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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

LM03 Probability Concepts

Part 3

4. Portfolio Expected Return and Variance of Return

Expected Return

A portfolio’s expected return can be calculated as:

$\rm{E(}{\mathrm{R}}_{\mathrm{P}}\mathrm{)\ =\ }{\mathrm{w}}_{\mathrm{1}}\mathrm{E(}{\mathrm{R}}_{\mathrm{1}}\mathrm{)\ +\ }{\mathrm{w}}_{\mathrm{2}}\mathrm{E(}{\mathrm{R}}_{\mathrm{2}}\mathrm{)\ +\ \dots +\ }{\mathrm{w}}_{\mathrm{n}}\mathrm{E(}{\mathrm{R}}_{\mathrm{n}}\mathrm{)}$

where:

w_n = portfolio weight of n^th security in the portfolio

R_n = expected return of n^th security in the portfolio

n = number of securities in the portfolio

We will discuss portfolio expected return and variance of return using a two-stock portfolio.

Example

40% of the portfolio is invested in Stock A and 60% is invested in Stock B. As shown in the table below, the expected return of each stock depends on the economic scenario.

Scenario	P(Scenario)	Expected returns of A	Expected returns of B
Recession	0.25	2%	4%
Normal	0.50	8%	10%
Boom	0.25	12%	16%

This information can also be presented as a joint probability function of A’s and B’s returns:

	R_B = 4%	R_B = 10%	R_B = 16%
R_A = 2%	0.25	0	0
R_A = 8%	0	0.50	0
R_A = 12%	0	0	0.25

Row 1 and Column 1 represent the returns of A and B respectively. The other cells contain probabilities. Calculate the expected return of A and B.

Solution:

Given the data presented above:

The expected return of A is: 0.25 x 2 + 0.50 x 8 + 0.25 x 12 = 7.5%.

The expected return of B is: 0.25 x 4 + 0.50 x 10 + 0.25 x 16 = 10%.

Expected return of the portfolio = weight of A in the portfolio x expected return of A + weight of stock B in the portfolio x expected return of B = 0.4 x 7.5 + 0.6 x 10 = 9%

The expected portfolio return is 9%. As the term implies, this is the expected return. The actual return will vary around 9%. The amount of variability is measured by the variance. In order to determine the variance of return, we must first calculate the covariance.

Covariance

Covariance tells us how movements in a random variable vary with movements in another random variable, whereas variance tells us how a random variable varies with itself. Assume there are two random variables R_i and R_j. The forward-looking, population covariance between R_i and R_j (used to measure how they move together) is given by:

$\rm Cov\left(R_i,\ R_j\right)=\ E\left(\left[R_i\ - \ {ER}_i\right]\left[\ R_j\ - \ {ER}_j\right]\right)$

where:

ER_i = expected return for variable R_i

ER_j = expected return for variable R_j

If the random variables are returns, the units of both forward-looking covariance and historical variance would be returns squared.

The sample covariance between two random variables Ri and Rj is the average value of the product of the deviations of observations on two random variables from their sample means. It is calculated as follows:

$\rm Cov}\left(R_{i} ,R_{j} \right)=\sum\limits_{i=1}^{n}{\left(R_{i,t} -\bar{R}_{i} \right)\left(R_{j,t} -\bar{R}_{j} \right)\mathord{\left/ {\vphantom {\left(R_{i,t} -\bar{R}_{i} \right)\left(R_{j,t} -\bar{R}_{j} \right) \left(n-1\right)}} \right. \kern-\nulldelimiterspace} \left(n-1\right)}$

Unlike the population covariance, the sample covariance is based on historical data set. This reading focuses on covariance in a forward-looking sense.

Example

Continuing with our previous example, calculate the covariance of returns between A and B.

