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LM01 Alternative Investment Features, Methods, and Structures.

LM01 Categories, Characteristics, and Compensation Structures of Alternative Investments

LM01 Corporate Structures and Ownership

LM01 Derivative Instrument and Derivative Market Features

LM01 Derivative Instrument and Derivative Market Features.

LM01 Ethics and Trust in the Investment Profession

LM01 Ethics and Trust in the Investment Profession.

LM01 Fixed-Income Instrument Features.

LM01 Fixed-Income Securities: Defining Elements

LM01 Introduction to Financial Statement Analysis

LM01 Introduction to Financial Statement Analysis.

LM01 Market Organization & Structure.

LM01 Market Organization and Structure

LM01 Organizational Forms, Corporate Issuer Features, and Ownership.

LM01 Portfolio Management Overview

LM01 Portfolio Management: An Overview.

LM01 Rates and Returns.

LM01 The Firm & Market Structures.

LM01 Time Value of Money

LM01 Topics in Demand and Supply Analysis

LM02 Alternative Investment Performance and Returns.

LM02 Analyzing Income Statements.

LM02 Code of Ethics and Standards of Professional Conduct Profession

LM02 Code of Ethics and Standards of Professional Conduct.

LM02 Financial Reporting Standards

LM02 Fixed Income Markets - Issuance Trading and Funding

LM02 Fixed-Income Cash Flows and Types.

LM02 Forward Commitment and Contingent Claim Features and Instruments

LM02 Forward Commitment and Contingent Claim Features and Instruments.

LM02 Introduction to Corporate Governance and Other ESG Considerations

LM02 Investors and Other Stakeholders.

LM02 Organizing, Visualizing, and Describing Data

LM02 Performance Calculation and Appraisal of Alternative Investments

LM02 Portfolio Risk & Return: Part I.

LM02 Portfolio Risk and Return Part I

LM02 Security Market Indexes

LM02 Security Market Indexes.

LM02 The Firm and Market Structures

LM02 Time Value of Money in Finance.

LM02 Understanding Business Cycles.

LM03 Aggregate Output, Prices and Economic Growth

LM03 Analyzing Balance Sheets.

LM03 Business Models & Risks

LM03 Corporate Governance: Conflicts, Mechanisms, Risks, and Benefits.

LM03 Derivative Benefits, Risks, and Issuer and Investor Uses

LM03 Derivative Benefits, Risks, and Issuer and Investor Uses.

LM03 Fiscal Policy.

LM03 Fixed-Income Issuance and Trading.

LM03 Guidance for Standards I-VII

LM03 Guidance for Standards I–VII.

LM03 Introduction to Fixed Income Valuation

LM03 Investments in Private Capital: Equity and Debt.

LM03 Market Efficiency

LM03 Market Efficiency.

LM03 Portfolio Risk & Return: Part II.

LM03 Portfolio Risk and Return Part II

LM03 Private Capital, Real Estate, Infrastructure, Natural Resources, and Hedge Funds

LM03 Probability Concepts

LM03 Statistical Measures of Asset Returns.

LM03 Understanding Income Statements

LM04 An Introduction to Asset-Backed Securities

LM04 Analyzing Statements of Cash Flows I.

LM04 Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

LM04 Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives.

LM04 Basics of Portfolio Planning & Construction

LM04 Basics of Portfolio Planning and Construction.

LM04 Capital Investments

LM04 Common Probability Distributions

LM04 Fixed-Income Markets for Corporate Issuers.

LM04 Introduction to the Global Investment Performance Standards (GIPS)

LM04 Introduction to the Global Investment Performance Standards (GIPS).

LM04 Monetary Policy.

LM04 Overview of Equity Securities

LM04 Overview of Equity Securities.

LM04 Probability Trees and Conditional Expectations.

LM04 Real Estate and Infrastructure.

LM04 Understanding Balance Sheets

LM04 Understanding Business Cycles

LM04 Working Capital and Liquidity.

LM05 Analyzing Statements of Cash Flows II.

LM05 Capital Investments and Capital Allocation.

LM05 Company Analysis: Past and Present.

LM05 Fixed-Income Markets for Government Issuers.

LM05 Introduction to Geopolitics.

LM05 Introduction to Industry and Company Analysis

LM05 Monetary and Fiscal Policy

LM05 Natural Resources.

LM05 Portfolio Mathematics.

LM05 Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

LM05 Pricing and Valuation of Forward Contracts.

LM05 Sampling and Estimation

LM05 The Behavioral Biases of Individuals

LM05 The Behavioral Biases of Individuals.

LM05 Understanding Cash Flow Statements

LM05 Understanding Fixed-Income Risk and Return

LM05 Working Capital & Liquidity

LM06 Analysis of Inventories.

LM06 Capital Structure.

