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Module 1The investment settingand Modern

portfolio Theory

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Portfolio Management

- Purpose maximization of wealth by reaching a

heuristic Reward-to-risk - How? Allocate, Select and Protect
- Illustration realized and expected wealth?
- Realized wealth Expected wealth Error
- Heuristic Reward to risk Allocation

Selection protection - It always starts with the Policy
- Ask the right question!? what risk? ?Thus, what

allocation? - Set the right allocation target in terms of

objectives, constraints and weight range

monitoring

Choose a Portfolio strategy Passive or Active

Asset allocation Security Selection

Active (for pros) Market timing Stock/Bond picking

Passive (for ind.) Fixed weights Indexing

- No matter what, an investment strategy is based

on four decisions - What asset classes to consider for investment
- What normal or policy weights to assign to each

eligible class - The allowable allocation ranges based on policy

weights - What specific securities to purchase for the

portfolio - Most (85 to 95) of the overall investment

return is due to the first two decisions, not the

selection of individual investments

First, set the rules the policy statement

- TOTAL RETURN INCOME YIELD CAPITAL GAIN YIELD
- Objectives Think in terms of risk and return to

find the best weightsi.e., - Capital preservation (high income, low capital

gain)? Low to moderate risk - Balanced return (Balanced capital gains and

income reinvestment)?moderate to high risk - Pure Capital appreciation (high capital gains,

low to no income)?High risk - Constraints - liquidity, time horizon, tax

factors, legal and regulatory constraints, and

unique needs and preferences - Management - Define an allowable allocation

ranges based on policy weights - Selection - Define guideline to pick securities

to purchase for the portfolio (optional)

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Objectives ?Age/Risk Matrix

Risk tolerance/ Time Horizon 0-5years (C/B/S) 6-10 (C/B/S) 11 (C/B/S)

Higher 10/30/60 0/20/80 0/0/100

Moderate 20/40/40 10/40/50 10/30/60

Lower 50/40/10 30/40/30 10/50/40

- C stands for CASHi.e. money market securities
- B stands for Bondsi.e. corporate, municipal or

treasury securities - S stands for Stocksi.e. value, growth,

international equity securities - Color code
- Capital preservation
- Balanced return
- Capital appreciation
- Where do you fit?
- ?Retirement simulation comes next

In Sum (1),

- Active strategy will likely outperform a passive

strategy - Asset allocation is more powerful than stock

selection - Everything starts with the policy statementi.e.,

(1) types and weights of assets to include in

portfolio and (2) guidelines on how to manage

the allocation and selection process. - Risk of a strategy depends on the investors

goals and time horizon - Over long time periods sizable allocation to

equity will improve results - Over short time periods sizable allocation to

fixed income securities will shield against

uncertainty

Example Case

- Mr. Bob is 70 years of age, is in excellent

health pursues a simple but active lifestyle, and

has no children. He has interest in a private

company for 90 million and has decided that a

medical research foundation will receive half the

proceeds now it will also be the primary

beneficiary of his estate upon his death. Mr. Bob

is committed to the foundation s well-being

because he believes strongly that , through it, a

cure will be found for the disease that killed

his wife. He now realizes that an appropriate

investment policy and asset allocation are

required if his goals are to be met through

investment of his considerable assets. Currently

the following assets are available for building

an appropriate portfolio - 45 million Cash (from the sale of the private

company interest, net of 45 million gift to the

foundation) - 10 million stocks and bonds (5 million each)
- 9 million warehouse property not fully leased)
- 1 Million Bob residence
- Build a policy statement for Mr. Bob!

Objectives (return)

- Large liquid wealth from selling interest in the

private company - Income from leasing warehouse
- Not burdened by large or specific needs for

current income nor liquidity. - He has enough spendable income.
- He will leave his estate to a Tax-exempted

foundation - He has already offered a large gift to the

foundation - Thus, an inflation-adjusted enhancement of the

capital base for the benefit of the foundation

will the primary minimum return goal. - He is in the highest tax bracket (not mentioned

but apparent) - Tax minimization should be a collateral goal.

Objectives (risk)

- Unmarried, Childless, 70 years old but in good

health - ? Still a long actuarial life (10), thus long

term return goal. - Likely free of debt (not mentioned, but neither

the opposite) - Not skilled in the management of a large

portfolio - Yet, not a complete novice since he owned stocks

and bonds prior to his wifes death. - His heirthe foundationhas already received a

large asset base. - ?Long term return goal with a portfolio bearing

above average risk.

Constraints

- Time--Two things (1) long actuarial life and (2)

beneficiary of his estatethe foundation has a

virtually perpetual life - Taxes highest tax brackets, investment should

take this into consideration tax-sheltered

investments. - Unique circumstances Large asset base, a

foundation as a unique recipient? some freedom in

the building of the portfolio

Adapted Strategy

- Majority in stocks (shield against inflation,

above average risk tolerance, and no real income

or liquidity needs) - He already has 15 in real estate (house

warehouse)? no more needed, diversification

effect achieved. - Additional freedom Non-US stocks? additional

diversification - ? Target 75 equity (including Real Estate)
- Fixed Income used to minimize income taxesi.e.,

municipal and treasury securities. No need to

look for YIELD nor downgrade the quality of the

issues used. - Additional freedom Non-US fixed-income?

additional diversification effect. - ? Target 25 in fixed income

Proposed Allocation

Current Proposed Range

Cash / Money Market 70 0 0-5

US Stocks--LC 30 30-40

US StocksSC 15 15-25

Non US Stocks 15 15-25

Total 7.5? 60 60-80

Real Estate 15 15 10-15

US Fixed Income 15 10-20

Non-US Fixed Income 10 5-15

Total Fixed Income 7.5? 25 15-35

What is Investments?

