Department of Banking and Finance SPRING 2007-08 Efficient Diversification Efficient Diversification by Asst. Prof. Sami Fethi.

Department of Banking and Finance

SPRING 2007-08

Efficient DiversificationEfficient Diversification

by

Asst. Prof. Sami Fethi

2

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved.

Ch 6: Efficient Diversification

Diversification and Portfolio riskDiversification and Portfolio risk

Recall: portfolio is a collection of assets and risk is the chance of financial loss. What are the sources of risk affecting a portfolio? 1) The first type risk is associated with general

economic conditions such as the business cycle, the inflation rate, interest rate, exchange rate and so forth.

None of them are predicted with certainty so all conditions affect a company’s the rate of return.

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2) The second one is the firm-specific factors that affect a firm without noticeably affecting other firms.

If you have one stock in your portfolio, this means that you cannot reduce risk factor. However, you need to consider a diversification strategy such as naïve diversification half of your portfolio in a company and leaving the other half in an other company. This precaution reduce portfolio risk.

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For instance, If you invest half of your risky portfolio in Mobil company and leaving the other half in Dell company, what happens to portfolio risk?

Assume that if computer prices increases, this helps Dell company and when oil prices fall, this hurts Mobil company.

The two effects are offsetting which stabilizes portfolio return.

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When all risk is firm-specific, diversification can reduce risk to low level. This reduction of risk to very low levels because of independent risk sources is called the insurance principle.

When common sources of risk affect all firms, even extensive diversification cannot eliminate risk. Graphically, as portfolio standard deviation falls, the number of securities increases, but it is not reduced to zero.

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The risk that remains even after diversification is called market risk. This risk is attributable to market-wide risk sources. They are also called systematic or non-diversifiable risk.

The risk that can be eliminated by diversification is called unique risk, firm-specific risk, non-systematic risk, or diversifiable risk.

It is important to note that portfolio risk decreases as diversification increases.

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Diversification and Portfolio riskDiversification and Portfolio riskGraphically PresentedGraphically Presented

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Diversification and Portfolio riskDiversification and Portfolio riskGraphically PresentedGraphically Presented

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Asset allocation with two Risky AssetsAsset allocation with two Risky Assets

Portfolio risk depends on the correlation between the returns of the assets in the portfolio.

Asset allocation across the three key asset classes: stocks, bonds, and risk-free money market securities.

Example 1: suppose there are three possible scenarios for an economy: a recession period, a normal growth period, and a boom period. The stock fund will have a rate of return of –11% in recession, 13% in normal period and 27% in boom period.

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Example 1 cont..Example 1 cont..

Suppose that a bond fund will provide ROR of 16% in the recession, 6% in the normal period and –4% in the boom period. What is the expected or mean return for both stock and bond funds?

The expected return on each fund equals the probability-weighted average of outcomes in the scenarios.

The variance is the probability-weighted average across all scenarios of the squared deviation between the actual returns of the fund and its expected return.

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Capital market expectations for the Capital market expectations for the stock and bondstock and bond

stock fund stock fund bond fund bond fund(A) (B) (C) (D) (E) (F)

Col. B Col. B

Rate of Rate of

Scenario Probability Return Col. C Return Col. E

Recession 0.3 -11 -3.3 16 4.8

Normal 0.4 13 5.2 6 2.4

Boom 0.3 27 8.1 -4 -1.2

ExpectedExpected Return = sum 10 sum 6

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Example 2Example 2

Suppose, we form a portfolio with 60% invested in the stock fund and 40% in the bond fund.

Calculate portfolio return in recession portfolio return in each scenario is the weighted

average of the returns on the two funds. Calculate portfolio return in recession =

0.60 (-11%) + 0.40 (16%)

-0.20%

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Covariance and correlationCovariance and correlation

If we compute the probability-weighted average of the products across all scenarios, we obtain a measure of the extent to which the returns tend to vary with each other, that is to co-vary, it is called the covariance.

