Portfolio Management Capital Market Theory

1
Portfolio ManagementCapital Market Theory
Asset Pricing Theory
Hady Abdul Khalek, CFA
2
An Introduction to Portfolio Management

• Objective investors maximize returns for a given
  level of risk.
• The optimum portfolio for an investor is not just
  a collection of investments that are good by
  themselves.
• Investors are risk averse assuming returns are
  equal, they will prefer the less risky asset.
• Risk aversion implies a positive relationship
  between expected return and risk.
• Risk is a measure of uncertainty regarding an
  investments outcome. Alternatively, risk can be
  considered the probability of a bad outcome.

3
A- The Portfolio Management ProcessElements of
Portfolio Management

• Evaluating Investor and Market characteristics
• Determine the objectives and constraints of the
  investor
• Evaluate the economic environment
• Developing an investment policy statement (IPS)
• Determining an asset allocation strategy
• Implementing the portfolio decisions
• Measuring and evaluating performance
• Monitoring dynamic investor objectives and
  capital market conditions
• The ongoing portfolio management process can be
  detailed with the integrative steps described by
  planning, execution, and feedback.

4
Investment Objectives

• Investment objectives are concerned with risk and
  return considerations.
• Risk tolerance is the combination of willingness
  and ability to take risk.
• Risk aversion indicates an investors inability
  and unwillingness to take risk.
• For an individual, risk tolerance may be
  determined by behavioral and psychological
  factors, whereas for an institution, these
  factors are primarily determined by portfolio
  constraints.
• Risk objectives can be either absolute (standard
  deviation of total return) or relative (tracking
  risk).

5
Return Objectives

• Required return can be classified as either a
  desired or a required return.
• Desired return how much the investor wishes to
  receive from the portfolio.
• Required return some level of return that must
  be achieved by portfolio.
• Required return serves as a much stricter
  benchmark than desired return.
• The level of return needs to be consistent with
  the risk objectives.
• Return should be evaluated on a total return
  basis capital gains and current income.

6
Investment Constraints

• Investment constraints are those factors limiting
  the universe of available choices. They include
• Liquidity expected or unexpected cash outflows
  that will be needed at some specified time.
• Time horizon the time period(s) during which a
  portfolio is expected to generate returns to meet
  major life events. Longer time horizons often
  indicate a greater ability to take risk, even if
  willingness is not evident.
• Tax concerns differential tax treatments are
  applied to investment income and capital gains.
• Legal and regulatory factors are externally
  generated constraints that mainly impact
  institutional investors.
• Unique circumstances special concerns of the
  investor.

7
Diversification and Portfolio Risk

• There are two broad classes of risk that affect
  portfolios
• Systematic Risk or market risk or
  non-diversifiable risk determined by
  Macroeconomic factors (affect whole economy),
  such as
• Business cycle
• Inflation rate
• Interest rate
• Exchange rate

8
Diversification and Portfolio Risk

• Unsystematic Risk or unique risk or
  firm-specific risk or diversifiable risk
  determined by Firm-specific factors, such as
• Firms successful RD
• Managements style and philosophy
• Unsystematic risk can be eliminated with
  diversification, i.e., spreading out the risk of
  a portfolio by investing in a variety of
  securities.
• The Total Risk Systematic Risk Unsystematic
  Risk

9
Diversification and Portfolio Risk
Graph
10
Utility Function Indifference Curves

• Indifference curves represent different
  combinations of risk and return, which provide
  the same level of utility to the investor.
• An investor is indifferent between any two
  portfolios that lie on the same indifference
  curve.
• Flat indifference curves indicate that an
  individual has a higher tolerance for risk. Very
  steep indifference curves belong to highly
  risk-averse investors.
• The optimal portfolio offers the greatest amount
  of utility to the individual investor.

11
return
Highly risk averse
risk
12
Highly risk tolerant
return
risk
13
Utility Values

• Utility is a measure of ranking portfolios
• Investors select portfolios with greatest utility
• Utility can be referred to as certainty
  equivalent rate

14
Calculating Rate of Return on Risky Assets

• E(r) (r1xp1) (r2xp2)(rtxpt)
• Where
• Ri return in state of the world i
• Pi probability of state i occurring
• t number of states

E(r) 0.255 0.5015 0.2525 15
15
Calculating the Variance of a Risky Asset
So Variance 0.0050 And the Standard Deviation
is v0.0050 0.0707 7.07
16
Calculating the Covariance of Two Risky Assets
17
Markowitz Portfolio Theory

• Any asset or portfolio can be described by two
  characteristics
• The expected return
• The risk measure (variance)
• Portfolios variance is a function of not only
  the variance of returns on the individual
  investments in the portfolio, but also of the
  covariance between returns of these individual
  investments.
• In a large portfolio, the covariances are much
  more important determinants of the total
  portfolio variance than the variances of
  individual investments.

