Making Sense of Mutual Fund Risk

By C. Michele Matherly and J. Howard Finch

In Brief

A Picture Is Worth a Thousand Words

Visual aids are always powerful communications tools.They are no less valuable in conveying the risk and return performance of various investment alternatives. Specifically, mutual fund risk and return performance are expressed through numerical measures that can be difficult to understand and compare.

Given the increasing demand for financial planning services, CPAs can help their clients with sound, clear advice about mutual fund risk and return measures explained visually through bar charts and graphs. understanding and communicating investment-related risk is challenging. Many people relate better to pictures than words when it comes to understanding complex statistical concepts. Accordingly, graphs and charts can enhance interpretability of investment risk measures and should be used frequently to improve understanding. Applied to common mutual funds, these risk measures include a fund’s mean return, standard deviation, Sharpe ratio, R-squared, beta, and alpha, all of which are reported in the risk analysis section of the Morningstar Mutual Funds standard fund report.

Standard Deviation and Mean of Expected Returns

Consider the Blue Fund and the Red Fund, imaginary mutual funds, which have performed as follows: Over the last three years, the Blue Fund achieved an average annual return of 20%, while the Red Fund averaged 15% over the same time period. By this measure, the Blue Fund achieved superior performance and currently represents the better investment alternative. Using return as a single performance measure without any adjustment for risk, however, is a classic example of poor economic reasoning, because it fails to consider the possible scope of future returns.

Volatility in a mutual fund’s periodic rate of return is a common measure of its risk. If the historical mean is used as the basis for the fund’s expected rate of return, then the standard deviation of historical returns can form the basis for expected volatility (from any cause) of its return. Standard deviation provides a measure from which a range of likely future returns can be extrapolated. Over the most recent three-year period, the Blue Fund had a mean return of 20% and a standard deviation of 17%, whereas the Red Fund had a mean return of 15% and a standard deviation of 10%. Exhibit 1 and Exhibit 2 illustrate the mean return and standard deviation of the Blue Fund and Red Fund, respectively.

Over extended time periods fund returns follow an approximate normal distribution, or bell curve, as seen in Exhibits 1 and 2. Given a normal distribution of random outcomes, there is approximately 68% probability that the fund’s actual return will fall within one standard deviation of the mean return. That is, there is a 68% chance that the Red Fund’s return will fall between 5% (15% – 10%) and 25% (15% + 10%).

A graphical analysis of the mean return and standard deviation compares the relative riskiness of two funds that exhibit the assumed risk and return characteristics. The graphs found in Exhibits 1 and 2 could help convey the idea that there is no such thing as a free lunch, because the higher expected return of the Blue Fund comes at the cost of greater volatility exposure. Exhibit 3 explains how to graph a normal distribution using a spreadsheet.

The concepts of mean return and standard deviation can also be communicated by graphing each fund’s historical performance in a risk-return space (where each variable is an axis of the graph). This type of graph allows the CPA to compare the relative performance of many funds and provides visual evidence of underperformance. Exhibit 4 presents five hypothetical funds in risk-return space and illustrates the concept of dominance.

Fund	Mean	Standard Deviation
U	19%	20%
V	25%	22%
X	17%	26%
Y	14%	8%
Z	24%	38%

As seen above and in Exhibit 4, Fund U has a higher average return for less total risk exposure than Fund X; therefore, U dominates X. Fund V has a higher average return with less total risk exposure than Fund Z, so V dominates Z. Fund Y is not dominated by any fund as none of the others have a higher average return with less risk exposure. Therefore, funds Y, U, and V constitute the efficient investments. Deciding to invest in any of these funds or allocating money amongst the three depends upon the investor’s personal risk preference.

The Sharpe Measure

The measures previously covered highlight the fallacy of evaluating mutual fund performance solely on the basis of average return. In the preceding example, a ranking solely by return would rank Fund V first, Fund Z second, and Fund Y last. Yet a sensible investor would not invest in Fund Z because Fund V offers higher average return for less total risk exposure.

The Sharpe measure combines a fund’s mean return, standard deviation, and the average risk-free return (usually the 90-day Treasury bill rate) and calculates a ranking number that represents excess return per unit of total risk exposure. To calculate the Sharpe measure, use the following equation:

S = m – rf/ s

Where:
m = the mean return (the annualized monthly average fund return)
rf = the risk-free return (the annualized average return of 90-day Treasury bill)
s = the standard deviation (the annualized monthly standard deviation of returns).

Morningstar computes these figures for the preceding 36-month period. Using Exhibit 4, assume the annualized average Treasury bill rate over the past 36 months has been 5.5%. The Sharpe measure for Fund X is computed at 0.44:

Sx = 17% – 5.5%/ 26%

Similar computations for all five hypothetical funds yield the following rankings of risk-adjusted performance:

Fund	Sharpe Measure
Y	1.06
V	0.89
U	0.68
Z	0.49
X	0.44

Funds Y, V, and U provide the highest excess return for their risk. Ranking comparative performance by the Sharpe measure effectively communicates both risk and return in a single number, facilitating the identification of superior performance.

