Modern portfolio theory

Modern portfolio theory, or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. The variance of return is used as a measure of risk, because it is tractable when assets are combined into portfolios. Often, the historical variance and covariance of returns is used as a proxy for the forward-looking versions of these quantities, but other, more sophisticated methods are available.
Economist Harry Markowitz introduced MPT in a 1952 paper, for which he was later awarded a Nobel Memorial Prize in Economic Sciences; see Markowitz model.
In 1940, Bruno de Finetti published the mean-variance analysis method, in the context of proportional reinsurance, under a stronger assumption. The paper was obscure and only became known to economists of the English-speaking world in 2006.

Mathematical model

Risk and Expected Return Analysis

Modern Portfolio Theory assumes that risk averse investors will only accept higher volatility if compensated by higher expected returns. The return of an individual asset is defined as the Total Net Return.
Depending on the asset class, the income component and the price are defined specifically:For Stocks: represents dividends.For Bonds: represents coupon payments, and the prices are treated as Dirty Prices.

Definition of Variables (in Order of Formula)

To reflect realistic net performance, the components of the return formula are defined as follows: : The quoted price of the asset at the end of the period and the beginning. : Calculated as, where is the nominal value, is the coupon rate, and is the day-count fraction. / : Includes brokerage commissions, exchange fees, financial transaction taxes, and custody fees prorated over the holding period. : The universal symbol for periodic income, such as dividends for stocks or coupon payments for bonds.

Complexity	Expected Return	Variance
One-Asset
Two-Asset
Three-Asset
N-Asset

Practical Application: Bonds vs. Stocks

While the mathematical structure of MPT is identical for all assets, the calculation of for bonds must account for the pull-to-par effect and day-count conventions. This ensures that the portfolio weights reflect the true Fair Market Value of the holdings at any given time.

Diversification

An investor can reduce portfolio risk by holding combinations of instruments that are not perfectly positively correlated. 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.

Correlation Scenario	Mathematical Result	Risk Implication
Perfect Positive		No Risk Reduction: Risk is simply the weighted average of individual volatilities.
Zero Correlation		Idiosyncratic Risk Elimination: As, portfolio variance approaches zero.
Partial Correlation		Diversification Benefit: Provides a "free lunch" by reducing risk without sacrificing return.

In reality, most assets have a correlation. Markowitz proved that as long as, 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 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 is described by standard deviation and it serves as a measure of risk.
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. Combinations along this upper edge represent portfolios 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, 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", the efficient frontier is found by minimizing the following expression:
where
is a vector of portfolio weights and ;
is the covariance matrix for the returns on the assets in the portfolio;
is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and 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.
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.
An alternative approach to specifying the efficient frontier is to do so parametrically on the expected portfolio return This version of the problem requires that we minimize
subject to
and
for parameter. This problem is easily solved using a Lagrange multiplier which leads to the following linear system of equations:

Two mutual fund theorem

A fundamental result of Markowitz's analysis is the two mutual fund 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 and are two efficient portfolios, then any third efficient portfolio 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 composition depends on the target location relative to the two funds:Long Positions : If the target portfolio lies on the frontier segment between and, the investor allocates a positive fraction to Fund 1 and 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 : To achieve a return higher than both and, the investor sells Fund 2 short and invests more than 100% of their capital into Fund 1.
* Shorting Fund 1 : To achieve a return lower than both funds, 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, 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 vertical intercept of this line represents a portfolio allocated 100% to the risk-free asset. The tangency point denotes a portfolio with 100% investment in risky assets, while segments between the intercept and tangency represent lending portfolios. 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, expressed by the formula:

Component	Equation
Expected Return

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 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. 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

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. 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. 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, the tangent point varies as well, but always falling on a single line.
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 axis. However, only the branch with is meaningful. By symmetry, the two asymptotes of the hyperbola intersect at a point on the axis. The point is the height of the leftmost point of the hyperbola, and can be interpreted as the expected return of the global minimum-variance portfolio.

Tangency portfolio

The tangency portfolio exists if and only if.
In particular, if the risk-free return is greater or equal to, then the tangent portfolio does not exist. The capital market line 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 approaches 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, such that is a random variable with zero variance—that is, it is not random at all.
Suppose and, 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. 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

The above analysis describes the optimal behavior of an individual investor. Asset pricing theory builds on this analysis, allowing MPT to derive the required expected return for a correctly priced asset in this context.
Intuitively, in a perfect market with rational investors, if a security was expensive relative to others—providing too much risk for the price—demand would fall and its price would drop. Conversely, if an asset were cheap, demand and price would increase. This process continues until the market reaches a state of "market equilibrium". In this equilibrium, relative supplies equal relative demands. Since everyone holds the risky assets in identical proportions, the risky assets' prices and expected returns adjust until the ratios in the tangency portfolio match the ratios of assets supplied to the market.