18
Markowitzs Assumptions

• Investors consider investments as the probability
  distribution of expected returns over a holding
  period.
• Investors seek to maximize expected utility
• Investors measure portfolio risk on the basis of
  expected return variability
• Investors make decisions only on the basis of
  expected return and risk
• For a given level of risk, investors prefer
  higher return to lower returns.

19
Two-Security Portfolio Return
rp W1r1 W2r2 W1 Proportion of funds in
Security 1 W2 Proportion of funds in Security
2 r1 Expected return on Security 1 r2
Expected return on Security 2
20
Two-Security Portfolio Risk
sp2 w12s12 w22s22 2W1W2 Cov(r1r2)
21
Covariance
Cov(r1r2) r1,2s1s2
r1,2 Correlation coefficient of
returns
s1 Standard deviation of returns for
Security 1 s2 Standard deviation of
returns for Security 2
22
Correlation Coefficients Possible Values
Range of values for r 1,2 -1.0 lt r lt 1.0
If r 1.0, the securities would be perfectly
positively correlated If r - 1.0, the
securities would be perfectly negatively
correlated
23
Three-Security Portfolio
rp W1r1 W2r2 W3r3
s2p W12s12
W22s22
W32s32
2W1W2
Cov(r1r2)
Cov(r1r3)
2W1W3
Cov(r2r3)
2W2W3
24
The Efficient Frontier

• The efficient frontier consists of the set
  portfolios that has the maximum expected return
  for a given risk level.
• Optimal portfolio the portfolio that lies at the
  point of tangency between the efficient frontier
  and his/her utility (indifference) curve.
• An investors optimal portfolio is the efficient
  portfolio that yields the highest utility.
• A risk averse investor has steep utility curves.

25
The Risk-Return Trade-Off with Two-Risky Asset
Portfolios

• An investment opportunity set can be created to
  show all attainable combinations of risk and
  return offered by portfolios using available
  assets in differing proportions.
• Example
• E(rstocks)20, sstocks 30
• E(rbonds)10, sbonds 15
• ?stock, bonds .3
• Consider the following portfolios
• wstocks wbonds
  E(rportfolio) s portfolio
• 0 1
  10 15.00 (only bonds)
• .2 .8 12 14.94
• .4 .6 14 17.02
• .6 .4 16 20.61
• .8 .2 18 25.06
• 1 0 20 30.00
  (only stocks)
• Graph the investment opportunity set of possible
  combinations.

26
The Risk-Return Trade-Off with Two-Risky Asset
Portfolios

• Graph

27
The Risk-Return Trade-Off with Two-Risky Asset
Portfolios

• Graphassume ?stock, bonds -0.50

28
The Risk-Return Trade-Off with Two-Risky Asset
Portfolios

• Which portfolio would you prefer?

29
The Risk-Return Trade-Off with Two-Risky Asset
Portfolios

• Graph
• For ? 1, diversification is ineffective
• For ? -1, there is a combination that results
  in zero risk and high expected return
• For any ? lt 1, there will be a combination that
  dominates bonds and stocks taken alone
• In practice
• Historical data is used to build the investment
  opportunity set
• A perfectly negative correlation (? -1) is
  rarely found
• Investors prefer high expected returns and low
  risk (northwest section of the Investment
  Opportunity set)

30
The minimum-variance frontier of risky assets
E(r)
Efficient frontier
Individual assets
Global minimum variance portfolio
Minimum variance frontier
St. Dev.
31
Extending to Include Riskless Asset

• The optimal combination becomes linear
• A single combination of risky and riskless assets
  will dominate

32
Choosing the Optimal Risky Portfolio

• The tangency portfolio O will be chosen as The
  Optimal Portfolio
• It yields the CAL with the highest feasible
  reward-to-variability ratio (steepest slope)
• Following their personal risk aversion, investors
  will allocate their investment funds somewhere on
  CAL

33
Efficient Frontier With Only Risky Assets

• Based on the portfolio variance, we can
  calculate the volatility and expected returns for
  all the possible portfolios that can be
  constructed from N assets by varying the
  portfolio weights of the assets.

