Modern portfolio theory

"Portfolio analysis" redirects here. For the text book, see Portfolio Analysis. For theorems about the mean-variance efficient frontier, see Mutual fund separation theorem. For non-mean-variance portfolio analysis, see Marginal conditional stochastic dominance.

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. ^[1] Often, the historical variance and covariance of returns is used as a proxy for the forward-looking versions of these quantities, ^[2] but other, more sophisticated methods are available. ^[3]

Economist Harry Markowitz introduced MPT in a 1952 paper, ^[1] for which he was later awarded a Nobel Memorial Prize in Economic Sciences; see Markowitz model.

In 1940, Bruno de Finetti published ^[4] 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. ^[5]

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. ^[6] The return of an individual asset ${\displaystyle R_{i}}$ is defined as the Total Net Return.

Depending on the asset class, the income component ${\displaystyle D_{t}}$ and the price ${\displaystyle P}$ are defined specifically:

• For Stocks:${\displaystyle D_{t}}$ represents dividends.
• For Bonds:${\displaystyle D_{t}}$ represents coupon payments, and the prices ${\displaystyle P}$ are treated as Dirty Prices (Clean Price + Accrued interest ${\displaystyle S=K\cdot {\frac {f}{100}}\cdot {\frac {A}{D}}}$).

Total Net Return Calculation

Component	Formula
Total Net Return (${\displaystyle R_{i}}$)	${\displaystyle R_{i}={\frac {(P_{t}+S_{t}-C_{sale})-(P_{t-1}+S_{t-1}+C_{buy})+D_{t}}{P_{t-1}+S_{t-1}+C_{buy}}}}$

Definition of Variables (in Order of Formula)

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

• ${\displaystyle P}$ (Market Price): The quoted price of the asset at the end of the period (${\displaystyle P_{t}}$) and the beginning (${\displaystyle P_{t-1}}$).
• ${\displaystyle S}$ (Accrued Interest): Calculated as ${\displaystyle K\cdot {\frac {f}{100}}\cdot {\frac {A}{D}}}$, where ${\displaystyle K}$ is the nominal value, ${\displaystyle f}$ is the coupon rate, and ${\displaystyle A/D}$ is the day-count fraction. ^[7]
• ${\displaystyle C_{sale}}$ / ${\displaystyle C_{buy}}$ (Transaction Costs): Includes brokerage commissions, exchange fees, financial transaction taxes, and custody fees prorated over the holding period. ^[8]
• ${\displaystyle D_{t}}$ (Distributions): The universal symbol for periodic income, such as dividends for stocks or coupon payments for bonds.

Portfolio Risk and Return Metrics

Complexity	Expected Return ${\displaystyle \operatorname {E} (R_{p})}$	Variance (Risk) ${\displaystyle \sigma _{p}^{2}}$
One-Asset	${\displaystyle \operatorname {E} (R_{A})}$	${\displaystyle \sigma _{A}^{2}}$
Two-Asset	${\displaystyle w_{A}\operatorname {E} (R_{A})+w_{B}\operatorname {E} (R_{B})}$	${\displaystyle \underbrace {w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}} _{\text{Variance Component}}+\underbrace {2w_{A}w_{B}\sigma _{A}\sigma _{B}\rho _{AB}} _{\text{Correlation Component}}}$
Three-Asset	${\displaystyle \sum _{i=A}^{C}w_{i}\operatorname {E} (R_{i})}$	${\displaystyle \underbrace {\sum w_{i}^{2}\sigma _{i}^{2}} _{\text{Variance}}+\underbrace {2w_{A}w_{B}\sigma _{AB}+2w_{A}w_{C}\sigma _{AC}+2w_{B}w_{C}\sigma _{BC}} _{\text{Pairwise Covariances}}}$
N-Asset	${\displaystyle \mathbf {w} ^{\intercal }{\boldsymbol {\mu }}}$	${\displaystyle \mathbf {w} ^{\intercal }\Sigma \mathbf {w} }$

Practical Application: Bonds vs. Stocks

While the mathematical structure of MPT is identical for all assets, the calculation of ${\displaystyle R_{i}}$ for bonds must account for the pull-to-par effect and day-count conventions. This ensures that the portfolio weights ${\displaystyle w_{i}}$ reflect the true Fair Market Value (Dirty Price) of the holdings at any given time ${\displaystyle t}$. ^[9]

Diversification

An investor can reduce portfolio risk (specifically the portfolio standard deviation ${\displaystyle \sigma _{p}}$) by holding combinations of instruments that are not perfectly positively correlated (${\displaystyle \rho _{ij}<1}$). 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. ^[6]

Impact of correlation on the risk-return profile: As correlation (${\displaystyle \rho }$) decreases, the diversification benefit increases, allowing for lower risk at the same level of return.

