Extract the criticisms that modern portfolio theory faces from this link https://en.wikipedia.org/wiki/Modern_portfolio_theory, place them in a bullet list

Despite its theoretical importance, critics of MPT question whether it is an ideal investment tool, because its model of financial markets does not match the real world in many ways

The risk, return, and correlation measures used by MPT are based on [expected values](https://en.wikipedia.org/wiki/Expected_value), 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](https://en.wikipedia.org/wiki/Variance)
 and [covariance](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Log-normal_distribution)) and can give rise to, besides reduced [volatility](https://en.wikipedia.org/wiki/Volatility_(finance)), also inflated growth of return. In practice, investors must substitute predictions based on historical measurements of asset return and volatility for these values in the equations. Very often such expected values fail to take account of new circumstances that did not exist when the historical data were generated.

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](https://en.wikipedia.org/wiki/Probability) in nature, not structural. This is a major difference as compared to many engineering approaches to [risk management](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Variance) is a symmetric measure that counts abnormally high returns as just as risky as abnormally low returns. The psychological phenomenon of [loss aversion](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Coherent_risk_measure)) might better reflect investors' true preferences.

Modern portfolio theory has also been criticized because it assumes that returns follow a [Gaussian distribution](https://en.wikipedia.org/wiki/Normal_distribution). Already in the 1960s, [Benoit Mandelbrot](https://en.wikipedia.org/wiki/Benoit_Mandelbrot) and [Eugene Fama](https://en.wikipedia.org/wiki/Eugene_Fama) showed the inadequacy of this assumption and proposed the use of more general [stable distributions](https://en.wikipedia.org/wiki/Stable_distributions) instead. [Stefan Mittnik](https://en.wikipedia.org/wiki/Stefan_Mittnik) and [Svetlozar Rachev](https://en.wikipedia.org/wiki/Svetlozar_Rachev) presented strategies for deriving optimal portfolios in such settings.

[Contrarian investors](https://en.wikipedia.org/wiki/Contrarian_investing) and [value investors](https://en.wikipedia.org/wiki/Value_investing) typically do not subscribe to Modern Portfolio Theory. One objection is that the MPT relies on the [efficient-market hypothesis](https://en.wikipedia.org/wiki/Efficient-market_hypothesis)
 and uses fluctuations in share price as a substitute for risk. [Sir John Templeton](https://en.wikipedia.org/wiki/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.

- The risk, return, and correlation measures used by Modern portfolio theory (MPT) are based on expected values, such expected values fail to take account of new circumstances that did not exist when the historical data were generated
- The risk measurements used in MPT are probabilistic because MPT models risk in terms of the likelihood of losses based on past market data and not why the losses occur.
- MPT attempts to minimize risks in the form of variance. However, this metric doesn’t reflect investors’ true concerns in practice. Variance is symmetric, so it punishes high returns the same way as high losses. However, investors care more about loss than gain, so the perception of risk is asymmetric in nature. Coherent risk measures should be a better metric that reflects investors’ preference
- MPT assumes that returns follow Gaussian distribution. However, many scholars suggest that returns might follow other distributions such as stable distributions
- MPT relies on the efficient-market hypothesis and assumes that share price fluctuation is a risk measure. However, building a portfolio based on historical volatility is a questionable premise criticized by Sir John Templeton