When considering a fund's volatility, an investor may find it difficult to decide which fund will provide the optimal risk-reward combination. Many websites provide various volatility measures for mutual funds free of charge; however, it can be hard to know not only what the figures mean but also how to analyze them.

Furthermore, the relationship between these figures is not always obvious. Read on to learn about the four most common volatility measures and how they are applied in the type of risk analysis based on modern portfolio theory.

### Key Takeaways

- The relationship between portfolio returns and risk can be represented by the efficient frontier, a curve that is a part of modern portfolio theory.
- Another way to measure risk is standard deviation, which reports a fund's volatility, indicating the tendency of the returns to rise or fall drastically in a short period of time.
- Beta, another useful statistical measure, compares the volatility (or risk) of a fund to its index or benchmark.
- The R-squared of a fund shows investors if the beta of a mutual fund is measured against an appropriate benchmark.
- Alpha measures how much, if any, extra risk helped the fund outperform its corresponding benchmark.

## Optimal Portfolio Theory and Mutual Funds

One examination of the relationship between portfolio returns and risk is the efficient frontier, a curve that is a part of the modern portfolio theory. The curve forms from a graph plotting return and risk indicated by volatility, which is represented by the standard deviation. According to the modern portfolio theory, funds lying on the curve are yielding the maximum return possible, given the amount of volatility.

As standard deviation increases, so does the return. Once expected returns of a portfolio reach a certain level, an investor must take on a large amount of volatility for a small increase in return. Obviously, portfolios with a risk/return relationship plotted far below the curve are not optimal since the investor is taking on a large amount of instability for a small return. To determine if the proposed fund has an optimal return for the amount of volatility acquired, an investor needs to do an analysis of the fund's standard deviation.

Modern portfolio theory and volatility are not the only means investors use to analyze the risk caused by many different factors in the market. And things like risk tolerance and investment strategy affect how an investor views his or her exposure to risk. Here are four other measures.

## 1. Standard Deviation

As with many statistical measures, the calculation for standard deviation can be intimidating, but because the number is extremely useful for those who know how to use it, there are many free mutual fund screening services that provide the standard deviations of funds.

The standard deviation essentially reports a fund's volatility, which indicates the tendency of the returns to rise or fall drastically in a short period of time. A volatile security is also considered a higher risk because its performance may change quickly in either direction at any moment. The standard deviation of a fund measures this risk by measuring the degree to which the fund fluctuates in relation to its mean return.

A fund with a consistent four-year return of 3%, for example, would have a mean, or average, of 3%. The standard deviation for this fund would then be zero because the fund's return in any given year does not differ from its four-year mean of 3%. On the other hand, a fund that in each of the last four years returned -5%, 17%, 2%, and 30% would have a mean return of 11%. This fund would also exhibit a high standard deviation because each year, the return of the fund differs from the mean return. This fund is, therefore, riskier because it fluctuates widely between negative and positive returns within a short period.

Remember, because volatility is only one indicator of the risk affecting a security, a stable past performance of a fund is not necessarily a guarantee of future stability. Since unforeseen market factors can influence the volatility, a fund with a standard deviation close or equal to zero this year may behave differently the following year.

To determine how well a fund is maximizing the return received for its volatility, you can compare the fund to another with a similar investment strategy and similar returns. The fund with the lower standard deviation would be more optimal because it is maximizing the return received for the amount of risk acquired. Consider the following graph:

With the S&P 500 Fund B, the investor would be acquiring a larger amount of volatility risk than necessary to achieve the same returns as Fund A. Fund A would provide the investor with the optimal risk/return relationship.

## 2. Beta

While standard deviation determines the volatility of a fund according to the disparity of its returns over a period of time, beta, another useful statistical measure, compares the volatility (or risk) of a fund to its index or benchmark. A fund with a beta very close to one means the fund's performance closely matches the index or benchmark. A beta greater than one indicates greater volatility than the overall market, and a beta less than one indicates less volatility than the benchmark.

If, for example, a fund has a beta of 1.05 in relation to the S&P 500, the fund has been moving 5% more than the index. Therefore, if the S&P 500 increased by 15%, the fund would be expected to increase by 15.75%. On the other hand, a fund with a beta of 2.4 would be expected to move 2.4 times more than its corresponding index. So if the S&P 500 moved 10%, the fund would be expected to rise 24%, and if the S&P 500 declined 10%, the fund would be expected to lose 24%.

Investors expecting the market to be bullish may choose funds exhibiting high betas, which increase investors' chances of beating the market. If an investor expects the market to be bearish in the near future, the funds with betas less than one are a good choice because they would be expected to decline less in value than the index. For example, if a fund had a beta of 0.5, and the S&P 500 declined by 6%, the fund would be expected to decline only 3%.

Beta by itself is limited and can be skewed due to factors other than the market risk affecting the fund's volatility.

## 3. R-Squared

The R-squared of a fund shows investors if the beta of a mutual fund is measured against an appropriate benchmark. Measuring the correlation of a fund's movements to that of an index, R-squared describes the level of association between the fund's volatility and market risk, or, more specifically, the degree to which a fund's volatility is a result of the day-to-day fluctuations experienced by the overall market.

R-squared values range between 0 and 100, where 0 represents the least correlation, and 100 represents full correlation. If a fund's beta has an R-squared value close to 100, the beta of the fund should be trusted. On the other hand, an R-squared value close to 0 indicates the beta is not particularly useful because the fund is being compared against an inappropriate benchmark.

If, for example, a bond fund was judged against the S&P 500, the R-squared value would be very low. A bond index such as the Bloomberg Barclays US Aggregate Bond Index would be a much more appropriate benchmark for a bond fund so that the resulting R-squared value would be higher. Obviously, the risks apparent in the stock market are different than those associated with the bond market. Therefore, if the beta for a bond were calculated using a stock index, the beta would not be trustworthy.

An inappropriate benchmark will skew more than just beta. Alpha is calculated using beta, so if the R-squared value of a fund is low, it is also wise not to trust the figure given for alpha. We'll go through an example in the next section.

## 4. Alpha

Up to this point, we have learned how to examine figures measuring risk posed by volatility, but how do we measure the extra return rewarded to you for taking on the risk posed by factors other than market volatility? Enter alpha, which measures how much if any of this extra risk helped the fund outperform its corresponding benchmark. Using beta, alpha's computation compares the fund's performance to that of the benchmark's risk-adjusted returns and establishes if the fund outperformed the market, given the same amount of risk.

For example, if a fund has an alpha of one, it means that the fund outperformed the benchmark by 1%. Negative alphas are bad in that they indicate the fund underperformed for the amount of extra, fund-specific risk the fund's investors undertook.

## The Bottom Line

This explanation of these four statistical measures provides you with the basic knowledge for using them to apply the premise of the optimal portfolio theory, which uses volatility to establish risk and offers a guideline for determining how much of a fund's volatility carries a higher potential for return. These figures can be difficult to understand, so if you use them, it is important to know what they mean.

These calculations only work within one type of risk analysis. If you are deciding on buying mutual funds, it is important to be aware of factors other than volatility that affect and indicate the risk posed by mutual funds.