Impact of Correlation on Portfolio Risk

Correlation Scenario	Mathematical Result	Risk Implication
Perfect Positive (${\displaystyle \rho _{ij}=1}$)	${\displaystyle \sigma _{p}=\sum _{i=1}^{n}w_{i}\sigma _{i}}$	No Risk Reduction: Risk is simply the weighted average of individual volatilities. [8]
Zero Correlation (${\displaystyle \rho _{ij}=0}$)	${\displaystyle \sigma _{p}^{2}=\sum _{i=1}^{n}w_{i}^{2}\sigma _{i}^{2}}$	Idiosyncratic Risk Elimination: As ${\displaystyle n\to \infty }$, portfolio variance approaches zero. [9]
Partial Correlation (${\displaystyle 0<\rho <1}$)	${\displaystyle \sigma _{p}<\sum w_{i}\sigma _{i}}$	Diversification Benefit: Provides a "free lunch" by reducing risk without sacrificing return.

In reality, most assets have a correlation ${\displaystyle 0<\rho <1}$. Markowitz proved that as long as ${\displaystyle \rho <1}$, 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

Main article: Efficient frontier

See also: Portfolio optimization

Efficient Frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet', and is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier.

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. ^[10]

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, ^[11] 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" ${\displaystyle q\in [0,\infty )}$, the efficient frontier is found by minimizing the following expression:

${\displaystyle w^{T}\Sigma w-qR^{T}w}$

where

• ${\displaystyle w\in \mathbb {R} ^{N}}$ is a vector of portfolio weights and ${\displaystyle \sum _{i=1}^{N}w_{i}=1.}$ (The weights can be negative);
• ${\displaystyle \Sigma \in \mathbb {R} ^{N\times N}}$ is the covariance matrix for the returns on the assets in the portfolio;
• ${\displaystyle q\geq 0}$ is a "risk tolerance" factor, where 0 results in the portfolio with minimal risk and ${\displaystyle \infty }$ results in the portfolio infinitely far out on the frontier with both expected return and risk unbounded; and
• ${\displaystyle R\in \mathbb {R} ^{N}}$ is a vector of expected returns.
• ${\displaystyle w^{T}\Sigma w\in \mathbb {R} }$ is the variance of portfolio return.
• ${\displaystyle R^{T}w\in \mathbb {R} }$ 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, ^[12] 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, ^[13] in JavaScript ^[14] 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 ${\displaystyle R^{T}w.}$ This version of the problem requires that we minimize

${\displaystyle w^{T}\Sigma w}$

subject to

${\displaystyle R^{T}w=\mu }$

and

${\displaystyle \sum _{i=1}^{N}w_{i}=1}$

for parameter ${\displaystyle \mu }$. This problem is easily solved using a Lagrange multiplier which leads to the following linear system of equations:

${\displaystyle {\begin{bmatrix}2\Sigma &-R&-{\bf {1}}\\R^{T}&0&0\\{\bf {1}}^{T}&0&0\end{bmatrix}}{\begin{bmatrix}w\\\lambda _{1}\\\lambda _{2}\end{bmatrix}}={\begin{bmatrix}0\\\mu \\1\end{bmatrix}}}$

Two mutual fund theorem

A fundamental result of Markowitz's analysis is the two mutual fund theorem (also known as the separation theorem). ^[15] 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 ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ are two efficient portfolios, then any third efficient portfolio ${\displaystyle P_{target}}$ can be expressed as:

$${\displaystyle P_{target}=\alpha P_{1}+(1-\alpha )P_{2}}$$where ${\displaystyle \alpha }$ is the weighting factor. Because the underlying assets in ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ are valued based on their Total Net Return (including capital gains, dividends, and interest, net of transaction costs), the resulting combination ${\displaystyle P_{target}}$ 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 (${\displaystyle 0<\alpha <1}$): If the target portfolio ${\displaystyle P_{target}}$ lies on the frontier segment between ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$, the investor allocates a positive fraction ${\displaystyle \alpha }$ to Fund 1 and ${\displaystyle 1-\alpha }$ 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 (${\displaystyle \alpha >1}$): To achieve a return higher than both ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ (assuming ${\displaystyle E(R_{1})>E(R_{2})}$), the investor sells Fund 2 short (negative weight) and invests more than 100% of their capital into Fund 1.
  • Shorting Fund 1 (${\displaystyle \alpha <0}$): 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. ^[9]