- Purpose maximization of portfolio wealth through

adequate Portfolio management - Fair Reward-to-risk? Ask the right question!
- Optimal portfolio management Allocation

Selection Risk protection

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Historical Return

- The additional cents on the dollar invested
- R(profitadditional cash flows)/initial

investment - Over a period of timeaverage return
- Average returnS(all returns)/nb of observations
- Why do returns matter?
- does not mean muchalone
- Cross-comparison between markets
- Are normally distributed

Historical Risk

- We need to think in terms of estimates in an

uncertain world - Estimateaverage return /- some volatility
- Uncertainty or volatility of returns
- Standard deviation of returns
- Measured in
- What does it mean?

Example Historical Returns and Risks

- Computation of Monthly Rates of Return

Variance (Standard Deviation) of Expected Returns

for an Individual Investment

Standard deviation is the square root of the

variance Variance is a measure of the variation

of possible rates of return Ri, from the expected

rate of return E(Ri)

- where Pi is the probability of the possible rate

of return, Ri

Example Expected Return and Risk

Variance ( 2) .00050 Standard Deviation (

) .02236

Covariance and Correlation

- Covariance is a measure of the degree to which

two variables move together relative to their

individual mean values over time - The correlation coefficient is obtained by

standardizing (dividing) the covariance by the

product of the individual standard deviations

(Correlation coefficient varies from -1 to 1)

Example case

- Look at the following data () and tell the

difference between an arithmetic and a geometric

average. Also, which of these asset classes is

the most attractive? What is the chance that you

will break-even if you invest in stocks only?

What is the chance that you make more than 10

return (arithmetic) if you invest in corporate

bonds only? If you hold only a real estate

portfolio, what would be the rational for adding

other classes?

Ar. Average Geo. Average Standard dev.

Stocks 10.28 8.81 16.9

T-bills 6.54 6.49 3.2

LT Gov Bonds 6.1 5.91 6.4

LT Corp Bonds 5.75 5.35 9.6

Real Estate 9.49 9.44 3.5

Correlations Stocks T-bill LT Gov Bonds LT Corp. Bonds Real estate

Stocks 1

T-bills 0.11 1

LT Gov Bonds 0.27 0.89 1

LT Corp Bonds 0.33 0.46 0.76 1

Real Estate 0.29 0.06 -0.02 0.08 1

Solution

- The arithmetic average assumes the presence of

simple interest, while the geometric average

assumes compounding (interest-on-interest).

Ranking is best accomplished by using the

coefficient of variation (standard

deviation/arithmetic mean, multiplied by 100) - 1 - Real Estate 36.88
- 2 - Treasury Bills 48.93
- 3 - Long Gov't Bonds 104.92
- 4 - Common Stocks 164.37
- 5 - Long Corp. Bond 166.96

- To get the probability of success or failure,

compute a Z-score to be transformed into a

probability - P(Rstockgt0)gt z(Rstock 0)/standard deviation

stock 10.28/16.90.608 - P(Rstockgt0)P(0.608) 72.9
- P(Rcorp bondsgt10)gt z(Rbond 10)/standard

deviation bond (5.75-10)/9.6-0.44 - P(Rcorp bondsgt10) P(-0.44)1-P(0.44)1-0.6733
- It seems at first that government bonds offer

less return and more risk than real estate.

However, real estate and government bonds will

provide a good combination as they do not

fluctuate in a similar fashion, so that the

variability of the portfolio is less than the

variability of the individual investments (the

correlation coefficient applicable to this pair

of investments is known and is slightly negative.

- A note on real estate investments a risk

component of low liquidity is not included in the

standard deviationso, Real estate might not be

so much a better investment.

Risk and Return

- How to compare assets?
- Coefficient of variation measure of relative

risk - CV Total risk/return
- CS 1.56
- SCS 1.91
- CB 1.41
- TB 1.64
- Rf 0.84
- Which one do you pick?
- What is the problem here?

Portfolio effect

- Portfolio Return is the weighted average return

of each asset in the portfolio - Portfolio Risk is not the weighted average risk

of each asset in the portfolio. - Portfolio risk has to do with each assets weight

and risk, but also the degree to which they move

together (?)

Portfolio Standard Deviation Formula (with

covariance)

- Any asset of a portfolio may be described by two

characteristics - The expected rate of return
- The expected standard deviations of returns
- The correlation, measured by covariance, affects

the portfolio standard deviation - Low correlation reduces portfolio risk while not

affecting the expected return

Mathematical Explanation

Summary Portfolio effect

- Portfolio return (RP)
- Average return of all securities
- Portfolio risk (sP)
- Average risk of all securities
- Minus
- the propensity of those securities to be

unrelated (returnwise!)

