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Chapter 6: Problem 67

Statistics for investing \((3.2)\) 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 (S\&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 had higher returns?

Short Answer

Expert verified
(a) Fidelity Technology Fund has a closer relationship to the market. (b) Correlation doesn't indicate which fund had higher returns.

Step by step solution

01

Understanding the Correlation Coefficient

The correlation coefficient, denoted as \( r \), measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where 1 means a perfect positive relationship, -1 a perfect negative relationship, and 0 no relationship.
02

Determine Closer Relationship to the Market

For part (a), compare the correlation coefficients of the Fidelity Technology Fund and the Fidelity Real Estate Fund with the S&P 500 index. The Fidelity Technology Fund has \( r = 0.77 \) and the Fidelity Real Estate Fund has \( r = 0.37 \). The higher the correlation coefficient to 1, the closer the relationship to the stock market.
03

Conclusion on Closer Relationship

Since \( r = 0.77 \) (Fidelity Technology Fund) is greater than \( r = 0.37 \) (Fidelity Real Estate Fund), the Fidelity Technology Fund has a closer relationship to the returns of the stock market as a whole.
04

Understanding Information on Returns

For part (b), understand that the correlation coefficient does not convey information about which fund has higher returns. It only measures how closely each fund tracks the index.
05

Conclusion on Returns Information

The correlation values provided do not tell Joe anything about which fund has higher returns. Correlation indicates the relationship but not the magnitude of returns.

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Key Concepts

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

S&P 500 Index
The S&P 500 Index is a widely recognized benchmark of the overall U.S. stock market. It comprises 500 of the largest publicly traded companies in the United States. These companies span various industries, reflecting the general economic health of the country.

The index provides investors with a gauge of the stock market's performance. It is often used as a baseline to compare other investments, such as mutual funds.

  • An increase in the S&P 500 Index indicates an overall rise in the market value of the included securities.
  • A decrease suggests a market downturn or slower growth in those companies.
Understanding how investments correlate with the S&P 500 is crucial for investors. A higher correlation with this index suggests that an investment's performance closely follows the broader market movements.
Mutual Funds
Mutual funds are financial vehicles that pool money from various investors to invest in a diversified portfolio of assets. These assets often include stocks, bonds, and other securities. Mutual funds are managed by professional fund managers who make investment decisions on behalf of investors.

Mutual funds come in many forms, and their objectives can vary significantly:
  • Index Funds: These funds aim to replicate the performance of a specific index, like the S&P 500. They often have lower management fees due to their passive style.
  • Actively Managed Funds: These funds employ fund managers to actively make investment decisions with the goal of outperforming the market index.
  • Sector Funds: Focus on specific sectors, such as technology or real estate, offering targeted exposure for investors.
When selecting a mutual fund, examining the correlation with major indices like the S&P 500 can provide insight into how volatile or stable a fund might be relative to the broader market.
Stock Market
The stock market represents the collection of exchanges and other venues where shares of publicly held companies are bought and sold. The market functions as a platform for companies to raise capital by offering shares, and for investors to generate returns through dividends and capital appreciation.

The stock market is influenced by numerous factors, including:
  • Economic indicators, such as employment rates and GDP growth.
  • Company earnings and financial performance.
  • Global events, which can impact investor sentiment and market stability.
Investors watch the stock market closely to gain insights into economic trends and to make informed investment decisions. The S&P 500 Index, as part of the stock market, serves as a critical tool for evaluating market direction and investment opportunities.

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Most popular questions from this chapter

Blood types In the United States, 44\(\%\) of adults have type O blood. Suppose we choose 7 U.S. adults at random. Let \(X=\) the number who have type \(O\) blood. Use the binomial probability formula to find \(P(X=4)\) . Interpret this result in context.

How many close friends do you have? An opinion poll asks this question of an SRS of 1100 adults. Suppose that the number of close friends adults claim to have varies from person to person with mean \(\mu=9\) and standard deviation \(\sigma=2.5\) We will see later that in repeated random samples of size 1100 , the mean response \(\overline{x}\) will vary according to the Normal distribution with mean 9 and standard deviation 0.075 . What is \(P(8.9 \leq \overline{x} \leq 9.1)\) , the probability that the sample result \(\overline{x}\) estimates the population truth \(\mu=9\) to within \(\pm 0.1 ?\)

Exercises 39 and 40 refer to the following setting. Ms. Hall gave her class a 10 -question multiple-choice quiz. Let \(X=\) the number of questions that a randomly selected student in the class answered correctly. The computer output below gives information about the probability distribution of \(X .\) To determine each student's grade on the quiz (out of \(100 ),\) Ms. Hall will multiply his or her number of correct answers by \(10 .\) Let \(G=\) the grade of a randomly chosen student in the class. N Mean Median StDev Min Max Q1 Q3 30 7.6 8.5 1.32 4 10 8 9 Easy quiz (a) Find the median of G. Show your method. (b) Find the \(I Q R\) of \(G\) . Show your method. (c) What shape would the probability distribution of G have? Justify your answer.

A deck of cards contains 52 cards, of which 4 are aces. You are offered the following wager: Draw one card at random from the deck. You win \(\$ 10\) if the card drawn is an ace. Otherwise, you lose \(\$ 1 .\) If you make this wager very many times, what will be the mean amount you win? (a) About \(-\$ 1\) , because you will lose most of the time. (b) About \(\$ 9,\) because you win \(\$ 10\) but lose only \(\$ 1\) . (c) About \(-\$ 0.15 ;\) that is, on average you lose about 15 cents. (d) About \(\$ 0.77 ;\) that is, on average you win about 77 cents. (e) About \(\$ 0,\) because the random draw gives you a fair bet.

Random digit dialing When an opinion poll calls residential telephone numbers at random, only 20\(\%\) of the calls reach a live person. You watch the random digit dialing machine make 15 calls. Let \(X=\) the number of calls that reach a live person. (a) Find and interpret \(\mu_{X}\) (b) Find and interpret \(\sigma_{X}\)
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