Risk-free asset and the capital allocation line

Main article: Capital allocation line

The risk-free asset is the theoretical asset that pays a deterministic risk-free rate. ^[8] 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. ^[7] By definition, the risk-free asset has zero variance in returns if held to maturity and remains uncorrelated with any risky asset or portfolio. ^[9] 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. ^[16]

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. ^[8] The vertical intercept of this line represents a portfolio allocated 100% to the risk-free asset ${\displaystyle R_{F}}$. ^[17] The tangency point denotes a portfolio with 100% investment in risky assets, while segments between the intercept and tangency represent lending portfolios (long ${\displaystyle R_{F}}$). ^[9] Points extending beyond the tangency point represent borrowing portfolios, where the investor leverages the risky tangency portfolio by shorting the risk-free asset. ^[8] This efficient locus is defined as the capital allocation line (CAL), expressed by the formula:

Capital Allocation Line (CAL) Formula

Component	Equation
Expected Return (${\displaystyle E(R_{C})}$)	${\displaystyle E(R_{C})=R_{F}+\sigma _{C}{\frac {E(R_{P})-R_{F}}{\sigma _{P}}}}$

In this context, P represents the tangency portfolio of risky assets, F denotes the risk-free asset, and C is the combined portfolio. ^[16] The introduction of ${\displaystyle R_{F}}$ improves the investment opportunity set because the CAL provides a higher expected return for every level of risk compared to the risky-only hyperbola. ^[8] 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. ^[17]

Geometric intuition

The efficient frontier can be pictured as a problem in quadratic curves. ^[15] On the market, we have the assets ${\displaystyle R_{1},R_{2},\dots ,R_{n}}$. 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 ${\displaystyle w_{1},w_{2},\dots ,w_{n}}$, such that ${\displaystyle \sum _{i}w_{i}=1}$, and we hold the assets according to ${\displaystyle w^{T}R=\sum _{i}w_{i}R_{i}}$.

Markowitz bullet

The ellipsoid is the contour of constant variance. The ${\displaystyle x+y+z=1}$ 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:$${\displaystyle {\begin{cases}E[w^{T}R]=\mu \\\min \sigma ^{2}=Var[w^{T}R]\\\sum _{i}w_{i}=1\end{cases}}}$$Portfolios are points in the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. The third equation states that the portfolio should fall on a plane defined by ${\displaystyle \sum _{i}w_{i}=1}$. The first equation states that the portfolio should fall on a plane defined by ${\displaystyle w^{T}E[R]=\mu }$. The second condition states that the portfolio should fall on the contour surface for ${\displaystyle \sum _{ij}w_{i}\rho _{ij}w_{j}}$ 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 ${\displaystyle \rho _{ij}}$ is invertible). Therefore, we can solve the quadratic optimization graphically by drawing ellipsoidal contours on the plane ${\displaystyle \sum _{i}w_{i}=1}$, then intersect the contours with the plane ${\displaystyle \{w:w^{T}E[R]=\mu {\text{ and }}\sum _{i}w_{i}=1\}}$. 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 ${\displaystyle \mu }$, 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 ${\displaystyle \{w+w't:t\in \mathbb {R} \}}$. We find that along the line,$${\displaystyle {\begin{cases}\mu &=(w'^{T}E[R])t+w^{T}E[R]\\\sigma ^{2}&=(w'^{T}\rho w')t^{2}+2(w^{T}\rho w')t+(w^{T}\rho w)\end{cases}}}$$giving a hyperbola in the ${\displaystyle (\sigma ,\mu )}$ plane. The hyperbola has two branches, symmetric with respect to the ${\displaystyle \mu }$ axis. However, only the branch with ${\displaystyle \sigma >0}$ is meaningful. By symmetry, the two asymptotes of the hyperbola intersect at a point ${\displaystyle \mu _{MVP}}$ on the ${\displaystyle \mu }$ axis. The point ${\displaystyle \mu _{mid}}$ 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

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 ${\displaystyle \mu _{RF}<\mu _{MVP}}$.