Portfolio risk and returnin English

- Portfolio return
- (weighted) average assets return
- Portfolio risk
- (weighted) average assets risk
- (weighted) average assets prices propensity to

move in opposite direction - Or
- Portfolio risk
- (weighted) average assets risk
- - Benefits from diversification

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Example Combining Stocks with Different Returns

and Risk

1 .10 .50

.0049 .07 2

.20 .50 .0100 .10

- Case Correlation Coefficient

Covariance - a 1.00

.0070 - b 0.50

.0035 - c 0.00

.0000 - d -0.50

-.0035 - e -1.00

-.0070

Combining Stocks with Different Returns and Risk

- Assets may differ in expected rates of return and

individual standard deviations - Negative correlation reduces portfolio risk
- Combining two assets with -1.0 correlation

reduces the portfolio standard deviation to zero

only when individual standard deviations are equal

Example Combining Stocks with Different Returns

and Risk

1 .10 .07 2

.20 .1

Portfolio Risk-Return Plots for Different Weights

E(R)

2

With two perfectly correlated assets, it is only

possible to create a two asset portfolio with

risk-return along a line between either single

asset

Rij 1.00

1

Standard Deviation of Return

Portfolio Risk-Return Plots for Different Weights

E(R)

f

2

g

With uncorrelated assets it is possible to create

a two asset portfolio with lower risk than either

single asset

h

i

j

Rij 1.00

k

1

Rij 0.00

Standard Deviation of Return

Portfolio Risk-Return Plots for Different Weights

E(R)

f

2

g

With correlated assets it is possible to create a

two asset portfolio between the first two curves

h

i

j

Rij 1.00

Rij 0.50

k

1

Rij 0.00

Standard Deviation of Return

Portfolio Risk-Return Plots for Different Weights

E(R)

With negatively correlated assets it is

possible to create a two asset portfolio with

much lower risk than either single asset

Rij -0.50

f

2

g

h

i

j

Rij 1.00

Rij 0.50

k

1

Rij 0.00

Standard Deviation of Return

Portfolio Risk-Return Plots for Different Weights

Exhibit 7.13

E(R)

f

Rij -0.50

Rij -1.00

2

g

h

i

j

Rij 1.00

Rij 0.50

k

1

Rij 0.00

With perfectly negatively correlated assets it is

possible to create a two asset portfolio with

almost no risk

Standard Deviation of Return

Numerous Portfolio Combinations of Available

Assets

Efficient Frontier for Alternative Portfolios

Efficient Frontier In Practice (all equity

markets of the world 1981-2001)

Application 9 different Institutional efficient

Benchmarks

Asset Allocation and cultural Differences

- Mindset, Social, political, and tax environments
- U.S. institutional investors average 45

allocation in equities - In the United Kingdom, equities make up 72 of

assets - In Germany, equities are 11
- In Japan, equities are 24 of assets

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Example What is the use of an efficient set?

- Goal find an optimal mix (weight) so that the

ratio of compensation for risk to risk (or reward

to risk) is optimal for your level of risk

tolerance. - Your inputs Expected returns, standard

deviations and correlations (for each asset

class) - Your output Optimal weight in each asset class

(how much should you put in each asst class?)

Can we do better than the efficient set?

- Imagine two portfolio (1) a risky best of the

best portfolio with an expected return of Rm and

a standard deviation of sm and (2) a riskless

portfolio of t-bills with an expected of Rf and a

standard deviation close to zero. - You allocate Wrf in the riskless portfolio and

(1-Wrf) in the risky (best of the best portfolio) - The standard deviation and expected return of

this portfolio shall be - sp(1-Wrf) x sm or Wrf1- sp/sm, then
- RpWrf x Rf (1-Wrf) x Rp replace Wrf by 1-

sp/sm - RP Rf (Rm Rf) /sm x sp?Capital Market

Line (CML) - Rp intercept slope x sp

What does it mean?

Example Illustration of the separation theorem

Wrfgt0

Wrflt0

- How is the concept of leverage included in the

CML?

Wrf0

It means that

- We know how to get the composition of the

best-of-the-best portfolio (M)? It has the

highest reward to risk i.e., (Rm Rf)/sm - Then, we know how to get Rm and sm
- Finally, for the risk we are willing to take

(indifference curve? policy statement), we can

find our optimal asset allocation by mixing the

best of the best portfolio with cash! - Cool (I mean sweeeeet) huh?
- Application efficient frontier analysis

Example The CML and the allocation process

- Step 1 Estimate the expected returns, standard

deviations and correlations for each asset

classes - Step2 Find the the allocation mix as well as the

expected return and standard deviation of the

M--i.e., M should be optimized so that its

reward-to-risk (Rm-Rf)/sigma_M is optimal - Step 3 Choose an allocation for your utility for

risk using the CML. You will invest Wrf in the

risk-free rate and (1-Wrf) in M. - ? Application Say that Rf has an expected return

of 5 M has a standard deviation of 20 and an

expected return of 30 and it consists of an

allocation of 40 in foreign stocks, 15 in

foreign bonds, 20 in domestic stocks and 25 in

domestic bonds. What is the expected return, risk

and allocation in the following 2 cases (1) you

are willing to accept 15 risk and (2) you are

willing to accept 25 risk.

Solution

- For a standard deviation of 15
- Allocation
- Wrf1-(15/20)25
- W(foreign stocks) (1-25) x 4030
- W(foreign bonds)(1-25) x 15 11.25
- W(domestic stocs)(1-25) x 2015
- W(domestic bonds)(1-25) x 2518.75
- Expected return 25 x 5 75 x 3023.75
- For a standard deviation of 25
- Allocation
- Wrf1-(25/20)-25 (borrow 25 at the risk free

rate and use it to invest in M) - W(foreign stocks) (125) x 4050
- W(foreign bonds)(125) x 15 18.75
- W(domestic stocs)(125) x 2025
- W(domestic bonds)(125) x 2531.25
- Expected return -25 x 5 125 x 3036.25

The selection process Intuitive Risk for

individual assets

- Return(asset) expected unexpected
- Risk (return) 0 market risk business risk
- The trick if you hold many securities, the

particularities of each security becomes

irrelevantthus, in a well diversified portfolio

business-specific risk is irrelevant!

