optimal risky portfolios asset allocations n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Optimal Risky Portfolios- Asset Allocations PowerPoint Presentation
Download Presentation
Optimal Risky Portfolios- Asset Allocations

Loading in 2 Seconds...

play fullscreen
1 / 79

Optimal Risky Portfolios- Asset Allocations - PowerPoint PPT Presentation


  • 107 Views
  • Uploaded on

Optimal Risky Portfolios- Asset Allocations. BKM Ch 7. Asset Allocation. Idea from bank account to diversified portfolio principles are the same for any number of stocks Discussion A. bonds and stocks B. bills, bonds and stocks C. any number of risky assets.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Optimal Risky Portfolios- Asset Allocations' - ailish


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
asset allocation
Asset Allocation
  • Idea
      • from bank account to diversified portfolio
      • principles are the same for any number of stocks
  • Discussion
      • A. bonds and stocks
      • B. bills, bonds and stocks
      • C. any number of risky assets

Bahattin Buyuksahin, JHU , Investment

diversification and portfolio risk
Diversification and Portfolio Risk
  • Market risk
    • Systematic or nondiversifiable
  • Firm-specific risk
    • Diversifiable or nonsystematic

Bahattin Buyuksahin, JHU , Investment

figure 7 1 portfolio risk as a function of the number of stocks in the portfolio
Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio

Bahattin Buyuksahin, JHU , Investment

figure 7 2 portfolio diversification
Figure 7.2 Portfolio Diversification

Bahattin Buyuksahin, JHU , Investment

covariance and correlation
Covariance and Correlation
  • Portfolio risk depends on the correlation between the returns of the assets in the portfolio
  • Covariance and the correlation coefficient provide a measure of the way returns two assets vary

Bahattin Buyuksahin, JHU , Investment

two security portfolio return
Two-Security Portfolio: Return

Bahattin Buyuksahin, JHU , Investment

two security portfolio risk

= Variance of Security D

= Variance of Security E

= Covariance of returns for

Security D and Security E

Two-Security Portfolio: Risk

Bahattin Buyuksahin, JHU , Investment

two security portfolio risk continued
Two-Security Portfolio: Risk Continued
  • Another way to express variance of the portfolio:

Bahattin Buyuksahin, JHU , Investment

covariance
Covariance

Cov(rD,rE) = DEDE

D,E = Correlation coefficient of

returns

D = Standard deviation of

returns for Security D

E = Standard deviation of

returns for Security E

Bahattin Buyuksahin, JHU , Investment

correlation coefficients possible values
Correlation Coefficients: Possible Values

Range of values for 1,2

+ 1.0 >r> -1.0

If r = 1.0, the securities would be perfectly positively correlated

If r = - 1.0, the securities would be perfectly negatively correlated

Bahattin Buyuksahin, JHU , Investment

table 7 1 descriptive statistics for two mutual funds
Table 7.1 Descriptive Statistics for Two Mutual Funds

Bahattin Buyuksahin, JHU , Investment

three security portfolio
Three-Security Portfolio

2p = w1212

+ w2212

+ w3232

+ 2w1w2

Cov(r1,r2)

Cov(r1,r3)

+ 2w1w3

+ 2w2w3

Cov(r2,r3)

Bahattin Buyuksahin, JHU , Investment

asset allocation1
Asset Allocation
  • Portfolio of 2 risky assets (cont’d)
      • examples
          • BKM7 Tables 7.1 & 7.3
          • BKM7 Figs. 7.3 (return), 7.4 (risk) & 7.5 (trade-off)
          • portfolio opportunity set (BKM7 Fig. 7.5)
      • minimum variance portfolio
        • choose wD such that portfolio variance is lowest
          • optimization problem
        • minimum variance portfolio has less risk
          • than either component (i.e., asset)

Bahattin Buyuksahin, JHU , Investment

table 7 2 computation of portfolio variance from the covariance matrix
Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

Bahattin Buyuksahin, JHU , Investment

table 7 3 expected return and standard deviation with various correlation coefficients
Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

Bahattin Buyuksahin, JHU , Investment

figure 7 3 portfolio expected return as a function of investment proportions
Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