The negative value for the covariance indicates that the two asset vary inversely, that is when one assets performs well, the other tends to perform poorly.

It is really difficult to interpret the magnitude of covariance. An easier statistics to interpret is the correlation coefficient.

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Covariance and correlationCovariance and correlation

The correlation coefficient is simply defined as the covariance divided by the product of the standard deviation of the returns on each fund.

Correlation coefficient (ρ)= covariance/ σSTOCK σBOND

Correlation can range from values of –1 to 1. Correlation of zero indicate that the returns on the

two assets are unrelated to each other. Positive correlated shows two series move in the

same direction while negative moves in opposite directions.

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Let us have the following table which shows covariance Let us have the following table which shows covariance between the returns of the stock and bond funds:between the returns of the stock and bond funds:

A B C D E F G H I J1 Stock fund Stock fund Bond fund Bond fund

23 Scenario Probability ROR Dev from m re SQ DEV B x E ROR Dev from m re SQ DEV B x I

4 Recession 0.3 -11 -21 441 132.3 16 10 100 30

5 Normal 0.4 13 3 9 3.6 6 0 0 0

6 Boom 0.3 27 17 289 86.7 -4 -10 100 30

7 Variance sum 222.6 sum 60

8 st.dev=SQRT(Var) 14.92 7.75

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Let us have the following table which shows covariance Let us have the following table which shows covariance between the returns of the stock and bond funds:between the returns of the stock and bond funds:

(A) (B) (C) (D) (E) (F)

1

2 Scenario Probability stock fund bond fund Prod of dev BXE

3 Recession 0.3 -21 10 -210 -63

4 Normal 0.4 3 0 0 0

5 Boom 0.3 17 -10 -170 -51

6 covariance sum -114

7 -0.99

Deviation from mean returnDeviation from mean return

correlation coefficient

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Correlation coefficientCorrelation coefficient

The correlation coefficient is simply defined as the covariance divided by the product of the standard deviation of the returns on each fund.

Correlation coefficient (ρ)=covariance/ σSTOCK σBOND

=-114/(14.92x7.75)i.e.,(-21)2x(0.3)+(3)2x(0.4)+(17)2x(0.3)=SQRT of VAR=14.92i.e.,(-10)2x(0.3)+(0)2x(0.4)+(-10)2x(0.3)=SQRT of VAR=7.75

=-0.99 This confirms the overwhelming tendency of the

returns on the stock and bond funds to vary inversely in the scenario analysis.

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Example 3Example 3

The rates of return of the bond portfolio in the three scenarios based on the previous table are 10% in a recession, 7% in a normal period, and 2% in a boom. The stock returns in the three scenarios are –12% (recession), 10% (normal), and 28% (boom). What are the covariance and correlation coefficient between the rates of return on the two portfolios?

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Ch 6: Efficient DiversificationExample 3Example 3(A) (B) (C) (D) (E) (F)

Scenario Probability ROR Col. B ROR Col. B

Col. C Col. E

Recession 0.3 -12 -3.6 10 3Normal 0.4 10 4 7 2.8Boom 0.3 28 8.4 2 0.6

Sum 8.8 sum 6.4

(A) (B) (C) (D) (E) (F)

Scenario Probability SQ De Mea Col. B SQ De Mea Col. B

Col. C Col. E

Recession 0.3 432.64 129.792 12.96 3.888Normal 0.4 1.44 0.576 0.36 0.144Boom 0.3 386.64 110.592 19.36 5.808

sum 240.96 sum 9.84

Stdev 15.52 3.14

(A) (B) (C) (D) (E) (F)

Scenario Probability stock fund bond fund Prod of dev BXE

Recession 0.3 -20.8 3.6 -74.88 -22.464

Normal 0.4 1.2 0.6 0.72 0.288Boom 0.3 19.2 -4.4 -84.48 -25.344

covariance sum -47.52

-0.98correlation coefficient

Bond FundStock fund

Expected or mean return

Stock fund Bond Fund

Variance

Deviation from mean returnCovariance

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rrpp = W = W11rr1 1 ++ WW22rr22

