OPTIMAL RISKY PORTFOLIOS

1
OPTIMAL RISKY PORTFOLIOS

• CHAPTER 7

2
Outline of the Chapter

• Sources of risk and advantages of diversification
• Forming a portfolio, P with two risky assets
• Expected value and variance of the portfolio, P
• Correlation between two risky assets
• Finding optimum weights for the minimum-variance
  portfolio P
• Define portfolio opportunity sets
• Forming optimal portfolio with two risky and a
  risk-free asset
• Finding optimum weights of risky assets in
  Portfolio P (when there is a risk-free asset)
• Finding the optimum weight invested in the risky
  portfolio P
• The Markowitz portfolio selection model
• How to construct an optimum portfolio with many
  risky securities and a risk-free asset

3
Diversification and Portfolio Risk

• Sources of risk (uncertainty)
• Market risk
• Risk that comes from conditions in the general
  economy
• Business cycle, inflation, interest rates, and
  exchange rates...
• Systematic or nondiversifiable
• Firm-specific risk
• A companys success in research and development
  and personnel changes...
• Unique, diversifiable or nonsystematic

4
Diversification and Portfolio Risk (Continued)
5
Diversification and Portfolio Risk (Continued)

• Panel A
• All risk is firm-specific
• A diversification (including additional
  securities in the portfolio) can reduce the risk
  (portfolio standard deviation) to low levels
• All risk sources are independent and
  diversification reduces the exposure to any
  particular source of risk to a negligible level

6
Diversification and Portfolio Risk (Continued)

• Panel B
• Common sources of risk (market risk) affects all
  firms
• Diversification can reduce the risk but can not
  eliminate risk , can not decrease the risk to
  negligible level
• On average portfolio risk reduces with
  diversification, but the power of diversification
  is limited by the market risk

7
Portfolios of Two Risky Assets

• Efficient diversification
• Construct risky portfolios to provide the lowest
  possible risk for any given level of expected
  return
• Portfolio of two risky assets
• Asset allocation decision
• Two mutual funds
• D a bond portfolio (long-term debt securities)
• E a stock portfolio (equity securities)

8
Portfolios of Two Risky Assets (Continued)

• wD proportion invested in the bond fund
• wE1- wD proportion invested in the equity
  fund
• rP rate of return on the portfolio
• rD rate of return on the debt fund
• rE rate of return on the equity fund
• rP wD rD wErE
• E(rP) wD E(rD) wEE(rE)
• The expected return on the portofolio is a
  weighted average of expected returns on the
  component securities with portfolio proportions
  as weights

9
Portfolios of Two Risky Assets (Continued)

• s2Pw2Ds2D w2Es2E2wDwECov(rD,rE)
• where
• s2D variance of the debt fund
• s2E variance of the equity fund
• Cov(rD,rE) covariance of the returns on the
  debt and equity fund
• The variance of the portfolio is not a weighted
  average of the individual asset variances

10
Portfolios of Two Risky Assets (Continued)

• Variance of the portfolio is reduced if the
  covariance term between two risky assets is
  negative
• What about when covariance term is positive?
• Note that Cov(rD,rE)?DEsDsE
• where ?DE is the correlation coefficient
  between D and E
• s2Pw2Ds2D w2Es2E2wDwE ?DEsDsE

11
Portfolios of Two Risky Assets (Continued)

• If ?DE1
• Perfect positive correlation
• The variance equation simplifies to
• s2P(wDsD wEsE)2
• sP(wDsD wEsE)
• The standard deviation of the portfolio is the
  weighted average of the component standard
  deviations

12
Portfolios of Two Risky Assets (Continued)

• If ?DE-1
• Perfect negative correlation
• The variance equation simplifies to
• s2P(wDsD -wEsE)2
• sPwDsD wEsE
• The standard deviation of the portfolio is the
  absolute value of the weighted average of the
  component standard deviations
• with a negative sign

13
Portfolios of Two Risky Assets (Continued)

• -1 ?DE1
• Portfolio risk depends on the correlation between
  the returns of the assets in the portfolio
• As the correlation coefficient between two risky
  assets increases (decreases), the portfolio
  variance increases (decreases)
• As long as the correlation coefficient is less
  than 1, the portfolio standard deviation is less
  than the weighted average of the component
  standard deviations

