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Efficient Diversification - Marriott School

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					     Bm410 Investments




Theory 2: Modern Portfolio
Theory (MPT) and Efficient
          Frontiers

         How to reduce
   the risk of your portfolio
             Real Objectives

 Remember “Portfolio Theory” is a
  difficult subject to understand. It is in
  essence the attempt to answer the two
  critical questions:
     •  1. How do you build an optimal
        portfolio?
     • 2. How do you price assets?
       • The next few class periods will be
           devoted to answering those two
           questions! Hang in there!!!
           Other Objectives

A. Show how covariance and correlation
 affect the power of diversification to
 reduce portfolio risk (to give the
 “optimal” portfolio)
B. Construct efficient portfolios
C. Understand the theory behind factor
 models
                                                         CAL:
            E(r)      Review of the CAL                 (Capital
   This graph is the risk return combination           Allocation
   available by choosing different values of y.          Line)
   Note we have E(r) and variance on the axis.
                                          P
E(rp) = 15%
                                                      Risk premium
                                                     E(rp) - rf = 8%
                     S = 8/22
    rf = 7%
              FSlope: Reward to variability ratio:
               ratio of risk premium to std. dev.

          0                              P = 22%         s
  A Portfolio Theory Timeline




CAL MPT CAPM SML                      APT


1920s   1950s      1960s    1960s     1970s
        Note: Dates are approximate
     Risk Aversion and Allocation

Key concepts regarding risk:
  • Greater levels of risk aversion lead to larger
    proportions of the risk free rate
  • Lower levels of risk aversion lead to larger
    proportions of the portfolio of risky assets
  • Willingness to accept high levels of risk for high
    levels of returns would result in leveraged
    combinations
    A. Show how Covariance and
     Correlation affect Diversification
• Previously, investors assumed that there was
  no risk in investing in government securities
      • While there was no “default” risk, other types of
        risk became apparent
   • There came a change in this “risk” view
      • People started realizing that there was risk to the
        so-called risk-free asset—other risks
          • The risk-free asset wasn’t so risk-free
            anymore
      • Analysts started using variance (standard
        deviation) as a measure of risk or volatility
          • This opened up a world of opportunities
Harry Markowitz—the father of Modern
       Portfolio Theory (MPT)
  Harry Markowitz came up with Modern
   Portfolio Theory in 1952
    • Key was his point of view regarding portfolios
         • He looked at the assets and their contribution to
           the overall portfolio risk, not just individual risk!
         • He noted that investors must be compensated for
           risk that cannot be diversified away, but not for
           risk that was diversifiable!
    • And he did it all without computers or calculators
  He was later awarded the Nobel Prize in Economics for
   this important work. This chapter is based on his work
                MPT Notations I
                 – (Just memorize them)

 1. The return (or expected return E(rp)) of a
  portfolio is the weighted return of each of the
  assets.
           Return:            Expected Return:
      rp   = w1r1 +w2r2    E(rp) = w1E(r1) +w2E(r2)
                      Note 1: w1 +w2 = 1

             w1=weight asset1, r1 = return asset1,
   E(r) = Expected Return
       • Note 1 means that the portfolio is subject to a
           “no short-selling” constraint that says the
           weights of all assets must equal 100%
         MPT Notations II (continued)

 2. The risk (σp2 or variance) of a portfolio is
  the squared weighted risk (variance) of each
  asset plus the cross-product
          σp2 = w12σ12 + w22σ22 + 2w1w2 (ρ1,2σ1σ2 )
                                    cross-term

   •      Note: It is not just the weighted risk. It is the
          squared weighted risk plus the cross-product
              (This is a two security portfolio)
   σ1 is the standard deviation (σ12 variance) of asset 1
 ρ1,2 is the correlation coefficient between assets 1 and 2
                Problem #1

What is the relationship of the portfolio
 standard deviation to the weighted
 average of the standard deviations of the
 component assets?

