Portfolio Frontier Mathematics
Introduction
Over time, investors face two critical decisions. Every period, they need to decide how much to consume and, therefore, how much to save. Then, given how much they have decided to save, they need to determine how to invest.
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).
Markowitz (1952) was the first to establish a solid foundation for analyzing investment problems. He demonstrated that by combining different risky assets, investors can diversify and reduce the individual risk contributed by each asset. However, some risk, known as systematic risk, cannot be diversified away. Therefore, for a specific target expected return, it is possible to identify a combination of risky assets that minimizes the portfolio’s overall variance.
The resulting mean-variance analysis is a cornerstone of modern finance. Without short-selling constraints, a region inside a 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).
In this note, I review standard results in portfolio mathematics. The analysis follows Chapter 3 in Huang and Litzenberger (1988), although some derivations are original. The following section explores the minimum-variance frontier spanned by risky assets without a riskless asset. In this scenario, no combination of risky assets has a perfect positive or negative correlation with another risk asset.
N-Risky Assets
Portfolio Statistics
The investment opportunity set of the economy is spanned by n-risky assets with returns \{ r_{1}, r_{2}, \cdots, 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}, \cdots, 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.
With this notation, the return r of a portfolio is r = \sum_{i = 1}^{n} w_{i} r_{i}.
Throughout these notes 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}'.
We denote by \mathbf{r} = (r_{1}, r_{2}, \ldots, r_{n})' the vector of risky returns. That is, \mathbf{r} is a column vector since it is the transpose of a row vector. Similarly, we can write the vector of portfolio weights as \mathbf{w} = (w_{1}, w_{2}, \ldots, w_{n})'. 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.
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.
The covariance matrix of returns is denoted by \mathbf{V} so that \mathbf{V}_{i, j} = \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} \operatorname{E}\left( ((\mathbf{r} - \mathbf{e})' \mathbf{y})^{2} \right) & = \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} \geq 0. \\ \end{aligned}
Definition 1 (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}.
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. Intuitively, this implies that it is not possible to combine the risky assets to build a portfolio that has zero variance.
Mathematically, since the covariance matrix is symmetric, all its eigenvalues are real numbers. Therefore, being positive semidefinite implies that all its eigenvalues must be non-negative. Finally, assuming that \mathbf{V} is invertible implies that all eigenvalues must be different from zero. In other words, all the eigenvalues of the covariance matrix of the basis risky assets must be 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}
Taking expectations in (1), the expected return of portfolio p is computed as \operatorname{E}(r_{p}) = \sum_{i = 1}^{n} w_{i, p} \operatorname{E}(r_{i}) = \mathbf{w}_{p}' \mathbf{e}. Applying (1) to portfolios p and q, the covariance between these two portfolios is \operatorname{Cov}(r_{p}, r_{q}) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i, p} w_{j, q} \operatorname{Cov}(r_{i}, r_{j}) = \mathbf{w}_{q}' \mathbf{V} \mathbf{w}_{p} Finally, the variance of portfolio p is just \sigma^{2}(r_{p}) = \operatorname{Cov}(r_{p}, r_{p}) = \mathbf{w}_{p}' \mathbf{V} \mathbf{w}_{p}
The Minimum-Variance Frontier
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.
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} & = \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})
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} 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}, which implies that \begin{pmatrix} \lambda_{1} \\ \lambda_{2} \end{pmatrix} = \pmb{\Gamma}^{-1} \begin{pmatrix} \mu \\ 1 \end{pmatrix}. \tag{4}
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} 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}.
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 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.
Consider two frontier portfolios, p and q with expected returns \mu_{p} and \mu_{q}. Their portfolio weights 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} Consider now forming a portfolio b consisting of 1 - \beta of p and \beta of q. Its expected return is \mu_{b} = (1 - \beta) \mu_{p} + \beta \mu_{q}, and its portfolio weights are given by \begin{aligned} \mathbf{w}_{b} & = (1 - \beta) \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix} + \beta \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{q} \\ 1 \end{pmatrix} \\ & = \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} (1 - \beta) \mu_{p} \\ 1 - \beta \end{pmatrix} + \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \beta \mu_{q} \\ \beta \end{pmatrix} \\ & = \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} (1 - \beta) \mu_{p} + \beta \mu_{q} \\ 1 - \beta + \beta \end{pmatrix} \\ & = \mathbf{V}^{-1} \begin{pmatrix} \mathbf{e} & \pmb{\iota} \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{b} \\ 1 \end{pmatrix} \\ \end{aligned} which shows that b is also a frontier portfolio.
