Quantitative Methods: Applications

Question 1

P Distributions: Discrete and Continuous (3)

Answer 1

• Probability distribution: gives the probabilities of all possible outcomes of a random variable.
• Discrete distribution: has a finite number of possible outcomes
• Continuous distribution: has a finite number of possible outcomes

Question 2

P Distributions: Probability Functions

Answer 2

• Discrete random variable: the numbers of days it will rain next week than can take on values : {0,1,2,3,4,5,6,7}
• Continuous random variable: the amount of rain that will fall tonight
• Probability function: p(x) gives the probability that a discrete random variable will take on the value x

Question 3

P Distributions: Cumulative Distribution Function

Answer 3

A cumulative distribution function cdf, F(x), gives the probability that a random variable will be less than or equal to a given value. For the probability function:

• p(x) = x/ 15 for X = {1,2,3,4,5}
• F(3) = 1/15 + 2/15 + 3/15 = 6/15 = 40%

Question 4

P Distributions: CDF for a Continuous Distribution

Answer 4

(1) Example: The %ROE, x,for a firm is defined over (-29,+30) and has a Cdf of F(x) = (x + 20) /59. What is the probability that the ROE will be positive and less than or equal to 15? Prob (0

Question 5

P Distributions: Discrete Uniform

Answer 5

(1) A discrete uniform distribution has a finite number of possible outcomes, all of which are equally likely. For example, p(x) = .2 for X = {1,2,3,4,5} p(2) = 20% F(3) = 69% Prob(2

Question 6

P Distributions: Binomial Random Variable (2)

Answer 6

(1) The probability of exactly x successes in n trials, given just two possible outcomes (success and failure)
(2) Probability of success on each trial (p) is constant, and all trials are independent

Question 7

P Distributions: Binomial - Example

Answer 7

What is probability of drawing exactly two white marbles from a bowl of black and white marbles in six tries if the probability of selecting white is .4 each time?

Question 8

P Distributions: Bernouli random variable

Answer 8

Question 9

P Distributions: Continuous Uniform Distribution

Answer 9

(1) Probability distributed evenly over an interval.
(2) Example: random variable is continuous uniform over the interval 2 to 10.

Question 10

P Distributions: Properties of Normal Distribution (6)

Answer 10

1. Completely described by mean and variance
2. Symmetric around the mean (skewness = 0)
3. Kurtosis (a measure of peakedness) = 3
4. Linear combination of normally distributed random variables is also normally distributed
5. Probabilities decrease further from the mean, but the tails go on forever.
6. Multivariate normal: more than one random variable, need means, variance and correlation coefficient.

Question 11

P Distributions: Confidence Interval: Normal Distribution (@)

Answer 11

(1) Confidence Interval: A range of values around an expected outcome. A random variable is expected to be within this range a certain percentage of the time.
(2) Example: the mean annual return (normally distributed) on a portfolio over many years is 11%, and the standard deviation of returns is 8%. Calculate a 95% confidence interval on next years return. —90% conf. int = Xbar+- 1.65s —95% conf. int = Xbar+- 1.96s —99% conf. int - Xbar +- 2.58s 11% +- (1.96((8%)=-4.7% to 26.7%

Question 12

P Distributions: Standard Normal Distribution

Answer 12

(1) A normal distribution that has been standardized has a mean of 0 and a standard deviation of 1. (2) To standardize a random variable, calculate the z-value. (3) Subtract the mean and divide by the standard deviation. z=X-mean/sd

Question 13

P Distributions: Calculating Probabilities Using the Standard Normal Distribution

Answer 13

Example: The EPS for a large group of firms are normally distributed an d have a u=$4.00 and a o=$1.50. Find the probability that a selected firm’s earnings are less than $3.70. z= 3.70-4.00/1.50-/20 3.60 is .2 sd below 4.00 mean. Check z table at .2 and .00. For negative z-table, calculate 10 - table value. There is a 42.07% probability that the EPS of a randomly selected form will be more than .20 sd below the mean Iess than $3.70.

Question 14

P Distributions: Shortfall risk and Roy’s Safety-First Ratio

Answer 14

(1) Shortfall risk: Probability that a portfolio return or value will be below a target return or value. (2) Rou’s Safety-First Ratio: Number of std. dev target is below expected expected return/value. (3) Example: Given the two portfolios, which has the lower probability of generating a return below 5%? —15-5)/12=/93 —18-4/25=.25

Question 15

P Distributions: Lognormal Distribution

Answer 15

(1) If x is normal, then e^x is lognormal. (2) Lognormal is always positive, used for modeling price relatives –> (1 +return= e^x

Question 16

P Distributions: Continuous Compounding

Answer 16

(1) Continuously compounding rate = ln(1 + HPR) (2) EAY with continuous compounding = e^i-1 (3) Example: 1-year holding period return = 8% —Continuous compounded rate of return = ln (1.08) = 7.7% —7.7% rate with continuous compounding, EAY = e^.077-1 = 8%

Question 17

P Distributions: Monte Carlo Simulation (5)

Answer 17

Simulation can be used to estimate a distribution of derivatives prices of NPVs (1) Specify distributions of random variables such as interest rates, underlying stock prices (2) Use computer random generation of variables (3) Value the derivative using those values (4) Repeat steps 2 and 3 1000s of times (5) Calculate mean/variance of all values.