Remember:
     σp2 = w12σ12 + w22σ22 + 2w1w2 (ρ1,2σ1σ2 )
               Answer #1

In the special case that all assets are
 perfectly correlated, the portfolio
 standard deviation will be equal to the
 weighted average of the component
 standard deviations. Otherwise, the
 portfolio standard deviations will be less
 than the weighted average of the
 component standard deviations.
        MPT Notations III (continued)

 3. The measure of the way the assets in the
  portfolio move together is given by its
  covariance.
                Covariance = ( ρ1,2 ) * σ1σ2
  • By itself, the covariance is just a number.
     However, if we divide the covariance by the
     standard deviation of each asset, we get the
     traditional correlation coefficient
     Correlation Coefficient (ρ1,2 ) = covariance / σ1σ2
 The goal for an MPT “optimal” portfolio:
  • To maximize return for a given level of risk
  • To minimize risk for a given level of return
           MPT Notations (continued)

 The Key to diversification is the correlation =
  r (rho)
   sp2 = w12s12 + w22s22 + 2w1w2 (r1,2s1s2 )


   This 2w1w2 (r1,2s1s2 ) is called the cross product
 What happens to the cross produce when the r1,2 = x?
      If r1,2 is +1 = The cross product is added
      If r1,2 is 0 = The cross product is dropped
      If r1,2 is –1 = The cross product is subtracted
     E(r)
               The Impact of Correlation
13%
               r = -1

                        r=0           r = .3
            r = -1
8%                              r=1



            Two security portfolios with different
                        correlations

                          12%             20%        St. Dev
               Problem #2

An investor is consider adding another
 investment to a portfolio. To achieve the
 maximum diversification benefits, the
 investor should add, if possible, an
 investment that has which of the
 following correlation coefficients with
 the other investments in the portfolio?
  • a. -1.0 b. -0.5 c. 0.0 d. +1.0
               Answer #2

a. –1.0. This is perfectly negative
 correlation, which helps achieve
 maximum diversification. However,
 adding any company will a correlation
 less than 1 will help reduce portfolio risk.
               Questions

Do you understand the importance of the
 correlation coefficient in determining
 portfolio risk?
             B. Efficient Frontiers

What is the efficient frontier?
  • It is a graphical representation of a combination of
    all assets that will give either the highest return for a
    given level of risk, or the lowest risk for a given
    level of return
 Why do we care?
  • If our portfolio is on the efficient frontier, we will
    have the highest return for our risk level (or lowest
    risk for our return level) and is our “optimal
    portfolio”
  • Being on the optimal frontier is a goal to strive for
         Efficient Frontiers (continued)

We have calculated a two security portfolio. What
about a three security portfolio?
   • The return is: rp = w1r1 + w2r2 + w3r3
   • The risk is: sp2 = w12s12 + w22s22 + w32s32 +
                    2w1w2 Cov(r1r2) +
                    2w1w3 Cov(r1r3) +
                    2w2w3 Cov(r2r3)
   • Note with 3 variables you have 3 cross terms
      Remember, Cov(r1r2) = (r1,2s1s2 )
         Efficient Frontiers (continued)

• In General, For an “n” Security Portfolio:
  rp = Weighted average of the “n” securities
      You must have “n” forecasts for expected returns
   sp2 = Standard deviation of the portfolio
      You must consider all pair wise covariance
      measures, or (n*(n-1))/2 calculations
   • If you have 50 stocks, you will have 50 expected
      returns and (50*50-1)/2) or:
        • 1,225 correlation coefficients (or covariance)
          measures.
        • That is a lot to calculate.
          Efficient Frontiers (continued)

 What is the Mean Variance Criterion?
   • Mean variance criterion (the criteria of return and risk)
     states:
       • Portfolio A dominates Portfolio B if:
           • E(rA) >= E(rB)
              • [the expected return of asset A is greater or
                 equal to the expected return of asset B]
                   and
           • σA <= σB
              • [the variance of A is less than or equal to the
                 variance of B]
          Efficient Frontiers (continued)