Property 4 (Global Minimum Variance Portfolio) There is a portfolio that minimizes the variance among all portfolios called the global minimum variance portfolio, which is denoted by g. The covariance of the global minimum variance portfolio with any other portfolio, not necessarily a frontier portfolio, is always the same and equal to the variance of the global 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})
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}_{g} & = \lambda \mathbf{V}^{-1} \pmb{\iota}, \\ 1 = \mathbf{w}_{g}' \pmb{\iota} & = \lambda \pmb{\iota}' \mathbf{V}^{-1} \pmb{\iota} = \lambda C, \\ \mathbf{w}_{g} & = \dfrac{1}{C} \mathbf{V}^{-1} \pmb{\iota}. \end{aligned} The covariance of the global minimum variance portfolio with any other portfolio i is given by \begin{aligned} \operatorname{Cov}(r_{i}, r_{g}) & = \mathbf{w}_{i}' \mathbf{V} \mathbf{w}_{g} \\ & = \mathbf{w}_{i}' \mathbf{V} \left( \frac{1}{C} \mathbf{V}^{-1} \pmb{\iota} \right) \\ & = \frac{1}{C}. \end{aligned} Since \operatorname{V}(r_{g}) = \operatorname{Cov}(r_{g}, r_{g}) = 1 / C, we have that \operatorname{Cov}(r_{i}, r_{g}) = \operatorname{V}(r_{g}) for any portfolio i.
Note that the standard deviation of the global minimum variance portfolio corresponds to the \sigma-coordinate of the vertex of the hyperbola described by (5).
Beta Pricing
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. It turns out that the idiosyncratic risk of any asset is uncorrelated with all frontier portfolios.
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. This seemingly marginal property is at the heart of modern asset pricing. Given a frontier portfolio, what determines the expected return of any asset is how much it covaries with the frontier portfolio regardless of its total risk.
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. That is, the residual is unrelated to the returns of frontier portfolios. 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 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}.
Using (6), we can compute the derivative of \sigma with respect to \mu, \begin{aligned} \dfrac{d \sigma}{d \mu} & = \dfrac{d \sqrt{\sigma^{2}}}{d \mu} \\ & = \dfrac{1}{2 \sigma} 2 \begin{pmatrix} 1 & 0 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu \\ 1 \end{pmatrix} \\ & = \dfrac{1}{\sigma} \begin{pmatrix} 1 & 0 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu \\ 1 \end{pmatrix} \\ \end{aligned} \tag{9} Denote by m_{p} the slope coefficient of the minimum variance frontier at the point (\sigma_{p}, \mu_{p}). Equation (9) implies that \frac{\sigma_{p}}{m_{p}} = \begin{pmatrix} 1 & 0 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix}, since d \sigma / d \mu = 1 / (d \mu / d \sigma) = 1 / m.
The equation of the tangent line to the minimum variance frontier at point (\sigma_{p}, \mu_{p}) is given by \mu - \mu_{p} = m_{p} (\sigma - \sigma_{p}). Denote by \mu_{z} the intercept of this line with the vertical axis. Then we have that \mu_{z} = \mu_{p} - m_{p} \sigma_{p}. We can now compute the covariance of r_{p} and a frontier portfolio r_{z} with expected return \mu_{z} \begin{aligned} \operatorname{Cov}(r_{z}, r_{p}) & = \begin{pmatrix} \mu_{z} & 1 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix} \\ & = \begin{pmatrix} \mu_{p} - m_{p} \sigma_{p} & 1 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix} \\ & = \begin{pmatrix} \mu_{p} & 1 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix} - m_{p} \sigma_{p} \begin{pmatrix} 1 & 0 \end{pmatrix} \pmb{\Gamma}^{-1} \begin{pmatrix} \mu_{p} \\ 1 \end{pmatrix} \\ & = \sigma_{p}^{2} - m_{p} \sigma_{p} \left( \frac{\sigma_{p}}{m_{p}} \right) \\ & = 0. \\ \end{aligned} Obviously, the covariance of the minimum variance portfolio with any other frontier portfolio is equal to its variance, so it is impossible to find a frontier portfolio that has zero-covariance with it.