In particular, if the risk-free return is greater or equal to ${\displaystyle \mu _{MVP}}$, 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 ${\displaystyle \mu _{RF}}$ approaches ${\displaystyle \mu _{MVP}}$ 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 ${\displaystyle v}$, such that ${\displaystyle v^{T}R}$ is a random variable with zero variance—that is, it is not random at all.

Suppose ${\displaystyle \sum _{i}v_{i}=0}$ and ${\displaystyle v^{T}R=0}$, 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 ${\displaystyle \sum _{i}v_{i}=0}$ and ${\displaystyle v^{T}R\neq 0}$, then that means there is free money, breaking the no arbitrage assumption.

Suppose ${\displaystyle \sum _{i}v_{i}\neq 0}$, then we can scale the vector to ${\displaystyle \sum _{i}v_{i}=1}$. This means that we have constructed a risk-free asset with return ${\displaystyle v^{T}R}$. 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

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 tangency portfolio), the risky assets' prices and expected returns adjust until the ratios in the tangency portfolio match the ratios of assets supplied to the market. ^[18]

Equilibrium Condition (The Treynor Ratio)

Concept	Formula	Description
Equilibrium Slope	${\displaystyle {\frac {E(R_{i})-R_{f}}{\beta _{i}}}=E(R_{M})-R_{f}}$	In equilibrium, the risk premium per unit of systematic risk (${\displaystyle \beta _{i}}$) is equal for all assets.

Systematic risk and specific risk

The Security Market Line illustrates that the market only compensates for systematic risk (${\displaystyle \beta }$). While total risk includes idiosyncratic factors, the risk premium is determined solely by the asset's relationship to the broader market.

The total risk of an individual asset or portfolio is decomposed into two distinct components: specific risk and systematic risk. ^[19]

• 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. ^[20]
• 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. ^[21]

Because rational investors can eliminate unique risk at no cost through diversification, the market only provides a risk premium for bearing systematic risk. ^[22] This implies that an asset's expected return is not determined by its total variance, but specifically by its covariance with the market portfolio. ^[23] Consequently, the equilibrium price of an asset must adjust until its risk-adjusted return aligns with the Security Market Line shown in the diagram. ^[24] Under these assumptions, assets with the same Beta must offer the same expected return, regardless of their individual specific risk profiles. ^[25] 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. ^[26]

Advanced Risk Decomposition (Single-Index Model)

Component (Symbol)	Mathematical Formula	Description
Total Risk (${\displaystyle \sigma _{i}^{2}}$)	${\displaystyle \sigma _{i}^{2}=\beta _{i}^{2}\sigma _{M}^{2}+\sigma ^{2}(\epsilon _{i})}$	Sum of systematic and idiosyncratic variance components.
Systematic Component (${\displaystyle \beta _{i}^{2}\sigma _{M}^{2}}$)	${\displaystyle \underbrace {\left({\frac {\operatorname {Cov} (R_{i},R_{M})}{\sigma _{M}^{2}}}\right)^{2}} _{\beta _{i}^{2}}\sigma _{M}^{2}}$	Risk attributed to the asset's sensitivity to market movements (Market Variance ${\displaystyle \sigma _{M}^{2}}$).
Specific Component (${\displaystyle \sigma ^{2}(\epsilon _{i})}$)	${\displaystyle \sigma ^{2}(\epsilon _{i})}$	Residual variance (error term ${\displaystyle \epsilon _{i}}$ unique to the firm; the diversifiable portion of risk).
Beta Factor (${\displaystyle \beta _{i}}$)	${\displaystyle \beta _{i}={\frac {\operatorname {Cov} (R_{i},R_{M})}{\sigma _{M}^{2}}}=\rho _{i,M}{\frac {\sigma _{i}}{\sigma _{M}}}}$	Asset sensitivity relative to the market (determined by covariance ${\displaystyle \sigma _{i,M}}$ and correlation ${\displaystyle \rho _{i,M}}$). [8]
Systematic Proportion (${\displaystyle R^{2}}$)	${\displaystyle R^{2}={\frac {\beta _{i}^{2}\sigma _{M}^{2}}{\sigma _{i}^{2}}}}$	The Coefficient of determination (showing the percentage of total risk that is systematic). [9]

Capital asset pricing model

Main article: 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 (SML):

The CAPM Equation (Security Market Line)
$${\displaystyle \operatorname {E} (R_{i})=R_{f}+\beta _{i}(\operatorname {E} (R_{m})-R_{f})}$$
Variable Description ${\displaystyle R_{f}}$ The risk-free rate (the return on an investment with zero risk, such as a bank deposit or a government-backed bond). ${\displaystyle \beta _{i}}$ (Beta) Measure of asset sensitivity to the overall market; found via regression analysis on historical data. ${\displaystyle E(R_{m})-R_{f}}$ The market premium (the expected excess return of the market portfolio over the risk-free rate ${\displaystyle R_{f}}$).

