Course Content
THE TIME VALUE OF MONEY
This chapter is covered in reading 6 of study session 2 of the material provided by the institute. After reading this chapter, a student should be able to: a) interpret interest rates as required rates of return, discount rates, or opportunity costs; b) explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk; c) calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding; d) solve time value of money problems for different frequencies of compounding; e) calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, perpetuity (PV only), and a series of unequal cash flows; f) demonstrate the use of a timeline in modeling and solving time value of money problems.
0/2
Interest Rates
00:00
Future Value & Present Value
00:00
STATISTICAL CONCEPTS AND MARKET RETURNS
This chapter is covered in reading 7 of study session 2 of the material provided by the Institute. After reading this chapter, a student should be able to: a. distinguish between descriptive statistics and inferential statistics, between a population and a sample, and among the types of measurement scales; b. define a parameter, a sample statistic, and a frequency distribution; c. calculate and interpret relative frequencies and cumulative relative frequencies, given a frequency distribution; d. describe the properties of a data set presented as a histogram or a frequency polygon; e. calculate and interpret measures of central tendency, including the population mean, the sample mean, arithmetic mean, weighted average or mean, geometric mean, harmonic mean, median, and mode; f. calculate and interpret quartiles, quintiles, deciles, and percentiles; g. calculate and interpret 1) a range and a mean absolute deviation and 2) the variance and standard deviation of a population and of a sample; h. calculate and interpret the proportion of observations falling within a specified number of standard deviations of the mean using Chebyshev’s inequality; i. calculate and interpret the coefficient of variation and the Sharpe ratio; j. explain skewness and the meaning of a positively or negatively skewed return distribution; k. describe the relative locations of the mean, median, and mode for a unimodal, nonsymmetrical distribution; l. explain measures of sample skewness and kurtosis; m. compare the use of arithmetic and geometric means when analyzing investment returns.
0/9
Statistical Concepts
00:00
Summarizing Data Using Frequency Distribution
00:00
Measures of Central Tendencies
00:00
Measures of Dispersion
00:00
Chebyshev’s Inequality
00:00
Coefficient of Variations
00:00
Skewness of Return Distribution
00:00
Kurtosis
00:00
Geometric Mean Vs. Arithmetic Mean
00:00
PROBABILITY CONCEPTS
This chapter is covered in reading 8 of study session 2 of the material provided by the Institute. After reading this chapter, a student should be able to: a. define a random variable, an outcome, an event, mutually exclusive events, and exhaustive events; b. state the two defining properties of probability and distinguish among empirical, subjective, and a priori probabilities; c. state the probability of an event in terms of odds for and against the event; d. distinguish between unconditional and conditional probabilities; e. explain the multiplication, addition, and total probability rules; f. calculate and interpret 1) the joint probability of two events, 2) the probability that at least one of two events will occur, given the probability of each and the joint probability of the two events, and 3) a joint probability of any number of independent events; g. distinguish between dependent and independent events; h. calculate and interpret an unconditional probability using the total probability rule; i. explain the use of conditional expectation in investment applications; j. explain the use of a tree diagram to represent an investment problem; k. calculate and interpret covariance and correlation; l. calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio; m. calculate and interpret covariance given a joint probability function; n. calculate and interpret an updated probability using Bayes’ formula; o. identify the most appropriate method to solve a particular counting problem and solve counting problems using factorial, combination, and permutation concepts.
0/9
Some Important Terminologies
00:00
Conditional & Unconditional Probabilities
00:00
Total Probability Rule
00:00
Expected Value
00:00
Conditional Expected Value
00:00
Portfolio Expected Return & Variance
00:00
Joint Probability Function
00:00
Bayes’ Formula
00:00
Principles of Counting
00:00
COMMON PROBABILITY DISTRIBUTIONS
This chapter is covered under reading 9, study session 3, as provided by the institute. After reading this chapter, a student shall be able to: a. define a probability distribution and distinguish between discrete and continuous random variables and their probability functions; b. describe the set of possible outcomes of a specified discrete random variable; c. interpret a cumulative distribution function; d. calculate and interpret probabilities for a random variable, given its cumulative distribution function; e. define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable; f. calculate and interpret probabilities given the discrete uniform and the binomial distribution functions; g. construct a binomial tree to describe stock price movement; h. calculate and interpret tracking error; i. define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution; j. explain the key properties of the normal distribution; k. distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution; l. determine the probability that a normally distributed random variable lies inside a given interval; m. define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution; n. define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety-first criterion; o. explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices; p. distinguish between discretely and continuously compounded rates of return and calculate and interpret a continuously compounded rate of return, given a specific holding period return; q. explain Monte Carlo simulation and describe its applications and limitations; r. compare Monte Carlo simulation and historical simulation.
0/6
Probability Distribution
00:00
Discrete Random Variables
00:00
Continuous Random Variables
00:00
Shortfall Risk
00:00
Lognormal Distribution
00:00
Monte-Carlo Simulation
00:00
SAMPLING AND ESTIMATION
This chapter is covered under reading 10, study session 3, as provided by the institute. After reading this chapter, a student shall be able to: a. define simple random sampling and a sampling distribution; b. explain sampling error; c. distinguish between simple random and stratified random sampling; d. distinguish between time-series and cross-sectional data; e. explain the central limit theorem and its importance; f. calculate and interpret the standard error of the sample mean; g. identify and describe desirable properties of an estimator; h. distinguish between a point estimate and a confidence interval estimate of a population parameter; i. describe properties of Student’s t-distribution and calculate and interpret its degrees of freedom; j. calculate and interpret a confidence interval for a population mean, given a normal distribution with 1) a known population variance, 2) an unknown population variance, or 3) an unknown variance and a large sample size; k. describe the issues regarding selection of the appropriate sample size, data mining bias, sample selection bias, survivorship bias, look-ahead bias, and time-period bias.
0/7
Sampling
00:00
Data Types
00:00
Central Limit Theorm
00:00
Estimators
00:00
Confidence Intervals
00:00
Student’s t-Distribution
00:00
Sampling Issues
00:00
HYPOTHESIS TESTING
This chapter is covered under reading 11, study session 3, as provided by the institute. After reading this chapter, a student shall be able to: a. define a hypothesis, describe the steps of hypothesis testing, and describe and interpret the choice of the null and alternative hypotheses; b. distinguish between one-tailed and two-tailed tests of hypotheses; c. explain a test statistic, Type I and Type II errors, a significance level, and how significance levels are used in hypothesis testing; d. explain a decision rule, the power of a test, and the relation between confidence intervals and hypothesis tests; e. distinguish between a statistical result and an economically meaningful result; f. explain and interpret the p-value as it relates to hypothesis testing; g. identify the appropriate test statistic and interpret the results for a hypothesis test concerning the population mean of both large and small samples when the population is normally or approximately normally distributed and the variance is 1) known or 2) unknown; h. identify the appropriate test statistic and interpret the results for a hypothesis test concerning the equality of the population means of two at least approximately normally distributed populations, based on independent random samples with 1) equal or 2) unequal assumed variances; i. identify the appropriate test statistic and interpret the results for a hypothesis test concerning the mean difference of two normally distributed populations; j. identify the appropriate test statistic and interpret the results for a hypothesis test concerning 1) the variance of a normally distributed population, and 2) the equality of the variances of two normally distributed populations based on two independent random samples; k. formulate a test of the hypothesis that the population correlation coefficient equals zero and determine whether the hypothesis is rejected at a given level of significance; l. distinguish between parametric and nonparametric tests and describe situations in which the use of nonparametric tests may be appropriate.
0/8
Statistical Inference
00:00
Hypothesis Testing
00:00
P-Value
00:00
Test of a Single Mean
00:00
Differences in Mean or Mean Differences
00:00
Variance Test
00:00
Testing for Non-Zero Correlation
00:00
Non-Parametric Inference
00:00
Quantitative Methods
About Lesson

