{p}, \varepsilon_{i}) = \operatorname{Cov}(r_{p}, r_{i}) - \operatorname{Cov}(r_{p}, r_{p, i}) = 0.
• Since we just saw that the expected return of an asset depends only on how it covaries with a frontier portfolio, the residual is not priced, hence the name idiosyncratic risk.

Property 5 (Idiosyncratic Risk) Consider an asset whose expected return is \mu_{i}. We define the idiosyncratic risk of the asset as the difference between its return, and the return of a frontier portfolio that has the same expected return. The idiosyncratic risk of any asset is uncorrelated with all frontier portfolios.

Property 6 (Zero-Covariance) For a given frontier portfolio p with expected return \mu_{p} and variance \sigma_{p}^2, we can always find (except for the minimum variance portfolio) another frontier portfolio z with expected return \mu_{z} that is uncorrelated with p, i.e.  \operatorname{Cov}(r_{z}, r_{p}) = 0. Just draw a line that is tangent to the minimum-variance frontier at the point (\mu_{p}, \sigma_{p}). The intercept of this line with the vertical axis gives \mu_{z}.

Defining Beta

• Let \mu_{i} = \operatorname{E}(r_{i}). We start by re-writing (8) as r_{i} = r_{p, i} + \varepsilon_{i}, and note that r_{p, i} is a frontier portfolio.
• Pick an arbitrary frontier portfolio p (different from the MVP) with expected return \mu_{p}, and find its associated zero-covariance frontier portfolio z.
• We then form a portfolio composed of both frontier portfolios such that r_{p, i} = (1 - \beta_{i}) r_{z} + \beta_{i} r_{p}, with \beta_{i} = \frac{\mu_{i} - \mu_{z}}{\mu_{p} - \mu_{z}}.

Redefining Beta

• The choice for \beta_{i} guarantees that \operatorname{E}(r_{p, i}) = \mu_{i}.
• Note that \varepsilon_{i} is uncorrelated with both r_{z} and r_{p}, and that r_{z} and r_{p} are also uncorrelated.
• The covariance of r_{i} and r_{p} is given by \begin{aligned} \operatorname{Cov}(r_{i}, r_{p}) & = \operatorname{Cov}((1 - \beta_{i}) r_{z} + \beta_{i} r_{p} + \varepsilon_{i}, r_{p}) \\ & = (1 - \beta_{i}) \underbrace{\operatorname{Cov}(r_{z}, r_{p})}_{0} + \beta_{i} \underbrace{\operatorname{Cov}(r_{p}, r_{p})}_{\sigma^{2}(r_{p})} + \underbrace{\operatorname{Cov}(\varepsilon_{i}, r_{p})}_{0} \\ & = \beta_{i} \sigma^{2}(r_{p}), \end{aligned} which yields that \beta_{i} = \dfrac{\operatorname{Cov}(r_{i}, r_{p})}{\sigma^{2}(r_{p})}.

Property 7 (Beta-pricing with Frontier Portfolios) Frontier portfolios contain all the information we need to price assets and carry all the systematic risk of the economy. Just pick any frontier portfolio p with return r_{p}, and compute its associated zero-covariance portfolio r_{z}. Then for any asset or portfolio i we have that r_{i} = (1 - \beta_{i}) r_{z} + \beta_{i} r_{p} + \varepsilon_{i}, where \begin{gather*} \beta_{i} = \dfrac{\operatorname{Cov}(r_{i}, r_{p})}{\sigma^{2}(r_{p})}, \\ \operatorname{E}(\varepsilon_{i}) = \operatorname{Cov}(r_{p}, r_{z}) = \operatorname{Cov}(r_{p}, \varepsilon_{i}) = \operatorname{Cov}(r_{z}, \varepsilon_{i}) = 0. \end{gather*}

Adding a Risk-Free Asset

A Portfolio with a Risk-Free Asset

• In this section we add a risk-free asset denoted by r_{f} to the investment opportunity set.
• Investors can allocate \mathbf{w} to the risky assets and 1 - \mathbf{w}' \pmb{\iota} to the risk-free asset, so that the returns of any portfolio can be expressed as \begin{aligned} r & = \sum_{i = 1}^{n} w_{i} r_{i} + (1 - \sum_{i = 1}^{n} w_{i}) r_{f} \\ & = \mathbf{w}' \mathbf{r} + (1 - \mathbf{w}' \pmb{\iota}) r_{f}. \\ \end{aligned}
• Note that the weights of the risky assets do not need to sum up to one since any amount not invested in the risky assets can be invested in the risk-free asset.

Definition 2 (Economy) We consider an economy spanned by n-risky assets with returns \mathbf{r} and a risk-free asset r_{f}. The risk-asset returns are such that no linear combination among them can synthesize a risk-free asset, i.e. \mathbf{V}^{-1} exists.

The returns r of any portfolio can be expressed as r = \mathbf{w}' \mathbf{r} + (1 - \mathbf{w}' \pmb{\iota}) r_{f}.

Notes

• The condition that \mathbf{V} is invertible guarantees that the n-risky assets are linearly independent, i.e. no combination of them generates a risk-free asset.
• However, there is now a risk-free asset to invest.
• If there is no risk-free asset and \mathbf{V} is not invertible, it means that we can synthesize the risk-free asset from the existing risky assets.
• In that case we can compute the implied risk-free rate, reduce the dimension of the risky assets by one, and proceed as if there is a risk-free asset.