WW11 = Proportion of funds in Security 1 = Proportion of funds in Security 1

WW22 = Proportion of funds in Security 2 = Proportion of funds in Security 2

rr11 = Expected return on Security 1 = Expected return on Security 1

rr22 = Expected return on Security 2 = Expected return on Security 2

Two-Security Portfolio: ReturnTwo-Security Portfolio: Return

WWiiii=1=1

nn

= 11

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p2

= w121

2 + w222

2 + 2W1W2 Cov(r1r2)p2

= w121

2 + w222

2 + 2W1W2 Cov(r1r2)

12 = Variance of Security 112 = Variance of Security 1

22 = Variance of Security 222 = Variance of Security 2

Cov(r1r2) = Covariance of returns for Security 1 and Security 2Cov(r1r2) = Covariance of returns for Security 1 and Security 2

Two-Security Portfolio: RiskTwo-Security Portfolio: Risk

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E(rp) = W1r1 + W2r2E(rp) = W1r1 + W2r2

Two-Security PortfolioTwo-Security Portfolio

p2

= w121

2 + w222

2 + 2W1W2 Cov(r1r2)

p = [w1

212 + w2

222 + 2W1W2 Cov(r1r2)]1/2

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Example 4Example 4

Suppose that for some reason you are required to invest 50% of your portfolio in bonds and 50% in stocks. r1=6%, r2=10%, σ1=12%, σ2=25%, w1=0.5, and w2=1-0.5=0.5.

A) If the standard deviation of your portfolio is 15%, what must be the correlation coefficient between stock and bond returns?

B) What is the expected rate of return on your portfolio?

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Example 4Example 4

A) A) p2

= w121

2 + w222

2 + 2W1W2 12

151522= (0.5x12)= (0.5x12)22+ (0.5x25)+ (0.5x25)22 +2 +2 (0.5x12)(0.5x12)

(0.5x25) (0.5x25) 12

12=0.21183

B) E (rB) E (rpp) = W) = W1 1 E (rE (r11)+)+ WW2 2 E (rE (r22))

= = (0.5x6)+ (0.5x10)(0.5x6)+ (0.5x10)

= 8%= 8%

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CovarianceCovariance

1,2 = Correlation coefficient of returns

1,2 = Correlation coefficient of returns

Cov(r1r2) = 12Cov(r1r2) = 12

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2

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Correlation Coefficients: Possible ValuesCorrelation Coefficients: Possible Values

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

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

Range of values for 1,2

-1.0 < < 1.0

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2p = W1

2122

p = W121

2 + W22

+ W22

+ 2W1W2+ 2W1W2

rp = W1r1 + W2r2 + W3r3rp = W1r1 + W2r2 + W3r3

Cov(r1r2) Cov(r1r2)

+ W323

2+ W323

2

Cov(r1r3) Cov(r1r3)+ 2W1W3+ 2W1W3

Cov(r2r3) Cov(r2r3)+ 2W2W3+ 2W2W3

Three-Security PortfolioThree-Security Portfolio

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rp = Weighted average of the n securities

rp = Weighted average of the n securities

p2 = (Consider all pair-wise

covariance measures)

p2 = (Consider all pair-wise

covariance measures)

In General, For an n-Security Portfolio:In General, For an n-Security Portfolio:

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= 0= 0

E(r)E(r)

= 1= 1

= -1= -1

= .3= .3

13%13%

8%8%

12%12% 20%20% St. DevSt. Dev

TWO-SECURITY PORTFOLIOS WITH TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONSDIFFERENT CORRELATIONS The figure shows the

opportunity set with perfect positive correlation. No portfolio can be discarded as inefficient and the choice among portfolios depends only on risk preference. Diversification in the case of perfect positive correlation is not effective.

Perfect positive correlation is the only case in which there is no benefit from diversification.

In the case of negative correlation, there are benefits to diversification.