Concept	Formula	Description
Equilibrium Slope		In equilibrium, the risk premium per unit of systematic risk is equal for all assets.

Systematic risk and specific risk

The total risk of an individual asset or portfolio is decomposed into two distinct components: specific risk and systematic risk.Specific risk is associated with individual assets. Within a portfolio, these risks can be mitigated through diversification, as the unique price movements of uncorrelated assets tend to offset each other.Systematic risk refers to the risk common to all securities in a given market, driven by macroeconomic factors.
Because rational investors can eliminate unique risk at no cost through diversification, the market only provides a risk premium for bearing systematic risk. This implies that an asset's expected return is not determined by its total variance, but specifically by its covariance with the market portfolio. Consequently, the equilibrium price of an asset must adjust until its risk-adjusted return aligns with the Security Market Line shown in the diagram. Under these assumptions, assets with the same Beta must offer the same expected return, regardless of their individual specific risk profiles. This fundamental distinction serves as the basis for modern portfolio management, where the goal is to optimize the exposure to rewarded systematic factors while neutralizing unrewarded idiosyncratic noise.

Component	Mathematical Formula	Description
Total Risk		Sum of systematic and idiosyncratic variance components.
Systematic Component		Risk attributed to the asset's sensitivity to market movements.
Specific Component		Residual variance.
Beta Factor		Asset sensitivity relative to the market.
Systematic Proportion		The Coefficient of determination.

Capital asset pricing model

The CAPM derives the theoretical required expected return for an asset given the risk-free rate and the market risk as a whole. It is formally defined by the Security Market Line :

The CAPM Equation

Derivation of the CAPM Equation

The following table outlines the marginal impact of adding an asset a to the market portfolio m. In equilibrium, the marginal gain in expected return per unit of marginal risk must be identical for all assets.

Step	Formula	Logic / Assumption
1. Marginal Risk		Since the weight is very small, the quadratic term vanishes.
2. Marginal Return		The additional return gained by the allocation .
3. Equilibrium Ratio		The improvement from asset a must match the market's reward-to-risk ratio.
4. Solving for		Algebraic rearrangement.
5. Final Beta Form		Substituting the Beta definition.

Estimation and Application

The CAPM equation is estimated statistically using the Security Characteristic Line , which regresses the excess return of a stock against the excess return of the market:

The SCL Equation

Once the required expected return is established, it is used as the discount rate to determine the asset's intrinsic value based on future cash flows :

Fundamental Valuation

An asset is considered **undervalued** if its calculated is higher than the current market price, and **overvalued** if the price exceeds this intrinsic value.

Criticisms

Despite its theoretical importance, critics of MPT question whether it is an ideal investment tool, because its model of financial markets does not match the real world in many ways.
The risk, return, and correlation measures used by MPT are based on expected values, which means that they are statistical statements about the future. Such measures often cannot capture the true statistical features of the risk and return which often follow highly skewed distributions and can give rise to, besides reduced volatility, also inflated growth of return. In practice, investors must substitute predictions based on historical measurements of asset return and volatility for these values in the equations. Very often such expected values fail to take account of new circumstances that did not exist when the historical data was generated. An optimal approach to capturing trends, which differs from Markowitz optimization by utilizing invariance properties, is also derived from physics. Instead of transforming the normalized expectations using the inverse of the correlation matrix, the invariant portfolio employs the inverse of the square root of the correlation matrix. The optimization problem is solved under the assumption that expected values are uncertain and correlated. The Markowitz solution corresponds only to the case where the correlation between expected returns is similar to the correlation between returns.
More fundamentally, investors are stuck with estimating key parameters from past market data because MPT attempts to model risk in terms of the likelihood of losses, but says nothing about why those losses might occur. The risk measurements used are probabilistic in nature, not structural. This is a major difference as compared to many engineering approaches to risk management.
Mathematical risk measurements are also useful only to the degree that they reflect investors' true concerns—there is no point minimizing a variable that nobody cares about in practice. In particular, variance is a symmetric measure that counts abnormally high returns as just as risky as abnormally low returns. The psychological phenomenon of loss aversion is the idea that investors are more concerned about losses than gains, meaning that our intuitive concept of risk is fundamentally asymmetric in nature. There many other risk measures might better reflect investors' true preferences.
Modern portfolio theory has also been criticized because it assumes that returns follow a Gaussian distribution. Already in the 1960s, Benoit Mandelbrot and Eugene Fama showed the inadequacy of this assumption and proposed the use of more general stable distributions instead. Stefan Mittnik and Svetlozar Rachev presented strategies for deriving optimal portfolios in such settings. More recently, Nassim Nicholas Taleb has also criticized modern portfolio theory on this ground, writing:
Contrarian investors and value investors typically do not subscribe to Modern Portfolio Theory. One objection is that the MPT relies on the efficient-market hypothesis and uses fluctuations in share price as a substitute for risk. Sir John Templeton believed in diversification as a concept, but also felt the theoretical foundations of MPT were questionable, and concluded : "the notion that building portfolios on the basis of unreliable and irrelevant statistical inputs, such as historical volatility, was doomed to failure."
A few studies have argued that "naive diversification", splitting capital equally among available investment options, might have advantages over MPT in some situations.
When applied to certain universes of assets, the Markowitz model has been identified by academics to be inadequate due to its susceptibility to model instability which may arise, for example, among a universe of highly correlated assets.