Beta and R-Squared

Another risk measure found in the Morningstar report is a fund’s beta. While standard deviation represents the volatility of a mutual fund’s returns caused by any factor, beta measures a mutual fund’s systematic movement with the overall stock market. Beta addresses the question, “How risky is a fund’s return relative to the overall market?” An index such as the Standard & Poor’s 500 usually serves as a benchmark for the market and, by construction, the beta for the market benchmark portfolio equals 1.0. Individual fund betas are compared to the benchmark. For example, a mutual fund with a beta of 1.2 is 20% more volatile than the relative benchmark index return. The returns from this fund will move up and down with the overall market, but at a 20% greater rate.

A standard bar chart can communicate these complex ideas clearly. Exhibit 5 compares the expected return for a fund with a beta of 1.2 to the market index return in both a rising and falling market. Investors should realize that increased volatility is a mixed blessing. During rising markets, higher-beta funds will, on average, earn higher returns than market indices. During down markets, higher-beta funds will, on average, lose more.

However, a beta needs to be interpreted in light of the R-squared measurement that reflects the reliability of the beta risk statistic. R-squared measures the percentage of the fund’s return volatility that is explained by volatility in the overall market return. A high R-squared, defined by Morningstar as 85–100, means a reliable beta as a market risk measure. In contrast, a low R-squared, defined by Morningstar as less than 70, indicates that the beta fails to explain much of the fund’s volatility.

The validity of the beta measure also depends upon the suitability of the index used to compute it. The S&P 500 is a poor benchmark for computing and comparing the beta of a small-cap equity fund. For this reason, Morningstar reports two betas and R-squared statistics for each fund it covers. One beta is based on the S&P 500, generally the most common benchmark for broad-based investment performance. The second beta is based on a “best-fit” index that Morningstar believes is the most appropriate benchmark for that fund. The validity of the “best-fit” index is usually indicated by a higher R-squared.

Bar charts illustrate the comparative risk associated with different mutual fund choices. For example, if the fund in Exhibit 6 is a small-cap equity fund that has a “best-fit” R-squared of 91% against the Russell 2000 index and the S&P 500 beta has an R-squared of 55%, comparative bar charts show that most of this fund’s return volatility (91%) will follow the small-cap equity market, rather than the S&P 500. This demonstration could comfort investors worried about a fund’s underperformance with respect to a widely quoted index.

Alpha

Alpha is the final figure reported under “other measures” in a Morningstar report. Alpha represents a fund’s excess return as compared to the return predicted by its beta. Investors can use diversification to eliminate many of the specific risks to which individual securities are exposed, such as a labor strike against an airline. By combining the airline stock with securities from different industries, the labor strike risk is offset by different factors that affect the returns of the other portfolio components. Investors holding well-diversified portfolios should only be exposed to, and only compensated for, risk that cannot be diversified away.

In a properly diversified portfolio, company-specific risk is elimianated, leaving only market risk, which is captured by the fund’s beta. The capital asset pricing model (CAPM) predicts the expected return of a fund as a function of its beta.

The CAPM is a predictive model that computes expected returns based upon different projections of market returns. Suppose the expected overall market return three years ago for the S&P 500 index was an average of 14%, while the average risk-free return was 5.5%. According to the CAPM, the predicted return for a fund with a beta of 1.2 would have been 15.7%.

Er = rf + [Rm – rf ]B
Er = 5.5% + [14% – 5.5%] ¥ 1.2
Er = 15.7%.

Where:

Er = the expected return of the mutual fund
rf = the risk-free return (the annualized average return of 90-day Treasury bill)
Rm = the market return (the S&P 500 index or a “best-fit” index)
B = the fund’s beta.

However, suppose over the past 36 months the actual annualized fund return was 17%. The fund’s alpha (a) would be 1.3%:

a = 17% – 15.7%
a = 1.3%.

Alpha is the difference between the expected return based on market risk exposure (beta) and the actual return earned. Exhibit 7 illustrates the concept of alpha. In the exhibit, a line connects the point representing the risk-free return (5.5%) to the point indicating the fund’s expected return (15%), given its beta. The vertical difference between the fund’s expected return (15.7%) and actual return (17%) equals alpha.

A positive alpha indicates the fund earned a higher than expected return based on its market risk exposure. Negative alphas indicate underperformance. Of course, the alpha measure is only as good as the beta used to compute it. A low R-squared implies a poor alpha just as it does a poor beta. Alpha and beta statistics that are derived from the “best-fit” index should be used in these calculations.