= .3 = .3 is a lot better than = 1 = 1 and quite a bit worse than ( = 0) = 0) zero correlation.

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Ch 6: Efficient DiversificationTWO-SECURITY PORTFOLIOS WITH TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONSDIFFERENT CORRELATIONS

The figure shows the opportunity set with perfect positive correlation. No portfolio can be discarded as inefficient and the choice among portfolios depends only on risk preference. Diversification in the case of perfect positive correlation is not effective.

Perfect positive correlation is the only case in which there is no benefit from diversification.

In the case of negative correlation, there are benefits to diversification.

= .3 = .3 is a lot better than = 1 = 1 and quite a bit worse than ( = 0) = 0) zero correlation.

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Portfolio Risk/Return Two Securities: Portfolio Risk/Return Two Securities: Correlation EffectsCorrelation Effects

Relationship depends on correlation coefficient

-1.0 < < +1.0The smaller the correlation, the greater the

risk reduction potentialIf= +1.0, no risk reduction is possible

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Ch 6: Efficient DiversificationMinimum Variance CombinationMinimum Variance Combination

Investment opportunity setInvestment opportunity setE(r)E(r)

10%10%

7%7%

11%11% 16%16%St. DevSt. Dev

StockStock

.. .. portfolio Zportfolio Z

.. The mean variance portfolioThe mean variance portfolio

.. BondsBonds

26%26% 31%31%

6%6%

A mean-var criterion indicates A mean-var criterion indicates higher mean return and lower var. higher mean return and lower var. In this case, the stock fund In this case, the stock fund dominates portfolio Z so has higher dominates portfolio Z so has higher expected return and lower volatility. expected return and lower volatility. If portfolios lie below the min-var If portfolios lie below the min-var portfolio, they can be rejected as portfolio, they can be rejected as inefficient. This is valid for the case inefficient. This is valid for the case of zero correlation between the of zero correlation between the funds. funds.

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11

11 22

- Cov(r1r2) - Cov(r1r2)

W1W1 ==++ - 2Cov(r1r2) - 2Cov(r1r2)

22

W2W2 = (1 - W1)= (1 - W1)

Minimum Variance Combination-Example

2

2

22

22E(r2) = .14E(r2) = .14 = .20= .20Sec 2Sec 2 1212 = .2= .2E(r1) = .10E(r1) = .10 = .15= .15Sec 1Sec 1

2

2

Suppose, we invest some proportions in both stocks and in Suppose, we invest some proportions in both stocks and in bonds and the other relevant input data as follows. bonds and the other relevant input data as follows. Compute the proportions of the funds and the portfolio Compute the proportions of the funds and the portfolio variance.variance.

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WW11==

(.2)(.2)22 - (.2)(.15)(.2) - (.2)(.15)(.2)

(.15)(.15)22 + (.2) + (.2)22 - 2(.2)(.15)(.2) - 2(.2)(.15)(.2)

WW11 = .6733= .6733

WW22 = (1 - .6733) = .3267= (1 - .6733) = .3267

Example cont….Example cont….

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rrpp = .6733(.10) + .3267(.14) = .1131 = .6733(.10) + .3267(.14) = .1131

pp = [(.6733)= [(.6733)22(.15)(.15)2 2 + (.3267)+ (.3267)22(.2)(.2)2 2 ++

2(.6733)(.3267)(.2)(.15)(.2)]2(.6733)(.3267)(.2)(.15)(.2)]1/21/2

pp= [.0171]= [.0171] 1/21/2 = .1308= .1308

Example cont….Example cont….

