Mutual funds and mathematics

Summary: Many mathematicians attempt to develop mathematical models that forecast the future direction of the stock market and thus to produce better investment results for mutual funds.

Mutual funds are a type of investment in which large numbers of people pool their money and a fund manager invests these funds in one or more types of security. Investors own shares in the fund, and the value of those shares is determined by the total value of all the securities owned by the fund. Mutual funds are a popular investment vehicle because they allow people to achieve a varied investment portfolio with a relatively small investment, thus limiting their risk in comparison to buying individual stocks, bonds, or other assets.

Many types of mutual funds are available, depending on the desires of the investor. For instance, are they more interested in a riskier fund that may produce a higher yield for their investment, or a safer fund that is more likely to preserve the value of their capital? Some mutual funds specialize in a single type of investment—for instance, international stocks, health sector stocks, U.S. government bonds, or real estate—while others invest in a variety of securities in order to achieve a desired balance between yield and risk. Although mutual funds are often perceived as a safe investment, they are not guaranteed by the Federal Deposit Insurance Corporation (FDIC) as are bank deposits, and it is possible to lose money by investing in mutual funds. Economists, statisticians, actuaries, and others frequently try to predict the stock market using time series analyses and other mathematical methods. Prediction has historically proven to be quite challenging because of the complexities of time series data and the different socioeconomic variables and human psychological factors that appear to influence the stock market.

History and Growth

Although the first mutual funds were offered in the United States in the 1920s, the modern mutual fund industry dates from 1940 when the Investment Company Act established a body of rules regarding financial investments. In 1949, less than $2 billion were invested in mutual funds, but they became a more popular investment vehicle in the 1960s. By 1973, $47 billion was invested in mutual funds. By 1987, this amount had grown to $4 trillion, and by 2000, to $6 trillion, representing the investments of over 83 million investors. One factor in the growth of individual investments in mutual funds is the shift in the United States from guaranteed pension plans to retirement savings plans like the 401(k) in which an individual worker is responsible for choosing how to invest his or her retirement funds.

In 2008, there were over 8000 mutual funds in the United States versus about 3000 stocks listed on the NASDAQ stock exchange and a similar number on the New York Stock Exchange. It may at first be counter intuitive that there should be more funds than stocks, but this fact is not surprising if one considers any mutual fund as a composite made up of individual stocks or a subset of the total number of stocks (although, of course, a mutual fund may also include bonds and other components). Any set of n elements has 2n possible subsets, so a set of 10 elements has 1024 subsets and a set of 25 elements has over 33 million.

Risk Minimization

One appeal of mutual funds is that they allow people to reduce their risk through diversification. Modern portfolio theory attempts to select assets to minimize risk, maximize return, or some combination of those two (in general, higher risk is associated with higher return, although this does not hold absolutely). The basic concept behind the theory is that stocks or other assets, such as bonds, in the fund are evaluated in the context of other assets, and the goal is to maximize return or minimize risk for the total collection of assets, called a “portfolio.” American economist Harry Markowitz developed portfolio theory beginning in the 1950s, and in 1990, was awarded the Nobel Prize in Economics for this achievement.

Management

Because the performance of a mutual fund is often related to that of the economy as a whole, the performance of specific mutual funds as well as mutual funds as a class is often evaluated against the performance of indices such as the Dow Jones Industrial Average (a scaled average of the stocks of 30 large, publicly owned companies) or the S&P 500 (a weighted index of 500 large-cap common stocks). There are always pitfalls in making these types of comparisons; for instance, the return of mutual funds as a whole appears larger than it really is because funds that do poorly often go out of existence and are thus dropped from the average (survivorship bias). Interestingly, over time most individual funds produce somewhat worse results than a large index, such as the S&P 500, suggesting that the talent of individual managers (who choose when to buy and sell the stocks or other investments that comprise a mutual fund) are less efficient than the stock market as a whole. For this reason there are mutual funds today that are not “actively managed” in the sense that an individual manager makes buying and selling decisions. Instead, such funds simply own the stocks that comprise some index, such as the S&P 500, with buying and selling decisions motivated by changes in the makeup of the index (for instance, because of mergers or to new stocks joining or leaving the index).

This method is not a criticism of mutual funds per se but simply an argument for the efficiency of the market. Studies of the stock picks of professional analysts also tend to perform only marginally better than those selected randomly—most famously by throwing darts at a dartboard. Despite this well-known result, many individuals and investment firms have developed complex mathematical models that attempt to forecast the future direction of the stock market and thus produce better investment results. In addition, people have tried to predict the movement of the stock market with other types of data; for instance, in 2010, two graduate students found that the emotional content of tweets (messages sent on Twitter, a social networking Web site that can receive and send text messages from mobile devices, such as mobile phones) from the general public could be used to predict movement of the Dow Jones Industrial Average several days in advance.

Bibliography

Fink, Matthew P. The Rise of Mutual Funds: An Insider’s View. New York: Oxford University Press, 2008.

Grossman, Lisa. “Twitter Can Predict the Stock Market.” Wire Science (October 19, 2010). http://www.wired.com/wiredscience/2010/10/twitter-crystal-ball.

Malkiel, Burton G. A Random Walk Down Wall Street. Rev. ed. New York: W. W. Norton, 2007.

Paulos, John A. A Mathematician Plays the Stock Market. New York: Basic Books, 2003.

Wepsic, Eric, Sendehil Revuluri, and Chris Welty. “Mathematical Modeling in the World of Finance.” Math Horizons 6 (September 1998).