A Risk Measure for the individual assets a more

rigorous Explanation

- Because all individual risky assets are part of

the M portfolio, an assets rate of return in

relation to the return for the M portfolio may be

described using the following linear model

where Rit return for asset i during period

t ai constant term for asset i bi slope

coefficient for asset i RMt return for the M

portfolio during period t random error

term

Variance of Returns for a Risky Asset

Measuring Components of Risk

- ?i2 bi2 ?m2 ?2(ei)
- where
- ?i2 total variance
- bi2 ?m2 systematic variance
- ?2(ei) unsystematic variance

Examining Percentage of Variance

- Total Risk Systematic Risk Unsystematic Risk
- The proportion of systematic risk is Systematic

Risk/Total Risk R2 - so ßi2 ? m2 / ?i2 R2
- The proportion of diversifiable risk is then

measured by (1- R2 )

Risk Reduction with Diversification

St. Deviation

Unique Risk s2(eP)s2(e) / n

bP2sM2

Market Risk

Number of Securities

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Risk and Return for a single asset

- The higher the risk, the greater the expected

return. - RiReal rate Inflation premium Risk premium
- Ririsk free rate compensation for risk
- Compensation for riskrisk premiumcompensation

for a high systematic risk

Risk that matters

- If only market risk matters, then the risk

premium of a security should be related (somehow)

to the market risk premium! - Lets assume that those risk premiums are

proportional - security risk premiumß x market risk premium
- This ß is a multiplier which has to do with the

relative risk premium of a security to the market

risk premiumit is a relative Market (systematic)

Risk

SML

- RiRF RRP, then
- Security risk premium (Ri- RF)
- Market risk premium (Rm- RF)
- If security risk premiumß x market risk premium
- Then, (Ri- RF) ß x (Rm- RF)
- That is,
- Ri RF ß x (Rm - RF)
- This is also known as the SML (market

equilibrium), a component of the CAPM - As a result, any securitys return can calculated

using ß, RRF, and Rm

Formal derivation of CAPM

Some things to know about beta

- What is betas formula
- Ans Beta COV(I,M)/VAR(M)
- What is the market relative risk (ß)?
- Ans 1
- What does a ß of 2 mean?
- Ans twice more risky that the market
- What does a ß of 1 mean?
- Ans negatively correlated with the market
- What is the beta of a risk-free security?
- Ans 0
- How do we get ß?
- Ans regression or above formula
- What is the ß of a portfolio?
- Ans The weighted average of each stocks beta

Graph of SML

- What if the observed returns are different from

the theoretical returns? - The Alpha-strategy consists of finding securities

with abnormal excess return.

Example A simple illustration

- Assume RFR 6
- RM 12

E(RA) 0.06 0.70 (0.12-0.06) 0.102

10.2 E(RB) 0.06 1.00 (0.12-0.06) 0.120

12.0 E(RC) 0.06 1.15 (0.12-0.06) 0.129

12.9 E(RD) 0.06 1.40 (0.12-0.06) 0.144

14.4 E(RE) 0.06 -0.30 (0.12-0.06) 0.042

4.2

Comparison of Required Rate of Return to

Estimated Rate of Return

Plot of Estimated Returnson SML Graph

.22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02

C

SML

A

E

B

D

.20 .40 .60 .80

1.20 1.40 1.60 1.80

-.40 -.20

Question How can we use the SML to select

underpriced securities?

- According to the SML
- Ri-Rf 0 ß x (Rm-Rf)
- In a regression format (Ri-Rf) a ß x (Rm-Rf)

e - Then (alpha strategy)
- if a is not significantly different from 0,

security is fairly priced - if a is significantly greater 0, security is

underpriced - if a is significantly smaller than 0, security is

overpriced

Illustration Alpha Strategy

x A

x B

a

Alpha-strategy example

- SML (Ri Rf) alpha beta x (Rm-Rf) e
- Example EXTR

Alpha Beta R2

EXTR (t-stat) 0.019 (2.28) 1.47 (4.57) 0.25

What is the amount of diversifiable risk in EXTR

if its standard deviation is 20? Ans 20 x

(75)1/2 17.32

In Sum,

- What is the difference between the CML and SML?

Why are the measures of risk different? - RP Rf (Rm Rf) /sm x sp ?CML (allocation)
- Ri Rf (Rm-Rf) x si/ sm x ri,m ?SML

(selection) - Ri Rf (Rm-Rf) x ßi,m

Lets conclude and summarize now

- Develop an investment policy statement
- Identify investment needs, risk tolerance, and

familiarity with capital markets - Identify objectives and constraints
- Investment plans are enhanced by accurate

formulation of a policy statement - ALLOCATION determine the market/sector weights
- Asset allocation determines long-run returns and

risk, which success depends on construction of

the policy statement - (1) EFFICIENT FRONTIER and (2) CML
- CML EFFICIENT FRONTIER when T-Bill is included

in the efficient set - SELECTION determine undervalued securities
- Actual (observed of predicted) Return Vs. SML

(fair) return - Alpha Analysis Is the SML significantly

violated? - Optimal allocation between selected securities

with the efficient frontier

Example

- You are a new analyst with a small mutual fund.

You have been assigned to make a presentation to

clients on portfolio management. You need to

discuss two main issues (1) diversification

(relationship between portfolio variance and

correlation) and (2) asset pricing (difference

between the CML and SML). Be ready to answer both

issues.