34
The Optimal Risky Portfolio with a Risk-Free Asset
35
Efficient Frontier With a Risk-Free Asset
(1-Month T-bill), Optimal portfolio (SP 500)
Expected Return
CML
Efficient Frontier
SP 500
Port D
Port C
S
Risk-Free Asset (1-month T-bill)
Volatility s
36
Single Index Model
(
)
(
)
b
a
e
r
r
r
r

-


-
f
m
f
i
i
i
i
Risk Prem
Market Risk Prem
or Index Risk Prem
a
the stocks expected return if the markets
excess return is zero
i
(rm - rf) 0
ßi(rm - rf) the component of return due to
movements in the market index
ei firm specific component, not due to market
movements
37
Risk Premium Format
38
Efficient Frontier With the Risk-Free Asset

• When a risky portfolio is combined with some
  allocation to a risk-free asset, the resulting
  risk/return combinations will lie on a straight
  line between the two. (Markowitz efficient
  frontier is converted from a curve into a
  straight.
• This straight line is called the Capital Market
  Line (CML) and improves investors risk-return
  trade-off.
• CML dominates the efficient frontier in the sense
  that for every point on the efficient frontier
  (except for the point where the CML intersects
  the efficient frontier), there is another point
  on the CML with the same risk and a higher
  expected return.

39
An Introduction to Asset Pricing Models

• Capital market theory extends portfolio theory
  and yields a model for pricing all risky assets.
• The application of capital market theory, the
  capital asset pricing model (CAPM), allows the
  determination of the required return for any
  asset.
• Assumptions
• All investors target points on the efficient
  frontier depending on individual risk-return
  utility functions.
• Investors can borrow or lend at risk-free rate
• Investors have homogeneous expectations
• Investors have the same one period investment
  horizon
• Investments are infinitely divisible
• There are no taxes or transaction costs
• There is no inflation and interest rates remain
  constant
• Capital markets are in equilibrium

40
The Risk-free Asset

• The assumption of risk-free asset is essential to
  the economy.
• The standard deviation of the risk-free assets
  return is zero because the return is certain.
• The risk-free rate should equal the expected long
  run growth rate of the economy with an adjustment
  for short-term liquidity.
• The covariance and correlation of the risk-free
  asset with any other asset or portfolio will
  always equal zero.

41
Risk-free Asset and Risky Portfolios

• E(Rport) (1-WA) (RFR) WAE(Ri) RFR
  WAE(Ri)-RFR
• sport WAs(Ri)
• E(Rport) RFR sportE(Ri)-RFR /s(Ri)
• Capital Market Line (CML) is the line of
  tangency between the RFR point on the vertical
  axis and the efficient frontier.
• All portfolios on CML are perfectly positively
  correlated.

42

• Market portfolio is a diversified portfolio where
  the unsystematic risk or the risk attributable to
  individual assets is eliminated.
• The remaining risk is systematic risk, which the
  variability in returns of all risky assets caused
  by macroeconomic variables.
• Market Portfolio is the line of tangency between
  the RFR point on the vertical axis and the
  efficient frontier.
• The market portfolio contains all risky assets.
  Any asset not contained in it will have no demand
  and no value.

43
Capital Asset Pricing Model (CAPM)

• CAPM is a model that predicts the expected return
  on each risky asset.
• Security Market Line (SML) visually represent
  the relationship between systematic risk and the
  expected or required rate of return on an asset.
• The risk measure of the asset is its systematic
  risk measured using beta (ß).
• E(Ri) RFR ßi(RM-RFR)
• ß is standardized because it divides an assets
  covariance Cov(i,M) with the market portfolio by
  the variance of the market portfolio (sM2).
• RM-RFR is the market risk premium

44
The Security Market Line

• The systematic risk is calculated as the
  covariance of the returns on security or
  portfolio i with the returns on the market
  portfolio, Cov (Ri, RM), divided by the variance
  of the returns on the market portfolio, s2M
• Betai Cov (Ri,RM)/ s2M

45
Using the SML for Security Selection

• The SML will tell us assets required returns
  from the SML, given their level of systematic
  risk (as measured by beta). We can compare this
  to the assets expected returns (given our
  forecasts of future prices and dividends) to
  identify undervalued assets and create the
  appropriate trading strategy.
• An asset with an expected return greater than its
  required return from the SML is undervalued we
  should buy it.
• An asset with an expected return less than the
  required return from the SML is overvalued we
  should sell it (or short sell it if were
  inclined to be aggressive).
• An asset with an expected return equal to its
  required return from the SML is properly valued
  were indifferent between buying and selling it.

46
Example Using the SML

• The following table contains information based on
  analysts forecasts for three stocks. The
  risk-free rate is 7 percent and the expected
  market return is 15 percent.
• Compute the expected and required return on each
  stock, determine whether each stock is
  undervalued, overvalued, or properly valued, and
  outline an appropriate trading strategy.