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WW11==

(.2)(.2)22 - (.2)(.15)(.2) - (.2)(.15)(.2)

(.15)(.15)22 + (.2) + (.2)22 - 2(.2)(.15)(-.3) - 2(.2)(.15)(-.3)

WW11 = .6087= .6087

WW22 = (1 - .6087) = .3913= (1 - .6087) = .3913

Minimum Variance Combination-example2: Minimum Variance Combination-example2: = -.3 = -.3

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rrpp = .6087(.10) + .3913(.14) = .1157 = .6087(.10) + .3913(.14) = .1157

pp = [(.6087)= [(.6087)22(.15)(.15)2 2 + (.3913)+ (.3913)22(.2)(.2)2 2 ++

2(.6087)(.3913)(.2)(.15)(-.3)]2(.6087)(.3913)(.2)(.15)(-.3)]1/21/2

pp= [.0102]= [.0102] 1/21/2= .1009= .1009

Minimum Variance: example2: Minimum Variance: example2: = -.3 cont….. = -.3 cont…..

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Example 4-2Example 4-2

Suppose, you invest 50% in both stocks and in bonds and Suppose, you invest 50% in both stocks and in bonds and the other relevant input data as follows: st.devthe other relevant input data as follows: st.devBB=12, =12,

st.devst.devSS=25.=25.

• (a) Compute the correlation coefficient between stock and (a) Compute the correlation coefficient between stock and bond returns if st.devbond returns if st.devpp=15.=15.• (b) What is the expected ROR on your portfolio if expected (b) What is the expected ROR on your portfolio if expected RORs for stock and bond are 6 and 10 respectively. RORs for stock and bond are 6 and 10 respectively. • (c) Are you likely to be better or worse off if the correlation (c) Are you likely to be better or worse off if the correlation coefficient between stock and bond returns is 0.22 coefficient between stock and bond returns is 0.22 compared to part (a).compared to part (a).

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Example 4-2 cont..Example 4-2 cont..

21

BSBS2B

2B

2S

2SP )r,r(Covww2ww

152= [(0.5x12)2 + (0.5 25)2 + 2 (0.5x12) (0.5 25 )]ρSB

ρSB= 0.2138

E(rp) = (0.5 6%) + (0.5 10%) 8%

Smaller correlation implies greater benefits from diversification so there will be lower risk.

rp = W1r1 + W2r2 rp = W1r1 + W2r2

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Example 5Example 5

There are three mutual funds such as a stock fund, a long-term government fund and a T-bill money market fund and this yields a rate of 5.5%. The probability distributions of risky funds are:

The correlation between the fund returns is 0.15.

E.Returnst.devstock fund 15% 32%Bond fund 9 23

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Tabulate the investment opportunity set of the two risky funds (i.e., construct the covariance matrix ).

What are the expected return, standard deviation and minimum variance portfolio?

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The parameters of the opportunity set are:

E(rS) = 15%, E(rB) = 9%, S = 32%, B = 23%, =

0.15,rf = 5.5%

From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(rS, rB) = SB]:

Bonds Stocks

Bonds 529.0 110.4

Stocks 110.4 1024.0

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The minimum-variance portfolio proportions are:

)r,r(Cov2

)r,r(Cov)S(w

BS2B

2S

BS2B

Min

3142.0)4.1102(5291024

4.110529

wMin(B) = 0.6858

The mean and standard deviation of the minimum variance portfolio are:

E(rMin) = (0.3142 15%) + (0.6858 9%) 10.89%

21

BSBS2B

2B

2S

2SMin )r,r(Covww2ww

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= [(0.31422 1024) + (0.68582 529) + (2 0.3142 0.6858 110.4)]1/ 2

= 19.94%

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Ch 6: Efficient DiversificationExample 6Example 6 First draw the diagrams by using the figures of stock 1 versus stock 2. Second match up the diagrams (A-E) to the following list of correlation coefficients by choosing the correlation that best describes the relationship between the returns on the two stocks =-1, 0, 0.2, 0.5, 1.0.