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. ^[18]

Mathematical Derivation Steps

Step	Formula	Logic / Assumption
1. Marginal Risk	${\displaystyle \Delta \sigma _{P}\approx 2w_{m}w_{a}\rho _{am}\sigma _{a}\sigma _{m}}$	Since the weight ${\displaystyle w_{a}}$ is very small, the quadratic term (${\displaystyle w_{a}^{2}}$) vanishes.
2. Marginal Return	${\displaystyle \Delta E(R_{P})=w_{a}(E(R_{a})-R_{f})}$	The additional return gained by the allocation ${\displaystyle w_{a}}$ (relative to the risk-free rate ${\displaystyle R_{f}}$).
3. Equilibrium Ratio	${\displaystyle {\frac {w_{a}(E(R_{a})-R_{f})}{2w_{m}w_{a}\rho _{am}\sigma _{a}\sigma _{m}}}={\frac {E(R_{m})-R_{f}}{2\sigma _{m}^{2}}}}$	The improvement from asset a must match the market's reward-to-risk ratio.
4. Solving for ${\displaystyle E(R_{a})}$	${\displaystyle E(R_{a})=R_{f}+[E(R_{m})-R_{f}]{\frac {\rho _{am}\sigma _{a}\sigma _{m}}{\sigma _{m}^{2}}}}$	Algebraic rearrangement (isolating the expected return ${\displaystyle E(R_{a})}$).
5. Final Beta Form	${\displaystyle E(R_{a})=R_{f}+\beta _{a}(E(R_{m})-R_{f})}$	Substituting the Beta definition (${\displaystyle \beta _{a}={\frac {\sigma _{am}}{\sigma _{m}^{2}}}}$, 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:

The SCL Equation (Statistical Estimation)
$${\displaystyle {\text{SCL}}:R_{i,t}-R_{f}=\alpha _{i}+\beta _{i}(R_{M,t}-R_{f})+\epsilon _{i,t}}$$
Variable (Symbol) Description ${\displaystyle \alpha _{i}}$ (Alpha) The intercept representing the abnormal return. In theory, the expected alpha is zero (${\displaystyle \alpha =0}$). ${\displaystyle \beta _{i}}$ (Beta) The slope of the regression, representing the asset's systematic risk. ${\displaystyle R_{i,t},R_{M,t}}$ The returns of the asset and the market at a specific point in time (${\displaystyle t}$). ${\displaystyle R_{f}}$ The risk-free rate (e.g., bank deposit or government bond return). ${\displaystyle \epsilon _{i,t}}$ (Error Term) The residual or idiosyncratic return (specific risk) unique to the firm.

Once the required expected return ${\displaystyle E(R_{i})}$ is established, it is used as the discount rate to determine the asset's intrinsic value based on future cash flows (CF):

Fundamental Valuation (Present Value)
$${\displaystyle {\text{Value}}_{0}=\sum _{t=1}^{n}{\frac {E(CF_{t})}{(1+E(R_{i}))^{t}}}}$$
Variable Description ${\displaystyle Value_{0}}$ The theoretical fair price (Present Value) today. ${\displaystyle E(CF_{t})}$ The expected cash flow in period ${\displaystyle t}$. ${\displaystyle E(R_{i})}$ The CAPM-derived required return (discount rate). ${\displaystyle n}$ The total number of periods (time horizon).

An asset is considered **undervalued** if its calculated ${\displaystyle Value_{0}}$ 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. ^[2]

The risk, return, and correlation measures used by MPT are based on expected values, which means that they are statistical statements about the future (the expected value of returns is explicit in the above equations, and implicit in the definitions of variance and covariance). Such measures often cannot capture the true statistical features of the risk and return which often follow highly skewed distributions (e.g., the log-normal distribution) and can give rise to, besides reduced volatility, also inflated growth of return. ^[27] 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. ^[28] 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. ^[29] The optimization problem is solved under the assumption that expected values are uncertain and correlated. ^[30] 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.