What is the minimum variance portfolio?
  • It is the combination of assets in a portfolio which
     gives the lowest available risk
Why is it useful?
  • It gives a starting point of minimum portfolio risk.
  • From this point you can take on additional risk, but
     only if you want and are compensated for this risk

 What is the formula for calculating the MVP?
  W1 = [ σ22 - ρ1,2(σ1 * σ2 )] / [σ12 + σ22 - 2ρ1,2(σ1 * σ2 )]
                       W2 = (1-W1)
                     Problem #3

 Suppose we had two assets with ρ1,2 = .2:
   • Asset 1: E(r1) = .10        σ1 = .15
   • Asset 2: E(r2) = .14        σ2 = .20
 Calculate the minimum variance portfolio, i.e. the point of
  lowest risk. What are the:
   • a. Weights of the assets in the minimum variance
     portfolio (MVP),
   • b. The expected return of that portfolio, and
   • c. The standard deviation of that portfolio?
 Remember:
   W1 = [ σ22 - ρ1,2(σ1 * σ2 )] / [σ12 + σ22 - 2ρ1,2(σ1 * σ2 )]
                            W2 = (1-W1)
                     Answer #3

 a. The minimum variance portfolio weight is at
  W1 = [ σ22 - ρ1,2(σ1 * σ2 )] / [σ12 + σ22 - 2ρ1,2(σ1 *
  σ2 )]:
   • W1 = [ .22 - .2(.15*.2 )] / [.152+.22 -2*.2(.15*.2 )] or
      • W1 = .673 or 67.3%
   • W2 = (1-.673) or
      • W2 = .327 or 32.7%
 b. The expected return of the MVP is
   • rp = w1r1 +w2r2 :
   • rp = .673* .10 + .327 * .14 = 11.3%
                 Answer #3

 c. The minimum variance portfolio risk or
  standard deviation is:
    sp2 = w12s12 + w22s22 + 2w1w2 (r1,2s1s2 )
    sp = [.6732 * .152 + .3272 * .22 + 2 * .673
     *.327 * .2 * .15 * .2]1/2
    sp = .1308 or 13.08%
       Extending These Concepts
            to All Securities
Key Points:
   The optimal combinations result in lowest level of
    risk for a given return
   The optimal combinations dominant, i.e., are mean
    variance efficient and pass the mean variance
    criterion
   The optimal combinations is described as the
    efficient frontier
   The goal for an MPT “optimal” portfolio:
      To maximize return for a given level of risk
      To minimize risk for a given level of return
       The Efficient Frontier and Minimum
                Variance Portfolio
               Efficient
               frontier
E(r)
                                   Individual
        Global                       assets
       minimum
       variance
       portfolio                Minimum
                                variance
                                 frontier

                                 Standard Deviation
  Extending the Efficient Frontier to
       Include Riskless Asset
 What happens if we include a riskless asset in
  the portfolio?
   • The optimal combination goes from
     curvilinear to linear
   • A single combination of risky and riskless
     assets will dominate
    E(r)   Efficient Frontier with CALs
                         CAL (P)         CAL (A)
                                   M
M
                    P
P
                                           CAL (Global
                                        minimum variance)
A                A
                G

F


                     P     P&F M       A&F           s
         Dominant CAL with a
        Risk-Free Investment (F)
 What happens when you combine a risk-free
  asset with a dominant CAL?
   • CAL(FP) dominates other lines -- it has the
     best risk/return or the largest slope
      • Slope = (E(R) - Rf) / s
        [ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA]
      • Regardless of risk preferences,
        combinations of P & F dominate
                Implications for
             Portfolio Construction
 Determine your opportunity set
   • Determine your preferred set of assets, i.e., which
     assets are potential candidates for the portfolio
   • Calculate your efficient frontier (and discard any
     portfolios below the minimum variance portfolio
 Choose the optimal risky portfolio (ORP)
   • This dominates all alternative feasible lines
 Choose your appropriate mix between the risky (ORP)
  and the risk free (T-bills) assets
   • The result is called the separation property:
     portfolio choice is two independent decisions: the
     determination of the ORP and the personal choice
     of the best mix of the risky and risk free asset.
       Choosing a Single Portfolio

