现代投资组合理论的几个统计衡量指标——alpha, beta, standard deviation, R-squared and the Sharpe ratio

先转一篇investopedia的文章，了解一下概念：

Measuring Risk With Alpha, Beta and Sharpe
Richard Loth, Investopedia 11.05.07, 12:15 PM ET

 

There are five main indicators of investment risk that apply to the analysis of stocks, bonds and mutual fund portfolios. They are alpha, beta, r-squared, standard deviation and theSharpe ratio.

These statistical measures are historical predictors of investment risk/volatility and are all major components of modern portfolio theory(MPT). The MPT is a standard financial and academic methodology used for assessing the performance of equity, fixed-income and mutual fund investments by comparing them to market benchmarks. 

All of these risk measurements are intended to help investors determine the risk-reward parameters of their investments. 

 

Alpha 
Alpha is a measure of an investment's performance on a risk-adjusted basis. It takes the volatility (price risk) of a security or fund portfolio and compares its risk-adjusted performance to a benchmark index. The excess return of the investment relative to the return of the benchmark index is its alpha. 

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Simply stated, alpha is often considered to represent the value that a portfolio manager adds or subtracts from a fund portfolio's return. A positive alpha of 1 means the fund has outperformed its benchmark index by 1%. Correspondingly, a similar negative alpha would indicate an underperformance of 1%. For investors, the more positive an alpha is, the better it is. (To learn more, see "Adding Alpha Without Adding Risk.") 

Beta 
Beta, also known as the beta coefficient, is a measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole. Beta is calculated using regression analysis, and you can think of it as the tendency of an investment's return to respond to swings in the market. By definition, the market has a beta of 1 Individual security and portfolio values are measured according to how they deviate from the market. 

 

A beta of 1.0 indicates that the investment's price will move in lock-step with the market. A beta of less than 1 indicates the investment will be less volatile than the market, and correspondingly, a beta of more than 1 indicates the investment's price will be more volatile than the market. For example, if a fund portfolio's beta is 1.2, it's theoretically 20% more volatile than the market. 

Conservative investors looking to preserve capital should focus on securities and fund portfolios with low betas, whereas those investors willing to take on more risk in search of higher returns should look for high-beta investments. (Keep reading about beta in "Beta: Know the Risk.") 

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R-Squared 
R-Squared is a statistical measure that represents the percentage of a fund portfolio's or security's movements that can be explained by movements in a benchmark index. For fixed-income securities and their corresponding mutual funds, the benchmark is the U.S. Treasury Billand likewise, with equities and equity funds, the benchmark is the S&P 500 Index. 

R-squared values range from 0 to 100. According to Morningstar, a mutual fund with an R-squared value between 85 and 100 has a performance record that is closely correlated to the index. A fund rated 70 or less would not perform like the index. 

Mutual fund investors should avoid actively managed funds with high R-squared ratios, which are generally criticized by analysts as being "closet" index funds. In these cases, why pay the higher fees for so-called "professional management" when you can get the same or better results from an index fund? (To learn more, read "Understanding Volatility Measurements," "The Lowdown On Index Funds" and"Benchmark Your Returns With Indexes.") 

 

Standard Deviation 
Standard deviation measures the dispersion of data from its mean. In plain English, the more that data is spread apart, the higher the difference is from the norm. In finance, standard deviation is applied to the annual rate of return of an investment to measure its volatility (risk). A volatile stock would have a high standard deviation. With mutual funds, the standard deviation tells us how much the return on a fund is deviating from the expected returns based on its historical performance. 

Sharpe Ratio 
Developed by Nobel laureate economist William Sharpe, this ratio measures risk-adjusted performance. It is calculated by subtracting therisk-free rate of return ( U.S. Treasury Bond) from the rate of return for an investment and dividing the result by the investment's standard deviation of its return. 

The Sharpe ratio tells investors whether an investment's returns are due to smart investment decisions or are the result of excess risk. This assessment is very useful, because although one portfolio or security can reap higher returns than its peers, it is only a good investment if those higher returns do not come with too much additional risk. The greater an investment's Sharpe ratio, the better its risk-adjusted performance. (Keep reading about this subject in "Understanding The Sharpe Ratio" and "The Sharpe Ratio Can Oversimplify Risk.") 

Many investors tend to focus exclusively on investment return, with little concern for investment risk. The five risk measures we have just discussed can provide some balance to the risk-return equation. The good news for investors is that these indicators are calculated for them, and are available on several financial Web sites, as well as being incorporated into many investment research reports. As useful as these measurements are, keep in mind that when considering a stock, bond, or mutual fund investment, volatility risk is just one of the factors you should be considering that can affect the quality of an investment. 

 

下面一篇是怎样利用这些指标在Morningstar中来挑选基金：

Morningstar Mutual Fund Risk Measures: Alpha, Beta, and R-squared

What Is Risk

Before getting into the of Morningstar risk measures, let’s first take a look a the general definition of risk in investment. According to Investopedia, risk is

The chance that an investment’s actual return will be different than expected. This includes the possibility of losing some or all of the original investment. It is usually measured by calculating the standard deviation of the historical returns or average returns of a specific investment. A fundamental idea in finance is the relationship between risk and return. The greater the amount of risk that an investor is willing to take on, the greater the potential return. The reason for this is that investors need to be compensated for taking on additional risk.

