2. Quantitative Methods Flashcards
What is the maturity premium?
A premium for the increased sensitivity of the price of a financial instrument with a longer time to maturity to changes in market interest rates.
What is an interest rate?
An interest rate is the rate of return that levels cash flows occurring on different dates and can be perceived as the price of money.
How is an interest rate viewed as a required rate of return?
It represents the minimum return that investors expect for providing their capital.
What does an interest rate signify as a discount rate?
It is used to determine the present value of future cash flows by discounting them back to their present value.
How does an interest rate function as an opportunity cost?
It represents the potential benefit lost by choosing one alternative over another, highlighting the cost of forgone opportunities.
Why is an interest rate considered the price of money?
Because it reflects the cost of borrowing money or the return on investment for lending money.
What is the real risk-free interest rate?
The interest rate for postponing consumption, not adjusted for inflation, with no default risk or liquidity constraints.
What forms the nominal risk-free interest rate?
The sum of the real risk-free interest rate and the inflation premium.
What is the default risk premium?
A premium for the possibility that the entity may fail to meet its financial obligations.
Why is a liquidity premium added?
To compensate for reduced or no liquidity, making it harder to sell the financial instrument.
What is the time value of money (TVM)?
TVM shows the relationship between time, present value (PV), future value (FV), and interest rate
How can an interest rate be perceived?
As a required rate of return, a discount rate, or an opportunity cost.
What questions do interest rates help answer?
Expected future profits from an investment, present value of a future amount, and future profits forgone in favor of current consumption.
What components make up the interest rate?
Real risk-free interest rate, inflation premium, default risk premium, liquidity premium, and maturity premium.
What is the expected value of a random variable?
The average outcome of an event, weighted by the probabilities of the possible outcomes.
How is the expected value denoted?
It is denoted as E, for example, E(X)
represents the expected value of a random variable X.
What does the expected value of a random variable tell us?
It provides a measure of the central tendency, indicating the average result of an event based on the probabilities of various outcomes.
How is the expected value of a rate of return calculated?
By multiplying each possible return by its probability and summing the results.
Why is the expected value important in financial analysis?
It helps investors and analysts make informed decisions by providing an average outcome based on the probabilities of different returns or events.
What is conditional expected value?
The expected value of a random variable given a specific scenario or condition.
How does conditional expected value differ from regular expected value?
It incorporates a specific condition or scenario into the calculation, providing a more accurate and relevant expectation.
Why is conditional expected value important in financial analysis?
It allows for more precise expectations by considering specific conditions, helping investors make better-informed decisions.
What is a population in statistics?
A population consists of all elements of a group of interest and is described by parameters.
What is a sample in statistics?
A sample is a subset of a population, usually selected randomly, and is described by sample statistics.
What are measures of central tendency?
Measures that help determine the center of analyzed data, including the arithmetic mean, mode, and median.
How do measures of location differ from measures of central tendency?
Measures of location include measures of central tendency and provide information on data distribution in different locations.
Why are measures of location important in statistics?
They help characterize an entire population or sample by offering insights into where data are centered and how they are distributed.
What do measures of central tendency help identify?
The center point or typical value of a dataset.
Name three common measures of central tendency.
Arithmetic mean, median, and mode.
What do measures of location provide information on?
Data distribution in different locations, including measures of central tendency and other location-based measures.
What do measures of dispersion examine?
The spread or variability of the data, such as range, variance, and standard deviation.
What does skewness measure?
The asymmetry of the data distribution.
What does peakedness (kurtosis) assess?
The sharpness of the peak of the data distribution.
What is a random variable?
A variable that can take different numerical values depending on the case, influenced by chance.
What is a sample space?
The set of all possible values or outcomes a random variable can take. For example, the sample space for a dice includes 1, 2, 3, 4, 5, and 6.
What is an event in probability?
A subset of a sample space, representing one or more possible outcomes.
What are exhaustive events?
A set of events that covers all possible outcomes of a random variable.
What are mutually exclusive events?
Events that cannot occur at the same time. For example, rolling 2 and 3 on a dice are mutually exclusive events.
