Module 1 The investment setting and Modern portfolio Theory

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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

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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

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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

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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

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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!

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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.

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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.

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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

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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

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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
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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

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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?

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Example Historical Returns and Risks

• Computation of Monthly Rates of Return

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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

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Example Expected Return and Risk
Variance ( 2) .00050 Standard Deviation (
) .02236
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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)

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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
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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.

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• 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.

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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?

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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 (?)

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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

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Mathematical Explanation
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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!)

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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

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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

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Example Combining Stocks with Different Returns
and Risk
1 .10 .07 2
.20 .1
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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
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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
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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
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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
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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
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Numerous Portfolio Combinations of Available
Assets
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Efficient Frontier for Alternative Portfolios
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Efficient Frontier In Practice (all equity
markets of the world 1981-2001)
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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?)

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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

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What does it mean?
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Example Illustration of the separation theorem
Wrfgt0
Wrflt0

• How is the concept of leverage included in the
  CML?

Wrf0
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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

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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.

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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

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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!

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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
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Variance of Returns for a Risky Asset
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Measuring Components of Risk

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

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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 )

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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

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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

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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

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Formal derivation of CAPM
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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

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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.

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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
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Comparison of Required Rate of Return to
Estimated Rate of Return
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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
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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

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Illustration Alpha Strategy
x A
x B
a
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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
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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

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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

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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.

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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)

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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?

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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.
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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.

80
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)
81
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.
82
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.
83
Differential CML Using Market Proxy That Is
Mean-Variance Efficient
84
Differential Performance Based on an Error in
Estimating Systematic Risk
85
Differential SML Based on Measured Risk-Free
Asset and Proxy Market Portfolio
86
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?

87
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.

88
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.
89
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

90
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
91
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
92
Illiquidity and Average Returns
Average monthly return()
Bid-ask spread ()
93
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.
94
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

95
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)

96
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

97
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

98
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)

99
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

100
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

101
Capital Asset Pricing Model (review)
Security Market Line
M
E(Rm)
Market Risk Premium E(Rm) - rf
rf
102
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

103
Regression Approach Excel (review)
US Technology Index
US Utilities Index
104
Regression Approach - Excel
US Technology Index
US Utilities Index
105
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
106
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)

107
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.

108
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

109
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
110
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

111
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
112
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
113
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

114
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
115
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.

116
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

117
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

118
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
119
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

120
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
121
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)

122
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)

123
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)
124
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

125
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?

126
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