Extensions

Since MPT's introduction in 1952, many attempts have been made to improve the model, especially by using more realistic assumptions.
Post-modern portfolio theory extends MPT by adopting non-normally distributed, asymmetric, and fat-tailed measures of risk. This helps with some of these problems, but not others.
Black–Litterman model optimization is an extension of unconstrained Markowitz optimization that incorporates relative and absolute 'views' on inputs of risk and returns from.
The model is also extended by assuming that expected returns are uncertain, and the correlation matrix in this case can differ from the correlation matrix between returns.

Connection with rational choice theory

Modern portfolio theory is inconsistent with main axioms of rational choice theory, most notably with monotonicity axiom, stating that, if investing into portfolio X will, with probability one, return more money than investing into portfolio Y, then a rational investor should prefer X to Y. In contrast, modern portfolio theory is based on a different axiom, called variance aversion,
and may recommend to invest into Y on the basis that it has lower variance. Maccheroni et al. described choice theory which is the closest possible to the modern portfolio theory, while satisfying monotonicity axiom. Alternatively, mean-deviation analysis
is a rational choice theory resulting from replacing variance by an appropriate deviation risk measure.

Other applications

In the 1970s, concepts from MPT found their way into the field of regional science. In a series of seminal works, Michael Conroy modeled the labor force in the economy using portfolio-theoretic methods to examine growth and variability in the labor force. This was followed by a long literature on the relationship between economic growth and volatility.
More recently, modern portfolio theory has been used to model the self-concept in social psychology. When the self attributes comprising the self-concept constitute a well-diversified portfolio, then psychological outcomes at the level of the individual such as mood and self-esteem should be more stable than when the self-concept is undiversified. This prediction has been confirmed in studies involving human subjects.
Recently, modern portfolio theory has been applied to modelling the uncertainty and correlation between documents in information retrieval. Given a query, the aim is to maximize the overall relevance of a ranked list of documents and at the same time minimize the overall uncertainty of the ranked list.

Project portfolios and other "non-financial" assets

Some experts apply MPT to portfolios of projects and other assets besides financial instruments. When MPT is applied outside of traditional financial portfolios, some distinctions between the different types of portfolios must be considered.

The assets in financial portfolios are, for practical purposes, continuously divisible while portfolios of projects are "lumpy". For example, while we can compute that the optimal portfolio position for 3 stocks is, say, 44%, 35%, 21%, the optimal position for a project portfolio may not allow us to simply change the amount spent on a project. Projects might be all or nothing or, at least, have logical units that cannot be separated. A portfolio optimization method would have to take the discrete nature of projects into account.
The assets of financial portfolios are liquid; they can be assessed or re-assessed at any point in time. But opportunities for launching new projects may be limited and may occur in limited windows of time. Projects that have already been initiated cannot be abandoned without the loss of the sunk costs.

Neither of these necessarily eliminate the possibility of using MPT and such portfolios. They simply indicate the need to run the optimization with an additional set of mathematically expressed constraints that would not normally apply to financial portfolios.
Furthermore, some of the simplest elements of Modern Portfolio Theory are applicable to virtually any kind of portfolio. The concept of capturing the risk tolerance of an investor by documenting how much risk is acceptable for a given return may be applied to a variety of decision analysis problems. MPT uses historical variance as a measure of risk, but portfolios of assets like major projects do not have a well-defined "historical variance". In this case, the MPT investment boundary can be expressed in more general terms like "chance of an ROI less than cost of capital" or "chance of losing more than half of the investment". When risk is put in terms of uncertainty about forecasts and possible losses then the concept is transferable to various types of investment.