Slide bài giảng quản lý danh mục đầu tư (the investment process)

Portfolio management is the act of building and maintaining an appropriate investment mix for a given risk tolerance.  The key factors for any portfolio management strategy involve asset allocation, diversification, and rebalancing rules.  Active portfolio management seeks to “beat the market” through identifying undervalued assets, often through shortterm trades and market timing.  Passive (indexed) portfolio management seeks to replicate the broader market while keeping costs and fees to a minimum.

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Lecturer: Dr LINH D NGUYEN

FACULTY OF FINANCE BANKING UNIVERSITY OF HCMC

Chapter 1:

OVERVIEW OF THE INVESTMENT PROCESS

CONTENT

1 Introduction

2 Portfolio Management Process

3 Individual Investor Life Cycle

TS Nguyễn Duy Linh

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1 INTRODUCTION

A What is a portfolio?

B What is portfolio management

C What is portfolio’s asset classes

 Non-publicly tradable securities like real estate, art, and privateinvestments can also be included in a portfolio

 Asset allocation, risk tolerance, andthe individual's time horizon are all

critical factors when assembling and

adjusting an investment portfolio

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B WHAT IS PORTFOLIO MANAGEMENT

 Portfolio management is the act of building and maintaining an

appropriate investment mix fora given risk tolerance

 The key factors for any portfolio management strategy involve

asset allocation, diversification, and rebalancing rules.

 Active portfolio management seeks to “beat the market”

through identifying undervalued assets, often through short-term

trades and market timing

 Passive (indexed) portfolio management seeks to replicatethe

broader market while keeping costs and fees to a minimum

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C WHAT IS PORTFOLIO’S ASSET CLASSES

 An asset class is a grouping of investments that exhibit similarcharacteristics and are subject to the same laws and regulations

 Equities (stocks), fixed income (bonds), cash and cashequivalents, real estate, commodities, futures, and other financialderivatives are examples of asset classes

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 There is usually very little correlation,and in some cases a negativecorrelation, between different assetclasses

 Financial advisers focus on asset class

as a way to help investors diversifytheir portfolio

2 THE PORTFOLIO MANAGEMENT PROCESS

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The unified presentation of portfolio management as a

process represented an important advance in the

investment management literature.

Portfolio management is a process – an integrated set of

activities that combine in a logical, orderly manner to

produce a desired product.

2 THE PORTFOLIO MANAGEMENT PROCESS

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A CFA INSTITUTE INVESTMENT MANAGEMENT PROCESS

The elements of the investment process

Planning: Establishing all the elements necessary for

decision making (data about clients/capital markets)

Execution: Details of optimal asset allocation and security

selection

Feedback: Adapting to changes in expectations and

objectives and changes in portfolio composition

CFA INVESTMENT MANAGEMENT PROCESS

OVERVIEW

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Specification and quantification of

Investor’s objectives, constraints,

Monitoring related input factors

investor-Portfolio construction and revision

Security selection Port Implementation Port optimization

Capital market expectations

Monitoring economic and market input factors

Attainment of investor objectives Performance measurement

Feedback Planning

Strategic asset allocation Determining

target asset class weights

A Identifying and specifying the investor’s objectives and constraints

B Creating the Investment Policy Statement

C Forming capital market expectations

D Creating the strategic asset allocation (target minimum and maximum class weights)

II Execution: Portfolio construction and revision

A Asset allocation (including tactical) and portfolio optimization (combining assets to meet risk and return objectives)

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INVESTMENT OBJECTIVES

 The investor’s objectives are his investment goals, expressed in

terms of both risk and returns  Investment managers must

assess the level of risk that investors can tolerate in pursuit of

higher returns (risk–return trade-off)

 Risk tolerance:

 A function of an individual’s psychological makeup

 Also affected by other factors, such as a person’s current insurance

coverage, cash reserves, family situation, and age

 Influenced by one’s current net worth and income expectations

 Return objectives

 May be stated in terms of an absolute or a relative percentage return or a

general goal, such as capital preservation, current income, capital

appreciation, or total return

The return needs to be no less than the rate of inflation

Is an appropriate objective for investors who want the portfolio

to grow in real terms over time to meet some future need

Under this strategy, growth mainly occurs through capital gains

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Investors want to generate income rather than capital gains

Retirees may favor this objective for part of their portfolio to

help generate spendable funds

Investors want the portfolio to grow over time to meet a future

Assume that he holds a steady job, is a valued employee, has adequate insurance coverage, and has enough money in the bank to provide a cash reserve

Assume that his current long-term, high-priority investment goal is to build a retirement fund

He can select a strategy carrying moderate to high amounts of risk because the income stream from his job will probably grow over time

Further, given young age and income growth potential, a total return or capital appreciationobjective is appropriate

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Here’s a possible objective statement:

Invest funds in a variety of moderate- to higher-risk investments

The average risk of the equity portfolio should exceed that of a

broad stock market index, such as the NYSE stock index Foreign

and domestic equity exposure should range from 80 percent to 95

percent of the total portfolio Remaining funds should be invested

in short- and intermediate-term notes and bonds.