14
Portfolios of Two Risky Assets (Continued)

• A hedge asset
• an asset which has a negative correlation with
  the other assets in the portfolio
• Such assets decrease the total risk of the
  portfolio
• ?lt0
• When ?-1, a perfectly hedged position can be
  obtained (sP0) by choosing the portfolio
  proportions to solve
• wDsD wEsE0
• wD sE /(sDsE)
• wE sD /(sDsE)1- wD

15
Portfolios of Two Risky Assets (Continued)

• Since the expected return of the portfolio is not
  affected from the correlations, it is preffered
  to include assets with low or negative
  correlations in the portfolio in order to
  decrease the risk (standard deviation)
• The portfolio expected return is the weighted
  average of its component expected returns, but
  its standard deviation is less than the weighted
  average of the component standard deviations
• Portfolios of less than perfectly correlated
  assets always offer better risk-return
  opportunities than the individual component
  securities on their own

16
Portfolios of Two Risky Assets (Continued)

• Example
• E(rP) wD E(rD) wEE(rE)
• wD 8 wE13
• s2Pw2D122w2E 2022wDwE .32012
• 144w2D400w2E144wDwE
• In order to find the expected return and risk
  combinations of different portfolios with
  different proportions of debt and equity we
  change the weights in each portoflio

17
Portfolios of Two Risky Assets (Continued)
18
Portfolios of Two Risky Assets (Continued)

• When the proportion invested in debt, wD,
  changes from 0 to 1 (the proportion invested in
  equity, wE, changes from 1 to 0) the portfolio
  expected return goes from 13 (expected return of
  equity funds) to 8 (expected return of debt
  funds)

19
Portfolios of Two Risky Assets (Continued)

• What if wDgt1 and wE lt0
• Sell short the equity fund
• Invest the proceeds in the debt fund
• Decrease the expected return of the porfolio
• What if wEgt1 and wD lt0
• Sell short the debt fund
• Invest the proceeds in the equity fund
• Increase the expected return of the portfolio

20
Portfolios of Two Risky Assets (Continued)

• The changes in the proportions of debt and
  equity funds also have effects on the portfolio
  risk (variance)
• The figure shows the relationship between
  standard deviation and portfolio weights
• ?0.30
• As the weight of the equity funds (stocks)
  increases from 0 to 1 in the portfolio, the
  portfolio standard deviation first falls and then
  rises

21
Portfolios of Two Risky Assets (Continued)

• The portfolio standard deviation decreases with
  the first diversification between stocks and
  bonds but then increases since the portfolio
  becomes heavily concentrated in stocks
• What is the minimum level to which portfolio
  standard deviation can be held?
• Find the local minimum of the variance function
• s2Pw2Ds2D w2Es2E2wDwE Cov(rD,rE)
• Substitute 1-wD for wE
• Take the first derivative with respect to wD
• Set the derivative equal to 0
• wmin(D)(s2E - Cov(rD,rE))/(s2D s2E -
  2Cov(rD,rE)

22
Portfolios of Two Risky Assets (Continued)

• In our example
• wmin(D)0.82
• wmin(E)1-0.820.18
• The standard deviation of minimum-variance
  portfolio
• sMinw2min(D)s2D w2min(E) s2E2wmin(D)
  wmin(E)Cov(rD,rE)1/2
• (0.822122) (0.182202)(20.820.1872
  ) ½
• 11.45

23
Portfolios of Two Risky Assets (Continued)

• The lines in the Figure 7.4 plots the portfolio
  standard deviation as a function of investment
  proportions (for different weights of the debt
  and the stock (equity) funds) for different
  correlation coefficients (?)
• All the lines pass through the two undiversified
  portfolios
• when wD1 and wE 1
• The minimum-variance portfolio has a standard
  deviation smaller than that of either of the
  individual component assets
• Effect of diversification

24
Portfolios of Two Risky Assets (Continued)

• When ?1
• There is no advantage from diversification
• The portfolio standard deviation is the weighted
  average of the component asset standard
  deviations
• When ?0
• Assets are uncorrelated
• Lower the correlation between assets, lower the
  portfolio risk and more effective the
  diversification is