Question 18

Sampling and Estimation: Sampling (4)

Answer 18

(1) To make inferences about parameters of a population we will use a sample (2) A simple random sample is one where every population member has an equal chance of being selected (3) A sampling distribution is the distribution of sample statistics for repeated sample size n. (4) Sampling error is the difference between a sample statistic and true population parameter.

Question 19

P Distributions: Historical Simulation (3)

Answer 19

(1) Similar to Monte Carlo simulation, but generates random variables from distributions of historical data. (2) Advantage: Don’t have to estimate distribution of risk factors (3) Disadvantage: Future outcomes for risk factors may be outside the historical range.

Question 20

Sampling and Estimation: Stratified Random Sampling (2)

Answer 20

(1) Create subgroups from population based on important characteristics (e.g. identify bonds according to callable, ratings, maturity, and coupon. (2) Selected samples from each subgroup in proportion to the size of the subgroup. —Used to construct bond portfolios to match a bond index or to construct a sample that has certain characteristics in common with the underlying population.

Question 21

Sampling and Estimation: Time-Series vs. Cross-Sectional Data (2)

Answer 21

(1) Time-series data: for example, monthly prices for IBM stocks for five years. (2)Cross-sectional data: for example, returns on all health care stocks last month

Question 22

Sampling and Estimation: Central Limit Theorem (2)

Answer 22

(1) For any population with mean u and variance o^2, as the size of the random sample gets large, the distribution of sample means approaches normal distribution with a mean u and a varianve o2/n (2) Allows us to make inferences about and construct confidence intervals for population means based on sample means.

Question 23

Sampling and Estimation: Standard Error of the Sample Mean

Answer 23

(1) standard error of sample mean is the standard deviation of the distribution of sample means. (2) When population is known ox=o/sq.rt n When population is unknown sx =s/sq.rt n

Question 24

Sampling and Estimation: Standard Error of the Sample Mean-Example

Answer 24

(1) Example: The mean P/E for a sample of 41 firms is 19.0, and the standard deviation of the population is 6.6. What is the standard error of the sample mean? ox=o/sq.rt n = 6.6/sq.rt 41 = 1.03 (2) Interpretation: for sample size n=41, the distribution of the sample means would have a mean of 19.0 and a standard deviation of 1.03.

Question 25

Sampling and Estimation: Desirable Estimator Properties (3)

Answer 25

(1) Unbiased: expected value equal to parameter. (2) Efficient: sampling distribution has smallest variable of all unbiased estimators. (3) Consistent: larger sample –> better estimator, Standard error of estimate decrease with large sample size.

Question 26

Sampling and Estimation: Point Estimate and Confidence Interval

Answer 26

Example: The mean P/E point estimate for a sample of 41 firms is 19.0, and the standard error of the sample mean is 1.03, and the population is normal. —90% confidence interval is 19 +- 1.65(1.03) 17.3 < mean < 2.07 —95% confidence interval is 19 +- 1.96 (1.03 17.0 < mean < 21.0 —95 confidence interval for a randomly chosen firm (from population) is 19+- 1.96(6.6) or 6.06 < mean < 31.94

Question 27

Sampling and Estimation: Confidence Intervals for Mean

Answer 27

(1) When sampling from a normal distribution and known variance, always use z-statistic for reliability factors. (2) When sampling from a normal distribution and unknown variance, always use t-statistic for reliability factors.* (3) When sampling a nonnormal distribution and known variable, use z-statics for large samples (n>30). Reliability factors unknown for small samples. (4) When sampling a nonnormal distribution and unknown variance, use t-static for large sample. Reliability factors are unknown for small samples.* *z-statistics is theoretically acceptable, t is more conservative.