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Assume that she has adequate insurance coverage and a cashreserve Let’s also assume she is retiring this year

Depending on her income from social security and a pensionplan, she may need some current income from her retirementportfolio to meet living expenses She also needs protectionagainst inflation

A risk-averse investor will choose a combination of current income and capital preservationstrategies

A more risk-tolerant investor will choose a combination of

current income and total returnin an attempt to have principalgrowthoutpaceinflation

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Here’s an example of such an objective statement:

Invest in stock and bond investments to meet income needs (from

bond income and stock dividends) and to provide for real growth

(from equities) Fixed-income securities should comprise 55–65

percent of the total portfolio; of this, 5–15 percent should be

invested in short-term securities for extra liquidity and safety The

remaining 35–45 percent of the portfolio should be invested in

high-quality stocks whose risk is similar to the S&P 500 index.

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SMART GOALS

HOW TO MAKE YOUR GOALS ACHIEVABLE

 SMART is an acronym that you can use to guide your goal setting To make sure your goals are clear and reachable, each one should be:

Specific (simple, sensible, significant).

Measurable (meaningful, motivating).

Achievable (agreed, attainable).

Relevant (reasonable, realistic and resourced, results-based).

Time bound (time-based, time limited, time/cost limited,

timely, time-sensitive)

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Legal and regulatory constraints

Unique needs and preferences

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INVESTMENT CONSTRAINTS

Investment planning is complicated by taxes that can seriously

become overwhelming if international investments are part of

the portfolio

Taxable income from interest, dividends, or rents is taxable at

the investor’s marginal tax rate

A note regarding taxes:

The impact of taxes on investment strategy and final results is

clearly very significant

Consult a tax accountant for advice regarding tax regulations

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INVESTMENT CONSTRAINTS

The investment process and the financial markets are

highly regulated and subject to numerous laws

Regulations can constrain the investment choices available

to someone in a fiduciary role

A fiduciary, or trustee, supervises an investment portfolio

of a third party, such as a trust account or discretionary

Unique Needs and Preferences

Covers the unique concerns of each investor

Because each investor is unique, the implications of this final constraint differ for each person

Each individual must decide on and then communicate these specific needs and preferences in their policy statement

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B THE NEED FOR

AN INVESTMENT POLICY STATEMENT

Components of an Investment Policy Statement (IPS)

1 Brief client description

2 Purpose of establishing policies and guidelines

3 Duties and investment responsibilities of parties involved

4 Statement of investment goals, objectives, and constraints

5 Schedule for review of investment performance and the investment

policy statement

6 Performance measures and benchmarks

7 Any considerations in developing strategic asset allocation

8 Investment strategies and investment styles

9 Guidelines for rebalancing

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THE NEED FOR

AN INVESTMENT POLICY STATEMENT

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Important reasons for constructing an IPS:

 It helps the investor decide on realistic investment goals after learning about the financial markets and the risks of investing

 It creates a standard by which to judge the performance

of the portfolio manager

 Protects the client against a portfolio manager’s inappropriate investments or unethical behavior

 The first step before beginning any investment program is to construct a comprehensive IPS

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C THE IMPORTANCE OF ASSET ALLOCATION

 Four decisions involved in constructing an investment strategy:

What asset classes should be considered for investment?

What policy weights should be assigned to each eligible asset

The asset allocation decision involves the first three points

How important is the asset allocation decision to an

investor? In a word, VERY

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THE IMPORTANCE OF ASSET ALLOCATION

3 INDIVIDUAL INVESTOR LIFE CYCLE

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Accumulation phase

Early to middle years of working career

Long investment time horizon and future earning ability

Individuals typically willing to make relatively high-risk investments

in the hopes of making above-average nominal returns over time

Consolidation phase

Past midpoint of careers

Earnings greater than expenses

Typical investment horizon for this phase is still long (20 to 30

years), so moderately high-risk investmentsare attractive

Individuals in this phase are concerned about capital preservation and

do not want to take abnormally high risks

OVERVIEW

38

 Spending phase

Begins after retirement

Living expenses are covered by social security income and income from prior investments, including employer pension plans

The overall portfolio may be less risky than in the consolidation phase, but investors still need some risky growth investments, such as common stocks, for inflation protection

 Gifting phase

May be concurrent with the spending phase

Excess assets can be used to provide financial assistance to relatives

or to establish charitable trusts as an estate planning tool to minimize estate taxes

B BENEFITS OF INVESTING EARLY

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= × 1 +

= × 1 + − 1 / Giá trị tương lai của một số tiền

Giá trị tương lai của dòng tiền cuối kỳ

Giá trị hiện tại của dòng tiền cuối kỳ = × 1− 1

“no pain, no gain” “no risk, no reward.”