LOS G requires us to:

calculate and interpret 1) a range and a mean absolute deviation and 2) the
variance and standard deviation of a population and of a sample.

 

Dispersion is nothing but the variability around the central tendencies. There are different measures of dispersions, they are:

1.  Range

The range is the difference between the maximum and minimum values in data.

Range = Max – Min

The range gives a basic idea about the extent to which the limits within which the variation is possible. The shortcoming of using range as a measure of dispersion is that the extreme values or outliers may distort the range.

2.  Mean Absolute Deviation

Mean absolute deviation (MAD) is the average distance between each data value and mean. However, while calculating the mean, the negative deviations are converted to the positive ones by modifying them or taking their absolute values.

The formula for calculating MAD is:

Mean Absolute Deviation Quantitative Methods CFA level 1 Study Notes

3.  Variance and Standard Deviation

a.  Variance is the squared deviation of the random variables in a data set from its mean. The variance of the parameters, i.e. the population data can be calculated using the following formula:

Variance Formula Quantitative Methods CFA level 1 Study Notes

b.  However, if we take the sample from the population, considering it as a reflection of the population at large, then the formula for calculating the variance is:

Sample Variance Quantitative Methods CFA level 1 Study Notes

c.  While calculating the sample variance, we use ‘n-1’ as the number instead of ‘N’. ‘n-1’ represents the degree of freedom while using the sample data instead of population. It represents the minimum number of independent ways in which a system can vary. We know that Σ (Xi – x̅) = 0, therefore, if we know the values of (n-1) observations, we can also calculate the last. Thus, this value is not independent.

Example:

 Taking the data given in the previous example of the return on equity stock, we can calculate the variance and the standard deviation as follows:

Year

Return on Equity

  

Variance

1

-2.00%

=(-2-3.80)2

0.34%

2

8.00%

=(8-3.80)2

0.18%

3

27.00%

=(27-3.80)2

5.38%

4

-9.00%

=(-9-3.80)2

1.64%

5

-5.00%

=(-5-3.80)2

0.77%

Average:

3.80%

Variance:

2.08%

 

d.  The standard deviation, on the other hand, is the square root of the variance. It measures the dispersion around the arithmetic mean.

Thus, the standard deviation of the population data is:

Standard Deviation Quantitative Methods CFA level 1 Study Notes

And the standard deviation of a sample data is:

Sample Standard Deviation Quantitative Methods CFA level 1 Study Notes

e.  However, if we are given geometric mean instead of the arithmetic mean, we can calculate the dispersion around it using the following formula:

Dispersion around geometric mean Quantitative Methods CFA level 1 Study Notes

 

4.  Semi- Variance

a.  Semi-variance measures the dispersion of all the observations that are less than the mean.

b.  It can be calculated using the following steps:

i.  Find x̅

ii.  Select all the observations that fall below x̅

iii.  Calculate the semi-variance using the following formula:

Semi Variance formula Quantitative Methods CFA level 1 Study Notes

 

c.  The ‘n’ that we use in the calculation of semi-variance reflects the full data set and not just the observations that fall below the mean.

5.  Semi-Deviation

Semi-deviation is the square root of the semi-variance.

 

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