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. **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.