Bahattin Buyuksahin, JHU , Investment

figure 7 4 portfolio standard deviation as a function of investment proportions
Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

Bahattin Buyuksahin, JHU , Investment

minimum variance portfolio as depicted in figure 7 4
Minimum Variance Portfolio as Depicted in Figure 7.4

Bahattin Buyuksahin, JHU , Investment

Standard deviation is smaller than that of either of the individual component assets

Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk

figure 7 5 portfolio expected return as a function of standard deviation
Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

Bahattin Buyuksahin, JHU , Investment

correlation effects
The relationship depends on the correlation coefficient

-1.0 << +1.0

The smaller the correlation, the greater the risk reduction potential

If r = +1.0, no risk reduction is possible

Correlation Effects

Bahattin Buyuksahin, JHU , Investment

figure 7 6 the opportunity set of the debt and equity funds and two feasible cals
Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

Bahattin Buyuksahin, JHU , Investment

the sharpe ratio
The Sharpe Ratio

Bahattin Buyuksahin, JHU , Investment

Maximize the slope of the CAL for any possible portfolio, p

The objective function is the slope:

slide24
Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

Bahattin Buyuksahin, JHU , Investment

asset allocation2
Asset Allocation
  • Finding the optimal risky portfolio: II. Formally
    • Intuitively
        • BKM7 Figs. 7.6 and 7.7
        • improve the reward-to-variability ratio
        • optimal risky portfolio  tangency point (Fig. 7.8)
    • Formally:

Bahattin Buyuksahin, JHU , Investment

asset allocation 18
Asset Allocation 18
  • formally (continued)

Bahattin Buyuksahin, JHU , Investment

asset allocation 19
Asset Allocation 19
  • Example (BKM7 Fig. 7.8)
    • 1. plot D, E, riskless
    • 2. compute optimal risky portfolio weights
      • wD = Num/Den = 0.4; wE = 1- wD = 0.6
    • 3. given investor risk aversion (A=4), compute w*
    • bottom line: 25.61% in bills; 29.76% in bonds (0.7439x0.4); rest in stocks

Bahattin Buyuksahin, JHU , Investment

markowitz portfolio selection model
Markowitz Portfolio Selection Model

Bahattin Buyuksahin, JHU , Investment

  • Security Selection
    • First step is to determine the risk-return opportunities available
    • All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations
markowitz portfolio selection model1
Markowitz Portfolio Selection Model
  • Combining many risky assets & T-Bills
      • basic idea remains unchanged
        • 1. specify risk-return characteristics of securities
          • find the efficient frontier (Markowitz)
        • 2. find the optimal risk portfolio
          • maximize reward-to-variability ratio
        • 3. combine optimal risk portfolio & riskless asset
          • capital allocation

Bahattin Buyuksahin, JHU , Investment

markowitz portfolio selection model2
Markowitz Portfolio Selection Model
  • finding the efficient frontier
    • definition
      • set of portfolios with highest return for given risk
      • minimum-variance frontier
    • take as given the risk-return characteristics of securities
      • estimated from historical data or forecasts
      • n securities ->n return + n(n-1) var. & cov.
    • use an optimization program
      • to compute the efficient frontier (Markowitz)
      • subject to same constraints

Bahattin Buyuksahin, JHU , Investment

markowitz portfolio selection model3
Markowitz Portfolio Selection Model
  • Finding the efficient frontier (cont’d)
    • optimization constraints
      • portfolio weights sum up to 1
      • no short sales, dividend yield, asset restrictions, …
  • Individual assets vs. frontier portfolios
    • BKM7 Fig. 7.10
    • short sales -> not on the efficient frontier
    • no short sales -> may be on the frontier
      • example: highest return asset

Bahattin Buyuksahin, JHU , Investment

markowitz portfolio selection model continued
Markowitz Portfolio Selection Model Continued

Bahattin Buyuksahin, JHU , Investment

We now search for the CAL with the highest reward-to-variability ratio

figure 7 11 the efficient frontier of risky assets with the optimal cal
Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

Bahattin Buyuksahin, JHU , Investment

markowitz portfolio selection model continued1
Markowitz Portfolio Selection Model Continued