25
Portfolios of Two Risky Assets (Continued)

• When ?-1
• Perfect hedge potential
• In this case
• wmin(D) sE /(sDsE)
• 20/(1220)0.625
• wmin(E) 1- wmin(D)
• 1-0.6250.375
• sMin(P)0
• Maximum advantage from diversification

26
Portfolios of Two Risky Assets (Continued)

• When we combine Figure 7.3 and 7.4 we can
  show the relationship between portfolio risk and
  expected return
• Given the parameters of the available assets
• The values for expected returns and standard
  deviations are from Table 7.3
• The lines show the portfolio opportunity sets for
  different correlation coefficients

27
Portfolios of Two Risky Assets (Continued)

• Portfolio Opportunity Set
• Shows all the combinations of portfolio expected
  return and standard deviation that can be
  constructed from the two available assets
• To sum up,
• Expected return of any portfolio is the weighted
  average of the asset expected returns
• This is not the case for standard deviation
  (risk)
• There is a benefit in diversification
• The benefits of diversification depends on the
  correlation coefficient
• -1.0 lt ? lt 1.0
• The smaller the correlation, the greater the risk
  reduction potential

28
Asset Allocation with Stocks, Bonds and Bills

• In the last chapter we learnt how to divide the
  capital between risky and risk-free assets
• In the last section we analyse how the standard
  deviation and expected return of the risky
  portfolio changes with different proportions and
  correlation coefficients
• In this section we will examine how to divide the
  capital between three, two risky (stocks (equity)
  and bonds) and a risk-free, assets
• The optimal portfolio with two risky assets and a
  risk-free asset

29
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• E(rD)8, E(rE)13
• sD 12, sE20
• rf5
• Two possible CALs are drawn from the risk-free
  rate to the two feasible (on the portfolio
  oppotunity set of a given ?) portfolios A and B

30
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• Portfolio A
• Minimum-variance portfolio
• wD(A)0.82
• wE(A)0.18
• E(rP) wD E(rD) wEE(rE)
• 0.82 8 0.18138.90
• s2w2Ds2D w2Es2E2wDwE ?DEsDsE
• s(0.822122) (0.182202)(20.820.1872
  ) ½
• 11.45
• Reward-to-volatility ratio
• SAE(rA)-rf/ sA
• (8.9-5)/11.450.34

31
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• Portfolio B
• wD(B)0.70
• wE(B)0.30
• E(rP) wD E(rD) wEE(rE)
• 0.70 8 0.30139.50
• s2w2Ds2D w2Es2E2wDwE ?DEsDsE
• s(0.702122) (0.302202)(20.700.3072
  ) ½
• 11.70 (Table 7.3)
• Reward-to-volatility ratio
• SBE(rB)-rf/ sB
• (9.50-5)/11.700.38

32
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• Thus the reward-to-volatility is higher in
  portfolio B. B dominates A (minimum-variance
  portfolio)
• Where is the highest feasible reward-to-volatility
  ratio with the given expected returns and
  standard deviations of risky assets and the risk
  free asset?
• Tangency point of CAL to portfolio opportunity
  set

33
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• The tangency portfolio P is the optimal risky
  portfolio to mix with T-Bills
• Maximize SPE(rP)-rf/ sP with respect to wi
• subject to Swi1

34
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• Weights of the optimal risky portfolio, P
• where
• R denotes the excess rates of return
• RE(r)-rf

35
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• Example
• E(rD)8, E(rE)13, rf5
• sD 12, sE20, Cov(rD, rE)72
• wD(8-5)400-(13-5)72/(8-5)400(13-5)144-(8-51
  3-5)720.40
• wE1-0.400.60
• E(rP) wD E(rD) wEE(rE)
• 0.4 8 0.61311
• s2Pw2D122w2E 2022wDwE 0.32012
• 1440.424000.621440.40.6
• sP14.2
• SP(11-5)/14.20.42

36
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• Now we have the optimal risky portfolio P
• We know the optimal weights of debt and equity
  (stock) in the portfolio
• We have to decide now how much of the capital
  should an investor invest in the portfolio P
  given his coefficient of risk aversion
• The capital allocation between risky and
  risk-free portfolio
• y?

