Modern portfolio theory

thumb|right|250px|Modern portfolio theory suggests a diversified portfolio of [[shares and other asset classes (such as debt in corporate bonds, treasury bonds, or money market funds) will realise more predictable returns if there is prudent market regulation.]]

Modern portfolio theory (MPT), 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 (or its transformation, the standard deviation) is used as a measure of risk, because it is tractable when assets are combined into portfolios. but other, more sophisticated methods are available.

Economist Harry Markowitz introduced MPT or mean variance framework more specifically 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. Later work linked portfolio risk and expected return to asset pricing in equilibrium. 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 (MPT) assumes that risk averse investors will only accept higher volatility if compensated by higher expected returns. The return of an individual asset <math>R_i</math> is defined as the Total Net Return.

Depending on the asset class, the income component <math>D_t</math> and the price <math>P</math> are defined specifically:

For Stocks: <math>D_t</math> represents dividends.
For Bonds: <math>D_t</math> represents coupon payments, and the prices <math>P</math> are treated as Dirty Prices (Clean Price + Accrued interest <math>S = K \cdot \frac{f}{100} \cdot \frac{A}{D}</math>).

{| class="wikitable" style="max-width: 480px; font-size: 90%; clear: left;"

|+ Total Net Return Calculation

! Component !! Formula

| Total Net Return (<math>R_i</math>)

| <math>R_i = \frac{(P_{t} + S_t - C_{sale}) - (P_{t-1} + S_{t-1} + C_{buy}) + D_t}{P_{t-1} + S_{t-1} + C_{buy</math>

Definition of Variables (in Order of Formula)

To reflect realistic net performance, the components of the return formula are defined as follows:

<math>P</math> (Market Price): The quoted price of the asset at the end of the period (<math>P_t</math>) and the beginning (<math>P_{t-1}</math>).
<math>S</math> (Accrued Interest): Calculated as <math>K \cdot \frac{f}{100} \cdot \frac{A}{D}</math>, where <math>K</math> is the nominal value, <math>f</math> is the coupon rate, and <math>A/D</math> is the day-count fraction.
<math>C_{sale}</math> / <math>C_{buy}</math> (Transaction Costs): Includes brokerage commissions, exchange fees, financial transaction taxes, and custody fees prorated over the holding period.
<math>D_t</math> (Distributions): The universal symbol for periodic income, such as dividends for stocks or coupon payments for bonds.

{| class="wikitable" style="max-width: 550px; font-size: 90%; clear: left;"

|+ Portfolio Risk and Return Metrics

! Complexity !! Expected Return <math>\operatorname{E}(R_p)</math> !! Variance (Risk) <math>\sigma_p^2</math>

| One-Asset

| <math>\operatorname{E}(R_A)</math>

| <math>\sigma_A^2</math>

| Two-Asset

| <math>w_A \operatorname{E}(R_A) + w_B \operatorname{E}(R_B)</math>

| <math>\underbrace{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2}_{\text{Variance Component + \underbrace{2 w_A w_B \sigma_A \sigma_B \rho_{AB_{\text{Correlation Component</math>

| Three-Asset

| <math>\sum_{i=A}^C w_i \operatorname{E}(R_i)</math>

| <math>\underbrace{\sum w_i^2 \sigma_i^2}_{\text{Variance + \underbrace{2w_A w_B \sigma_{AB} + 2w_A w_C \sigma_{AC} + 2w_B w_C \sigma_{BC_{\text{Pairwise Covariances</math>

| N-Asset

| <math>\mathbf{w}^\intercal \boldsymbol{\mu}</math>

| <math>\mathbf{w}^\intercal \Sigma \mathbf{w}</math>

Practical Application: Bonds vs. Stocks

While the mathematical structure of MPT is identical for all assets, the calculation of <math>R_i</math> for bonds must account for the pull-to-par effect and day-count conventions. This ensures that the portfolio weights <math>w_i</math> reflect the true Fair Market Value (Dirty Price) of the holdings at any given time <math>t</math>.