Question 28

Sampling and Estimation: Student’s t-distribution and degrees of freedom (4)

Answer 28

Properties of Student’s t-distribution (1) symmetrical (bell shaped) (2) Fatter tails than a normal distribution (3) Defined by a single parameter, degrees of freedom (df), where df=n-1 (4) As df increase, t-distribution approaches normal distribution (5) Lower degrees of freedom –> fatter tails

Question 29

Sampling and Estimation: T-Distributions

Answer 29

Question 30

Sampling and Estimation: Constructing Confidence Intervals

Answer 30

Question 31

Sampling and Estimation: Confidence Interval for Mean - Example

Answer 31

Example: Normal distribution, unknown variance: the sample mean is 19.0, the sample sd is 6.6, and n=41. establish a 90% confidence interval for the population mean. —t-table reliability factor is 1.684(df=40, a/2=.05) std. error of mean=s/sqrt of n= 6.6/sqrt 41= 1.03 19+-1.684(1.03=17.27 < mean < 20.73

Question 32

Sampling and Estimation: Sample Size Issues

Answer 32

We’ve seen that larger samples produce better estimates and small confidence intervals but: (1) Cost can be a factor- obtaining more data can increase costs, so there is a trade-off. (2) Including more data points from a population (time-period) with different parameters will not improve your estimate.

Question 33

Sampling and Estimation: Types of Bias (5)

Answer 33

(1) Data mining bias: from repeatedly doing tests on same data sample. (2) Sample selection bias: sample not really random. (3) Survivorship bias: sampling only surviving firms, mutual funds, hedge funds (4) Look-ahead bias: using information not available at the time to construct sample. (5) Time-period bias: relationship exists only during time period of sample data.

Question 34

Hypothesis Testing: Steps (7)

Answer 34

(1) State the hypothesis-relation to be tested (2) Select a test statistic (3) Specify the level of significance (4) State the decision rule for the hypothesis (5) Collect the sample and calculate statistics (6) Make a decision about the hypothesis (7) Make a decision based on test results.

Question 35

Hypothesis Testing: Null and Alternative Hypotheses (2)

Answer 35

(1) Null hypothesis H0 —The hypothesis to be tested —Researcher wants to reject it —Always includes the equal sign (2) Alternative hypothesis Ha —What the researcher would like to conclude —What is concluded if the researcher rejects the null hypothesis

Question 36

Hypothesis Testing: Test statistic and Critical Values (3)

Answer 36

(1) A test statistic is calculated from sample data, and compared to critical values to test H0 (2) If test statistics exceeds the critical value ( or is outside the range of critical values, the researcher rejects H0 (3) Critical values are like a confidence interval

Question 37

Hypothesis Testing: Two-tailed test

Answer 37

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Question 38

Hypothesis Testing: One-tailed test

Answer 38

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Question 39

Hypothesis Testing: Type I and Type II Errors

Answer 39

Type I Error:

• Rejecting H₀ when it is actually true
• Significance level is Prob of type I Error

Type II Error:

• Failing to reject H₀ whenit is actually false
• Power of test is 1- Prob of Type II Error

Question 40

Hypothesis Testing: Statistically vs. Economically Meaningful Result.

Answer 40

Statistical significance doe snot necessarily imply economic significance:

• Tranactions costs
• Taxes
• Risk

Question 41

Hypothesis Testing: p-value Example

Answer 41

A p-value is the smallest level of significance at which the null can be rejected, the probability of getting the test statistic by chance if the null is true.

Question 42

Hypothesis Testing: Test Statistics: T-statistic

Answer 42

Test of mean of normal population when variance is unknown, use a t-statistic

Question 43

Hypothesis Testing: Test Statistics: Z-statistic

Answer 43

Test of mean of normal population when variance is known, use a z-statistic.

Question 44

Hypothesis Testing: Example

Answer 44

Test the hypothesis that fund’s mean return is equal to 1% per month at the 95% confidence (5% significance) level

Data provided:

• Sample mean : 1.5%
• Sample size: 45
• Population standard deviation: 1.4%
• Population distribution is non-normal

Question 45

Hypothesis Testing: Test Statistics: Difference in Means

Answer 45

Test of whether the means of two normal populations are equal- independent samples

Question 46

Hypothesis Testing: Paired Comparisons Test

Answer 46

Test of the difference between the means of two different populations- dependent samples

Paired Comparison Test

Question 47

Hypothesis Testing: Test Statistics

Answer 47

Note that the LOS only require you to:

Odentify the appropriate test statistic and interpret results for a hypothesis test concerning:

1. The difference in means and the mean difference tests are t-tests, reject if t-stat is greater than critical value
2. Use difference in means tests for samples from two independent normal populations
3. Use mean differences test for two dependent, equal-size, samples from normal populations

Question 48

Hypothesis Testing: Test Statistics: Variance

Answer 48

Question 49

Hypothesis Testing: Test Statistics: F-Test

Answer 49

Test of whetehr the variances of two normal populations are equal is an F-test.

Putting the larger sample variance inthe numerator allows us to consider only upper critical value - although F-test is a two-tailed test.

Question 50

Hypothesis Testing: Parametric and Nonparametric Tests (2)

Answer 50

• Parametric tests are based on assumptions about population distributions and popultion parameters (e.g. t-test, z-test, F-test)
• Nonparametric tests make few if any assumptions about the popultaion distribution and test things other than parameter values (e.g., runs tests, rank correlation tests)