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Chapter 6: Problem 67

Statistics for investing (3.1) Joe's retirement plan invests in stocks through an "index fund" that follows the behavior of the stock market as a whole, as measured by the Standard \(\&\) Poor's \((\mathrm{S} \& \mathrm{P}) 500\) stock index. Joe wants to buy a mutual fund that does not track the index closely. He reads that monthly returns from Fidelity Technology Fund have correlation \(r=0.77\) with the S\&P 500 index and that Fidelity Real Estate Fund has correlation \(r=0.37\) with the index. (a) Which of these funds has the closer relationship to returns from the stock market as a whole? How do you know? (b) Does the information given tell Joe anything about which fund has had higher returns?

Short Answer

Expert verified
(a) Fidelity Technology Fund; it has a higher correlation (0.77). (b) No, correlation doesn't indicate return magnitude.

Step by step solution

01

Understand Correlation

The correlation coefficient \(r\) measures the strength and direction of the linear relationship between two variables. In this context, \(r\) indicates how closely the returns of the mutual fund follow the S\&P 500 index. The value of \(r\) ranges from -1 to 1. A value closer to 1 means a stronger positive association with the index.
02

Analyze Correlation Coefficients

Joe is considering two funds with different correlation coefficients: the Fidelity Technology Fund has \(r=0.77\) and the Fidelity Real Estate Fund has \(r=0.37\). A higher correlation implies that the fund mimics the movements of the S\&P 500 more closely.
03

Determine Which Fund Tracks the Index Closely

Since \(r=0.77\) is higher than \(r=0.37\), the Fidelity Technology Fund has a closer relationship to the returns of the S\&P 500. Thus, it tracks the index more closely than the Fidelity Real Estate Fund.
04

Evaluate Information on Returns

The correlation coefficient does not provide information about the returns' magnitude. A higher or lower \(r\) only indicates the degree of linear relationship with the S\&P 500, not the actual returns themselves. Therefore, Joe cannot determine which fund has had higher returns based solely on the given correlations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mutual Funds
A mutual fund is a type of investment vehicle that pools money from many investors to purchase securities. This professional management offers several benefits, especially for investors who may not have the time or expertise to manage their own portfolios.
Mutual funds provide broad diversification, which means spreading investment risk across various stocks or bonds. This approach reduces the impact of any one security's poor performance on the overall portfolio. For example, if you invest in a diverse mix of technology, real estate, and other sectors, the downturn in one may not dramatically affect your overall returns.
Investors in mutual funds also benefit from liquidity. This means that the units of the fund can be easily bought and sold. Furthermore, they save time and effort as the fund is professionally managed, allowing investors to rely on the expertise of fund managers without managing each asset themselves.
  • Diversification: Reduces risk by spreading investments.
  • Liquidity: Easy buying and selling of fund units.
  • Professional Management: Experts managing your investments.
Stock Market Index
A stock market index is a tool used to gauge the performance of a select segment of the stock market. It gives investors an idea of the overall market condition and helps in making informed investment decisions.
These indices are made up of a selected group of stocks and represent a portion of the total market. Take the S&P 500, for instance; it includes 500 of the largest companies listed in the United States, providing a broad picture of market activity.
Investors often look at indices like the S&P 500 to compare their own portfolio's performance or to make strategic investment choices. By following an index, investors can appreciate the general trends of the market and understand whether their chosen stocks or funds are moving in tandem with broader market patterns.
  • Performance Gauge: Indicates how a segment of the market is doing.
  • Market Trends: Helps identify current market conditions.
  • Investment Benchmark: Used for comparison and decision-making.
S&P 500
The S&P 500, short for Standard & Poor's 500, is one of the most well-known stock market indices in the world. It includes 500 of the largest companies by market capitalization in the United States, which covers a broad spectrum of industries.
As a widely used benchmark, the S&P 500 provides a snapshot of the stock market's overall health and is often synonymous with "the market." When someone says "the market went up," they usually mean the S&P 500 increased in value. However, it is important to note that this index only represents about 80% of the entire US stock market, so while it's broad, it isn't all-encompassing.
Investors often use the S&P 500 for designing investment portfolios or for understanding economic trends. Its diverse portfolio makes it less volatile than more concentrated indices.
  • Widely Known: Includes 500 large U.S. companies.
  • Benchmark: Synonymous with overall market performance.
  • Diverse Portfolio: Covers various industries, reducing volatility.
Statistics in Finance
Statistics play a crucial role in finance, aiding investors in making informed decisions based on data analysis and trends. One key statistical measure is the correlation coefficient, symbolized as \( r \), which describes the degree to which two variables move in relation to each other.
In the context of mutual funds and stock indices, a high correlation coefficient means that the mutual fund's performance is closely tied to the index. Thus, a fund with an \( r \) closer to 1 with the S&P 500 indicates it tends to rise and fall with the index more predictably.
It's critical for investors to understand that correlation does not imply causation or the magnitude of returns. Just because a fund follows market movements closely, it doesn't predict the scale of returns. Other statistical concepts like mean, standard deviation, and regression analysis are also fundamental in evaluating financial data for trends and risk assessment.
  • Correlation Coefficient: Measures how closely related two variables are.
  • Data Analysis: Helps in understanding market relationships.
  • Risk and Trends: Statistics identify investment risks and market trends.