How do you choose a single portfolio
 among all the options on the efficient
 frontier?
  • One way is to use iso-utility curves, two-
    dimensional graphs of investors preferences

 More risk Moderately    Less risk
  averse risk averse      averse


                                      Risk
                                     Loving
           CAL and Utility Curves
    E(r)
                    CAL (P)



                              P
P
                                           CAL (Global
                                        minimum varianc




F


                P     P&F M       A&F          s
         Implementation of MPT

 What has happened since the development of
  Modern Portfolio Theory in 1956?
  • It has not caught on as quickly as would have been
    expected.
      • This was due to two problems:
          • 1. Too much information was needed
          • 2. Too much computational power was
            required to calculate optimal portfolios
   Implementation of MPT (continued)

 What was needed:
  • Simplification of the amount and type of
    information
      For one portfolio (i.e. one point on the frontier)
       of 100 securities, you need:
         100 expected returns and 4,950 covariance
           calculations (n * (n-1) / 2 ) (or 100 standard
           deviations and 4,950 correlation
           coefficients)
  • Simplification of computational procedures
      Perhaps there are other ways to forecast that
       covariance matrix, i.e. single and multi index-
       models, grouping techniques, etc.
              Questions

Do we understand the concept of
 efficient frontiers?
        C. Understand the Theory
           behind Factor Models
 What about if, instead of doing all the calculations for
  your portfolio as suggested by MPT, you used instead
  an optimal proxy for all the stocks in the market, at P
  (kind of like an optimal index fund)?
    You could compare your stock to this proxy, and
     your portfolio risk standard would be the weighted
     risk standard for all your stocks
       It would simplify dramatically the amount of
         work to be done by analysts and portfolio
         managers (analysts like that!)
    You would create, in essence, a single- (or multi-)
     factor model, depending on your assumptions
           Prelude to Factor Models
    E(r)
                    CAL (P)
                               M

                P
P                             P becomes our optimal
                              proxy for the market,
                              similar to an index
                              fund



F
                              Standard deviation
                 Factor Models

 What are factor models?
   Statistical models designed to estimate both the
    systematic (marketed related) and firm-specific
    (individual firm related) risk with the goal of
    relating risk (both types) to investment returns on
    assets.
 What are single factor models?
   Factor models which assume that stock prices
    move together because of a common movement,
    which is assumed to be the market
 What are multi-factor models?
   Factor models which assume that stock prices
    move based on more factors than the common
    movement, the market, alone.
             Single Factor Model

 The Single Factor Asset Model:
                       Ri = E(ri) + ßiM + ei
 What is it saying?
   • The excess return on a security (over cash) is equal
     to the expected excess holding period return plus
     the impact of:
       • a. systematic factors or market related surprises
         (zero assumes there are “no surprises”), and
       • b. firm-specific factors, or firm-specific
         surprises
 Note that both M and ei have zero expected values as
  each represent the impact of unanticipated events
       Single Factor Model (continued)

 Single Factor Model notation and terms:
   • Notation: Ri = E(ri) + ßiM + ei
      • ßi = index of a securities’ sensitivity to the
         supposed market factor
      • M= some macro factor, which in this case, is the
         unanticipated movement of some macro factor
 Key assumption:
   • The common factor is a broad market index, such
     as the S&P500, and is at P on the efficient frontier.
      • Is it really a broad index? Is it at P?
           • This is a major simplification of reality
             Single Factor Model:
                 Specification