Options theory and MPT have at least one important conceptual difference from the probabilistic risk assessment done by nuclear power [plants]. A PRA is what economists would call a structural model. The components of a system and their relationships are modeled in Monte Carlo simulations. If valve X fails, it causes a loss of back pressure on pump Y, causing a drop in flow to vessel Z, and so on.

But in the Black–Scholes equation and MPT, there is no attempt to explain an underlying structure to price changes. Various outcomes are simply given probabilities. And, unlike the PRA, if there is no history of a particular system-level event like a liquidity crisis, there is no way to compute the odds of it. If nuclear engineers ran risk management this way, they would never be able to compute the odds of a meltdown at a particular plant until several similar events occurred in the same reactor design.

—  Douglas W. Hubbard, The Failure of Risk Management, p. 67, John Wiley & Sons, 2009. ISBN   978-0-470-38795-5

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. ^{[31] [32] [33]} More recently, Nassim Nicholas Taleb has also criticized modern portfolio theory on this ground, writing:

After the stock market crash (in 1987), they rewarded two theoreticians, Harry Markowitz and William Sharpe, who built beautifully Platonic models on a Gaussian base, contributing to what is called Modern Portfolio Theory. Simply, if you remove their Gaussian assumptions and treat prices as scalable, you are left with hot air. The Nobel Committee could have tested the Sharpe and Markowitz models—they work like quack remedies sold on the Internet—but nobody in Stockholm seems to have thought about it.

— Nassim N. Taleb, The Black Swan: The Impact of the Highly Improbable, p. 277, Random House, 2007. ISBN   978-1-4000-6351-2

Contrarian investors and value investors typically do not subscribe to Modern Portfolio Theory. ^[34] 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 (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." ^[35]

A few studies have argued that "naive diversification", splitting capital equally among available investment options, might have advantages over MPT in some situations. ^[36]

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. ^[37]

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. ^[38] 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. ^{[29] [30]}

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, ^[39] and may recommend to invest into Y on the basis that it has lower variance. Maccheroni et al. ^[40] described choice theory which is the closest possible to the modern portfolio theory, while satisfying monotonicity axiom. Alternatively, mean-deviation analysis ^[41] 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^{[ citation needed ]} 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. ^[42]

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. ^[43]

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. ^[44]

Project portfolios and other "non-financial" assets

Some experts apply MPT to portfolios of projects and other assets besides financial instruments. ^{[45] [46]} When MPT is applied outside of traditional financial portfolios, some distinctions between the different types of portfolios must be considered.

1. 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.
2. 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. ^[45]

See also

• Outline of finance § Portfolio theory
• Beta (finance)
• Bias ratio (finance)
• Black–Litterman model
• Financial economics § Portfolio theory
• Financial economics § Uncertainty
• Financial risk management § Investment management
• Intertemporal portfolio choice
• Investment theory
• Kelly criterion
• Marginal conditional stochastic dominance
• Markowitz model
• Mutual fund separation theorem
• Omega ratio
• Post-modern portfolio theory
• Sortino ratio
• Treynor ratio
• Two-moment decision models
• Universal portfolio algorithm