Problems With CAPM

- 1. Beta coefficients are not stable for

individual securities. - Performance evaluation depends upon the choice

of the market proxybenchmark error. - T-bills are not exactly risk-free
- Unpleasantries have been neglected (taxes and

transaction cost)

Empirical Tests of the CAPM

- A theory should not be judged on the basis of the

reasonableness of its assumptions. Instead, the

test of any model is how well it explains the

observed facts, and then how well it can predict. - There are two main questions about the CAPM that

need to be answered - 1) How stable is beta, and
- 2) Is the risk -return relationship linear as

suggested by the SML? That is, how well do the

returns conform to the equation of the security

market line?Positive linear between beta and

risky assets returns? E(Ri) RFR bi(Rm -

RFR) - Specific questions might include
- Does the intercept approximate the RFR that

prevailed during the test period? - Was the slope of the line positive and was it

consistent with the slope implied by the risk

premium (Rm - RFR) that prevailed during the test

period?

Stability of Beta

Individual stocks versus portfolios?Individual

stock betas have been found to be generally more

volatile over time whereas portfolio betas were

stable. In fact, Levy and Blume concluded that

portfolio betas are much more stable than betas

for individual stocks. Also, betas have a

tendency to regress toward one. Tole observed

greater stability of betas for larger portfolios

and that benefits are realized in having over a

100 securities in a portfolio. Time period?Basel

found that the stability of beta estimates rose

with larger estimation periods. Roenfeldt,

Grienpentrog and Pflamm found that 48 month betas

were not good for estimating subsequent 12-month

betas, but were good for estimating 24, 36, and

48-month betas. Trading volume? Carpenter and

Upton contended that that volume adjusted betas

were slightly better than ordinary predictions.

Comparability of Published Estimates of Beta

- Merrill Lynch?60 monthly returns with SP 500
- Value Line? 260 weekly returns with NYSE

Composite - Ri RFR biRm Et ? badjusted Merrill

Lynch 0.127 badjusted Value Line R2 .55

- It is often convenient to use the betas published

by Value Line or Merrill Lynch. The computation

of the betas, however, use different data.

Merrill Lynch uses 60 monthly observations and

SP 500 for market proxy while. Value Line uses

260 weekly observations and NYSE composite series

as the market proxy. Both services use the same

regression model and follow slightly different

adjustment procedures to deal with regression

tendencies. Given these relatively minor

differences, you would expect the published betas

to be quite similar. - Statman compared these published betas for both

individual stocks and portfolios and found small

but significant differences between the two beta

estimates. For 195 individual stocks he found

the relationship with an R-square value of .55.

Reilly and Wright also found that difference was

due to the alternative time intervals? The

securitys market value affects both the size and

the direction of the interval effect. Therefore,

when estimating beta or using published sources,

you must consider the interval used and the

firms relative size.

Relationship Between Systematic Risk and Return

The ultimate question regarding CAPM ? is there a

positive linear relationship between systematic

risk and the rates of return on risky assets?

Sharpe and Cooper (1972)?Positive relationship

between risk/return...not completely linear.

Black, Jensen, and Scholes (1972), Fama and

Macbeth (1973) ?Positive linear relationship

between monthly excess returns and betas. Douglas

(1969)? Systematic risk measures not significant

in explaining returns. Effect of Skewness on

Relationship investors prefer stocks with high

positive skewness that provide an opportunity for

very large returns Effect of Size, P/E, and

Leverage size, and P/E have an inverse impact on

returns after considering the CAPM. Financial

Leverage also helps explain cross-section of

returns ?Effect of Book-to-Market Value. For EX,

Fama and French found that Size and

book-to-market ratios explain returns on

securities. Also, Beta is not a significant

variable when other variables are included?Study

results support multifactor models)

Summary of CAPM Risk-Return Empirical Results

Early studies supported CAPM but not without some

doubts regarding the higher-than-expected

intercept. Subsequently, the efficient markets

literature has provided extensive evidence that

both size and the P/E ratio were variables that

could help explain cross-sectional variation

addition to beta. Recent studies also have found

that financial leverage and the book-to-market

value of equity ratio (i.e., B/P ratio) have

explanatory power regarding returns beyond beta.

The Fama-French study concluded that between 1963

and 1990, beta was not related to average returns

on stocks when included with other variables, or

when considered alone. Moreover, the two

dominant variables were size and the book value

to market value ratio, which was even stronger

than size although both variables were

significant. Subsequent studies have supported

the Fama-French results. However, more recent

studies that have used longer time periods for

computing betas, and adjusting for expected

returns have provided support for the original

CAPM.

The Market Portfolio Theory Vs Practice

Recall that the CAPM market portfolio is assumed

to include all the risky assets in the economy.

This would include art, real estate, stocks,

bonds, jewels, etc. In practice it is not

possible to choose a proxy that contains all

risky assets, so the SP 500, the NYSE composite

index, or some other market indicator must be

used as a proxy. Most academicians recognize

this problem and conclude that it is not

serious. Richard Roll concluded that the use of

indexes as a market proxy has a very serious

implication for tests of the model and especially

for using the model when evaluating portfolio

performance. Roll refers to it as a benchmark

error. With respect to tests of the CAPM, Roll

contends that prior to testing, an analysis is

required of whether the proxy for the market

portfolio is mean variance efficient (on the

Markowitcz efficient frontier) and whether it is

the true market portfolio.

Differential CML Using Market Proxy That Is

Mean-Variance Efficient

Differential Performance Based on an Error in

Estimating Systematic Risk

Differential SML Based on Measured Risk-Free

Asset and Proxy Market Portfolio

Relaxing the Assumptions

- The CAPM made a series of assumptions that may

appear to be unrealistic. One might ask what

happens to the results of the CAPM if we change

the assumptions upon which it was developed. - The assumptions that we will examine are
- Differential lending and borrowing rates

gtLending and borrowing at the risk-free rate? - Liquidity Risk Premium?
- What effect will transaction costs and taxes have?

