The Investment Advisor Body of Knowledge + Test Bank - IMCA

Скачать книгу

for each state's securities administrator as well as for U.S. territories, Canada, and Mexico.

      CHAPTER 3

      Statistics and Methods

      Statistical concepts and calculations form an important foundation for understanding applied financial methods and formulae. Investment advisors and consultants should have a firm grasp on quantitative concepts in order to analyze historical data, calculate and analyze investment risk and returns, draw accurate conclusions, and make appropriate recommendations to clients. This chapter provides explanations of key elementary statistical concepts.

      Understanding and applying familiar concepts like compounding, discount factors, averages, measures of dispersion, and confidence intervals should come easily to investment advisors and consultants. More advanced statistical concepts and calculations however may take additional study to master. In practice, simply understanding the intuition of more advanced concepts and applications like multicollinearity and multivariate regression models is often sufficient. The readings in the chapter explain each concept from a mathematical framework and apply them to common financial and investment challenges to help one better understand and apply each tool in a more meaningful way.

      Part I Mathematics and Statistics for Financial Risk Management: Some Basic Math

      Learning Objectives

      ■ Describe the concept of compounding and compute compound returns.

      ■ Discuss the concept of limited liability.

      ■ Graph log returns and discuss how logarithms are useful for charting times series that grow exponentially.

      ■ Explain continuously compounded returns and convert the nominal rate into an annualized rate.

      ■ Discuss the significance of discount factors.

      ■ Describe the concept of an infinite series and determine the present value of a perpetuity that pays a stated coupon payment annually.

      ■ Describe the concept of a finite series and determine the present value of a newly issued bond that matures at a time certain and pays a stated coupon payment annually.

      Part II Mathematics and Statistics for Financial Risk Management: Probabilities

      Learning Objectives

      ■ Define and explain discrete random variables.

      ■ Define and explain continuous random variables.

      ■ Describe the concept of mutually exclusive events and calculate the probability that an investment will fall within a given range of returns.

      ■ Describe the concept of independent events and determine the probability that an investment will earn a return above a stated figure.

      ■ Explain probability matrices.

      Part III Mathematics and Statistics for Financial Risk Management: Basic Statistics

      Learning Objectives

      ■ Describe a population and sample data, and calculate the mean, median, and mode of a data set.

      ■ Describe discrete random variables and calculate the mean, median, and mode of a year-end portfolio value based on facts given, including default proba- bilities.

      ■ Describe continuous random variables.

      ■ Determine the price of a newly issued bond based on facts given including default probabilities.

      ■ Explain and calculate variance and standard deviation.

      ■ Describe and calculate variance and correlation.

      ■ Explain the concept of moments in statistical analysis.

      ■ Describe and illustrate skewness and kurtosis graphically.

      Part IV Mathematics and Statistics for Financial Risk Management: Distributions

      Learning Objectives

      ■ Describe a normal distribution and its implications for measuring returns and risk.

      ■ Explain the concept and application of Monte Carlo simulations for investment advisors and consultants.

      Part V Mathematics and Statistics for Financial Risk Management: Hypothesis Testing and Confidence Intervals

      Learning Objectives

      ■ Review and calculate sample mean.

      ■ Review and calculate sample variance.

      ■ Explain the concept of confidence intervals and discuss the terms: t-distribution,t-statistic, and t-stat.

      ■ Discuss hypothesis testing and confirm or reject a hypothesis based on various confidence levels.

      ■ Discuss and calculate value-at-risk (VaR) and illustrate VaR graphically.

      ■ Describe back-testing.

      ■ Explain expected shortfall.

      Part VI Mathematics and Statistics for Financial Risk Management: Linear Regression Analysis

      Learning Objectives

      ■ Describe linear regression and explain the formula (y = a + bx).

      ■ Define coefficient of determination (R-squared) and explain how it is used.

      ■ Explain the difference between univariate and multivariate regression models.

      ■ Describe multicollinearity and discuss how increasingly integrated global financial markets cause challenges in managing risk.

      Part VII Mathematics and Statistics for Financial Risk Management: Time Series Models

      Learning Objectives

      ■ Describe the concept of “random walks” and how it might apply to investment markets.

      ■ Explain autocorrelation as it relates to measuring risk/return and random walks.

      ■ Discuss why over-estimations and under-estimations of risk occur based on the assumptions a) that variance is linear in time and b) that no serial correlation exists.

      Part I Some Basic Math

      Compounding

      Log returns might seem more complex than simple returns, but they have a number of advantages over simple returns in financial applications. One of the most useful features of log returns has to do with compounding returns. To get the return of a security for two periods using simple returns, we have to do something that is not very intuitive, namely adding one to each of the returns, multiplying, and then subtracting one:

(3.1)

      Here the first subscript on R denotes the length of the return, and the second subscript is the traditional time subscript. With log returns, calculating multiperiod returns is much simpler; we simply add:

(3.2)

      It is fairly straightforward to generalize this notation to any return length.

      Sample Problem

      Question:

Using Equation 3.1, generalize Equation 3.2 to returns of any length.

      Answer:

      Note that to get to the last line, we took the logs of both sides of the previous equation, using the fact that the log of the product of any two variables is equal to the sum of their logs.

      Limited

Скачать книгу