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CHAPTER 7. Optimal Risky Portfolios. 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.

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chapter 7


Optimal Risky Portfolios

diversification and portfolio risk
Diversification and Portfolio Risk

Market risk

Systematic or nondiversifiable

Firm-specific risk

Diversifiable or nonsystematic

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

two security portfolio risk
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 continued
Two-Security Portfolio: Risk Continued

Another way to express variance of the portfolio:


Cov(rD,rE) = DEDE

D,E = Correlation coefficient of


D = Standard deviation of

returns for Security D

E = Standard deviation of

returns for Security E

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

three security portfolio
Three-Security Portfolio

2p = w1212

+ w2212

+ w3232

+ 2w1w2



+ 2w1w3

+ 2w2w3


minimum variance portfolio as depicted in figure 7 4
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

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
the sharpe ratio
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
markowitz portfolio selection model
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 continued
Markowitz Portfolio Selection Model Continued

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

markowitz portfolio selection model continued29
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

capital allocation and the separation property
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

the power of diversification
The Power of Diversification


If we define the average variance and average covariance of the securities as:

We can then express portfolio variance as:

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

Consider the following:

Loss: payout = $100,000

p = .001

No Loss: payout = 0

1 − p = .999

risk pooling and the insurance principle
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
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
Risk Sharing

What does explain the insurance business?

Risk sharing or the distribution of a fixed amount of risk among many investors