Optimal Risky Portfolios- Asset Allocations

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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.

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### 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

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

Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio

Figure 7.2 Portfolio Diversification

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

Two-Security Portfolio: Return

= Variance of Security D

= Variance of Security E

= Covariance of returns for

Security D and Security E

Two-Security Portfolio: Risk

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

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

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

Table 7.1 Descriptive Statistics for Two Mutual Funds

Three-Security Portfolio

2p = w1212

+ w2212

+ w3232

+ 2w1w2

Cov(r1,r2)

Cov(r1,r3)

+ 2w1w3

+ 2w2w3

Cov(r2,r3)

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)

Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

Minimum Variance Portfolio as Depicted in Figure 7.4

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

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

Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

The Sharpe Ratio

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

The objective function is the slope:

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

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:

Asset Allocation 18
• formally (continued)

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

Markowitz Portfolio Selection Model

• 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 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

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

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

Markowitz Portfolio Selection Model Continued

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

Markowitz Portfolio Selection Model Continued

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

Capital Allocation and the Separation Property

• 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

The Power of Diversification

• 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

Risk Pooling, Risk Sharing and Risk in the Long Run

Loss: payout = \$100,000

p = .001

No Loss: payout = 0

1 − p = .999

Consider the following:

Risk Pooling and the Insurance Principle

• 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

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

• 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 2
• Perfect hedges (portfolio of 2 risky assets)
• perfectly positively correlated risky assets
• requires short sales
• perfectly negatively correlated risky assets

An Asset Allocation Problem 3

### CHAPTER 8

Index Models

Factor Model

• 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

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

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

Easier for security analysts to specialize

Advantages of the Single Index Model

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

Single-Index Model

Regression Equation:

Expected return-beta relationship:

Single-Index Model Continued

• 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

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

Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic Line (SCL) for HP

Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

Figure 8.4 Excess Returns on Portfolio Assets

Alpha and Security Analysis

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

• 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
Single-Index Model Input List

• 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

• Maximize the Sharpe ratio
• Expected return, SD, and Sharpe ratio:
Optimal Risky Portfolio of the Single-Index Model Continued

• 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 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

Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance Models

Index Model: Industry Practices

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

• Example (Merrill Lynch differences, Table 8.3)
• total (not excess) returns
• slopes are identical
• smallness
• percentage price changes
• dividends?
• S&P 500
• 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 8.4 Industry Betas and Adjustment Factors

Using Index Models