References

1. Markowitz, H.M. (March 1952). "Portfolio Selection" . The Journal of Finance. 7 (1): 77–91. doi:10.2307/2975974. JSTOR   2975974.
2. Wigglesworth, Robin (11 April 2018). "How a volatility virus infected Wall Street". The Financial Times.
3. Luc Bauwens, Sébastien Laurent, Jeroen V. K. Rombouts (February 2006). "Multivariate GARCH models: a survey". Journal of Applied Econometrics. 21 (1): 79–109. doi:10.1002/jae.842.
4. de Finetti, B. (1940): Il problema dei “Pieni”. Giornale dell’ Istituto Italiano degli Attuari 11, 1–88; translation (Barone, L. (2006)): The problem of full-risk insurances. Chapter I. The risk within a single accounting period. Journal of Investment Management 4(3), 19–43
5. Pressacco, Flavio; Serafini, Paolo (May 2007). "The origins of the mean-variance approach in finance: revisiting de Finetti 65 years later". Decisions in Economics and Finance. 30 (1): 19–49. doi: 10.1007/s10203-007-0067-7 . ISSN   1593-8883.
6. Markowitz, H.M. (1952). "Portfolio Selection". The Journal of Finance. 7 (1): 77–91.
7. Fabozzi, Frank J. (2021). Bond Markets, Analysis, and Strategies. Pearson. ISBN   978-0135962442.
8. Bodie, Zvi (2020). Investments. McGraw-Hill. ISBN   978-1260013832.
9. Elton, Edwin J.; Gruber, Martin J. (2014). Modern Portfolio Theory and Investment Analysis. John Wiley & Sons. ISBN   978-1118469996.
10. Portfolio Selection, Harry Markowitz - The Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91
11. see bottom of slide 6 here
12. Markowitz, H.M. (March 1956). "The Optimization of a Quadratic Function Subject to Linear Constraints". Naval Research Logistics Quarterly. 3 (1–2): 111–133. doi:10.1002/nav.3800030110.
13. Markowitz, Harry (February 2000). Mean-Variance Analysis in Portfolio Choice and Capital Markets. Wiley. ISBN   978-1-883-24975-5.
14. "PortfolioAllocation JavaScript library". github.com/lequant40. Retrieved 2018-06-13.
15. Merton, Robert C. (September 1972). "An Analytic Derivation of the Efficient Portfolio Frontier". The Journal of Financial and Quantitative Analysis. 7 (4): 1851–1872. doi:10.2307/2329621.
16. Sharpe, William F. (1964). "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk". The Journal of Finance. 19 (3): 425–442.
17. Merton, Robert C. (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica. 41 (5): 867–887.
18. Tim Bollerslev (2019). "Risk and Return in Equilibrium: The Capital Asset Pricing Model (CAPM)"
19. Sharpe, William F. (1964). "Capital Asset Prices". Journal of Finance. 19 (3): 425–442.
20. Markowitz, H. (1952). "Portfolio Selection," Journal of Finance.
21. Bodie, Z., et al. (2021). Investments, McGraw-Hill.
22. Sharpe, William F. (1964). "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk". Journal of Finance. 19 (3): 425–442.
23. Lintner, J. (1965). "The Valuation of Risk Assets," Review of Economics and Statistics.
24. Mossin, J. (1966). "Equilibrium in a Capital Asset Market," Econometrica.
25. Ross, S. A. (1976). "The Arbitrage Theory of Capital Asset Pricing," Journal of Economic Theory.
27. Hui, C.; Fox, G.A.; Gurevitch, J. (2017). "Scale-dependent portfolio effects explain growth inflation and volatility reduction in landscape demography". Proceedings of the National Academy of Sciences of the USA. 114 (47): 12507–12511. Bibcode:2017PNAS..11412507H. doi: 10.1073/pnas.1704213114 . PMC   5703273 . PMID   29109261.
28. Low, R.K.Y.; Faff, R.; Aas, K. (2016). "Enhancing mean–variance portfolio selection by modeling distributional asymmetries" (PDF). Journal of Economics and Business. 85: 49–72. doi:10.1016/j.jeconbus.2016.01.003.
29. Benichou, R.; Lemperiere, Y.; Serie, E.; Kockelkoren, J.; Seager, P.; Bouchaud, J.-P.; Potters, M. (2017). "Agnostic Risk Parity: Taming Known and Unknown-Unknowns". Journal of Investment Strategies. 6 (3): 1–12. arXiv: 1610.08818 . doi:10.21314/JOIS.2017.083.
30. Valeyre, S. (2024). "Optimal trend-following portfolios". Journal of Investment Strategies. 12. arXiv: 2201.06635 . doi:10.21314/JOIS.2023.008.
31. Rachev, Svetlozar T. and Stefan Mittnik (2000), Stable Paretian Models in Finance, Wiley, ISBN   978-0-471-95314-2.
32. Risk Manager Journal (2006), "New Approaches for Portfolio Optimization: Parting with the Bell Curve — Interview with Prof. Svetlozar Rachev and Prof.Stefan Mittnik" (PDF).
33. Doganoglu, Toker; Hartz, Christoph; Mittnik, Stefan (2007). "Portfolio Optimization When Risk Factors Are Conditionally Varying and Heavy Tailed" (PDF). Computational Economics. 29 (3–4): 333–354. doi:10.1007/s10614-006-9071-1. S2CID   8280640.
34. Seth Klarman (1991). Margin of Safety: Risk-averse Value Investing Strategies for the Thoughtful Investor. HarperCollins, ISBN   978-0887305108, pp. 97-102
35. Alasdair Nairn (2005). "Templeton's Way With Money: Strategies and Philosophy of a Legendary Investor." Wiley, ISBN 1118149610, p. 262
36. Duchin, Ran; Levy, Haim (2009). "Markowitz Versus the Talmudic Portfolio Diversification Strategies". The Journal of Portfolio Management. 35 (2): 71–74. doi:10.3905/JPM.2009.35.2.071. S2CID   154865200.
37. Henide, Karim (2023). "Sherman ratio optimization: constructing alternative ultrashort sovereign bond portfolios" . Journal of Investment Strategies. doi:10.21314/JOIS.2023.001.
38. Stoyanov, Stoyan; Rachev, Svetlozar; Racheva-Yotova, Boryana; Fabozzi, Frank (2011). "Fat-Tailed Models for Risk Estimation" (PDF). The Journal of Portfolio Management. 37 (2): 107–117. doi:10.3905/jpm.2011.37.2.107. S2CID   154172853.
39. Loffler, A. (1996). Variance Aversion Implies μ-σ²-Criterion. Journal of economic theory, 69(2), 532-539.
40. MacCheroni, Fabio; Marinacci, Massimo; Rustichini, Aldo; Taboga, Marco (2009). "Portfolio Selection with Monotone Mean-Variance Preferences" (PDF). Mathematical Finance. 19 (3): 487–521. doi:10.1111/j.1467-9965.2009.00376.x. S2CID   154536043.
41. Grechuk, Bogdan; Molyboha, Anton; Zabarankin, Michael (2012). "Mean-Deviation Analysis in the Theory of Choice". Risk Analysis. 32 (8): 1277–1292. Bibcode:2012RiskA..32.1277G. doi:10.1111/j.1539-6924.2011.01611.x. PMID   21477097. S2CID   12133839.
42. Chandra, Siddharth (2003). "Regional Economy Size and the Growth-Instability Frontier: Evidence from Europe". Journal of Regional Science . 43 (1): 95–122. Bibcode:2003JRegS..43...95C. doi:10.1111/1467-9787.00291. S2CID   154477444.
43. Chandra, Siddharth; Shadel, William G. (2007). "Crossing disciplinary boundaries: Applying financial portfolio theory to model the organization of the self-concept". Journal of Research in Personality. 41 (2): 346–373. doi:10.1016/j.jrp.2006.04.007.
44. Portfolio Theory of Information Retrieval July 11th, 2009 (2009-07-11). "Portfolio Theory of Information Retrieval | Dr. Jun Wang's Home Page". Web4.cs.ucl.ac.uk. Retrieved 2012-09-05.
45. Hubbard, Douglas (2007). How to Measure Anything: Finding the Value of Intangibles in Business. Hoboken, NJ: John Wiley & Sons. ISBN   978-0-470-11012-6.
46. Sabbadini, Tony (2010). "Manufacturing Portfolio Theory" (PDF). International Institute for Advanced Studies in Systems Research and Cybernetics.