Differential Borrowing Lending Rates

- One of the assumptions of the CAPM was that

investors could both borrow and lend unlimited

amounts at the risk-free rate. - It seems reasonable to assume that investors can

lend unlimited amounts at the risk-free rate by

buying government securities (e.g., T-bills). - However, it is questionable whether investors can

borrow at the risk-free rate since it is usually

below the prime rate and most investors cant

even borrow at the prime rate.

Investment Alternatives When the Cost of

Borrowing Is Higher Than the Cost of Lending

If borrowing rate (Rb) gt lending rate (RFR),

there are two different lines going to the

Markowitz efficient frontier. The line segment

RFR - F represents the investments opportunities

available we an investor lends at the RFR and

invests in Portfolio F. Opportunities on this

line beyond point F are unavailable because you

cannot borrow at the risk-free rate. Assuming

that borrowing takes place at Rb, the portfolio

that can be purchased with borrowed funds is

Portfolio K. Notice that Portfolio K is the

point of tangency between the borrowing rate, Rb,

and the efficient frontier. Due to the

differential lending and borrowing rates, the CML

becomes RFR - F - K - Rb.

Zero-Beta Model? coping with Rfr

- If the market portfolio is mean-variance

efficient, then we can find several portfolios

that have no systematic risk (beta 0) and the

assumption that a risk-free rate exists is not

necessary. - From among the zero-beta portfolios, we can

select the one with the smallest variance. This

will not affect the CML but it will allow the

construction of a linear SML where the intercept

is the minimum variance zero-beta portfolio. - The combination of this zero-beta portfolio and

the market portfolio will be a linear

relationship in return and risk described by this

equation. - where, E(Ri), E(Rz), and E(Rm) are the expected

returns for security i, the zero-beta portfolio,

and the market portfolio, respectively

Security Market Line With a Zero-Beta Portfolio

- Absence of a risk-free asset
- Combinations of portfolios on the efficient

frontier are efficient - All frontier portfolios have companion portfolios

that are uncorrelated - Returns on individual assets can be expressed as

linear combinations of efficient portfolios

Return

Zero beta

Risk

CAPM Liquidity

- Liquidity
- Illiquidity Premium
- Research supports a premium for illiquidity

f (ci) liquidity premium for security i f (ci)

increases at a decreasing rate

Illiquidity and Average Returns

Average monthly return()

Bid-ask spread ()

Security Market with Transaction Costs

The assumption of no transaction costs in the

development of the original CAPM was so that

investors will buy or sell mispriced securities

until they plot on the SML line. When transaction

costs exist investors will not correct all

mispricing. Instead they will only correct

mispricing to the point that the excess return

due to the correction process is exactly offset

by the cost. As shown below, transaction costs

result in the SML becoming a band of securities.

Transaction costs also affect the level of

diversification that investors are willing to pay

to achieve. The additional transactions needed

to extract the last 10 of non-systematic risk in

a portfolio may be more expensive than the

benefits derived.

GeneralizationWhat is a factor model?

- Factor models are used to operationalize

portfolio theory for equity portfolios - To construct portfolios that meet specific

objectives - To measure and control portfolio risk
- To evaluate portfolio performance
- They are all statistically based and designed to

reduce the portfolio selection problem - They require additional theoretical structure to

justify different approaches

Types of Factor Models

- Single Factor Models
- Capital Asset Pricing Model (CAPM) and Market

Model - Multi-factor Models
- Macroeconomic Factor Models prespecify

macroeconomic indicators such as interest rates,

inflation, exchange rates (e.g. Arbitrage Pricing

Theory, APT, and BIRR) - Statistical Factor Models extract from

historical time-series and cross-section of stock

returns (e.g. Principal Components or Factor

Analysis) - Fundamental Factor Models use firm-level

attributes and market data, like P/B, P/E,

industry, momentum, trading activity, etc. (e.g.

Fama-French, Carhart, BARRA, Vestek)

Examples of Factor Models

- E(Ri) rf
- ?i E(Rm) rf
- ?i,F1 E(RF1) rf ?i,F2 E(RF2) rf .

?i,FK-1 E(RFK-1) rf ?i,FK E(RFK) rf - ?i,m E(Rm) rf ?i,SMB ESMB ?i,HML

EHML ?i,Mom EMomentum

Single Factor Model CAPM (Review)

- The CAPM is a centerpiece of modern finance that

gives predictions about the relationship between

risk expected return - Based on original work of Harry Markowitz by Bill

Sharpe, John Lintner Jan Mossin in mid-1960s - Portfolio Theory (Markowitz, 1952) how an

investor could select an optimum portfolio - CAPM (Sharpe, Lintner, Mossin, 1964) predicts

how aggregate of investors will behave how

prices will be set so that markets clear

CAPM Aggregation (review)

- Begins with simplistic assumptions for

hypothetical world of investors builds into

comprehensive model - Investors are price takers
- Same one-period investment horizon (myopic)
- Fixed quantities of assets and all marketable
- No taxes, transactions costs, regulations, etc
- Risk-averse utility-maximizing Markowitz

investors - All investors analyze securities in same way with

same probabilistic forecasts for each - Since they must all choose the same optimal risk

(tangency) portfolio, this portfolio must be M,

the market portfolio, for market to clear (share

prices set to balance supply/demand)

Portfolio Problem 3 ? CAPM (Review)