LM06 Cost of Capital-Foundational Topics

LM06 Equity Valuation: Concepts and Basic Tools

LM06 Financial Analysis Techniques

LM06 Fixed-Income Bond Valuation: Prices and Yields.

LM06 Fundamentals of Credit Analysis

LM06 Hedge Funds.

LM06 Hypothesis Testing

LM06 Industry and Competitive Analysis.

LM06 International Trade.

LM06 Introduction to Geopolitics

LM06 Introduction to Risk Management

LM06 Introduction to Risk Management.

LM06 Pricing and Valuation of Futures Contracts

LM06 Pricing and Valuation of Futures Contracts.

LM06 Simulation Methods.

LM07 Analysis of Long-Term Assets.

LM07 Business Models.

LM07 Capital Flows and the FX Market.

LM07 Capital Structure

LM07 Company Analysis: Forecasting.

LM07 Estimation and Inference.

LM07 International Trade and Capital Flows

LM07 Introduction to Digital Assets.

LM07 Introduction to Linear Regression

LM07 Inventories

LM07 Pricing and Valuation of Interest Rate and Other Swaps.

LM07 Pricing and Valuation of Interest Rates and Other Swaps

LM07 Technical Analysis

LM07 Yield and Yield Spread Measures for Fixed-Rate Bonds.

LM08 Currency Exchange Rates

LM08 Equity Valuation: Concepts and Basic Tools.

LM08 Exchange Rate Calculations.

LM08 Fintech in Investment Management

LM08 Hypothesis Testing.

LM08 Long Lived Assets

LM08 Measures of Leverage

LM08 Pricing and Valuation of Options

LM08 Pricing and Valuation of Options.

LM08 Topics in Long-Term Liabilities and Equity.

LM08 Yield and Yield Spread Measures for Floating-Rate Instruments.

LM09 Analysis of Income Taxes.

LM09 Income Taxes

LM09 Option Replication Using Put-Call Parity

LM09 Option Replication Using Put–Call Parity.

LM09 Parametric and Non-Parametric Tests of Independence.

LM09 The Term Structure of Interest Rates: Spot, Par, and Forward Curves.

LM10 Financial Reporting Quality.

LM10 Interest Rate Risk and Return.

LM10 Non-current (Long-Term) Liabilities

LM10 Simple Linear Regression.

LM10 Valuing a Derivative Using a One-Period Binomial Model

LM10 Valuing a Derivative Using a One-Period Binomial Model.

LM11 Financial Analysis Techniques.

LM11 Financial Reporting Quality

LM11 Introduction to Big Data Techniques.

LM11 Yield-Based Bond Duration Measures and Properties.

LM12 Applications of Financial Statement Analysis

LM12 Introduction to Financial Statement Modeling.

LM12 Yield-Based Bond Convexity and Portfolio Properties.

LM13 Curve-Based and Empirical Fixed-Income Risk Measures.

LM14 Credit Risk.

LM15 Credit Analysis for Government Issuers.

LM16 Credit Analysis for Corporate Issuers.

LM17 Fixed-Income Securitization.

LM18 Asset-Backed Security (ABS) Instrument and Market Features.

LM19 Mortgage-Backed Security (MBS) Instrument and Market Features.

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IFT Notes for Level I CFA^® Program

LM07 Introduction to Linear Regression

Part 3

7. Measures of Dispersion

In this segment we look at measures that tell us how spread out or dispersed our data might be.

7.1 The Range

The range is the difference between the maximum and minimum values in a data set. It is expressed as:

Range = Max value – Min Value

If the annual returns data is: 10%, -5%, 10%, 25%. What is the range?

Here the maximum return is 25% and the minimum return is -5%. The range is 25% – (-5%) = 30%.

7.2 The Mean Absolute Deviation

It is the average of the absolute values of deviations from the mean. It is expressed as:

$MAD=\left[\sum\limits_{i=1}^{n}{\left|X_i-\ \overline{X}\right|}\right]/n$

where: $\overline{X}$ is the sample mean and n is the number of observations in the sample.

Example

Consider the following data set: 8, 12, 10, 8 and 5. Calculate the mean absolute deviation.

Solution:

$\overline{X}$ = (8 + 12 + 10 + 8 + 5) / 5 = 8.6

$MAD=\frac{\left|8-8.6\right|+\left|12-8.6\right|+\left|10-8.6\right|+\left|8-8.6\right|+\left|5-8.6\right|}{5}$

$MAD=\frac{0.6\ +\ 3.4\ +\ 1.4\ +\ 0.6\ +\ 3.6}{5}=\ 1.92$

7.3 Population Variance and Population Standard Deviation

Population variance is the mean of the squared deviations from the mean. The population variance is computed using all members of a population. It is expressed as:

$\sigma^2=\ \sum^N_{i=0}{(X_i}-\ \mu)^2/N$

where: µ is the population mean and N is the size of the population

Population standard deviation is defined as the positive square root of the population variance.