For each fund, Morningstar offers two sets of data, Volatility Measurements and Modern Portfolio Theory Statistics, to help investors get a sense of the risk of owning a particular fund. For Volatility Measurements, you can use the following data to gauge a fund’s volatility compared to the broad market:

• Mean
• Standard Deviation
• Sharpe Ratio
• Bear Market Decile Rank

And for Modern Portfolio Theory Statistics, you will see these data provided by Morningstar:

• R-Squared
• Beta
• Alpha

In the following, I will explain mean, standard deviation, beta, R-squared, and alpha and how to use them to assess risk involved in investing a mutual fund.

 

Mean, Standard Deviation, Beta, R-squared, and Alpha

Simply speaking, mean is the mathematical average of a set of data. If, for example, a stock XYZ’s annual return in the past three years are 10%, 5% and 15%, respectively, then the arithmetic mean of the stock’s return is 10%, the average 10%, 5% and 15%. Once the mean is known, we can calculate stock XYZ’s standard deviation , which measures the dispersion of the stock’s annual returns (i.e., 10%, 5% and 15%) from the mean expected return (10%). Therefore, the further away an equity’s annual return from the mean, the higher the standard deviation. In finance, standard deviation is used to gauge an equity’s volatility, whether the equity is a stock or a mutual fund.

 

Since the recession more than three years ago, the majority of stocks followed the movement of the general market and turned lower, the only difference among stocks is the extent of the downturn as compared to the benchmark. The risk that a stock tends to go along with the general market is captured by beta, also known as systematic risk (or market risk), which measure how an individual stock or fund reacts to the general market fluctuations. By definition, a benchmark (or index) has a beta of 1.00 and the beta of an equity is relative to this value. If the movement of a stock or fund can be completely explained by the movements of the general market, then this stock or fund will have a R-squared of 100. According to Morningstar, R-squared, represented by a percentage number ranging from 0 to 100, characterizes an equity’s movement against a benchmark. A R-squared that equals to 100 means all the equity’s movements are in-line with the benchmark.

 

With the Greek letter beta, investors can have an sense of how sensitive an equity is in relation to the broad market. If investors decide to take on a higher risk by investing in a volatile equity that carries a larger beta, then in theory, they should be rewarded with a higher than average return. The difference between the realized return and the average expected return is measured by another Greek letter alpha. A positive alpha indicates that the equity exceeds its expectations against the respective benchmark.

 

How Risk Measures Work

Now we know what the risk measurements are, let’s see how we can use them to assess the risk/reward of an investment.

 

To illustrate, I use two funds, Dodge & Cox Stock Fund (DODGX) and CGM Focus Fund (CGMFX), that I own to show how they are measured up against each other in each category. Using S&P 500 index as the benchmark, the performance and risk data of the two funds are shown in the following table (obtained from Morningstar.com, trailing 3-year data through January 31, 2011):

Funds2008 Return2009 Return2010 ReturnMeanSTDR-squaredBetaAlphaDODGX-43.3131.2713.490.0626.4197.131.19-1.95CGMFX-48.1810.4216.94-0.4533.3262.151.20

 

 

 

• Mean: The mean represents the annualized average monthly return. Therefore, a higher mean suggests a higher return the fund has delivered. In this case, DODGX performed a little better in the past three years with a mean of 0.06.
• Standard deviation (STD): In this case, both funds have a quite high STD comparing to the S&P 500, which has a mean of 0.20 and STD 21.91. A higher STD of CGMFX indicates that the fund is more volatile than the DODGX.
• R-squared: If we recall that R-squared measures a fund’s movement against the benchmark and a value close to 100 means the fund follows the benchmark very closely. Also, R-squared can help investor assess the usefulness of a fund’s beta or alpha statistics. A higher R-squared means the fund’s beta is more trustworthy. In this case, CGMFX’s 62.15 R-squared value says that only 62.15% of its movements can be explained by the fluctuations of S&P 500 index. This means that S&P 500 may not be a good benchmark to measure CGMFX. On the other hand, DODGX’s 97.13 R-squared value indicates the fund is well represented by S&P 500 and its beta value can thus be trusted.
• Beta: Now we know S&P 500 may not be a good benchmark for CGMFX, its beta value, though higher than that of DODGX, is not particularly helpful in assessing the fund’s risk in comparison to the benchmark. Generally, beta measures a fund’s risk associated with the market and a low beta only means that the funds market-related risk is low. For both DODGX and CGFMX, which have almost identical betas, they tend to swing 20% more than the benchmark in the same direction.
• Alpha: With a R-square value that we can trust, beta can be used to predict the fund’s expected return and alpha is the yardstick for the difference between a fund’s actual return and the predication. A large, positive alpha then means a fund has performed better than what its beta would predict. For DODGX, its alpha of -1.95 means the fund has underperformed the benchmark (S&P 500 index) by 1.95%, better than CGMFX’s -8.10 alpha.

Conclusions

When evaluating an investment (mutual fund in particular), there are many obvious factors we should consider: returns, risks, expenses, and turn-over ratio, etc. Among them, the risk factor, when used properly, can help us gauge what we can expect from the investment, though past performance does not necessarily indicate future results.