Diversification

An investor can reduce portfolio risk (specifically the portfolio standard deviation <math>\sigma_p</math>) by holding combinations of instruments that are not perfectly positively correlated (<math>\rho_{ij} < 1</math>). 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" <math>q \in [0,\infty)</math>, the efficient frontier is found by minimizing the following expression:

:<math>w^T \Sigma w - q R^T w</math>

where

<math>w\in\mathbb{R}^N</math> is a vector of portfolio weights and <math>\sum_{i=1}^N w_i = 1.</math> (The weights can be negative);
<math>\Sigma\in\mathbb{R}^{N\times N}</math> is the covariance matrix for the returns on the assets in the portfolio;
<math>q \ge 0</math> is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and <math>\infty</math> results in the portfolio infinitely far out on the frontier with both expected return and risk unbounded; and
<math>R\in\mathbb{R}^N</math> is a vector of expected returns.
<math>w^T \Sigma w\in\mathbb{R}</math> is the variance of portfolio return.
<math>R^T w\in\mathbb{R}</math> 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 <math>R^T w.</math> This version of the problem requires that we minimize

:<math>w^T \Sigma w</math>

subject to

:<math>R^T w = \mu</math>

and

:<math>\sum_{i=1}^{N} w_i = 1</math>

for parameter <math>\mu</math>. This problem is easily solved using a Lagrange multiplier which leads to the following linear system of equations:

:<math>\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}</math>

</blockquote>

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 <math>P_1</math> and <math>P_2</math> are two efficient portfolios, then any third efficient portfolio <math>P_{target}</math> can be expressed as:

:<math display="block">P_{target} = \alpha P_1 + (1 - \alpha) P_2</math>where <math>\alpha</math> is the weighting factor. Because the underlying assets in <math>P_1</math> and <math>P_2</math> are valued based on their Total Net Return (including capital gains, dividends, and interest, net of transaction costs), the resulting combination <math>P_{target}</math> 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 (<math>0 < \alpha < 1</math>): If the target portfolio <math>P_{target}</math> lies on the frontier segment between <math>P_1</math> and <math>P_2</math>, the investor allocates a positive fraction <math>\alpha</math> to Fund 1 and <math>1-\alpha</math> 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 (<math>\alpha > 1</math>): To achieve a return higher than both <math>P_1</math> and <math>P_2</math> (assuming <math>E(R_1) > E(R_2)</math>), the investor sells Fund 2 short (negative weight) and invests more than 100% of their capital into Fund 1.
Shorting Fund 1 (<math>\alpha < 0</math>): 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 <math>R_F</math>).

Specific risk (also known as diversifiable, unique, or idiosyncratic 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 (also known as market risk or non-diversifiable 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.

{| class="wikitable" style="max-width: 650px; font-size: 90%; clear: left;"

|+ Mathematical Derivation Steps

! Step !! Formula !! Logic / Assumption

| 1. Marginal Risk

| <math>\Delta \sigma_P \approx 2 w_m w_a \rho_{am} \sigma_a \sigma_m</math>

| Since the weight <math>w_a</math> is very small, the quadratic term (<math>w_a^2</math>) vanishes.

| 2. Marginal Return

| <math>\Delta E(R_P) = w_a (E(R_a) - R_f)</math>

| The additional return gained by the allocation <math>w_a</math> (relative to the risk-free rate <math>R_f</math>).

| 3. Equilibrium Ratio

| <math>\frac{w_a (E(R_a) - R_f)}{2 w_m w_a \rho_{am} \sigma_a \sigma_m} = \frac{E(R_m) - R_f}{2 \sigma_m^2}</math>

| The improvement from asset a must match the market's reward-to-risk ratio.

| 4. Solving for <math>E(R_a)</math>

| <math>E(R_a) = R_f + [E(R_m) - R_f] \frac{\rho_{am} \sigma_a \sigma_m}{\sigma_m^2}</math>

| Algebraic rearrangement (isolating the expected return <math>E(R_a)</math>).

| 5. Final Beta Form

| <math>E(R_a) = R_f + \beta_a (E(R_m) - R_f)</math>

| Substituting the Beta definition (<math>\beta_a = \frac{\sigma_{am{\sigma_m^2}</math>, utilizing the covariance identity).