Further reading

• Lintner, John (1965). "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets". The Review of Economics and Statistics. 47 (1): 13–39. doi:10.2307/1924119. JSTOR   1924119.
• Sharpe, William F. (1964). "Capital asset prices: A theory of market equilibrium under conditions of risk". Journal of Finance. 19 (3): 425–442. doi:10.2307/2977928. hdl: 10.1111/j.1540-6261.1964.tb02865.x . JSTOR   2977928.
• Tobin, James (1958). "Liquidity preference as behavior towards risk" (PDF). The Review of Economic Studies. 25 (2): 65–86. doi:10.2307/2296205. JSTOR   2296205.
• DeMiguel, V., Garlappi, L., Nogales, F. J., & Uppal, R. (2009). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management science, 55(5), 798-812.

External links

• Macro-Investment Analysis, Prof. William F. Sharpe, Stanford University
• Portfolio Optimization ^{[ dead link ]}, Prof. Stephen P. Boyd, Stanford University
• "New Approaches for Portfolio Optimization: Parting with the Bell Curve" — Interview with Prof. Svetlozar Rachev and Prof. Stefan Mittnik
• "Bruno de Finetti and Mean-Variance Portfolio Selection Article by Mark Rubinstein on Bruno de Finetti's discovery and comments by Markowitz.