An investor can reduce portfolio risk (specifically the portfolio standard deviation \sigma_p) by holding combinations of instruments that are not perfectly positively
correlated (\rho_{ij} ). This occurs because the
variance of a diversified portfolio depends more on the
covariance between assets than on the individual variances of the assets themselves. The is sometimes called the space of 'expected return vs risk'. Every possible combination of risky assets, can be plotted in this risk-expected return space, and the collection of all such possible portfolios defines a region in this space. The left boundary of this region is hyperbolic, and the upper part of the hyperbolic boundary is the
efficient frontier in the absence of a risk-free asset (sometimes called "the Markowitz bullet"). Combinations along this upper edge represent portfolios (including no holdings of the risk-free asset) for which there is lowest risk for a given level of expected return. Equivalently, a portfolio lying on the efficient frontier represents the combination offering the best possible expected return for given risk level. The tangent to the upper part of the hyperbolic boundary is the
capital allocation line (CAL). **The vertex of the hyperbola represents the Global Minimum Variance Portfolio (GMVP), which is the portfolio with the lowest possible risk among all combinations of risky assets.**
Matrices are preferred for calculations of the efficient frontier. In matrix form, for a given "risk tolerance" q \in [0,\infty), the efficient frontier is found by minimizing the following expression: :w^T \Sigma w - q R^T w where • w\in\mathbb{R}^N is a vector of portfolio weights and \sum_{i=1}^N w_i = 1. (The weights can be negative); • \Sigma\in\mathbb{R}^{N\times N} is the
covariance matrix for the returns on the assets in the portfolio; • q \ge 0 is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and \infty results in the portfolio infinitely far out on the frontier with both expected return and risk unbounded; and • R\in\mathbb{R}^N is a vector of expected returns. • w^T \Sigma w\in\mathbb{R} is the variance of portfolio return. • R^T w\in\mathbb{R} is the expected return on the portfolio. The above optimization finds the point on the frontier at which the inverse of the slope of the frontier would be
q if portfolio return variance instead of standard deviation were plotted horizontally. The frontier in its entirety is parametric on
q.
Harry Markowitz developed a specific procedure for solving the above problem, called the
critical line algorithm, that can handle additional linear constraints, upper and lower bounds on assets, and which is proved to work with a semi-positive definite covariance matrix. Examples of implementation of the critical line algorithm exist in
Visual Basic for Applications, in
JavaScript and in a few other languages. Also, many software packages, including
MATLAB,
Microsoft Excel,
Mathematica and
R, provide generic
optimization routines so that using these for solving the above problem is possible, with potential caveats (poor numerical accuracy, requirement of positive definiteness of the covariance matrix...). An alternative approach to specifying the efficient frontier is to do so parametrically on the expected portfolio return R^T w. This version of the problem requires that we minimize :w^T \Sigma w subject to :R^T w = \mu and :\sum_{i=1}^{N} w_i = 1 for parameter \mu. This problem is easily solved using a
Lagrange multiplier which leads to the following linear system of equations: :\begin{bmatrix}2\Sigma &-R & -{\bf1}\\ R^T &0 & 0 \\ {\bf1}^T &0 &0 \end{bmatrix} \begin{bmatrix}w\\\lambda_1\\\lambda_2\end{bmatrix} = \begin{bmatrix}0\\\mu \\ 1\end{bmatrix}
Two mutual fund theorem A fundamental result of Markowitz's analysis is the
two mutual fund theorem (also known as the separation theorem). This theorem mathematically states that any portfolio on the
efficient frontier can be constructed as a linear combination of any two distinct portfolios already located on the frontier. Mathematically, if P_1 and P_2 are two efficient portfolios, then any third efficient portfolio P_{target} can be expressed as: :P_{target} = \alpha P_1 + (1 - \alpha) P_2where \alpha is the weighting factor. Because the underlying assets in P_1 and P_2 are valued based on their
Total Net Return (including capital gains, dividends, and interest, net of transaction costs), the resulting combination P_{target} inherently accounts for all income streams and expenses. This implies that in the absence of a risk-free asset, an investor can achieve any optimal risk-return profile using only two "mutual funds" (the basis portfolios). The composition depends on the target location relative to the two funds: •
Long Positions (0 ): If the target portfolio P_{target} lies on the frontier segment between P_1 and P_2, the investor allocates a positive fraction \alpha to Fund 1 and 1-\alpha to Fund 2. No borrowing or short-selling is required. •
Short Selling and Leverage: If the target lies on the frontier curve but outside the segment between the two funds, the investor must use
short-selling: •
Shorting Fund 2 (\alpha > 1): To achieve a return higher than both P_1 and P_2 (assuming E(R_1) > E(R_2)), the investor sells Fund 2 short (negative weight) and invests more than 100% of their capital into Fund 1. •
Shorting Fund 1 (\alpha ): To achieve a return lower than both funds (or to minimize risk beyond the span), the investor sells Fund 1 short and invests the proceeds into Fund 2. This theorem is significant because it simplifies the complex optimization problem: once the frontier is identified, an investor no longer needs to analyze every individual asset (stock, bond, etc.), but only needs to choose the right mix of two frontier portfolios to satisfy their specific risk tolerance.