 How you feel about risking your money will drive many ofyour investment decisions

 The risk-comfort scale extends from very conservative (you

don’t want to risk losing a penny regardless of how little your

money earns) to very aggressive (you’re willing to risk much

of your money for the possibility that it will growtremendously)

 As you might guess, most investors’ tolerance for risk fallssomewhere in between

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C HOW MUCH RISK IS RIGHT FOR YOU, OR

WHAT IS YOUR LEVEL OF RISK TOLERANCE ?

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1 You win $300 in an office

football pool You:

(a) spend it on groceries,

(b) purchase lottery tickets

(c) put it in a money market

account,

(d) buy some stock

2 Two weeks after buying 100 shares of a $20 stock, the price jumps to over $30 You decide to:

(a) Buy more stock; it’s obviously a winner, (b) Sell it and take your profits(c) Sell half to recoup some costs and hold the rest, (d) Sit tight and wait for it to advance even more

HOW MUCH RISK IS RIGHT FOR YOU?

4 You’re planning a vacation trip and can either lock in a fixed room-and-meals rate of $150 per day or book standby and pay anywhere from $100

to $300 per day You:

(a) take the fixed-rate deal(b) talk to people who have been there about the availability of last-minute accommodations, (c) book standby and also arrange vacation insurance because you’re leery of the tour operator,

(d) take your chances with standby

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5 The owner of your apartment building is converting the

units to condominiums You can buy your unit for $75,000

or an option on a unit for $15,000 (Units have recently sold

for close to $100,000, and prices seem to be going up.) For

financing, you’ll have to borrow the down payment and pay

mortgage and condo fees higher than your present rent You:

(a) buy your unit,

(b) buy your unit and look for another to buy,

(c) sell the option and arrange to rent the unit yourself,

(d) sell the option and move out because you think the

conversion will attract couples with small children.

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 6) You have been working three years for a rapidly growing company As an executive, you are offered the option of buying up to 2% of company stock: 2,000 shares at $10 a share Although the company is privately owned (its stock does not trade on the open market), its majority owner has made handsome profits selling three other businesses and intends to sell this one eventually You:

 (a) purchase all the shares you can and tell the owner you would invest more if allowed,

 (b) purchase all the shares,

 (c) purchase half the shares,

 (d) purchase a small amount of shares

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7 You go to a casino for the

first time You choose to

(a) read restaurant reviews in the local newspaper,

(b) ask coworkers if they know of a suitable place,

(c) call the only other person you know in this city, who eats out a lot but only recently moved there(d) visit the city sometime before your dinner to check out the restaurants yourself

10 Your attitude toward money is best described as:

(a) A dollar saved is a dollar earned,

(b) You’ve got to spend money to make money,

(c) Cash and carry only, (d) Whenever possible, use other people’s money

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What your total score indicates:

10–17: You’re not willing to take chances with your money,

even though it means you can’t make big gains

18–25: You’re semi-conservative, willing to take a small chance with enough information

26–32: You’re semi-aggressive, willing to take chances if you think the odds of earning more are in your favor

33–40: You’re aggressive, looking for every opportunity to make your money grow, even though in some cases the odds may be quite long You view money as a tool to make more money

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25 and 28 points: an aggressive investor

20 and 24 points: your risk tolerance is above average

15 and 19 points: a moderate investor This means you are willing to accept some risk in exchange for a potential higher rate of return

< 15 points: a conservative investor

< 10 points: a very conservative investor.

D INITIAL RISK AND INVESTMENT GOAL CATEGORIES

AND ASSET ALLOCATIONS

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INITIAL RISK AND INVESTMENT GOAL CATEGORIES

AND ASSET ALLOCATIONS

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Lecturer: Dr NGUYỄN DUY LINH

FACULTY OF FINANCE BANKING UNIVERSITY OF HCMC

Chapter 2:

THE MARKOWITZ PORTFOLIO THEORY

CONTENT

 I Measures of Return and Risk

 II Diversification and Portfolio Risk

 III The Markowitz Portfolio Selection Model

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I MEASURES OF RETURN AND RISK

1 Understanding Interest Rates

2 Measures of Return and Risk

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a) Real versus Nominal Interest Rates

b) Taxes and the Real Interest Rate

c) Comparing Rates of Return for Different Holding Periods

d) Effective Annual Rate (EAR) and Annual Percentage Rate (APR)

e) Bills and Inflation, 1926-2015

1 UNDERSTANDING INTEREST RATES

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 Nominalinterest rate: Growth rate of your money

 Realinterest rate: Growth rate of your purchasing power

A REAL VERSUS NOMINAL INTEREST RATES

= Nominal Interest Rate

= Real Interest Rate

If E(i) denotes current expectations

of inflation, the Fisher Equation: = +

 Tax liabilities are based on nominal income, so the real after-tax rate is:

 The after-tax real rate falls as the inflation rises

B TAXES AND THE REAL INTEREST RATE

= Nominal Interest Rate = Real Interest Rate = Inflation Rate = Tax Rate

C COMPARING RATES OF RETURN FOR

DIFFERENT HOLDING PERIODS

 Zero Coupon Bond:

 Suppose prices of zero-coupon Treasuries with $100 face value

and various maturities are as follows:

 Effective Annual Rate (EAR): the percentage increase in funds

invested over a 1-year horizon

 The relationship between EAR and the total return:

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Horizon, T Price, P(T ) Total Return for

Given Horizon EAR

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D EFFECTIVE ANNUAL RATE (EAR) AND

ANNUAL PERCENTAGE RATE (APR)

 Annualized rates on short-term investments (by convention, T < 1

year) often are reported using simplerather than compound

interest These are called annual percentage rates (APRs).