Bahattin Buyuksahin, JHU , Investment

Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

figure 7 12 the efficient portfolio set
Figure 7.12 The Efficient Portfolio Set

Bahattin Buyuksahin, JHU , Investment

capital allocation and the separation property
Capital Allocation and the Separation Property

Bahattin Buyuksahin, JHU , Investment

  • The separation property tells us that the portfolio choice problem may be separated into two independent tasks
    • Determination of the optimal risky portfolio is purely technical
    • Allocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preference
figure 7 13 capital allocation lines with various portfolios from the efficient set
Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

Bahattin Buyuksahin, JHU , Investment

the power of diversification
The Power of Diversification

Bahattin Buyuksahin, JHU , Investment

  • Remember:
  • If we define the average variance and average covariance of the securities as:
  • We can then express portfolio variance as:
table 7 4 risk reduction of equally weighted portfolios in correlated and uncorrelated universes
Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes

Bahattin Buyuksahin, JHU , Investment

risk pooling risk sharing and risk in the long run
Risk Pooling, Risk Sharing and Risk in the Long Run

Loss: payout = $100,000

p = .001

No Loss: payout = 0

1 − p = .999

Bahattin Buyuksahin, JHU , Investment

Consider the following:

risk pooling and the insurance principle
Risk Pooling and the Insurance Principle

Bahattin Buyuksahin, JHU , Investment

  • Consider the variance of the portfolio:
  • It seems that selling more policies causes risk to fall
  • Flaw is similar to the idea that long-term stock investment is less risky
risk pooling and the insurance principle continued
Risk Pooling and the Insurance Principle Continued

Bahattin Buyuksahin, JHU , Investment

When we combine n uncorrelated insurance policies each with an expected profit of $ , both expected total profit and SD grow in direct proportion to n:

risk sharing
Risk Sharing

Bahattin Buyuksahin, JHU , Investment

  • What does explain the insurance business?
    • Risk sharing or the distribution of a fixed amount of risk among many investors
an asset allocation problem
An Asset Allocation Problem

Bahattin Buyuksahin, JHU , Investment

an asset allocation problem 2
An Asset Allocation Problem 2
  • Perfect hedges (portfolio of 2 risky assets)
        • perfectly positively correlated risky assets
          • requires short sales
        • perfectly negatively correlated risky assets

Bahattin Buyuksahin, JHU , Investment

an asset allocation problem 3
An Asset Allocation Problem 3

Bahattin Buyuksahin, JHU , Investment

chapter 8

CHAPTER 8

Index Models

factor model
Factor Model

Bahattin Buyuksahin, JHU , Investment

  • Idea
      • the same factor(s) drive all security returns
  • Implementation (simplify the estimation problem)
      • do not look for equilibrium relationship
          • between a security’s expected return
          • and risk or expected market returns
      • look for a statistical relationship
          • between realized stock return
          • and realized market return
factor model 2
Factor Model 2

Bahattin Buyuksahin, JHU , Investment

  • Formally
      • stock return
          • = expected stock return
          • + unexpected impact of common (market) factors
          • + unexpected impact of firm-specific factors
index model
Index Model

Bahattin Buyuksahin, JHU , Investment

  • Factor model
      • problem
          • what is the factor?
  • Index Model
      • solution
          • market portfolio proxy
          • S&P 500, Value Line Index, etc.
advantages of the single index model
Reduces the number of inputs for diversification

Easier for security analysts to specialize

Advantages of the Single Index Model

Bahattin Buyuksahin, JHU , Investment

single factor model
ßi = index of a securities’ particular return to the factor

m = Unanticipated movement related to security returns

ei = Assumption: a broad market index like the S&P 500 is the common factor.