What is objective probability?
he probability that an event will occur, calculated by dividing the number of favorable outcomes by the sample space, independent of personal feelings or intuitions.
What is a priori probability?
A type of objective probability assuming each event is equally likely and can be determined before the event occurs. For example, the probability of rolling a 4 on a dice is always one-sixth.
What is empirical probability?
A type of objective probability calculated based on the relative frequency of events in a long sequence. For example, if 8,000 out of 10,000 children like chocolate, the empirical probability is 0.8.
Why is objective probability important?
It provides a consistent and unbiased measure of the likelihood of an event, based on factual data rather than personal beliefs or intuitions.
What is subjective probability?
Probability based on individual beliefs, judgments, intuitions, and experience.
How does subjective probability differ from objective probability?
Subjective probability is influenced by personal estimations, whereas objective probability is based on factual data and consistent calculations.
Can subjective probabilities vary between individuals?
Yes, subjective probabilities can vary based on different personal experiences and intuitions.
Provide an example of subjective probability.
A meteorologist estimating a 70% probability of rainfall tomorrow based on intuition, weather observations, and experience.
Why is understanding subjective probability important?
It helps recognize that different individuals may have varying estimations of the likelihood of an event based on their unique perspectives and experiences.
What is target semideviation?
A measure of risk that focuses on the dispersion of returns below a specified target, rather than the overall variability of returns.
How does target semideviation differ from standard deviation?
It concentrates only on the part of the results below a specific target, whereas standard deviation measures the dispersion of observations both below and above the mean.
Why is it called “target semideviation”?
Because it focuses on the downside deviation below a chosen target.
Can the target for semideviation be any value of interest?
Yes, the target can be any value chosen by the investor, not just the mean.
How is target semideviation useful for low-risk portfolios?
Investors can choose the risk-free interest rate as the target and calculate the dispersion below this rate to assess the risk of potential loss.
What is the coefficient of variation (CV)?
A relative statistical measure of data dispersion expressed as a ratio of the standard deviation of a sample and the sample mean.
Why is the coefficient of variation (CV) important?
It standardizes dispersion, allowing for meaningful comparisons between datasets with different scales.
How is the coefficient of variation (CV) calculated?
By dividing the standard deviation by the sample mean.
How does the coefficient of variation (CV) help in comparing datasets?
By providing a standardized measure of dispersion, making it easier to compare the relative variability between different datasets.
What is a simple random sample?
A subset of elements selected randomly from a population where each element has an equal chance of being included.
What is systematic sampling?
A type of sampling where selection occurs in a systematic way, such as choosing every tenth number from a phone book.
When is stratified random sampling used?
When it’s necessary to ensure the presence of certain groups in the sample. The population is divided into strata, and samples are selected proportionally from each stratum.
What are sample statistics and population parameters?
Sample statistics are numerical characteristics of a sample used to estimate population parameters, which are real values describing a characteristic of the entire population.
What is sampling error?
The difference between the observed value of the sample statistic and the population parameter estimated using this sample statistic.
Why is a larger sample size important in sampling?
A larger sample size reduces standard errors and uncertainty in estimates, leading to more accurate results.
What is data-mining bias?
It is a bias that arises from using the same data set repeatedly to find patterns, leading to potentially misleading results.
What causes sample selection bias?
It occurs when the sample is not representative of the population due to non-random selection of elements.
What is look-ahead bias?
Look-ahead bias happens when future data is used inappropriately in an analysis, making the results appear better than they actually are.
What is time-period bias?
Time-period bias results from analyzing data over a specific time period that may not be representative of other periods.
What is data mining?
Data mining is the method of extensively searching through a dataset for statistically significant patterns using computers to perform large numbers of computations.
What is data snooping and how can it be avoided?
Data snooping occurs when we adjust our model to fit the data we explore rather than finding genuine patterns. It can be avoided by conducting an out-of-sample test.
What is intergenerational data mining?
It involves using datasets from previous research to make predictions about future generations of data, such as determining the number and age of retired people based on research conducted 40 years ago.
What is sample selection bias?
It occurs when relevant data is systematically excluded from the dataset, often due to inaccessibility, leading to distorted overall results.
What is survivorship bias?