 If there are n compounding periods in a year, and the per-period

rate is r f (T), then:

APR = n × r f (T)

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Given Horizon APR

interest These are called annual percentage rates (APRs).

 If there are n compounding periods in a year, and the per-period rate is r f (T), then

APR = n × r f (T) = r f (T) × (1/T)

 Why is there a difference between APR ( 5.42) and EAR (5.49)?

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Given Horizon APR

EFFECTIVE ANNUAL RATE (EAR) AND

ANNUAL PERCENTAGE RATE (APR)

 The relationship among the compounding period, the EAR, and

(a) a monthly rate of 1%;

(b) an annually, continuously compounded rate, r cc, of 12%

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E BILLS AND INFLATION, 1926-2015

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E BILLS AND INFLATION, 1926-2015

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T-Bills Inflation Real T-Bill T-Bills Inflation Real T-Bill

compensate investors for increases in i.

Statistics for T-bill rates, inflation rates, and real rates, 1926–2015

2 MEASURES OF RETURN AND RISK

A Return and Risk of Individual Investment

B Return and Risk of a Portfolio

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A RETURN AND RISK OF INDIVIDUAL INVESTMENT

Measuring Historical Rates of Return

Computing Mean Historical Return

Measuring Risk of Historical Return

Calculating Expected Rates of Return

Measuring Risk of Expected Rate of Return

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A RETURN AND RISK FOR INDIVIDUAL INVESTMENT

MEASURING HISTORICAL RATES OF RETURN

Holding Period Return (HPR)

P0= Beginning price

P1= Ending price

D1= Dividend during period one

after being held for two years Calculate HPR and EAR.

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COMPUTING MEAN HISTORICAL RETURN

Arithmetic Mean Return (AM)

where  HPR = the product of all the annual HPRs

Arithmetic Mean vs Geometric Mean Return

GM is considered a superior measure of the

long-term mean rate of return because:

GM indicates the compound annual rate of return based on

the ending value of the investment versus its beginning

GM is considered a superior measure of the term mean rate of return because:

long-AM is biased upward if you are attempting to measure an asset’s long-term performance, especially for for a volatile security For example:

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 Both AM and GM are used because:

AM is best used as an expected value for an individual

year

GM is the best measure of long-term performance since it

measures the compound annual rate of return for the asset

being measured

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MEASURING RISK OF HISTORICAL RETURN

Variance: A measure of variability equal to the sum of the

squares of a return’s deviation from the mean, divided by the total number of returns

Standard deviation: A measure of variability equal to the

square root of the variance

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where:

σ2 : the variance of the series

HPR it : the holding period return during period t of asset i

HPR : the arithmetic mean of the series (AM) of asset i

n : the number of observations

= ∑ HPR − HPR

=

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following annual rates of return Compute variance and standard deviation of this stock.

of the investment

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A Relative Measure of Risk: What investment is better?

Regarding AM, investment B appears to be

Regarding σ, investment B appears to be _

Regarding CV, investment B has relative variability or

lower risk per unit of expected return

Coefficient of Variation (CV) =Standard Deviation of Return

Investment A Investment B

Arithmetic return 0.07 0.12

Standard deviation 0.05 0.07

MEASURING RISK OF HISTORICAL RETURN: EXAMPLE

a Compute the arithmetic mean annual rate of return for each stock

Which stock is most desirable by this measure?

b Compute the standard deviation of the annual rate of return for each stock By this measure, which is the preferable stock?

c Compute the coefficient of variation for each stock By this relative measure of risk, which stock is preferable?

d Compute the geometric mean rate of return for each stock

CALCULATING EXPECTED RATE OF RETURN

The expected return from an investment:

Equal to the sum of the potentialreturns multiplied with the corresponding probabilityof the returns

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1Expected Return (Probability of Return) (Possible Return)

n

i

Economic Conditions Probability Rate of Return

Strong economy, no inflation 0.15 0.20 Weak economy, above-average inflation 0.15 −0.20

No major change in economy 0.70 0.10

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MEASURING RISK OF EXPECTED RATE OF RETURN