 Is this a good model?
  •     A model is of little use unless we can test it. It is
        not testable in its above form. To make it
        testable, we:
      •     1. Use the rate of return on a broad index of
            securities (i.e., the S&P 500) for a proxy. So
            now ßiM becomes ßiRm, or M is equal to Rm,
            the return on the market
      •     2. The expected holding period return E(Ri)
            becomes ai, because that is the stock’s excess
            return assuming the market’s excess return is
            zero
            Single Factor Model:
            Specification (continued)
 So the new model, which is now testable,
  becomes:
   • Ri = ai + ßiRm + ei
   or substituting in the risk free rate becomes
              (ri-rf) = ai + ßi(rm - rf) + ei
                  Single Factor Model:
                  Specification (continued)

      (ri - rf) = a i + ßi(rm - rf) + ei

      Risk Premium       Market Risk Premium            Firm
                                                        specific
                          or Index Risk Premium
                                                        Risk
a i = the stock’s expected return if the
        market’s excess return is zero          (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
            Single Factor Model:
            Specification (continued)
 Using the Risk Premium format
  • Let:
     • Ri = (ri - rf)
     • Rm = (rm - rf)
  • The equation become
     • Ri = ai + ßi(Rm) + ei
                      Single Factor Model:
         Fitting the Data: Regress Ri with Rm
Excess Returns (i)
Or ri - rf
         . .  .         . .. .         . .. . Characteristic
                                                           Security
                     .                        . . Line or best fit
            . .                . .. .
          . . .
                           . . . Excess returns
      .  .. .. .
             . .          . . .                 on market index
                                      .. . .Or .r - r
              . . .. . .
                                                       m   f

       .   . Representation of the reg. line = E(R |R ) = a + ß R .
   Algebraic                                       i   m       i   i   m
   The slope is also called the slope coefficient or simply beta.
      Note that this line is average tendencies, not actual. It shows
      the effect of the index return on our expectations of Ri
        Single Factor Model:
       Key Components of Risk
 What are the key components of risk?
   • Market or systematic risk
      • Risk related to the macro economic
        factor or market index
   • Unsystematic or firm specific risk
      • Risk not related to the macro factor or
        market index
 Total risk = Systematic + Unsystematic
        Single Factor Model:
    Advantages and Disadvantages
Advantages
   Reduces the number of inputs for diversification
   Easier for security analysts to specialize
   Fewer calculations
Disadvantages
   May be too much of a simplification
   May be a far cry from reality
   There may be more than a single factor to explain
    market returns
               Problem #4

Investors expect the market rate of return
 to be 10%. The expected rate of return
 on the stock with a beta of 1.2 is
 currently 12%. If the market return this
 year turns out to be 8%, how would you
 review your expectations of the rate of
 return on the stock?
                   Answer #4

The expected return on the stock would be your beta
  (1.2) times the market return or:
      1.2 * 8% = 9.6%
Likewise, you could also determine how much the
  return would decrease by multiplying the beta times
  the change in the market return or:
      1.2 * (8%-10%) = -2.4% + 12% = 9.6%
              Questions

Do we understand how factor models came
 about?
         Review of Objectives

A. Can you see how covariance and
  correlation affect the power of
  diversification to reduce portfolio risk?
B. Can you construct efficient portfolios?
C. Do you understand the theory (and
  history) behind factor models?
                   Question

A three-asset portfolio has the following
 characteristics:
     • AssetE(r)   s.d.   Weight
       A           15%    22%      .5
       B           10%     8%      .4
       C            6%     3%      .1


  What is the expected return on this 3
  asset portfolio?
                  Answer

E(r) = waE(r)a + wb E(r)b + wcE(r)c

= .5 (.15) + .4(.10) + .1(.06)

= .075 + .04 + .006 = .121 or 12.1%
                        Problem

Consistent with capital markets theory,
     systematic risk:
i. Refers to the variability in all risky assets
     caused by macroeconomic and other
     aggregate market-related variables
ii. Is measured by the coefficient of variation of
     returns on the market portfolio
iii. Refers to non-diversifiable risk
a. i only b. ii only c. i and iii only d. ii and iii only
                  Answer

c. i and iii only. Systematic risk is non
   diversifiable, market-related risk.

				
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