Survivorship bias occurs when only current, up-to-date data is analyzed, ignoring past data such as banks that went bankrupt or were taken over.
What does the Central Limit Theorem state?
The Central Limit Theorem states that the distribution of the sample mean approaches a normal distribution, regardless of the population’s distribution, if the sample size is large enough.
What is the minimum sample size generally considered sufficient for the Central Limit Theorem to hold?
A sample size of at least 30 observations.
What are the population characteristics assumed in the Central Limit Theorem?
The population can have any distribution with a finite variance.
What happens to the mean and variance of the sample mean according to the Central Limit Theorem?
The sample mean will have a mean equal to the population mean
𝜇, and the variance of the sample mean will be
𝜎^2/𝑛
Why is the Central Limit Theorem important?
It allows the distribution of the sample mean to be approximated by a normal distribution, simplifying analysis and inference regardless of the population distribution.
What is an estimator?
An estimator is a formula used to estimate the value of a parameter of a distribution based on sample data.
What are the three properties of good estimators?
Good estimators should be unbiased, efficient, and consistent.
What does it mean for an estimator to be unbiased?
An estimator is unbiased when its expected value equals the parameter it is intended to estimate.
What does it mean for an estimator to be efficient?
An estimator is efficient when its variance is the smallest among all unbiased estimators of the same population parameter.
What does it mean for an estimator to be consistent?
An estimator is consistent when the probability that it will be close to the population parameter increases as the sample size increases.
Why is the sample mean considered a good estimator of the population mean?
The sample mean is considered a good estimator because it is unbiased, efficient, and consistent in estimating the population mean.
Why is the sample standard deviation considered a good estimator of the population standard deviation?
The sample standard deviation is considered a good estimator because it is unbiased, efficient, and consistent in estimating the population standard deviation.
What is point estimation?
Point estimation involves estimating a single value for a population parameter, such as using the sample mean to estimate the population mean.
What is interval estimation?
Interval estimation involves estimating a range of values that are expected to include the population parameter with a certain degree of confidence. This range is called the confidence interval.
What is a confidence interval?
A confidence interval is a range of values that we expect to include the population parameter with a particular degree of confidence.
How is the lower confidence limit calculated?
The lower confidence limit is calculated as the point estimate minus the reliability factor times the standard error.
How is the upper confidence limit calculated?
The upper confidence limit is calculated as the point estimate plus the reliability factor times the standard error.
What is a reliability factor?
A reliability factor is a number based on the assumed distribution and degree of confidence, found in statistical tables.
How is the standard error calculated?
The standard error is calculated as the population standard deviation divided by the square root of the sample size, or the sample standard deviation divided by the square root of the sample size.
What are the three factors that influence the choice of distribution for the reliability factor?
he three factors are the size of the sample, whether the population distribution is normal, and whether we know the population mean.
When is a sample size considered small?
A sample size is considered small if it is lower than 30.
When is a sample size considered large?
A sample size is considered large if it is at least 30.
Why is the reliability factor important in constructing confidence intervals?
The reliability factor, based on the chosen distribution, helps determine the range of the confidence interval.
How does knowing the population mean affect the choice of distribution for the reliability factor?
Whether we know the population mean influences the distribution we choose for calculating the reliability factor.
What is the z-statistic based on and when is it used?
The z-statistic is based on normal distribution and is used when the population variance is known.
When do we use the z-alternative?
The z-alternative is used when the population variance is unknown, and it uses the sample standard deviation in the formula for the reliability factor. It is applicable only for large samples.
What is the t-statistic based on?
The t-statistic is based on Student’s t-distribution and is used when the sample size is small or the population variance is unknown.
Why can the z-alternative be used only for large samples?
The z-alternative can be used only for large samples because, according to the Central Limit Theorem, the sample mean is approximately normally distributed for large samples.
What is a hypothesis in statistics?
A hypothesis is a statement about the values of parameters in one or more populations.
What are the two types of hypotheses formulated in hypothesis testing?
The two types are the null hypothesis (H₀) and the alternative hypothesis (H₁).
What is the significance level in hypothesis testing?
The significance level (α) is the probability of rejecting the null hypothesis when it is actually true.