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Variance

The larger the variance for an expected rate of return, the

greater the dispersion of expected returns and the greater the

uncertainty, or risk, of the investment

Ex: Calculate variance for the last example

2 2

MEASURING RISK OF EXPECTED RATE OF RETURN

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    2

1Standard Deviation

MEASURING RISK OF EXPECTED RATE OF RETURN: EXAMPLE

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Calculating variance of the following series of HPR:

0.08 (p=35%), 0.10 (30%), 0.12(20%), 0.14(15%)

B RETURN AND RISK FOR A PORTFOLIO

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Measuring historical rate of return

Measuring expected rate of return

Measuring risk of a portfolio

Covariance of Returns

Correlation coefficient (correlation)

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MEASURING HISTORICAL RATE OF RETURN (HPR)

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The mean historical rate of return (HPR) for a portfolio of

investments is measured as:

the weighted average of the HPRs for the individual

investments in the portfolio, or

the overall percent change in value of the original portfolio

Stock Number of

Shares

Beginning Price Beginning Market Value Ending Price Ending Market

Market Weight Weighted HPR

A 100,000 $10 1,000,000 12 1,200,000

B 200,000 $20 4,000,000 21

C 500,000 $30 33 16,500,000

Total

MEASURING EXPECTED RATE OF RETURN (HPR)

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 The mean expected rate of return (HPR) for a portfolio of investments equals to the weighted average of the expected rates of return for the individual investments in the portfolio

=

 wi= weight of an individual asset in the portfolio, or the percent of the portfolio in

Expected Portfolio Return (w i × R i )

MEASURING RISK OF A PORTFOLIO

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In statistics, there are two basic concepts that must be

understood before we discuss the formula for the

variance of the rate of return for a portfolio.

 The covariance of a variable with itself:

 Note: when applying to sample data, we divide the values by

(n – 1) rather than by n to avoid statistical bias.

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Barclays Capital U.S Aggregate Bond Index (Rj)

Ri - E(Ri) Rj - E(Rj) [Ri - E(Ri)] x

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

S&P 500 Total Return Stock Market Index (Ri) Barclays Capital U.S Aggregate Bond Index (Rj)

CORRELATION

The correlation coefficient is obtained by standardizing

(dividing) the covariance by the product of the individual

Note: when sample data is used, σis divided by (n – 1) to

avoid statistical bias

CORRELATION

 The coefficient can vary only in the range +1 to −1

 A value of +1 would indicate perfectly positive correlation

This means that returns for the two assets move together in a positively and completely linear manner

 A value of −1 would indicate perfectly negative correlation

This means that the returns for two assets move together in a completely linear manner, but in opposite directions

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TS Nguyễn Duy Linh 61

σ E(R)

Case 1: Perfectly positively correlated

Case 2: Perfectly negatively correlated

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where:

σ port = standard deviation of the portfolio

w i = weights of an individual asset in the portfolio; where weights are

determined by the proportion of value in the portfolio

σ 2

i = variance of rates of return for Asset i

Cov ij = covariance between the rates of return for Assets i and j

Cov ij = ρijσiσj

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The two-asset portfolio:

The three-asset portfolio:

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APPENDIX: COMPUTATION OF PORTFOLIO

VARIANCE FROM THE COVARIANCE MATRIX

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Note: Cov(rD, rE) = Cov(rE, rD).

MEASURING RISK OF A PORTFOLIO - EXAMPLE

II DIVERSIFICATION AND PORTFOLIO RISK

1 Portfolio Risk and Diversification

2 Portfolios of Two Risky Assets

3 Portfolios of Three Assets

4 Estimation Issues

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1 PORTFOLIO RISK AND DIVERSIFICATION

Market risk (comes from conditions in the general economy)

Marketwide risk sources: come from conditions in the general economy, such as the business cycle, inflation, interest rates, and exchange rates

Remains even after diversification

Also called: Systematic or Non-diversifiable

Firm-specific risk

Risk that can be eliminated by diversification

Also Called: Diversifiable or Nonsystematic

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PORTFOLIO RISK AND DIVERSIFICATION

Number of Stocks in Portfolio

Expected Standard Deviation of Annual Portfolio Returns

2 PORTFOLIOS OF TWO RISKY ASSETS

Any asset or portfolio of assets can be described by

two characteristics:

The expected rate of return

The standard deviation of returns

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2 PORTFOLIOS OF TWO RISKY ASSETS

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PORTFOLIOS OF TWO RISKY ASSETS

A EQUAL RISK AND RETURN—CHANGING CORRELATIONS

E(R1) = 0.20, E(σ1) = 0.10, w1= 0.50E(R2) = 0.20, E(σ2) = 0.10, w2= 0.50

Stock A Stock C Portfolio AC

σ E(R)

Case 1: Perfectly positively correlated

Case 2: Perfectly negatively correlated

EQUAL RISK AND RETURN—CHANGING CORRELATIONS

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0.20

0.10 0.05

ρ = 1

ρ = -1

σ

E(R)

Minimum variance portfolio

Summary:

 The expected return of the portfolio does not change because it is simply the weighted average of the individual expected returns

 Demonstrates the concept of diversification, whereby the risk of the

portfolio is lower than the risk of either of the assets held in the portfolio

 Risk reduction benefit occurs to some degree any time the assets combined in a portfolio are not perfectly positively correlated (that

is, whenever ρ i,j< +1)

 Diversification works because there will be investment periods when

a negative return to one asset will be offset by a positive return to the other, thereby reducing the variability of the overall portfolio return

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When ρ12equals -1.00:

The negative covariance term exactly offsets the individual

variance terms, leaving an overall standard deviation of the

portfolio of zero

This would be a risk-free portfolio, meaning that the average

combined return for the two securities over time would be a

constant value (that is, have no variability)

Thus, a pair of completely negatively correlated assets provides

the maximum benefits of diversification by completely

eliminating variability from the portfolio

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 The minimum variance portfolio is the portfolio composed

of risky assets with smallest standard deviation

 Risk reduction depends on the correlation:

If r = +1.0, no risk reduction is possible

If r = 0, σPmay be less than the standard deviation of either component asset

If r = -1.0, a riskless hedge is possible

82

B COMBINING STOCKS WITH DIFFERENT RETURNS AND RISK

E(R1) = 0.10, E(σ1) = 0.07, w1= 0.50E(R2) = 0.20, E(σ2) = 0.10, w2= 0.50

COMBINING STOCKS WITH DIFFERENT RETURNS AND RISK

85

0.20

0.10 0.05

0.10 0.15

1

2

σ E(R)

0.07 Minimum variance portfolio

Trang 28

COMBINING STOCKS WITH DIFFERENT RETURNS AND RISK

The expected return of the portfolio does not change

because it is simply the weighted average of the

individual expected returns (the investment weights are

always equal at 0.50 each)

With perfect negative correlation, the portfolio standard

deviation is not zero This is because the different

examples have equal weights, but the asset standard

deviations are not equal

86

C CONSTANT CORRELATION WITH CHANGING WEIGHTS

E(R1) = 0.10, E(σ1) = 0.07, E(R2) = 0.20, E(σ2) = 0.10

CONSTANT CORRELATION WITH CHANGING WEIGHTS

If the weights of the two assets are changed while

holding the correlation coefficient constant, a set of

combinations is derived that trace an ellipse.

We call ellipse the portfolio opportunity set because it

shows all combinations of portfolio expected return and

standard deviation that can be constructed from the two

available assets.

The benefits of diversification are critically dependent

on the correlation between assets

89

90

0.20

0.10 0.05

0.10 0.15

σ

E(R)

k j

Portfolio opportunity set

Minimum variance portfolio

Trang 29

3 A THREE-ASSET PORTFOLIO

The results presented earlier for the two-asset portfolio

can be extended to a portfolio of n assets

As more assets are added to the portfolio, more risk will

be reduced (everything else being the same)

The general computing procedure is still the same, but

the amount of computation has increase rapidly

For the three-asset portfolio, the computation has

doubled in comparison with the two-asset portfolio

Among entire set of assets

With n assets, (n2-n)/2 correlation estimates

Estimation risk refers to potential errors

95

III THE MARKOWITZ PORTFOLIO SELECTION MODEL

1. Assumptions

2. Efficient Frontier

3. Risk Aversion and Utility Function

4. Portfolios: Risky Asset and Risk-free Asset

5. Optimal Complete Portfolio

6. Optimal Risky Portfolio

96

Trang 30

1 ASSUMPTIONS

Investors want to maximize the returns from the total set

of investments for a given level of risk

The portfolio should include all of assets and liabilities,

not only marketable securities but also car, house, and less

marketable investments such as coins, stamps, art,

antiques, and furniture

Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth

Investors estimate the risk of the portfolio on the basis of the variability of potential returns

98

1 ASSUMPTIONS

Markowitz’s assumptions (2/2)

Investors base decisions solely on expected return and risk, so

their utility curves are a function of expected return and the

variance (or standard deviation) of returns only

For a given risk level, investors prefer higher returns to lower

returns Similarly, for a given level of expected return, investors

prefer less risk to more risk

Using these assumptions, a single asset or portfolio of assets is

considered to be efficient if no other asset or portfolio of assets

offers higher expected return with the same (or lower) risk or

lower risk with the same (or higher) expected return

99

2 THE EFFICIENT FRONTIER

100

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A THE MINIMUM-VARIANCE FRONTIER

101

 If we examined different two-asset

combinations and derived the curves

assuming all the possible weights,

we would have the beside exhibit

 The envelope curve that contains

the best (lowest possible variance)

of all these possible combinations is

referred to as the

minimum-variance frontierof risky assets

 This frontier is a graph of the lowest

possible variance that can be

attained for a given portfolio

expected return

Numerous Portfolio Combinations of Available Assets

E(R)

Standard Deviation of Return (σ)

THE MINIMUM-VARIANCE FRONTIER

 Minimum-variance portfolio: the portfolio

composed of risky assets with smallest standard deviation It has a standard deviation smaller than that

of either of the individual component assets

This illustrates the effect

of diversification

102

Individual Assets Efficient Frontier

Variance Frontier

Minimum-The minimum-variance frontier of

risky assets

Global Minimum- Variance Portfolio

B THE (MARKOWITZ) EFFICIENT FRONTIER

104

 The efficient frontier represents the set of portfolios that has the

maximumrate of return for every givenlevel of risk or the

minimumrisk for every level of return

 Portfolio A dominates

Portfolio C because it has an

equal rate of return but

substantially less risk.