Single Factor Model

Bahattin Buyuksahin, JHU , Investment

single index model
Single-Index Model

Bahattin Buyuksahin, JHU , Investment

Regression Equation:

Expected return-beta relationship:

single index model continued
Single-Index Model Continued

Bahattin Buyuksahin, JHU , Investment

  • Risk and covariance:
    • Total risk = Systematic risk + Firm-specific risk:
    • Covariance = product of betas x market index risk:
    • Correlation = product of correlations with the market index
index model and diversification
Index Model and Diversification

Bahattin Buyuksahin, JHU , Investment

  • Portfolio’s variance:
  • Variance of the equally weighted portfolio of firm-specific components:
  • When n gets large, becomes negligible
slide59
Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient βp in the Single-Factor Economy

Bahattin Buyuksahin, JHU , Investment

figure 8 2 excess returns on hp and s p 500 april 2001 march 2006
Figure 8.2 Excess Returns on HP and S&P 500 April 2001 – March 2006

Bahattin Buyuksahin, JHU , Investment

figure 8 3 scatter diagram of hp the s p 500 and the security characteristic line scl for hp
Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic Line (SCL) for HP

Bahattin Buyuksahin, JHU , Investment

table 8 1 excel output regression statistics for the scl of hewlett packard
Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

Bahattin Buyuksahin, JHU , Investment

figure 8 4 excess returns on portfolio assets
Figure 8.4 Excess Returns on Portfolio Assets

Bahattin Buyuksahin, JHU , Investment

alpha and security analysis
Alpha and Security Analysis

Bahattin Buyuksahin, JHU , Investment

Macroeconomic analysis is used to estimate the risk premium and risk of the market index

Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ2 ( e i )

Developed from security analysis

alpha and security analysis continued
Alpha and Security Analysis Continued

Bahattin Buyuksahin, JHU , Investment

  • The market-driven expected return is conditional on information common to all securities
  • Security-specific expected return forecasts are derived from various security-valuation models
    • The alpha value distills the incremental risk premium attributable to private information
  • Helps determine whether security is a good or bad buy
single index model input list
Single-Index Model Input List

Bahattin Buyuksahin, JHU , Investment

  • Risk premium on the S&P 500 portfolio
  • Estimate of the SD of the S&P 500 portfolio
  • n sets of estimates of
    • Beta coefficient
    • Stock residual variances
    • Alpha values
optimal risky portfolio of the single index model
Optimal Risky Portfolio of the Single-Index Model

Bahattin Buyuksahin, JHU , Investment

  • Maximize the Sharpe ratio
    • Expected return, SD, and Sharpe ratio:
optimal risky portfolio of the single index model continued
Optimal Risky Portfolio of the Single-Index Model Continued

Bahattin Buyuksahin, JHU , Investment

  • Combination of:
    • Active portfolio denoted by A
    • Market-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by M
    • Modification of active portfolio position:
    • When
the information ratio
The Information Ratio

Bahattin Buyuksahin, JHU , Investment

The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):

figure 8 5 efficient frontiers with the index model and full covariance matrix
Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix

Bahattin Buyuksahin, JHU , Investment

table 8 2 comparison of portfolios from the single index and full covariance models
Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance Models

Bahattin Buyuksahin, JHU , Investment

index model industry practices
Index Model: Industry Practices

Bahattin Buyuksahin, JHU , Investment

  • Beta books
      • Merrill Lynch
          • monthly, S&P 500
      • Value Line
          • weekly, NYSE
      • etc.
  • Idea
      • regression analysis
index model industry practices 2
Index Model: Industry Practices 2

Bahattin Buyuksahin, JHU , Investment

  • Example (Merrill Lynch differences, Table 8.3)
      • total (not excess) returns
          • slopes are identical
          • smallness
      • percentage price changes
          • dividends?
          • S&P 500
      • adjusted beta
          • beta = (2/3) estimated beta + (1/3) . 1
          • sampling errors, convergence of new firms
      • exploiting alphas (Treynor-Black)
table 8 3 merrill lynch pierce fenner smith inc market sensitivity statistics
Table 8.3 Merrill Lynch, Pierce, Fenner & Smith, Inc.: Market Sensitivity Statistics

Bahattin Buyuksahin, JHU , Investment

table 8 4 industry betas and adjustment factors
Table 8.4 Industry Betas and Adjustment Factors

Bahattin Buyuksahin, JHU , Investment

using index models
Using Index Models

Bahattin Buyuksahin, JHU , Investment

using index models 2
Using Index Models 2

Bahattin Buyuksahin, JHU , Investment

using index models 3
Using Index Models 3

Bahattin Buyuksahin, JHU , Investment

using index models 4
Using Index Models 4

Bahattin Buyuksahin, JHU , Investment