One of the most popular tools in financial analysis, the Sharpe ratio is a measurement of the expected excess return of an investment in relation to its return volatility. The Sharpe ratio measures the average return in excess of the risk-free rate per unit of uncertainty to determine how much additional return an investor can receive with the added volatility of holding riskier assets.

For example, the Sharpe ratio for large market cap stocks will initially outperform safer bonds. However, in the long run, a bond portfolio will inevitably outperform stocks. Efficient frontiers are derived from mean-variance analysis , which attempts to create more efficient investment choices. The typical investor prefers high expected returns with low variance.

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Portfolios below the frontier curve are considered sub-optimal while those above are considered unattainable. The optimal choice with respect to the efficient frontier approach occurs when the straight line of the risk-free assets lie tangent to the risky asset frontier. CAPM is an equilibrium theory built on the relationship between risk and expected return. In particular, investors must be compensated for the time value of money and risk. The risk-free rate , typically a Treasury bond or stock index, represents the time value of money for placing money in any investment. Simply put, the mean return of an asset should be linearly related to its beta coefficient — this shows that riskier investments earn a premium over the benchmark rate.

The value at risk VaR approach to portfolio management is a simple way to measure risk. VaR measures the maximum loss that cannot be exceeded at a given confidence level. Calculated based on time period, confidence level and pre-determined loss amount, VaR statistics provide investors with a worst case scenario analysis. Investing in financial markets is inherently risky.

Many individuals use financial advisors and wealth managers to increase returns and reduce the risk of investments. Modern portfolio theory uses five statistical indicators — alpha, beta, standard deviation, R-squared, and the Sharpe ratio — to do this. Likewise, the capital asset pricing model and value at risk are widely employed to measure the risk to reward tradeoff with assets and portfolios. Mutual Fund Essentials. Investing Essentials. Portfolio Management.

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## High Frequency

Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Risk and financial portfolio analytics - A technical Introduction 1. Information regarding potential future products is intended to outline our general product direction and it should not be relied on in making a purchasing decision.

The information mentioned regarding potential future products is not a commitment, promise, or legal obligation to deliver any material, code or functionality. Information about potential future products may not be incorporated into any contract. The development, release, and timing of any future features or functionality described for our products remains at our sole discretion. Performance is based on measurements and projections using standard IBM benchmarks in a controlled environment. Therefore, no assurance can be given that an individual user will achieve results similar to those stated here.

Created by: Dennis Buttera 8. Descriptive Analytics What has happened? 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. 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 s, Benoit Mandelbrot and Eugene Fama showed the inadequacy of this assumption and proposed the use of stable distributions instead. Stefan Mittnik and Svetlozar Rachev presented strategies for deriving optimal portfolios in such settings. After the stock market crash in , 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. This risk is only an opportunity to buy or sell assets at attractive prices inasmuch as it suits one's book.

Since MPT's introduction in , 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. 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.

With the advances in Artificial Intelligence , other information such as market sentiment and financial knowledge can be incorporated automatically to the 'views'. 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, [18] and may recommend to invest into Y on the basis that it has lower variance.

Maccheroni et al. Alternatively, mean-deviation analysis [20] is a rational choice theory resulting from replacing variance by an appropriate deviation risk measure. In the s, 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. 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.

Some experts apply MPT to portfolios of projects and other assets besides financial instruments. 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.

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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.

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MPT uses historical variance as a measure of risk, but portfolios of assets like major projects don't 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. From Wikipedia, the free encyclopedia. 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.

Main article: Efficient frontier. See also: Portfolio optimization.

## How Investment Risk Is Quantified

Main article: Capital allocation line. Main article: Capital asset pricing model. Hubbard , 'The Failure of Risk Management', p. The Financial Times. March The Journal of Finance. Naval Research Logistics Quarterly. Retrieved Wilmott Magazine. Journal of Economics and Business. Svetlozar Rachev and Prof. Stefan Mittnik" PDF.