37
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• Example
• A4
• y E(rP)-rf/As2P
• (0.11-0.05)/40.14220.7439
• Thus the investor invests 74.39 of his capital
  in portfolio P (risky assets) and 25.61 of his
  capital in risk-free assets (T-bills)
• The 40 of portfolio P is composed of bonds and
  60 of portfolio P is composed of stocks (equity)

38
Asset Allocation with Stocks, Bonds and Bills
(Continued)

• How do we find portfolio C?
• Specify the expected returns, vaiances, and
  covariances of all securties
• Establish the risky portfolio P
• Calculate the weights of debt and equity in the
  optimal risky portfolio
• Calculate the expected risk and return of
  portfolio P by using the weights computed in part
  a
• Calculate y. Decide on how much to invest in
  risky and risk-free asset

39
The Markowitz Portfolio Selection Model

• Portfolio construction problem with many risky
  securities and a risk-free asset
• How?
• Identify the risk and return combinations
  available from the set of risky assets
• Identify the optimal portfolio of risky assets by
  finding the portfolio weights that result in the
  steepest CAL
• Choose an appropriate complete portfolio by
  mixing the risk-free asset with the optimal risky
  portfolio

40
The Markowitz Portfolio Selection Model
(Continued)

• Security Selection
• In the first step of identifying risk-return
  combinations for different portfolios the
  portfolio manager needs to know a set of
  estimates for the expected returns of each
  security and a set of estimates for the
  covariance matrix
• Then the expected return and variance of any
  risky portfolio for different weights in each
  security , wi, can be computed

41
The Markowitz Portfolio Selection Model
(Continued)

• The risk and return opportunities available to
  the investor are summarized by the
  minimum-variance frontier of risky assets
• Minimum-variance frontier
• Graph of the lowest possible variance that can be
  attained for a given portfolio expected return
• Given the input for expected returns, variances
  and covariances the minimum-variance portfolio
  for any targeted expected return can be computed

42
The Markowitz Portfolio Selection Model
(Continued)

• All the individual assets lie to the right
  inside the frontier
• Diversification investments leads to portfolios
  with higher expected returns and lower standard
  deviations
• Efficient frontier
• The part of the minimum-variance frontier that
  lies above the global minimum-variance portfolio
• The portfolios on the efficient frontier are the
  ones that provide the best risk-return
  combinations and thus are candidates for the
  optimal portfolio

43
The Markowitz Portfolio Selection Model
(Continued)

• In 1952, Harry Markowitz published a model of
  portfolio selection by taking into account the
  diversification principle
• He identifies the efficient set of portfolio (the
  efficient frontier of risky assets)
• The efficient frontier shows the portfolios with
  the highest expected return for any risk level,
  or the portfolios that minimizes the variance for
  any targeted expected return
• The efficient frontier can be adjusted by the
  portfolio managers due to different constraints
  such as prohibiton of short selling or minimum
  level of expected dividend yield but these
  contraints will decrease the reward-to-volatility
  ratio offered

44
The Markowitz Portfolio Selection Model
(Continued)

• Capital Allocation
• Identify the optimal risky portfolio
• Includes risk-free asset
• Search for the CAL with the highest
  reward-to-volatility ratio (the steepest CAL-one
  with the highest slope)
• Find the CAL with the optimal portfolio P
• The one that starts from F, return of the
  risk-free asset, and is tangent to the efficient
  frontier

45
The Markowitz Portfolio Selection Model
(Continued)

• Portfolio P is the optimal risky portfolio
• Portfolio manager will offer the same risky
  portfolio P to all clients regardless of their
  risk aversion

46
The Markowitz Portfolio Selection Model
(Continued)

• Lastly, investor decides on the proportions to
  invest in optimal portfolio P and T-Bills (risky
  and risk-free assets).
• An investor has to decide where to stay along the
  CAL
• This is where the degree of risk aversion of the
  cleint comes into play
• The more risk averse investors will invest more
  in the risk-free asset and less in the optimal
  risky portfolio
• For all investors the optimal risky portfolio is P