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Most popular questions from this chapter

Exercises 27 to 29 refer to the following setting. Choose an American household at random and let the random variable \(X\) be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars: \begin{tabular}{lcccccc} \hline Number of cars \(\mathrm{X}:\) & 0 & 1 & 2 & 3 & 4 & 5 \\ Probability: & 0.09 & 0.36 & 0.35 & 0.13 & 0.05 & 0.02 \\ \hline \end{tabular} The standard deviation of \(X\) is \(\sigma_{X}=1.08\). If many households were selected at random, which of the following would be the best interpretation of the value \(1.08 ?\) (a) The mean number of cars would be about 1.08 . (b) The number of cars would typically be about 1.08 from the mean. (c) The number of cars would be at most 1.08 from the mean.

Running a mile A study of 12,000 able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal with mean 7.11 minutes and standard deviation 0.74 minute. \({ }^{10}\) Choose a student at random from this group and call his time for the mile \(Y\). Find \(P(Y<6)\) and interpret the result.

determine whether the given random variable has a binomial distribution. Justify your answer Taking the train According to New Jersey Transit, the 8: 00 A.M. weekday train from Princeton to New York City has a \(90 \%\) chance of arriving on time on a randomly selected day. Suppose this claim is true. Choose 6 days at random. Let \(W=\) the number of days on which the train arrives late.

Auto emissions The amount of nitrogen oxides (NOX) present in the exhaust of a particular type of car varies from car to car according to a Normal distribution with mean 1.4 grams per mile \((g / \mathrm{mi})\) and standard deviation \(0.3 \mathrm{~g} / \mathrm{mi}\). Two randomly selected cars of this type are tested. One has \(1.1 \mathrm{~g} / \mathrm{mi}\) of \(\mathrm{NOX}\) the other has \(1.9 \mathrm{~g} / \mathrm{mi}\). The test station attendant finds this difference in emissions between two similar cars surprising. If the NOX levels for two randomly chosen cars of this type are independent, find the probability that the difference is at least as large as the value the attendant observed.

According to Benford's law (Exercise \(5,\) page 359\(),\) the probability that the first digit of the amount of a randomly chosen invoice is an 8 or a 9 is \(0.097 .\) Suppose you examine randomly selected invoices from a vendor until you find one whose amount begins with an 8 or a \(9 .\) (a) How many invoices do you expect to examine until you get one that begins with an 8 or 9 ? Justify your answer. (b) In fact, you don't get an amount starting with an 8 or 9 until the 40 th invoice. Do you suspect that the invoice amounts are not genuine? Compute an appropriate probability to support your answer.
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