A D% Return % ReturnStock1 Stock2 Stock1 Stock2

5 1 5 51 1 1 34 3 4 32 3 2 03 5 3 5

B E% Return % ReturnStock1 Stock2 Stock1 Stock2

1 1 5 42 2 1 33 3 4 14 4 2 05 5 3 5

C% Return Match up the diagrams (A-E) to the following list of correlation coefficients by choosing the correlation that best describes the relationship between the returns on the two stocks rho =-1, 0, 0.2, 0.5, 1.0. Stock1 Stock2

1 52 43 34 25 1

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Ch 6: Efficient DiversificationExample 6 cont..Example 6 cont..Scatter diagram A

0

2

4

6

0 1 2 3 4 5 6

Stock 1

Stoc

k 2

Scatter diagram B

0

2

4

6

0 2 4 6

Stock 1

Stoc

k 2

Scatter diagram C

0

2

4

6

0 2 4 6

Stock 1

stoc

k 2

Scatter diagram D

0

2

4

6

0 2 4 6

Stock 1

Stoc

k 2

Scatter diagram E

0

2

4

6

0 2 4 6

Stock 1

Sto

ck 2

Diagram A shows exact conflict and in this case cc is zero. Diagram B shows perfect positive correlation and cc is 1.0. Diagram C shows perfect negative correlation and cc is -1.0. Diagram D and Diagram E show positive correlation but Diagram D is tighter. Therefore D is associated with a correlation of 0.5 and E is associated with a correlation of 0.2.

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Diagram A shows exact conflict and in this case cc is zero. Diagram B shows perfect positive correlation and cc is 1.0. Diagram C shows perfect negative correlation and cc is -1.0. Diagram D and Diagram E show positive correlation but Diagram D is tighter. Therefore D is associated with a correlation of 0.5 and E is associated with a correlation of 0.2.

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Extending Concepts to All SecuritiesExtending Concepts to All Securities

The optimal combinations result in lowest level of risk for a given return

The optimal trade-off is described as the efficient frontier

These portfolios are dominant

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E(r)E(r) The minimum-variance frontier of The minimum-variance frontier of risky assetsrisky assets

EfficientEfficientfrontierfrontier

GlobalGlobalminimumminimumvariancevarianceportfolioportfolio

MinimumMinimumvariancevariancefrontierfrontier

IndividualIndividualassetsassets

St. Dev.

Efficient frontier represents the set of portfolios that offers the highest possible expected rate of return for each level of portfolio standard deviation. These portfolios may be viewed as efficiently diversified. Expected return-standard deviation combinations for any individuals asset end up inside the efficient frontier, because single-asset portfolios are inefficient- they are not efficiently diversified. The real choice is among portfolios on the efficient frontier above the minimum-variance portfolio.

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Efficient frontierEfficient frontier

Efficient frontier represents the set of portfolios that offers the highest possible expected rate of return for each level of portfolio standard deviation. These portfolios may be viewed as efficiently diversified. Expected return-standard deviation combinations for any individuals asset end up inside the efficient frontier, because single-asset portfolios are inefficient- they are not efficiently diversified. The real choice is among portfolios on the efficient frontier above the minimum-variance portfolio.

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Extending to Include Riskless AssetExtending to Include Riskless Asset

The optimal combination becomes linearA single combination of risky and riskless

assets will dominate

52



E(r)E(r)

CAL (GlobalCAL (Globalminimum variance)minimum variance)

CAL (A)CAL (A)CAL (P)CAL (P)

MM

PP

AA

FF

PP P&FP&F A&FA&FMM

AA

GG

PP

MM

ALTERNATIVE CALSALTERNATIVE CALS

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Dominant CAL with a Risk-Free Dominant CAL with a Risk-Free Investment (F)Investment (F)

CAL(P) dominates other lines -- it has the best risk/return or the largest slope

Slope = (E(R) - Rf) / E(RP) - Rf) / PE(RA) - Rf) /

Regardless of risk preferences combinations of P & F dominate

54



Example 7Example 7

The correlation coefficient between X and M is –0.2. Weight in M and X are 0.26 and 0.74.

Find the optimal risky portfolio (o) and its expected return and standard deviation.

Find the slope of the CAL generated by T-bills and portfolio o.

Calculate the composition of complete portfolio (an investor consider 22.22% of complete portfolio in the risky p) o and the remainder in T-bills.