Example

Calculate the population variance and standard deviation for this dataset: 2%, 5%, 4%, 7%, 8%, 8%, 12%, 10%, 8%, and 5%.

Solution:

$\sigma^2 = \frac{\left[{\left(2\ -\ 6.9\right)}^2+{\left(5\ -\ 6.9\right)}^2{\ +{\left(4\ -\ 6.9\right)}^2+{\left(7\ -\ 6.9\right)}^2+\left(8\ -\ 6.9\right)}^2 \atop +{\left(8\ -\ 6.9\right)}^2+{\left(12\ -\ 6.9\right)}^2+{\left(10\ -\ 6.9\right)}^2+{\left(8\ -\ 6.9\right)}^2+\ {\left(5\ -\ 6.9\right)}^2\right]}{10}$

${\sigma }^2= 7.89\%$

Population standard deviation $(\sigma) = \sqrt{7.89} = \ 2.81\%$

7.4 Sample Variance and Sample Standard Deviation

Sample variance applies when we are dealing with a subset, or sample, of the total population. It is expressed as:

$s^2=\ \sum\limits_{i=0}^{n}{(X_i}-\ \overline{X}\ ){\ }^2\ /\ (n-1)$

where: $\overline{X}$ is the sample mean and n is the number of observations in the sample.

Sample standard deviation is defined as the positive square root of the sample variance.

Example

Calculate the sample variance for the following data set: 8, 12, 10, 8 and 5.

Solution:

$s^2=\frac{\left[{\left(8-8.6\right)}^2+{\left(12-8.6\right)}^2+{\left(10-8.6\right)}^2+{\left(8-8.6\right)}^2+{\left(5-8.6\right)}^2\right]}{5-1}$

$\ s^2=6.80\%$

The sample standard deviation is the positive square root of the sample variance. For the sample data given above, $s = \sqrt{6.80} = \ 2.61 \%$

Using a financial calculator to calculate variance and standard deviations

The population and sample standard deviation can easily be computed using a financial calculator. Assume the following data set: 10%, -5%, 10%, 25%, the calculator key strokes are shown below:

*Keystrokes*	*Description*	*Display*
[2nd] [DATA]	Enters data entry mode
[2nd] [CLR WRK]	Clears data register	X01
10 [ENTER]		X01 = 10
[↓] [↓] 5+/- [ENTER]		X02 = -5
[↓] [↓] 10 [ENTER]		X03 = 10
[↓] [↓] 25 [ENTER]		X04 = 25
[2nd] [STAT] [ENTER]	Puts calculator into stats mode
[2nd] [SET]	Press repeatedly till you see à	1-V
[↓]	Number of data points	N = 4
[↓]	Mean	X = 10
[↓]	Sample standard deviation	Sx = 12.25
[↓]	Population standard deviation	σx = 10.61

Notice that the calculator gives both the sample and the population standard deviation. On the exam we will have to determine whether we are dealing with population or sample data and choose the appropriate value.

7.5 Semivariance, Semideviation, and Related Concepts

Instructor’s Note: Semivariance and semideviation are not emphasized in the learning outcomes and have a very low probability of being tested on the Level I exam. Nevertheless, a brief explanation is given below.

Variance and standard deviation of returns take account of returns above and below the mean, but often investors are concerned only with downside risk, for example returns below the mean. As a result, analysts have developed semivariance, semideviation and related dispersion measures that focus on downside risk. Semivariance is defined as the average squared deviation below the mean. Semideviation is the positive square root of semivariance.

7.6 Chebyshev’s Inequality

According to Chebyshev’s inequality, the proportion of the observations within k standard deviations of the arithmetic mean is at least: $1\ - \frac{1}{k^2}$ for all k > 1.

To find out what percent of the observations must be within two standard deviations of the mean we simply plug into the formula and get: $1\ - \frac{1}{2^2}=\ 1\ - \frac{1}{4}\ = 0.75 = 75\%$ . Hence, at least 75% of the data will be between two standard deviations of the mean.

Chebyshev’s inequality can be used to measure maximum amount of dispersion, regardless of the shape of the distribution. Notice that here we do not make any assumptions about whether the distribution is normal or not normal. This inequality applies across all distributions.

7.7 Coefficient of Variation

Coefficient of variation expresses how much dispersion exists relative to the mean of a distribution and allows for direct comparison of dispersion across different data sets. It is used in investment analysis to compare relative risks. When evaluating investments, a lower value is better. Coefficient of variation is expressed as:

$CV = \frac{S}{\overline{X}}$

where: s = sample standard deviation of a set of observations and $\overline{X}$ = sample mean

Example

Investment A has a mean return of 7% and a standard deviation of 5%. Investment B has a mean return of 12% and a standard deviation of 7%. Calculate the coefficients of variation.

Solution

The coefficients of variation can be calculated as follows:

${CV}_A=\frac{5\%}{7\%}=0.71$