- Portfolio Problem 3 considers two steps
- Choose one optimal risk portfolio along efficient

frontier same for all investors, market M - Now, each investor (different wealth, different

risk tolerances) choose optimal capital

allocation with riskfree security along CAL for

portfolio M, or capital market line - CAPM equilibrium
- share prices balance supply and demand
- no pressure for prices to change
- rational investors passively hold market

portfolio

CAPM Key implication (review)

- There is a linear relationship between expected

return for a security and the market risk

premium - E(Ri) rf ?i E(Rm) rf
- Beta, ?i, is a standardized measure of covariance

with market portfolio M ?i

Cov(Ri,RM)/?m2 - Expected return for stock equals riskfree return

PLUS market risk premium scaled by beta of stock

Capital Asset Pricing Model (review)

Security Market Line

M

E(Rm)

Market Risk Premium E(Rm) - rf

rf

The Market Model (review)

- CAPM implies that security prices are governed by

their market risks and NOT their firm-specific

risks - How to decompose total risk of stock into two

components? A single-factor model Market model - Based on simple statistical regression framework

using T historical returns Rit ?i ?i

Rmt ?it

Regression Approach Excel (review)

US Technology Index

US Utilities Index

Regression Approach - Excel

US Technology Index

US Utilities Index

Decomposing Risk (Review)

- If IBM stock has an annual standard deviation of

21, where as that of the SP 500 is only 16,

then we can decompose total risk of IBM given its

beta of 1.15 - 212 1.15 x 162 102
- 0.0441 0.0339 0.0102
- R-squared Proportion of variance due to market

0.0339/0.0441 76.9

Beta of IBM x Market Variability

Specific Risk IBM

Variability of IBM

Macroeconomic Factor Models

- Early tests of the market model suggested

significant systematic, but extra-market factors - B. King, JB 1966, used statistical procedures to

identify strong extra-market, but within-industry

comovements - F. Black, M. Jensen and M. Scholes, 1972,

uncovered significant systematic patterns in

betas by market size - S. Ross in 1976 developed the Arbitrage Pricing

Theory (APT) as an alternative to the CAPM which

proposed a set of macroeconomic factors which

pervasively explain stock returns and which are

priced (matter for expected return relative to

risk)

Arbitrage Pricing Theory (APT)

- Three major assumptions
- Capital markets are perfectly competitive
- Investors always prefer more wealth to less

wealth with certainty - The stochastic process generating asset returns

can be expressed as a linear function of a set of

K factors or indexes. - Assumptions Not Required for APT
- Quadratic utility function.
- Normally distributed security returns.
- Market portfolio that contains all risky assets

and is mean variance efficient.

EXAMPLE CAPM VS APT

- CAPM Assumptions
- Investors want to invest on Efficient Frontier

(mean variance efficient) - Investors can lend and borrow at the risk free

rate - Investors have similar expectations
- Investors have the same one-period time horizon
- Investments are infinitely divisible ?Investors

can buy any amounts of any security - No taxes and no transaction costs
- No changes in interest rates (no inflation or it

is fully anticipated) - Capital markets are in equilibrium? investments

are properly priced for their risk levels

- APT Assumptions
- Capital Markets are perfectly competitive
- Investors prefer more wealth than less wealth

with certainty - Returns can be described linearly as a function

of K factors - It does not require
- Homogenous expectations and quadratic utility

function - Normality distributed returns
- A M portfolio that is mean-variance efficient and

contains all risky assets

Arbitrage Pricing Theory (APT)

- All security returns are pervasively affected by

several macroeconomic factors

- where
- the expected return on an asset with zero

systematic risk where

the risk premium related to each of the

common factors - for example the risk premium

related to interest rate risk

bi the pricing relationship between the risk

premium and asset i - that is how responsive

asset i is to this common factor K?Different

stocks have different sensitivities (b

coefficients) to different factors, e.g. banks

and inflation, auto companies and national income

growth, utilities and interest rates,

etc.Multi-factor models can quantify these

multiple sources of risk

Arbitrage Pricing Theory (APT)

- Multiple factors expected to have an

impact on all assets

- Multiple factors expected to have an impact on

all assets - Inflation
- Growth in GNP
- Major political upheavals
- Changes in interest rates
- And many more.
- Contrast with CAPM insistence that only beta is

relevant

Example Two Stocks and a Two-Factor Model

- changes in the rate of inflation. The risk

premium related to this factor is 1 percent for

every 1 percent change in the rate

percent growth in real GNP. The average risk

premium related to this factor is 2 percent for

every 1 percent change in the rate

the rate of return on a zero-systematic-risk

asset (zero beta boj0) is 3 percent

Example of Two Stocks and a Two-Factor Model

- the response of asset X to changes in the rate

of inflation is 0.50

the response of asset Y to changes in the rate

of inflation is 2.00

the response of asset X to changes in the

growth rate of real GNP is 1.50

the response of asset Y to changes in the

growth rate of real GNP is 1.75

Example of Two Stocks and a Two-Factor Model

- .03 (.01)bi1 (.02)bi2
- Ex .03 (.01)(0.50) (.02)(1.50)
- .065 6.5
- Ey .03 (.01)(2.00) (.02)(1.75)
- .085 8.5

Arbitrage Pricing Line

- No arbitrage implies
- E(Ri) rf ?i,F1 E(RF1) rf ?i,F2

E(RF2) rf . ?i,FK E(RFK) rf - How many factors?
- Which factors included?
- Is market portfolio a factor?

E(Ri)-rf

bF2

bF1

Statistical Factor Models

- Early implementation of multi-factor models

inspired by theoretical development of Ross APT

as green light for purely statistical approach - Process Take variance-covariance matrix of

security returns and find extract factors that

maximally explain the variances - Principal Components Analysis Roll and Ross,

1980 - Factor Analysis - Chen, 1983 Connor and

Korajczyk, 1988 - Problem Economic interpretation of purely

statistical artifacts, e.g. F1, F2, F3, etc.