47
Example Using the SML

• Answer
• Expected and required returns are shown in the
  figure below

• Stock A is overvalued. It is expected to earn
  12, but based on its systematic risk it should
  earn 15.
• Stock B is undervalued. It is expected to earn
  17.5, but based on its systematic risk it should
  earn 13.4.
• Stock C is properly valued. It is expected to
  earn 16.6, and based on its systematic risk it
  should earn 16.6.
• The appropriate trading strategy is Short sell
  A, buy B and buy, sell, or ignore C.

48
Relaxing the CAPM assumptions

• The CAPM requires a number of assumptions, many
  of which do not reflect the true nature of the
  investment process. Let us study the impact on
  the CAPM of relaxing some of the assumptions
  required in the derivation of the model.
• Differential borrowing and lending rates
• Assumption investors are able to lend and
  borrow at the risk free rate. This is what makes
  the Capital Market Line (CML) straight.

49

• With unequal borrowing and lending rates, the
  CML follows the Markowitz efficient frontier
  (i.e. the no risk-free asset efficient frontier).
  Essentially, this puts a kink in the CML.
• Conclusion The CAPM cannot be derived without
  equal borrowing and lending rates or some
  substitute for equal borrowing and lending rates.
  

50
Relaxing the CAPM assumptions

• When the assumptions of CAPM are relaxed, the
  location of the SML will change, and individual
  investors will have a new SML.
• Taxes if investors have high tax rates, then CML
  and SML could be significantly different among
  investors.
• Transaction costs The cost trading the security
  may offset any potential excess return resulting
  from the trade ? securities will plot close to
  SML but not exactly on it.

51

• Homogeneous Expectation if all investors had
  different expectations about risk and return,
  then each would have a unique graph as a result
  of their divergence of expectations.
• One-planning period if one investor uses a
  one-year planning period and another uses a
  one-month planning period, then the two investors
  have different SML.

52
Substitute The zero beta version of the CAPM

• The zero beta version of the CAPM drops the
  assumption of the risk-free rate. In its place,
  it assumes that investors can find a portfolio of
  securities that are uncorrelated with the market.
  Using these securities, a diversified portfolio
  can be constructed. The diversification process
  will eliminate the portfolios unsystematic risk.
  What good is this? Since the securities in this
  portfolio are uncorrelated with the market, the
  portfolio will have a beta of zero. This means
  the portfolio will have no systematic risk.

53

• Further, diversification eliminates unsystematic
  risk. If the portfolio has no systematic risk and
  no unsystematic risk, it must have a total risk
  of zero. In other words, the zero beta portfolio
  is a riskless portfolio. Combining this zero beta
  portfolio with the Markowitz efficient frontier
  will create a straight CML. A straight CML allows
  for risk to be separated into its systematic and
  unsystematic portions so the SML can be drawn and
  the CAPM derived.

54
Substitute The zero beta version of the CAPM

• E(Rstock) E(Rzero beta portfolio) (Beta
  stock) E(Rmarket) E(Rzero beta portfolio)
• The expected return on the zero-beta portfolio
  will be greater than the risk-free lending rate,
  and the resulting security market line will have
  a smaller risk premium (i.e., a flatter slope).
• Conclusion The zero beta portfolio overcomes
  the problem of unequal borrowing and lending rates

55
BETA Stability Comparability

• I. Beta Stability
• Portfolio betas are more stable than individual
  betas. The more stocks in the portfolio, and the
  longer the estimation period, the more stable the
  beta estimate.
• II. Beta Comparability
• The variability in beta estimates is created by
  the use of different
• Index proxies to represent the market portfolio
• Holding periods to calculate historical returns
  (e.g. weekly or monthly)
• Time periods over which betas are measured.
• Adjustment methods to account for the tendency of
  betas to regress toward the mean.

56
Calculating Systematic Risk

• The regression model yields the assets
  characteristics line with the market portfolio.
• The characteristic line (CL) is the best fit
  through a scatter plot of returns for the risky
  asset and the market portfolio over some period
  of time.

57

• The slope of this regression line is the
  systematic risk for the asset.
• Ri,t ai biRM,t et
• Ri,t Return on asset I during period t
• RM,t Market rate of return during period t
• ai Regression intercept
• bi Systematic risk or beta of the asset
• et Random error term for period t
• Theoretically, the market portfolio should
  include all risky assets such as stocks and
  bonds, non-US stocks and bonds, real estate, and
  any other marketable risky asset.