X 15% 50%M 10 20T-bills 5 0

Expected Return standard deviation

55



E(r)E(r)

CAL (GlobalCAL (Globalminimum variance)minimum variance)

CAL (X)CAL (X)

CAL (M)CAL (M)

1515

1010

5%5%

2020 3535 5050

GG

MM

XX

11.2811.28

17.5917.59

CAL (O)CAL (O)

OO

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Example 7Example 7 In this case, you need to generate data to find out

mean and st.dev for optimal risky portfolio (i.e., 11.28 and 17.59) and weight in X and M are (i.e., 0.26 and 0.74) respectively.

The slope of CAL is (11.28-5)/17.59=0.357 The mean of the complete portfolio 0.22x11.28+

0.7778x5=6.40% and its standard deviation is 0.22x17.59=3.91%. The composition of the complete portfolio is 0.22x0.26 (optimal pf calculated by using data for x) =0.06 (6%) in X. In M, 0.22x0.74 (optimal pf calculated by using data for M) =0.16 (16%) in M and 78% in T-bills.

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Single Factor ModelSingle Factor Model

ri = E(Ri) + ßiF + e

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

F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns

Assumption: a broad market index like the S&P500 is the common factor

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Single Index ModelSingle Index Model

Risk PremRisk Prem Market Risk PremMarket Risk Prem or Index Risk Premor Index Risk Prem

ii= the stock’s expected return if the= the stock’s expected return if the market’s excess return is zeromarket’s excess return is zero

ßßii(r(rmm - r - rff)) = the component of return due to= the component of return due to

movements in the market indexmovements in the market index

(r(rmm - r - rff)) = 0 = 0

eei i = firm specific component, not due to market= firm specific component, not due to market

movementsmovements

errrr ifmiifi

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Let: RLet: Ri i = (r= (rii - r - rff))

RRm m = (r= (rmm - r - rff))Risk premiumRisk premiumformatformat

RRi i = = ii + ß + ßii(R(Rmm)) + e+ eii

Risk Premium FormatRisk Premium Format

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Estimating the Index ModelEstimating the Index Model

ExcessExcess Returns (i) Returns (i)

SecuritySecurityCharacteristicCharacteristicLineLine

.. ...... ..

..

.. ..

.. ....

.. ....

.. ..

.. ....

......

.. ..

.. ....

.. ....

.. ..

.. ....

.. ....

.. ..

..

.. ...... .... .... ..

ExcessExcess returns returnson market indexon market index

RRii = = ii + ß + ßiiRRmm + e + eii

61



Components of RiskComponents 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

62



Measuring Components of RiskMeasuring Components of Risk

i2 = i

2 m2 + 2(ei)

where;

i2 = total variance

i2 m

2 = systematic variance

2(ei) = unsystematic variance

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Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk/Total Risk = 2

ßi2

m2 / 2 = 2

i2 m

2 / i2 m

2 + 2(ei) = 2

Examining Percentage of VarianceExamining Percentage of Variance

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Advantages of the Single Index ModelAdvantages of the Single Index Model

Reduces the number of inputs for diversification

Easier for security analysts to specialize

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Ch 6: Efficient DiversificationExample 8Example 8

Week ABC XYZ MKT INX RISK ABC XYZ MKT1 65.13 -22.55 64.4 5.23 59.9 -27.78 59.172 51.84 31.44 24 4.76 47.08 26.68 19.243 -30.82 -6.45 9.15 6.22 -37.04 -12.67 2.934 -15.13 -51.14 -35.57 3.78 -18.91 -54.92 -39.355 70.63 33.78 11.59 4.43 66.2 29.35 7.166 107.82 32.95 23.13 3.78 104.04 29.17 19.357 -25.16 70.19 8.54 3.87 -29.03 66.32 4.678 50.48 27.63 25.87 4.15 46.33 23.48 21.729 -36.41 -48.79 -13.15 3.99 -40.4 -52.78 -17.14