Solution:

Say R_i represents the return on A and R_j represents the return on B, we have already calculated the expected returns of A and B as 7.5% and 10% respectively. The covariance of returns is:

E [(Ri – 7.5) (Rj – 10)]

= 0.25(2% – 7.5%)(4% – 10%) + 0.5(8% – 7.5%)(10% – 10%) + 0.25(12% – 7.5%)(16% – 10%)

= 0.000825 + 0 + 0.000675 = 0.0015

Correlation

The problem with covariance is that it can vary from negative infinity to positive infinity which makes it difficult to interpret. To address this problem, we use another measure called correlation. Correlation is a standardized measure of the linear relationship between two variables with values ranging between -1 and +1.

A correlation of 0 (uncorrelated variables) indicates an absence of any linear (straight-line) relationship between the variables.
A correlation of +1 indicates a perfect positive relationship.
A correlation of -1 indicates a perfect negative relationship.

It is computed as:

ρ(R_i,R_j) = Cov (R_i,R_j) / [σ(R_i) σ(R_j)]

The above correlation measure is forward-looking as it is calculated by dividing the forward-looking covariance by the product of forward-looking standard deviations.

Historical or sample correlation = Historical or sample covariance between two variables / (Sample standard deviation of variable Ri * Sample standard deviation of variable Rj)

We will now apply this formula to calculate the correlation between the returns of A and B from our example. We have already shown that the covariance of returns is 0.0015. In order to calculate the correlation, we need the standard deviation of A and B. Using a financial calculator, we can determine that the standard deviation of A is 0.0357 and the standard deviation of B is 0.0424. The correlation,

$\rm{\ }\mathrm{\rho}\left(\mathrm{A,B}\right)\mathrm{=}\frac{\mathrm{Cov}\left(\mathrm{A,\ B}\right)}{\mathrm{\sigma}\left(\mathrm{A}\right)\mathrm{\sigma }\left(\mathrm{B}\right)}$ = $\mathrm{0.0015/(0.0357\ x\ 0.0424)}$ = 0.99$ .

The correlation of 0.99 (almost 1) implies a very strong positive relationship between the returns of A and B. This is more meaningful than the covariance number of 0.0015 which tells us that there is a positive relationship between the returns of A and B but does not give a sense for the strength of the relationship.

5. Covariance Given a Joint Probability Function

Given two random variables R_i and R_j, the covariance between R_i and R_j is given by:

Cov(R_i, R_j) = E[(R_i – ER_i) (R_j – ER_j)]

where:

ER_i = expected return for variable R_i

ER_j = expected return for variable R_j

Definition of Independence for Random Variables.

Two random variables X and Y are independent if and only if P(X,Y) = P(X)P(Y).

For example, given independence, P(5,6) = P(5)P(6). Joint probabilities are obtained by multiplying the individual probabilities. Independence is a stronger property than uncorrelatedness because correlation addresses only linear relationships.

The following condition holds for independent random variables and, therefore, also holds for uncorrelated random variables.

Multiplication Rule for Expected Value of the Product of Uncorrelated Random Variables.

The expected value of the product of uncorrelated random variables is the product of their expected values.

E(XY) = E(X)E(Y) if X and Y are uncorrelated.

Many financial variables, such as revenue (price times quantity), are the product of random quantities.

Variance of returns

Once we know the covariance, we can calculate the variance of a portfolio using this formula:

$\rm{\sigma }}^{\mathrm{2}}\left({\mathrm{R}}_{\mathrm{P}}\right)\mathrm{=\ }{\mathrm{w}}^{\mathrm{2}}_{\mathrm{1}}{\mathrm{\sigma }}^{\mathrm{2}}_{\mathrm{1}}\mathrm{\ }\left({\mathrm{R}}_{\mathrm{1}}\right)\mathrm{+\ }{\mathrm{w}}^{\mathrm{2}}_{\mathrm{2}}{\mathrm{\sigma }}^{\mathrm{2}}_{\mathrm{2}}\mathrm{\ \ }\left({\mathrm{R}}_{\mathrm{2}}\right)\mathrm{+\ 2}{\mathrm{w}}_{\mathrm{1}}{\mathrm{w}}_{\mathrm{2}}\mathrm{Cov\ }\left({\mathrm{R}}_{\mathrm{1}}{\mathrm{R}}_{\mathrm{2}}\right)$