 Portfolio B dominatesC

because it has equal risk but

a higher expected rate of

return

Markowitz Efficient Frontier

3 RISK AVERSION AND UTILITY FUNCTION

Trang 32

A RISK AVERSION

consumers and investors), who, when exposed to

uncertainty, attempt to lower that uncertainty.

asset with the lower level of risk given a choice between

two assets with equal rates of return.

106

B ESTIMATING RISK AVERSION

Increases with expected return

Decreases with risk

Each portfolio receives a utility score to assess the

investor’s risk/return trade off

 Utility function: is an important concept that measures

preferences over a set of goods and services Utility represents the satisfaction that consumers receive for choosing and consuming a product or service.

U = Utility

E(r) = Expected return on the asset or portfolio

A = Coefficient of risk aversion

Trang 33

UTILITY FUNCTION

 Utility scores of portfolios with varying degrees of risk aversion

110

Investor Risk

Aversion (A)

Utility Score of Portfolio L [E(r) = 0.07; σ = 0.05]

Utility Score of Portfolio M [E(r) = 0.09; σ = 0.10]

Utility Score of Portfolio H [E(r) = 0.13; σ = 0.20]

We can interpret the utility score of risky portfolios as a

certainty equivalent rate of return The certainty

equivalent is the rate that a risk-free investment would need to offer to provide the same utility score as the risky portfolio

σP

Q

P

Indifference Curve

Equally preferred portfolios will lie in the mean–standard deviation plane on an

indifference curve,

which connects all portfolio points with the same utility valueThe indifference Curve

U1 U2

Expected Return, E(r)

Standard Deviation, σ Utility = E(r) - ½ Aσ2

Trang 34

4 PORTFOLIOS: RISKY ASSET AND RISK-FREE ASSET

116

A Capital allocation line (CAL)

B Capital allocation line with leverage

A CAPITAL ALLOCATION LINE (CAL)

117

It’s possible to create a complete portfolio by splitting investment funds between safe assets (e.g.

Treasury bills) and risky (e.g stocks, bonds) assets

A complete portfolio consists of a risky portfolio and risk-free asset Let:

y = Portion allocated to the risky portfolio, P

(1 - y) = Portion to be invested in risk-free asset, F

CAPITAL ALLOCATION LINE (CAL)

118

The expected return on the complete portfolio:

The risk of the complete portfolio:

Trang 35

120 E(rc)

σcE(rp) = 15%

σP= 22%

F

P

Capital Allocation Line (CAL)

The investment opportunity set with a risky asset and

a risk-free asset in the expected return–standard deviation plane

121

 Capital allocation line (CAL) depicts all the risk– return

combinations available to investors

 The slope of the CAL, denoted by S, equals the increase in the expected return of the complete portfolio per unit of additional standard deviation – in other words, incremental return per

incremental risk The slope, the reward-to-volatility ratio, is

usually called the Sharpe ratio

= + ×

122

 CAL is the primary result of the capital market theory: Investors

who allocate their money between a riskless security and the risky

portfolio P can expect a return equal to the risk-free rate plus

compensation for the number of risk units (σ c ) they accept.

 When the risky portfolio P is the market portfolio (M) – contains

all risky assets held anywhere in the marketplaceand receives the

highest level of expected return (in excess of the risk-free rate)

per unit of risk for any available portfolio of risky assets, CAL

becomes thecapital market line (CML)

123 E(rc)

σcE(rp) = 15%

σP= 22%

F

P

The investment opportunity set with a risky asset and

a risk-free asset in the expected return–standard deviation plane

rf= 7%

= The reward−to−volatility ratio

Trang 36

B CAPITAL ALLOCATION LINE WITH LEVERAGE

124

 What about points on the CAL to the right of portfolio P? If

investors can borrow at the (risk-free) rate of rf = 7%, they can

construct portfolios that may be plotted on the CAL to the right of

P

borrows an additional $120,000 at the rate of 7%, investing the

total available funds in the risky asset Compute the Sharpe ratio

CONCEPT CHECK

 Can the Sharpe (reward-to-volatility) ratio, S = [E(r C ) − r f ]/σ C , of any combination of the risky asset and the risk-free asset be different from the ratio for the risky asset taken alone, [E(r P ) −

r f ]/σ P , which, in this case, is 0.36?

