Investing in financial markets can carry significant risks. Modern portfolio theory (MPT) assesses the maximum expected portfolio return for a given amount of portfolio risk. Within the framework of MPT, an optimal portfolio is constructed on the basis of asset allocation, diversification and rebalancing. Asset allocation togetherÂ with diversificationÂ is the strategy of dividing a portfolio among various asset classes. Optimal diversification involves holding multiple instruments that aren't positively correlated. Â

While diversification and asset allocation can improve returns, systematic and unsystematic risks are inherent in investing. The efficient frontier does its best to minimize an investorâ€™s exposure to such risk. Introduced by Harry Markowitz in 1952, the concept identifies an optimal level of diversification and asset allocation given the intrinsic risks of a portfolio.

Along with the efficient frontier, statistical measures and methods including value at risk (VaR) and capital asset pricing model (CAPM) can all be used to measure risk.

## Alpha and Beta

When it comes to quantifying value and risk, two statistical metrics, alpha and beta, are useful for investors. Both are risk ratios used in MPT and help to determine the risk/reward profile of investment securities.

Alpha measures the performance of an investment portfolio and compares it to a benchmark index, such as the S&P 500. The difference between the returns of a portfolio and theÂ benchmark is referred to as alpha. A positive alpha of oneÂ means the portfolio has outperformed the benchmark by 1 percent.Â Likewise, a negative alpha indicates theÂ underperformance of an investment.

Beta measures the volatility of a portfolio compared to a benchmark index. The statistical measure beta is used in the CAPM, which uses risk and return to price an asset. Unlike alpha, beta captures the movements and swings in asset prices. A beta greater than oneÂ indicates higher volatility whereas a beta under oneÂ means the security will be more stable.

For example, StarbucksÂ (SBUX), withÂ a beta coefficient of 0.53, represents a safer investment than Nvidia (NVDA), which has a beta of 1.53, as of April 6, 2018. A savvy financial advisor or fund manager would likely avoid high alpha and beta investments for risk-averse clients.

## R-squared

In statistics, R-squared represents a notable component of regression analysis. The coefficient R represents the correlation between two variablesÂ â€”Â for investment purposes, R-squared measures the explained movement of a fund or security in relation to a benchmark. Â A high R-squared shows that a portfolioâ€™s performance is in line with the index. Financial advisors can use R-squared in tandem with beta to provide investors with a comprehensive picture of asset performance.

## Standard Deviation

By definition, standard deviation is a statistical procedure used to quantify any variation from the average return of a data set. In finance, standard deviation uses the return of an investment to measure the investmentâ€™s volatility. The measurement differs slightly from beta as it compares volatility to the historical returns of the security rather than a benchmark index.

Standard deviation and other technical indicators insist that history tends to repeat itself. High standard deviations are indicative of volatility, whileÂ lower standard deviations are associated with stable assets.

## Sharpe Ratio

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.

A ratio of oneÂ or greater is considered to have a better risk-to-reward trade-off. For example, the Sharpe ratio for large market cap stocks will initially outperform safer bonds.Â However, in the long run, a bond portfolio will inevitably outperform stocks.

## Efficient Frontier

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. The efficient frontier is constructed accordingly byÂ using a set of optimal portfolios that offer the highest expected return for a specific risk level.

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. Â Â

## Capital Asset Pricing Model

CAPM is an equilibrium theory built on the relationship between risk and expected return. The theoryÂ helps investors measure theÂ risk and expected return of an investment to appropriately price the asset. 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 a security should be linearly related to its beta coefficient â€” this shows that riskier investments earn a premium over the benchmark rate. Following a risk-to-reward framework, the expected return (under a CAPM model)Â will be higher when the investor bears greater risks.

## Value-at-Risk

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. Â

If an investment has a fiveÂ percent VaR, the investor faces a fiveÂ percent chance of losing the entire investment in any given month. The VaR methodology isnâ€™t the most comprehensive measure of risk,Â however, due to its simplistic approach, it still remains one of the most popular measures in portfolio management.

## The Bottom Line

Investing in financial markets is inherently risky. Many individuals use financial advisors and wealth managers to increase returns and reduce the risk of investments. These financial professionals use statistical measures and risk/reward models to differentiate volatile assets from stable ones. Modern portfolio theory uses five statistical indicators â€” alpha, beta, standard deviation, R-squared, and the Sharpe ratio â€” 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.