Estimation and Application

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

{| class="wikitable" style="width: 100%; max-width: 650px; border: 2px solid #a2a9b1; background-color: #f8f9fa; margin-bottom: 1em;"

! style="background-color: #eaecf0; text-align: left; padding-left: 10px; font-size: 110%;" | The SCL Equation (Statistical Estimation)

| style="text-align: center; padding: 1.2em;" | <math display="block">\text{SCL} : R_{i,t} - R_{f} = \alpha_i + \beta_i (R_{M,t} - R_{f}) + \epsilon_{i,t}</math>

{| class="wikitable" style="width: 98%; margin: 5px auto; font-size: 95%; background-color: #ffffff;"

! Variable (Symbol) !! Description

| <math>\alpha_i</math> (Alpha)

| The intercept representing the abnormal return. In theory, the expected alpha is zero (<math>\alpha = 0</math>).

| <math>\beta_i</math> (Beta)

| The slope of the regression, representing the asset's systematic risk.

| <math>R_{i,t}, R_{M,t}</math>

| The returns of the asset and the market at a specific point in time (<math>t</math>).

| <math>R_f</math>

| The risk-free rate (e.g., bank deposit or government bond return).

| <math>\epsilon_{i,t}</math> (Error Term)

| The residual or idiosyncratic return (specific risk) unique to the firm.

Once the required expected return <math>E(R_i)</math> is established, it is used as the discount rate to determine the asset's intrinsic value based on future cash flows (CF):

{| class="wikitable" style="width: 100%; max-width: 650px; border: 2px solid #a2a9b1; background-color: #f8f9fa;"

! style="background-color: #eaecf0; text-align: left; padding-left: 10px; font-size: 110%;" | Fundamental Valuation (Present Value)

| style="text-align: center; padding: 1.2em;" | <math display="block">\text{Value}_0 = \sum_{t=1}^n \frac{E(CF_t)}{(1 + E(R_i))^t}</math>

{| class="wikitable" style="width: 98%; margin: 5px auto; font-size: 95%; background-color: #ffffff;"

! Variable !! Description

| <math>Value_0</math>

| The theoretical fair price (Present Value) today.

| <math>E(CF_t)</math>

| The expected cash flow in period <math>t</math>.

| <math>E(R_i)</math>

| The CAPM-derived required return (discount rate).

| <math>n</math>

| The total number of periods (time horizon).

An asset is considered **undervalued** if its calculated <math>Value_0</math> 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. 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 (like coherent 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 and was one of the earliest investors to strategically allocate between stocks, bonds and cash. However, Templeton also felt the theoretical foundations of MPT were questionable, and concluded (as described by a biographer): "the notion that building portfolios on the basis of unreliable and irrelevant statistical inputs, such as historical volatility, was doomed to failure."

Daniel Peris states the earliest version of MPT was an important innovation in attempting to provide a systematic approach to risk and portfolio construction. However, Peris also believes MPT shows influence from broader intellectual trends from the 1950s and '60s that may be outdated, and has been distorted far beyond Markowitz's intention and so focused on advanced mathematics and share prices it loses sight of the fact stocks are minority ownership of companies. Due in part to his training as a historian before working in the finance industry, Peris also believes MPT and many other important concepts are ahistorical and additionally notes many of those who work in the industry typically have little interest in understanding the roots and development of the theories that underlie their career or evaluating if the concepts still have merit.

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.

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 (i.e., there is little or no recovery/salvage value of a half-complete project).

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.