 In the expected return–standard deviation plane all portfolios that are constructed from the same risky and risk-free funds (with various proportions) lie on a line from the risk-free rate through the risky fund

 The slope of the CAL is the same everywhere; hence the to-volatility (Sharpe) ratio is the same for all of these portfolios

reward-TS Nguyễn Duy Linh

126

CONCEPT CHECK

 Formally, if you invest a proportion, y, in a risky fund with

expected return E(rP) and standard deviation σP, and the

remainder, 1 − y, in a risk-free asset with a sure rate rf, then the

portfolio’s expected return and standard deviation are:

 and therefore the Sharpe ratio of this portfolio is:

 which is independent of the proportion y

127

CAPITAL ALLOCATION LINE WITH LEVERAGE

128

 However, nongovernment investors cannotat the risk-free rate!

 Lend at r f = 7% and borrow at = 9%

Lending range slope

Borrowing range slope =

Trang 37

CAPITAL ALLOCATION LINE WITH LEVERAGE

129 E(rc)

σcE(rp) = 15%

σP= 22%

F

P

The opportunity set with differential borrowing and lending rates

rf= 7%

= 9%

> 1 = 0.27

< 1 = 0.36

5 THE OPTIMAL COMPLETE PORTFOLIO

(Intentionally blank slide)

130

THE OPTIMAL COMPLETE PORTFOLIO

131

The investor must choose one optimal complete

portfolio, C, from the set of feasible choices

Expected return of the complete portfolio:

Variance:

Investors choose the allocation to the risky portfolio, y,

that maximizes their utility function

= ×

= − 1

2

132

Utility levels for various positions in risky assets (y) for an investor with risk aversion A = 4

0.0 0.070 0.000 0.070 0.1 0.078 0.022 0.077 0.2 0.086 0.044 0.082 0.3 0.094 0.066 0.085 0.4 0.102 0.088 0.5 0.110 0.110 0.086 0.6 0.118 0.132 0.083 0.7 0.126 0.154 0.8 0.134 0.176 0.072 0.9 0.142 0.198 0.064 1.0 0.150 0.220 0.053

Utility as a function of allocation

to the risky asset, y

Trang 38

133

E(rp) 15%

σ p 22%

Utility levels for various positions in

risky assets (y) for an investor with risk

aversion A = 4

Utility as a function of allocation

to the risky asset, y

A 





CONCEPT CHECK

 Suppose that expectations about the U.S equity market and the

T-bill rate are the same as they were in 2016, but you find that a

greater proportion is invested in T-bills today than in 2016 What

can you conclude about the change in risk tolerance over the years

since 2016?

 If all the investment parameters remain unchanged, the only reason

for an investor to decrease the investment proportion in the risky

asset isan increase in the degree of risk aversion If you think that

this is unlikely, then you have to reconsider your faith in your

assumptions Perhaps the U.S equity market is not a good proxy for

the optimal risky portfolio Perhaps investors expect a higher real

rate on T-bills

135

 Utility value (U) =

 CAL is the tangent of this indifference curve

Trang 39

138 E(r)

σ E(rp) = 15%

which maximizes utility value of the investor, given the risk aversion A, P and F

How to get the optimal

risky portfolio P?

6 OPTIMAL RISKY PORTFOLIO

139

140

0.20

0.10 0.05

Portfolio opportunity set

E(RD) = 0.10, E(σD) = 0.07, E(RE) = 0.20, E(σE) = 0.10

FIND THE MINIMUM-VARIANCE RISKY PORTFOLIO

 To solvethe variance minimization problem:

 To find min, take derivative w.r.t to wDand set equal to 0:

Trang 40

EXAMPLE

 E(RD) = 0.10, E(σD) = 0.07, E(RE) = 0.20, E(σE) = 0.10, ρ = 0.00

Calculate the characteristics of minimum-variance portfolio

143

EXAMPLE

 E(RD) = 0.10, E(σD) = 0.07, E(RE) = 0.20, E(σE) = 0.10, ρ = -1

Calculate the characteristics of minimum-variance portfolio

145

B FIND THE OPTIMAL RISKY PORTFOLIO

 When choosing their capital allocation between risky and

risk-free portfolios, investors naturally will want to work with the

risky portfolio that offers the highest reward-to-volatility or

Sharpe ratio.

The higher the Sharpe ratio, the greater the expected return

corresponding to any level of volatility

 Another way to put this is that the best risky portfolio is the one

that results in the steepest capital allocation line (CAL).

Steeper CALs provide higher excess returns for any level of

Tiêu đề	Overview of the Investment Process
Tác giả	Nguyen Duy Linh
Người hướng dẫn	TS. Nguyễn Duy Linh
Trường học	University of Ho Chi Minh City
Chuyên ngành	Finance
Thể loại	Lecture notes
Năm xuất bản	2022
Thành phố	Ho Chi Minh City

Định dạng
Số trang	117
Dung lượng	5,54 MB