In reality, most assets have a correlation. Markowitz proved that as long as \rho < 1, the portfolio standard deviation will always be less than the weighted average of the individual assets' standard deviations, thereby creating a "free lunch" of risk reduction without necessarily sacrificing expected return.
Efficient frontier with no risk-free asset
The MPT is a mean-variance theory, and it compares the expected (mean) return of a portfolio with the standard deviation of the same portfolio. The image shows expected return on the vertical axis, and the standard deviation on the horizontal axis (volatility). Volatility is described by standard deviation and it serves as a measure of risk.[9]
The 'return - standard deviation space' 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,[10] 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.**
<blockquote style="border: 1px solid black; padding: 1em;"> Matrices are preferred for calculations of the efficient frontier.
In matrix form, for a given "risk tolerance", the efficient frontier is found by minimizing the following expression: where
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,[11] 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,[12] in JavaScript[13] 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 subject to and for parameter \mu. This problem is easily solved using a Lagrange multiplier which leads to the following linear system of equations:
- is a vector of portfolio weights and (The weights can be negative);
- 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
- is a vector of expected returns.
- is the variance of portfolio return.
- is the expected return on the portfolio.
Two mutual fund theorem
A fundamental result of Markowitz's analysis is the two mutual fund theorem (also known as the separation theorem).[14] 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: 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:
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.[8]
- where \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.
Risk-free asset and the capital allocation line
The risk-free asset is the theoretical asset that pays a deterministic risk-free rate.[7] 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.[6] By definition, the risk-free asset has zero variance in returns if held to maturity and remains uncorrelated with any risky asset or portfolio.[8] 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.[15]
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.[7] The vertical intercept of this line represents a portfolio allocated 100% to the risk-free asset R_F.
Geometric intuition
The efficient frontier can be pictured as a problem in quadratic curves.[14] On the market, we have the assets. 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, such that , and we hold the assets according to.
Markowitz bullet
[[File:Mean-variance analysis, quadratic optimization 3D.gif|thumb|The ellipsoid is the contour of constant variance. The x+y+z=1 plane is the space of possible portfolios. The other plane is the contour of constant expected return.
The ellipsoid intersects the plane to give an ellipse of portfolios of constant variance. On this ellipse, the point of maximal (or minimal) expected return is the point where it is tangent to the contour of constant expected return. All these portfolios fall on one line.]] Since we wish to maximize expected return while minimizing the standard deviation of the return, we are to solve a quadratic optimization problem:Portfolios are points in the Euclidean space \R^n. The third equation states that the portfolio should fall on a plane defined by. The first equation states that the portfolio should fall on a plane defined by. The second condition states that the portfolio should fall on the contour surface for 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, then intersect the contours with the plane. 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.
Markowitz bullet
[[File:Mean-variance analysis, quadratic optimization 3D.gif|thumb|The ellipsoid is the contour of constant variance. The x+y+z=1 plane is the space of possible portfolios. The other plane is the contour of constant expected return.
The ellipsoid intersects the plane to give an ellipse of portfolios of constant variance. On this ellipse, the point of maximal (or minimal) expected return is the point where it is tangent to the contour of constant expected return. All these portfolios fall on one line.]] Since we wish to maximize expected return while minimizing the standard deviation of the return, we are to solve a quadratic optimization problem:Portfolios are points in the Euclidean space \R^n. The third equation states that the portfolio should fall on a plane defined by. The first equation states that the portfolio should fall on a plane defined by. The second condition states that the portfolio should fall on the contour surface for 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, then intersect the contours with the plane. 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. We find that along the line,giving a hyperbola in the 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
The tangency portfolio exists if and only if.
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 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 and, then that means there is free money, breaking the no arbitrage assumption.
Suppose, then we can scale the vector to. 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.