47
The Markowitz Portfolio Selection Model
(Continued)

• The separation property tells us that the
  portfolio choice problem may be separated into
  two independent tasks
• Determination of the optimal risky portfolio
• Purely technical
• Given the expected returns, and covariances the
  optimal risky portofolio is same for all
  investors
• Allocation of the complete portfolio between
  T-bills and the risky portfolio
• Depends on personal preference
• An investor is the decision maker

48
The Markowitz Portfolio Selection Model
(Continued)

• The Power of Diversification
• The general formula for the variance of a
  portfolio
• Diversification strategy is the equally weighted
  portfolio, wi1/n
• n variance and n(n-1) covariance terms
• If we define average variance and covariance
  terms as presented in the next slide the
  portfolio variance can be written as the bottom
  equation in the next slide

49
The Markowitz Portfolio Selection Model
(Continued)
50
The Markowitz Portfolio Selection Model
(Continued)

• The effect of diversification
• When the average covariance among security
  returns is 0
• When all risk is firm-specific
• Portfolio variance can be driven down to 0
• Second term of the last equation becomes 0
• First term approaches to 0 as n becomes larger

51
The Markowitz Portfolio Selection Model
(Continued)

• When economy wide risk factors cause positive
  correlation among stock returns
• As n increases portfolio variance remains
  positive
• The firm-specific risk represented by the first
  term in the last equation diversified away
• The second term approaches the average covariance
• (n-1)/n1-1/n
• The irreducible risk of a diversified portfolio
  depends on the covariance of the returns of the
  component securities
• Function of the systematic factors in the economy
• When there is a diversified portfolio the
  contribution to the portfolio risk (variance) of
  a particular security will depend on the
  covariance of that securitys return with those
  of other securities

52
Risk Pooling, Risk Sharing and Risk in the Long
Run

• Example
• Insurance company offers a 1-year policy on
  residential property
• Value of the residential property100,000
• Probability p0.001 there is going to be a
  disaster and the insurance company will pay
  100,000
• Porbability (1-p)0.999 there is not going to
  happen anything so the insurance company will not
  loose anything and its payout is going to be 0
• The insurance company sets aside 100,000 to
  cover its potential payout

53
Risk Pooling, Risk Sharing and Risk in the Long
Run (Continued)

• rf5
• E(r)0.001100,0000.9990100
• The insurer premium 120
• The insurer can invest this amount in T-bills and
  get a year end income 126 (120(1.05))
• Insurers expected profit126-10026
• Risk premium the extra amount earned for bearing
  the risk
• 2.6 basis points0.026 (26/100,000)
• sr 0.001(100,000-100)20.999(0-100)21/2
• 3,160.70

54
Risk Pooling, Risk Sharing and Risk in the Long
Run (Continued)

• The standard deviation of the return is 3.16 of
  the 100,000 investment but the risk premium on
  this investment is 0.26
• The risk is too high for the return
• Diversification
• Risk Pooling
• Sale of additional independent policies
• i.e 10,000
• Expected rate of return on each independent
  policy is 0.026 expected rate of return on the
  collection of policies (equally weighted
  portfolio)
• s2P(1/n)s2
• sP(1/n)1/2s0.0316

55
Risk Pooling, Risk Sharing and Risk in the Long
Run (Continued)

• It looks like as the insurance company sells more
  independent policies the standard deviation of
  the rate of return on equity capital falls,
  keeping expected return constant
• However the problem is increasing the size of the
  bundle of policies does not make for
  diversification
• Diversification
• Dividing a fixed investment budget across more
  assets

56
Risk Pooling, Risk Sharing and Risk in the Long
Run (Continued)

• Risk Sharing
• Then what is the aim of the insurance companies
  when they are increasing their number of policies
• Risk sharing they try to distribute the fixed
  amount of risk among many investors
• The insurance companies try to limit the exposure
  to any single source of risk
• Not every home will be burnt in a fire
• The portfolio risk management is about the
  allocation of a fixed investment budget to assets
  that are not perfectly correlated
• Expected returns, variances and covariances are
  sufficient to find the optimal portfolios