Property 7 (Beta-pricing with MVPs) 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*}
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, and hence it can be represented as a portfolio of any two other frontier portfolios. Therefore, just pick an arbitrary frontier portfolio p (but different from the GMVP) 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}}. 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. Hence, 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})}.
N-Risky Assets and a Risk-Free Asset
Portfolio Statistics
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{align*} 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{align*}
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}.
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 expected return of portfolio p is given by \operatorname{E}(r_{p}) = \sum_{i = 1}^{N} w_{i, p} \operatorname{E}(r_{i}) + (1 - \sum_{i = 1}^{N} w_{i, p}) r_{f} = \mathbf{w}_{p}' \mathbf{e} + (1 - \mathbf{w}_{p}' \pmb{\iota}) r_{f}. The covariance between portfolio p and another portfolio q is \begin{aligned} \operatorname{Cov}(r_{p}, r_{q}) & = \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i, p} w_{j, q} \operatorname{Cov}(r_{i}, r_{j}) + \sum_{i = 1}^{n} w_{i, p} \left( 1 - \sum_{j = 1}^{n} w_{j, q} \right) \underbrace{\operatorname{Cov}(r_{i}, r_{f})}_{0} \\ & \quad + \sum_{j = 1}^{n} \left( 1 - \sum_{i = 1}^{n} w_{i, p} \right) w_{j, q} \underbrace{\operatorname{Cov}(r_{f}, r_{j})}_{0} + \left(1 - \sum_{i = 1}^{n} w_{i, p} \right) \left(1 - \sum_{j = 1}^{N} w_{j, q}\right) \underbrace{\operatorname{Cov}(r_{f}, r_{f})}_{0} \\ & = \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i, p} w_{j, q} \operatorname{Cov}(r_{i}, r_{j}) \\ & = \mathbf{w}_{q}' \mathbf{V} \mathbf{w}_{p}. \end{aligned} Hence, the variance of portfolio p is just \operatorname{Cov}(r_{p}, r_{p}) = \mathbf{w}_{p}' \mathbf{V} \mathbf{w}_{p}.
The Minimum-Variance Frontier
Property 9 (Portfolio Statistics with a Risk-Free Asset) 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.
By allocating \mathbf{w} to the risky assets, and whatever is left, which is given by 1 - \mathbf{w}' \pmb{\iota}, to the risk-free asset, we avoid one restriction. Hence, 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 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{10} From (10) we get that \mathbf{w} = \lambda \mathbf{V}^{-1} \pmb{\epsilon}, so that \begin{aligned} \mathbf{w}' \pmb{\epsilon} & = \lambda \pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon} = \xi. \end{aligned} We finally have that \mathbf{w} = \dfrac{\mathbf{V}^{-1} \pmb{\epsilon}}{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\epsilon}} \xi. \tag{11} Using (11) 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{12} 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 MVP 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.
Starting from (11) 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 (11) we find \mathbf{w}_{q} = \dfrac{\mathbf{V}^{-1} \pmb{\epsilon}}{\pmb{\epsilon}' \mathbf{V}^{-1} \pmb{\iota}}. \tag{13}
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{gather*} \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{gather*} In particular we obtain the following pricing relation \operatorname{E}(r_{i}) = r_{f} + \beta_{i} (\operatorname{E}(r_{p}) - r_{f}).
This result follows directly from our analysis on risky assets. When there is a risk-free asset available, the minimum variance frontier is a set of two lines that intersect with the vertical axis at the same point. The intersection point is the risk-free rate, and this is the only asset that is uncorrelated with the other frontier portfolios. For a given frontier portfolio its associated zero-covariance frontier portfolio is the risk-free asset.
All the systematic risk is characterized by the frontier portfolios and hence this is the only thing that matters to price all assets or portfolios in the economy. The big question is how to find these frontier portfolios. In the next chapter we will specialize the model and add preferences. This will lead to identify the market portfolio as a candidate frontier portfolio. This is the true essence of the Capital Asset Pricing Model (CAPM).