58
Calculating BETA

• Beta is a standardized measure of systematic
  risk. It is calculated as
• ßi covi,M / s²M (si / sM) x ?i,M
• Where
• covi,M covariance between stock i and the
  market portfolio
• si standard deviation of stock i
• sM standard deviation of the market
  portfolio
• ?i,M correlation coefficient between
  stock i and the market portfolio
• Note that the beta of the market portfolio is one
  by definition.
• ßM s²M / s²M 1

59
Example Calculating Beta

• The covariance of stock A with the market
  portfolio M (covA,M) is 0.11 and the standard
  deviation of the market is 26. Calculate the
  beta of stock A.
• Answer
• First, we need to find the variance for the
  market. The variance is the standard deviation
  squared or 0.0676 ( 0.26²). Hence, the beta of
  stock A is
• ßA 0.1100 / 0.0676 1.63

60
Arbitrage Pricing Theory

• APT is an alternative to CAPM.
• APT requires fewer assumptions and considers
  multiple factors to explain the risk of an asset,
  in contrast to single-factor CAPM, which just
  uses the market return.

61

• APT assumes
• Perfect competition in capital markets
• More wealth is always preferable to less wealth
• A multiple factor model represents the random
  process by which asset returns are generated.

62
Arbitrage Pricing Theory

• Unlike CAPM, APT does not assume
• Investors have quadratic utility functions
• Asset returns are normally distributed
• A market portfolio containing all risky assets
  which is mean-variance efficient.
• Ri Ei bi1d1 bi2d2 bikdk et
• i 1 to N where N is the number of assets
• bik the sensitivity in assets returns to
  movements in a common factor
• dk a common factor with a mean of zero that
  influences the return on all assets (GDP,
  interest rate,..)

63

• APT can not explain the differences in returns
  for various securities because the model does not
  specify which factors impact security returns.

64
The Process of Portfolio Management

• Investment Process
• Identify the investors objectives
• Identify the constraints that the limit the means
  to achieving those objectives
• Construct investment policies that meet
  investors objectives and conform to the
  investors constraints.

65

• Eight major types of investors can be identified
  individuals, personal trusts, mutual funds,
  pension funds, endowment funds, life insurers,
  other insurers (non-life), and banks.

66

• Asset allocation
• The investment policy statement provides guidance
  as to which asset classes will be held as well as
  the ranges of weights held in each asset class.
• Asset allocation refers to the weighting across
  major asset classes (e.g., stocks, bonds, cash,
  real estate, etc.) based on capital market
  expectations.
• Studies have shown that most (85 - 95) of the
  performance of portfolios is due to the asset
  allocation decision.
• In contrast, security selection (the selection
  of specific securities) contributes less to the
  performance of portfolios.

67
Portfolio Objectives

• Portfolio objectives are often expressed in terms
  of trade-off between assumed risk and expected
  return.
• Individual investors portfolio objectives will
  generally depend on the age and the circumstances
  of the individual. Individuals are thought to
  follow a life-cycle investment process.
• Personal trusts are created when a person
  transfers the ownership of assets to a trust
  (trustee) for the benefit of one or more
  beneficiaries.

68

• Portfolio objectives and the common types of
  portfolio constraints for individual and
  institutional investors
• The key for determining individual investor
  objectives and constraints is the life cycle
  approach, which refers to the determination of
  risk- return positions of individuals at various
  life cycle changes. For example, younger
  investors typically can accept more risk than
  older investors. The life cycle approach can be
  broken down to 4 phases
• accumulation,
• consolidation,
• spending,
• gifting.

69

• Two types of beneficiaries exist Income
  beneficiaries that receive distributions during
  their lifetime from the investment income
  generated by the trust assets. Remaindermen
  receive the principal amount of the assets after
  the death of the income beneficiaries.
• Investment objectives of a trust are often more
  conservative than those for individuals.

70

• Accumulation phase - for investors in early to
  middle years of their careers, with low current
  wealth relative to their peak wealth years long
  term retirement planning goals, high-risk
  objectives.
• Consolidation phase for investors in middle
  career, with average current wealth relative to
  their peak wealth years long term retirement
  planning moderate risk objectives.

71
Spending phase for investors in early
retirement, wealth is peaking a major goal is
capital preservation conservative risk
objectives.   Gifting phase- for investors in
early retirement to late life, major goal estate
planning low risk objectives.   Individual
investor risk and return objectives
considerations include   ? Clients with a
capital preservation objective have very low
risk tolerance. ? Clients with a current income
objective want to generate income to supplement
earnings for consumption. They have a low risk
tolerance (e.g., retirees)