91Ó°ÊÓ

The S\&P 500 is a collection of 500 stocks of publicly traded companies. Using data obtained from Yahoo! Finance, the monthly rates of return of the S\&P500 since 1950 are normally distributed. The mean rate of return is \(0.007233(0.7233 \%),\) and the standard deviation for rate of return is \(0.04135(4.135 \%)\). (a) What is the probability that a randomly selected month has a positive rate of return? That is, what is \(P(x>0) ?\) (b) Treating the next 12 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive? That is, with \(n=12,\) what is \(P(\bar{x}>0) ?\) (c) Treating the next 24 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive? (d) Treating the next 36 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive? (e) Use the results of parts (b)-(d) to describe the likelihood of earning a positive rate of return on stocks as the investment time horizon increases.

Step by step solution

01

Understanding the Problem

The mean monthly rate of return is given as 0.007233, and the standard deviation is 0.04135. The task involves calculating the probability of seeing a positive rate of return over different sample sizes.

02

Calculate P(x > 0) for a Single Month

To find the probability that a randomly selected month has a positive rate of return, we need to find P(x > 0). We can use the Z-score formula: \[ Z = \frac{x - \mu}{\sigma} \]For x = 0, \[ Z = \frac{0 - 0.007233}{0.04135} = -0.1749 \]Using the standard normal distribution table, find the probability corresponding to Z = -0.1749.

03

Convert Z-Score to Probability

Find the value of P(Z > -0.1749) using the normal distribution table. This value will be approximately 0.5693, which means: \( P(x > 0) = 0.5693 \)

04

Calculate P(\bar{x} > 0) for n = 12 Months

We will use the standard error (SE) formula: \[ SE = \frac{\sigma}{\sqrt{n}} \]For n = 12: \[ SE = \frac{0.04135}{\sqrt{12}} = 0.01194 \]Using the Z-score formula for the sample mean: \[ Z = \frac{0 - 0.007233}{0.01194} = -0.6059 \]Find P(Z > -0.6059) using the standard normal distribution table.

05

Convert Z-Score to Probability for n = 12

Find the value of P(Z > -0.6059) using the normal distribution table. This value will be approximately 0.7284, which means: \( P(\bar{x} > 0) = 0.7284 \) when n = 12 months.

06

Calculate P(\bar{x} > 0) for n = 24 Months

For n = 24: \[ SE = \frac{0.04135}{\sqrt{24}} = 0.008442 \]Using the Z-score formula for the sample mean: \[ Z = \frac{0 - 0.007233}{0.008442} = -0.857 \]Find P(Z > -0.857) using the standard normal distribution table.

07

Convert Z-Score to Probability for n = 24

Find the value of P(Z > -0.857) using the normal distribution table. This value will be approximately 0.8054, which means: \( P(\bar{x} > 0) = 0.8054 \) when n = 24 months.

08

Calculate P(\bar{x} > 0) for n = 36 Months

For n = 36: \[ SE = \frac{0.04135}{\sqrt{36}} = 0.006892 \]Using the Z-score formula for the sample mean: \[ Z = \frac{0 - 0.007233}{0.006892} = -1.0495 \]Find P(Z > -1.0495) using the standard normal distribution table.

09

Convert Z-Score to Probability for n = 36

Find the value of P(Z > -1.0495) using the normal distribution table. This value will be approximately 0.8532, which means: \( P(\bar{x} > 0) = 0.8532 \) when n = 36 months.

10

Interpret the Results

The probability of a positive rate of return increases with larger sample sizes (number of months). Hence, a longer investment horizon increases the likelihood of earning a positive rate of return on stocks.

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Key Concepts

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

Normal Distribution

The normal distribution, often known as the bell curve, is a fundamental concept in statistics. It describes how the values of a variable are distributed. For the S&P 500 monthly rates of return, having a normal distribution means:

• Most values are closer to the mean
• Values symmetrically decrease as they move further from the mean
• Approximately 68% of the data falls within one standard deviation from the mean
• 95% within two standard deviations
• 99.7% within three standard deviations

Understanding this distribution helps us predict the probability of certain outcomes, like a positive rate of return.

Probability

Probability quantifies the likelihood of an event happening. For instance, determining the probability of the S&P 500 having a positive rate of return in a randomly selected month helps investors make informed decisions. Important notes:

• Probabilities always range from 0 to 1
• A probability of 0 means the event will not occur
• A probability of 1 means the event will certainly occur

In this exercise, we calculated various probabilities related to monthly rates of return and longer periods to guide investment decisions.

Z-Score

A Z-score is a measure of how many standard deviations a data point is from the mean. It standardizes different data points for comparison. For the S&P 500 rates of return:

• Z = (x - μ) / σ
• μ (mean) for the given data = 0.007233
• σ (standard deviation) = 0.04135

By calculating Z-scores, we can find the probability associated with a specific rate of return. For example, P(x > 0) translates to finding the probability for a Z-score greater than a certain value.

Standard Error

Standard error gives an estimate of the standard deviation of the sample mean. It helps in understanding how much sample means can vary from the population mean. It's calculated using:

• SE = σ / √n

Where 'n' is the sample size. For example:

• For n = 12: SE = 0.01194
• For n = 24: SE = 0.008442
• For n = 36: SE = 0.006892

The smaller the standard error, the closer the sample mean is expected to be to the population mean. This concept is crucial when predicting returns over larger sample sizes.

Mean Rate of Return

The mean rate of return is an average measure of returns on investments over a period. For the S&P 500, it gives investors an idea of the expected return:

• Given mean: 0.007233 or 0.7233%

This value, along with the standard deviation, helps in determining the probability of various return scenarios (e.g., positive return). It's an essential metric for assessing past performance and predicting future returns in financial investments.