# Modern portfolio theory

portfolio theoryportfolio analysismean-variancemean-variance analysismean-variance optimizationminimum volatilityportfolioportfolio choiceportfolio optimizationcapital allocation line (CAL)
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.wikipedia
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### Harry Markowitz

Harry M. MarkowitzMarkowitzHarry M. Markowitz Award
Economist Harry Markowitz introduced MPT in a 1952 essay, for which he was later awarded a Nobel Prize in Economics; see Markowitz model.
He is best known for his pioneering work in modern portfolio theory, studying the effects of asset risk, return, correlation and diversification on probable investment portfolio returns.

### Markowitz model

proceeds as follows
Economist Harry Markowitz introduced MPT in a 1952 essay, for which he was later awarded a Nobel Prize in Economics; see Markowitz model.
It is foundational to Modern portfolio theory.

### Mutual fund separation theorem

mean-variance efficientMean variance efficiencymutual fund theorem
One key result of the above analysis is the two mutual fund theorem.
In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio.

### Systematic risk

unsystematic risknon-diversifiable risksystematic market risk
Systematic risk (a.k.a. portfolio risk or market risk) refers to the risk common to all securities—except for selling short as noted below, systematic risk cannot be diversified away (within one market).
In contrast, specific risk (sometimes called residual risk, unsystematic risk, or idiosyncratic risk) is risk to which only specific agents or industries are vulnerable (and is uncorrelated with broad market returns).

### Capital asset pricing model

CAPMCapital Asset Pricing Model (CAPM)CAPM model
The CAPM is a model that derives the theoretical required expected return (i.e., discount rate) for an asset in a market, given the risk-free rate available to investors and the risk of the market as a whole.
The CAPM was introduced by Jack Treynor (1961, 1962), William F. Sharpe (1964), John Lintner (1965a,b) and Jan Mossin (1966) independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory.

### Alpha (finance)

alphaexcess returnoutperform market averages
where α i is called the asset's alpha, β i is the asset's beta coefficient and SCL is the security characteristic line.
Alpha, along with beta, is one of two key coefficients in the capital asset pricing model used in modern portfolio theory and is closely related to other important quantities such as standard deviation, R-squared and the Sharpe ratio.

### Risk–return spectrum

risk-return spectrumrisk-returnrisk-reward
The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-expected return profile – i.e., if for that level of risk an alternative portfolio exists that has better expected returns.
This principle, called the separation property, is a crucial feature of modern portfolio theory.

### Eugene Fama

Eugene F. FamaFamaEugène Fama
Already in the 1960s, Benoit Mandelbrot and Eugene Fama showed the inadequacy of this assumption and proposed the use of stable distributions instead.
Eugene Francis "Gene" Fama (born February 14, 1939) is an American economist, best known for his empirical work on portfolio theory, asset pricing, and the efficient-market hypothesis.

### Beta (finance)

betabeta coefficient beta coefficient
where α i is called the asset's alpha, β i is the asset's beta coefficient and SCL is the security characteristic line.
Both coefficients have an important role in modern portfolio theory.

### Risk-free interest rate

risk-free raterisk free raterisk-free asset
The risk-free asset is the (hypothetical) asset that pays a risk-free rate.
The risk-free interest rate is highly significant in the context of the general application of modern portfolio theory which is based on the capital asset pricing model.

### Covariance matrix

variance-covariance matrixcovariance matricescovariance
The covariance matrix plays a key role in financial economics, especially in portfolio theory and its mutual fund separation theorem and in the capital asset pricing model.

### Diversification (finance)

diversificationdiversifieddiversify
In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets.

### Post-modern portfolio theory

Post-modern portfolio theory extends MPT by adopting non-normally distributed, asymmetric, and fat-tailed measures of risk.
Post-modern portfolio theory (or PMPT) is an extension of the traditional modern portfolio theory (MPT, which is an application of mean-variance analysis or MVA).

### Two-moment decision model

mean-variance analysisTwo-moment decision modelsmean-variance
For mean-variance portfolio theory, see Modern portfolio theory or Mutual fund separation theorem.''

### Sharpe ratio

It is tangent to the hyperbola at the pure risky portfolio with the highest Sharpe ratio.

### Covariance

covariantcovariationcovary
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).
Covariances play a key role in financial economics, especially in portfolio theory and in the capital asset pricing model.

### Svetlozar Rachev

Rachev, SvetlozarRachev, Svetlozar T
Stefan Mittnik and Svetlozar Rachev presented strategies for deriving optimal portfolios in such settings.
In mathematical finance, Rachev is known for his work on application of non-Gaussian models for risk assessment, option pricing, and the applications of such models in portfolio theory.

### Security characteristic line

where α i is called the asset's alpha, β i is the asset's beta coefficient and SCL is the security characteristic line.

### Black–Litterman model

Black-Litterman modelBlack-Litterman
Black-Litterman model optimization is an extension of unconstrained Markowitz optimization that incorporates relative and absolute 'views' on inputs of risk and returns from financial experts.
It seeks to overcome problems that institutional investors have encountered in applying modern portfolio theory in practice.

### Asset pricing

Investment theoryAsset pricesAsset pricing model
Asset pricing theory builds on this analysis in the following way.
These models are born out of modern portfolio theory, with the capital asset pricing model (CAPM) as the prototypical result.

### Bias ratio

Bias ratio (finance)

### William F. Sharpe

William SharpeBill SharpeSharpe, William Forsyth
During his undergraduate studies, two professors had a large influence on him: Armen Alchian, a professor of economics who became his mentor, and J. Fred Weston, a professor of finance who first introduced him to Harry Markowitz's papers on portfolio theory.

### Marginal conditional stochastic dominance

Efficiency (finance)efficient
Note that this context of portfolio optimization is not limited to situations in which mean-variance analysis applies.