Risk-free asset and the capital allocation line The risk-free asset is the theoretical asset that pays a deterministic
risk-free rate. In practice, short-term government securities, such as
US Treasury bills, serve as a proxy for the risk-free asset due to their fixed interest payments and negligible
default risk. By definition, the risk-free asset has zero variance in returns if held to maturity and remains uncorrelated with any risky asset or portfolio. Consequently, when combined with a risky portfolio, the resulting change in expected return is linearly related to the change in risk as the allocation proportions vary. The introduction of a risk-free asset transforms the efficient frontier into a linear half-line tangent to the Markowitz bullet at the portfolio with the highest
Sharpe ratio. The tangency point denotes a portfolio with 100% investment in risky assets, while segments between the intercept and tangency represent lending portfolios (long R_F). Points extending beyond the tangency point represent borrowing portfolios, where the investor leverages the risky tangency portfolio by shorting the risk-free asset. This efficient locus is defined as the
capital allocation line (CAL), expressed by the formula: In this context,
P represents the tangency portfolio of risky assets,
F denotes the risk-free asset, and
C is the combined portfolio. The introduction of R_F improves the investment opportunity set because the CAL provides a higher expected return for every level of risk compared to the risky-only hyperbola. The principle that all investors can achieve their optimal risk-return profile using only the risk-free asset and a single risky fund is known as the
Mutual fund separation theorem, specifically the one-fund theorem.
Geometric intuition The efficient frontier can be pictured as a problem in
quadratic curves. On the market, we have the assets R_1, R_2, \dots, R_n. We have some funds, and a portfolio is a way to divide our funds into the assets. Each portfolio can be represented as a vector w_1, w_2, \dots, w_n, such that \sum_i w_i = 1, and we hold the assets according to w^T R = \sum_i w_i R_i .
Markowitz bullet Since we wish to maximize expected return while minimizing the standard deviation of the return, we are to solve a quadratic optimization problem:\begin{cases} E[w^T R] = \mu \\ \min \sigma^2 = Var[w^T R ]\\ \sum_i w_i = 1 \end{cases}Portfolios are points in the Euclidean space \R^n. The third equation states that the portfolio should fall on a plane defined by \sum_i w_i = 1. The first equation states that the portfolio should fall on a plane defined by w^T E[R] = \mu. The second condition states that the portfolio should fall on the contour surface for \sum_{ij} w_i \rho_{ij} w_j that is as close to the origin as possible. Since the equation is quadratic, each such contour surface is an ellipsoid (assuming that the covariance matrix \rho_{ij} is invertible). Therefore, we can solve the quadratic optimization graphically by drawing ellipsoidal contours on the plane \sum_i w_i = 1, then intersect the contours with the plane \{w: w^T E[R] = \mu \text{ and } \sum_i w_i =1\}. As the ellipsoidal contours shrink, eventually one of them would become exactly tangent to the plane, before the contours become completely disjoint from the plane. The tangent point is the optimal portfolio at this level of expected return. As we vary \mu, the tangent point varies as well, but always falling on a single line (this is the
two mutual funds theorem). Let the line be parameterized as \{w + w' t : t \in \R\}. We find that along the line,\begin{cases} \mu &= (w'^T E[R]) t + w^T E[R]\\ \sigma^2 &= (w'^T \rho w') t^2 + 2 (w^T \rho w') t + (w^T \rho w) \end{cases} giving a hyperbola in the (\sigma, \mu) plane. The hyperbola has two branches, symmetric with respect to the \mu axis. However, only the branch with \sigma > 0 is meaningful. By symmetry, the two asymptotes of the hyperbola intersect at a point \mu_{MVP} on the \mu axis. The point \mu_{mid} is the height of the leftmost point of the hyperbola, and can be interpreted as the expected return of the
global minimum-variance portfolio (global MVP).
Tangency portfolio File:Mean-variance analysis.gif|thumb|Illustration of the effect of changing the risk-free asset return rate. As the risk-free return rate approaches the return rate of the global minimum-variance portfolio, the tangency portfolio escapes to infinity. Animated at source The tangency portfolio exists if and only if \mu_{RF} . In particular, if the risk-free return is greater or equal to \mu_{MVP}, then the tangent portfolio
does not exist. The capital market line (CML) becomes parallel to the upper asymptote line of the hyperbola. Points
on the CML become impossible to achieve, though they can be
approached from below. It is usually assumed that the risk-free return is less than the return of the global MVP, in order that the tangency portfolio exists. However, even in this case, as \mu_{RF} approaches \mu_{MVP} from below, the tangency portfolio diverges to a portfolio with infinite return and variance. Since there are only finitely many assets in the market, such a portfolio must be shorting some assets heavily while longing some other assets heavily. In practice, such a tangency portfolio would be impossible to achieve, because one cannot short an asset too much due to
short sale constraints, and also because of
price impact, that is, longing a large amount of an asset would push up its price, breaking the assumption that the asset prices do not depend on the portfolio.
Non-invertible covariance matrix If the covariance matrix is not invertible, then there exists some nonzero vector v, such that v^T R is a random variable with zero variance—that is, it is not random at all. Suppose \sum_i v_i = 0 and v^T R = 0, then that means one of the assets can be exactly replicated using the other assets at the same price and the same return. Therefore, there is never a reason to buy that asset, and we can remove it from the market. Suppose \sum_i v_i = 0 and v^T R \neq 0 , then that means there is free money, breaking the
no arbitrage assumption. Suppose \sum_i v_i \neq 0 , then we can scale the vector to \sum_i v_i = 1. This means that we have constructed a risk-free asset with return v^T R . We can remove each such asset from the market, constructing one risk-free asset for each such asset removed. By the no arbitrage assumption, all their return rates are equal. For the assets that still remain in the market, their covariance matrix is invertible. == Asset pricing ==