Property 8 (Portfolio Statistics with a Risk-Free Asset) For portfolios of risky assets and a risk-free asset we have the following relations \begin{aligned} \operatorname{E}(r_{p}) & = \mathbf{w}_{p}' \mathbf{e} + (1 - \mathbf{w}_{p}' \pmb{\iota}) r_{f}, \\ \sigma_{p}^{2} & = \mathbf{w}_{p}' \mathbf{V} \mathbf{w}_{p}, \\ \operatorname{Cov}(r_{p}, r_{q}) & = \mathbf{w}_{q}' \mathbf{V} \mathbf{w}_{p}. \\ \end{aligned}

The Problem

• We want to solve the following problem: \begin{aligned} \min_{\mathbf{w}} \quad & \dfrac{1}{2} \mathbf{w}' \mathbf{V} \mathbf{w} \\ \textrm{s.t.} \quad & \begin{aligned}[t] \mathbf{w}' \mathbf{e} + (1 - \mathbf{w}' \pmb{\iota}) r_{f} = \mu \end{aligned} \end{aligned}
• To simplify notation, we define \begin{aligned} \pmb{\epsilon} & = \mathbf{e} - \pmb{\iota} r_{f}, \\ \xi & = \mu - r_{f}. \\ \end{aligned}
• The Lagrangian in this case is \mathcal{L} = \dfrac{1}{2} \mathbf{w}' \mathbf{V} \mathbf{w} + \lambda (\xi - \mathbf{w}' \pmb{\epsilon}).

The Solution

• The first order conditions for this problem are \begin{aligned} \dfrac{\partial \mathcal{L}}{\partial \mathbf{w}} & = \mathbf{V} \mathbf{w} - \lambda \pmb{\epsilon} = 0, \\ \dfrac{\partial \mathcal{L}}{\partial \lambda} & = \xi - \mathbf{w}' \mathbf{e} = 0. \end{aligned} \tag{9}
• From (9) we get that \mathbf{w} = \lambda \mathbf{V}^{-1} \pmb{\epsilon}, implying \mathbf{w}' \pmb{\epsilon} = \lambda \pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon} = \xi.
• We finally have that \mathbf{w} = \dfrac{\mathbf{V}^{-1} \pmb{\epsilon}}{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon}} \xi. \tag{10}

Maximum Sharpe Ratio

• Using (10) in \begin{aligned} \sigma^{2} & = \mathbf{w} ' \mathbf{V} \mathbf{w} = \left(\dfrac{\xi}{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon}}\right)^{2} \pmb{\epsilon}' \mathbf{V}^{-1} \mathbf{V} \mathbf{V}^{-1} \pmb{\epsilon} \\ & = \left(\dfrac{\xi}{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon}}\right)^{2} \pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon} = \dfrac{\xi^{2}}{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon}}, \end{aligned} which implies that \sigma = \dfrac{|\mu - r_{f}|}{\sqrt{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon}}}. \tag{11}
• We note that the maximum Sharpe-ratio that is possible in this economy is \mathit{SR} = \sqrt{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon}}.

The Tangency Portfolio

• When there is a risk-free asset, there is only one frontier portfolio that is composed exclusively of risky assets.
• This portfolio is called the tangency portfolio since it is the only common point of the MVF obtained exclusively from the risky assets, and the MVF obtained by adding a risk-free asset.
• This question is equivalent to asking:
  • Can we find \mu such that nothing is invested in the risk-free asset?
  • Or equivalently, can we find \mu such that everything is invested into risky assets? If so, is this \mu unique?
  • We will call the tangency portfolio q.

Finding The Tangency Portfolio

• Starting from (10) and our constraint that we want to invest everything into risky securities \begin{aligned} 1 & = \mathbf{w}' \pmb{\iota} \\ & = \dfrac{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\iota}}{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon}} \xi. \end{aligned}
• We can now solve for \xi and see that the solution exists and is unique. Using this result into (10) we find \mathbf{w}_{q} = \dfrac{\mathbf{V}^{-1} \pmb{\epsilon}}{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\iota}}. \tag{12}

Property 9 (The Minimum-Variance Frontier) The investment opportunity set with a risk-free asset is a cone whose frontier is given by \sigma = \dfrac{|\mu - r_{f}|}{\mathit{SR}} where \mathit{SR} denotes the maximum Sharpe ratio attainable in the economy.

Beta-Pricing

• Beta-pricing simplifies when we introduce a risk-free asset in the economy since the risk-free asset is uncorrelated with all other frontier portfolios.
• For any asset or portfolio i and any frontier portfolio p we have that r_{i} = (1 - \beta_{i}) r_{f} + \beta_{i} r_{p} + \varepsilon_{i} where \begin{aligned} \beta_{i} & = \dfrac{\operatorname{Cov}(r_{i}, r_{p})}{\sigma^{2}(r_{p})}, \\ \operatorname{E}(\varepsilon_{i}) & = \operatorname{Cov}(r_{p}, \varepsilon_{i}) = 0. \end{aligned}

References

References

Cochrane, John. 2009. Asset Pricing: Revised Edition. Princeton university press.

Huang, Chi-fu, and Robert H Litzenberger. 1988. Foundations for Financial Economics. North-Holland.

Markowitz, Harry. 1952. “The Utility of Wealth.” Journal of Political Economy 60 (2): 151–58.