What is the role of the test statistic in hypothesis testing?
The test statistic is used to determine whether to reject or fail to reject the null hypothesis based on its probability distribution.
What is the decision rule in hypothesis testing?
The decision rule specifies the criteria for rejecting or failing to reject the null hypothesis.
Why is gathering data important in hypothesis testing?
Gathering data is crucial to compute the test statistic and make an informed statistical decision.
What is the final step in hypothesis testing?
The final step is to make an investment or economic decision based on the statistical decision and relevant data.
When can we use the t-test for hypothesis testing concerning a single population mean?
We use the t-test when the population variance is unknown, and either the sample is large or the sample is small but the population is normally distributed or approximately normally distributed.
What distribution is the t-test based on?
The t-test is based on Student’s t-distribution.
When do we use the z-test for hypothesis testing concerning a single population mean?
We use the z-test when the sample is drawn from a normally distributed population with known variance.
What distribution is the z-test based on?
The z-test is based on the standardized normal distribution.
When do we use the t-test for differences between population means?
We use the t-test for differences between population means when the populations are assumed to be normally distributed.
What is the pooled estimator of variance used for?
The pooled estimator of variance (s²p) is used when the unknown population variances are assumed to be equal.
What do we use when the unknown population variances are not equal?
When the unknown population variances are not equal, we use the sample standard deviations of the two samples instead of a pooled estimator of the variance.
How do the calculations differ when unknown variances are not equal?
When unknown variances are not equal, we need to calculate the degrees of freedom differently, and there is no pooled estimator of the common variance.
What is an important requirement for using these t-tests?
The two samples must be independent of each other.
What is the chi-square test statistic used for?
It is used for testing the variance of a normally distributed population.
How is the chi-square distribution defined?
It is defined by the number of degrees of freedom, is asymmetrical, and is bounded below by zero.
How are degrees of freedom calculated in the chi-square test?
Degrees of freedom are equal to the sample size minus 1.
Why should the chi-square distribution be used with caution?
It should be used with caution because it is sensitive to violations of its assumptions, such as the normal distribution of the population.
Where are chi-square test values typically found?
In tables for probability in the right tail.
When do we reject the null hypothesis in a two-sided hypothesis test using the chi-square statistic?
We reject the null hypothesis if the test statistic is greater than the chi-square value with a probability of α/2 in the right tail or less than the chi-square value with a probability of α/2 in the left tail.
When do we reject the null hypothesis in a right-tailed chi-square test?
We reject the null hypothesis if the test statistic is greater than the chi-square value with a probability of α in the right tail.
When do we reject the null hypothesis in a left-tailed chi-square test?
We reject the null hypothesis if the test statistic is less than the chi-square value with a probability of α in the left tail or with a probability of (1−α) in the right tail.
What is the F-test used for in hypothesis testing?
The F-test is used to test the equality or inequality of variances of two normally distributed populations.
What is the F-distribution?
The F-distribution is a family of asymmetrical distributions bounded from below by 0, defined by two values of degrees of freedom: the numerator and denominator degrees of freedom.
How are sample variances used in the F-test?
The F-test requires computing the sample variances for both populations, with the greater variance placed in the numerator and the lower variance in the denominator.
What is an important requirement for using the F-test?
The samples from the two populations must be independent.
How is the F-distribution defined?
The F-distribution is defined by two values of degrees of freedom (numerator and denominator), is asymmetrical, and is bounded from below by 0.
How should sample variances be placed in the F-test formula?
The greater variance should be placed in the numerator and the lower variance in the denominator.
When do we reject the null hypothesis in a two-sided F-test?
Reject the null hypothesis if the test statistic is greater than the F-distribution value with a significance level of α/2 (right tail) or less than the value with a significance level of (1−α/2) (left tail).
When do we reject the null hypothesis in a right-tailed F-test?
Reject the null hypothesis if the test statistic is greater than the F-distribution value with a significance level of α (right tail).
When do we reject the null hypothesis in a left-tailed F-test?
Reject the null hypothesis if the test statistic is less than the F-distribution value with a significance level of α (left tail) or less than the value with a significance level of (1−α) (right tail).