Prespecified Macro Factors

- Economic intuition drove researchers to develop a

multi-factor macroeconomic model with

prespecified factors (no empirical justification) - Growth rate in monthly industrial production
- Default premium (Yield spread of Baa Aaa

corporates) - Term premium (Long-short bond yield spread)
- Expected and unexpected changes in CPI
- Major contributions Chen, Roll and Ross (1986),

Fama French (1988, 89), Ferson Harvey (1991) - Burmeister, Ibbotson, Roll Ross built the BIRR

model (www.birr.com) Salomon Smith Barney RAM

(Risk Attribute Model) model is competitor from

equity strategy group

Comparing BIRR RAM

- RAM factors
- Economic Growth changes in industrial

production - Business Cycle default spread as proxy
- Long-term Interest Rates change in 10-year

Treasury note yield - Short-term Interest Rates changes in 1-month

Treasury bill - Inflation Shock unexpected changes
- US Dollar fluctuations in trade-weighted index

of currencies

- BIRR factors
- Confidence Risk tilted toward safe haven

stocks - Time Horizon Risk growth stocks have higher

positive exposure vs income stocks - Inflation Risk unexpected changes in inflation
- Market Timing Risk market risk, but as a

residual to other four - Business Cycle Risk unexpected changes in

growth of business activity

Using Macroeconomic Factors

- Each stock has an estimated factor risk exposure

using multivariate regression - What is expected excess return for Reebok stock?
- 0.73(2.59) 0.77(-0.66) -0.48(-4.32)

4.59(1.49) 1.50(3.61) 15.71 - Compare projected return with different risk

exposures

BIRR Model for Reebok

Factor Market Risk Factor Sensitivity Price of

Risk Confidence 0.73 2.59 Time

Horizon 0.77 -0.66 Inflation Risk -0.48 -4.32 Bu

siness Cycle 4.59 1.49 Market Timing 1.50 3.61

Fundamental Factor Models

- These models use company and industry attributes

and market data as factors, e.g. P/E, P/B ratios,

projected earnings growth, trading activity and

even technical factors, like price momentum - These are extra-market risk factors that

represent systematic comovements in historical

stock returns (e.g. Fama/French, 1992/93,

Carhart, 1997) - Commercially available fundamental factor models

are BARRA, Vestek Systems and Wilshires Atlas

Alternative Risk Decomposition

Total Risk Variability

Specific Risk Diversifiable Risk

Market Risk Beta

Unique Risk Firm Specific Risk

Extramarket Risk Common Factor Risk

Macro Factor Risk GDP, Inflation, etc

Micro Factor Risk Size, P/E, P/B etc

Industry Risk

Examples

- Fama-French (1993) Three-factor model
- E(Ri) rf ?i,m E(Rm) rf ?i,SMB ESMB

?i,HML EHML - SMB is return spread between small-cap and

large-cap triciles - HML is return spread between high P/B and low P/B

triciles - Carhart (1997)
- E(Ri) rf ?i,m E(Rm) rf ?i,SMB ESMB

?i,HML EHML ?i,Mom EMomentum - Momentum is return spread of stocks between

highest and lowest quintiles of returns

performance over (-1mo, -7mo)

BARRA Model

- BARRA E2 model has 13 risk indices and 55

industry groups, but some models have 85 factors - Some of the factors are very complex in

construction, but detailed, thereby marketing

transparency - 1. Variability in markets (VIM) 2. Success

(momentum) - 3. Size 4. Trading activity (turnover)
- 5. Growth (in earnings) 6. Earnings/price

ratio - 7. Book/price ratio 8. Earnings variability
- 9. Financial leverage 10. Foreign income
- 11. Labor intensity 12. Yield
- 13. LOCAP (extension of size)

An 87-Factor Model

Low Valuation (9 indices)

Small Firm Size (8 indices)

Earnings Variability (5 indices)

Growth Orientation (9 indices)

Factor Based Risk Estimates

Market Variability (9 indices)

Financial Risk (9 indices)

Industry Classification (39 indices)

New Developments

- Chan, Karceski Lakonishok 1998 sought to

identify which factors best capture systematic

return covariation using a common dataset

(NYSE/Nasdaq, 1963-93, monthly stock returns) - Fundamental factors P/B, P/E, P/CF, D/P, Size
- Technical factors 1-month, 6-month, 60-month

momentum - Macroeconomic factors Inflation, default

premium, term premium, slope, industrial

production - Statistical factors Principal component

analysis up to 4 factors - Market factor equally- and value-weighted

indexes - Random factors
- CKL find that fundamental and technical factors

are stronger than macro variables and most

statistical factors

Global and local risk factors

- Evidence shows international diversification

(Direct stock purchases, American depository

receipts, Mutual Funds--Open-end funds and

Closed-end funds, WEBS, Trusts) is beneficial - Possible to expand the efficient frontier above

domestic only frontier - Possible to reduce the systematic risk level

below the domestic only level - Then,
- What are the risks involved in investment in

foreign securities? - How do you measure benchmark returns on foreign

investments? - Are there benefits to diversification in foreign

securities?

What do we learn?

- Evidence of priced factors beyond market beta

is becoming clearer, so it is more appropriate to

use a multi-factor model for risk measurement - Which factors are relevant? Buyer beware! Factor

models can be a license to data mine - Risk analysis should focus on understanding

active management bets in portfolio and risk

control of overall portfolio