10 -42.2 52.63 20.21 4.01 -46.21 48.62 16.2

Average 15.196 7.547 9.395

ABC XYZ MKTABC 3020.933XYZ 442.114 1766.923MKT 773.306 396.789 669.01

coeff std. Err t statIntercept 4.33635 16.56427 0.261789MKT RET 1.155897 0.63042 1.833535

coeff std. Err t stat XYZ-MKTIntercept 3.930054 14.98109 0.262334MKT 0.581816 0.527517 1.102933

Excess Returns

COV MATRIX

Summary output of excel regression

ABC-MKT

R-square=0.296

R-square=0.132

Annualized rates of return

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Example 8Example 8

Calculate the slope and intercept of characteristic lines for ABC and XYZ using the variances and co-variances concepts.

What is the characteristic line of XYZ and ABC?

Does ABC or XYZ have greater systematic risk?

What percentage of variance of XYZ is firm specific risk

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Example 8Example 8

The slope of coefficient for ABC

ßABC=cov (RABC, RMRK)/var (RMRK)

=773.31/669.01=1.156 The intercept for ABC

αABC= AV.(RABC)- ßABC x AV.(RMRK)

=15.20-1.156 x 9.40=4.33 The security the characteristic line of ABC is

RABC=4.33 + 1.156 RMRK

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Example 8Example 8

The slope of coefficient for XYZ

ßXYZ=cov (RXYZ, RMRK)/var (RMRK)

=396.78.31/669.01=0.58 The intercept for XYZ

αXYZ= AV.(RXYZ)- ßXYZ x AV.(RMRK)

=7.64-0.582 x 9.40=3.93 The security the characteristic line of XYZ is

RXYZ=3.93 + 0.582 RMRK

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Example 8Example 8

The beta coefficient of ABC is 1.15 greater than XYZ’s 0.58 implying that ABC has greater systematic risk.

The regression of XYZ on the market index shows an R square of 0.132, the percent of unexplained variance (non-systematic risk) is 0.868 or 86.8%.

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Example 9Example 9 A pension fund manager is considering three mutual funds

such as a stock fund, a long-term government and corporate bond fund, and a T-bill money market fund that yields a sure rate of 4.5%. The probability distributions of the risky funds as follows: (note: The correlation between the fund return is 0.18).

Expected

returnStandard deviation

Stock fund (S)

18% 34%

Bond fund (B)

12 26

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Example 9Example 9 a) Tabulate and draw the investment opportunity set of the

two risky funds. b) Use investment proportions for the stock fund of 0 to

100% in increments of 20%. c)What expected return and the standard deviation does your

graph show for the minimum variance portfolio? d)Draw a tangent from the risk-free rate to the opportunity set e) What is the reward-to-variability ratio of the best feasible

CAL? f) What is the equation of the CAL? What is the standard

deviation of your portfolio if it yields an expected return of 15%?

g) What is the proportion invested in the T-bill fund and each of the two risky funds?

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Example 9-Answer Example 9-Answer -a-a

1156159.12Stocks

159.12676Bonds

StocksBonds

Cov(rS, rB) = [ρ σs σB]:

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Example 9-AnswerExample 9-Answer-b and c-b and c

% in stocks % in bonds Exp. return Std dev.

00.00 100.00 12.00 26.00

20.00 80.00

13.20 23

34.00 66.0014.04

22.34 minimum variance

40.00 60.00

14.40 22.46

60.00 40.00 15.60

24.50

70.80 29.20 16.20 26.54 tangency portfolio

80.00 20.00

16.80

28.59

100.00 00.00

18.00

34.00

E(rp) = W1r1 + W2r2 = 1 (12) + (0) (18) = 12 E(rp) = W1r1 + W2r2 = 1 (12) + (0) (18) = 12

p = [w1

212 + w2

222 + 2W1W2 Cov(r1r2)]1/2 = 1(26)2 + (0)(34)2+ 2(0)...= 26

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Example 9-AnswerExample 9-Answer-c-c

)r,r(Cov2

)r,r(Cov)S(w

BS2B

2S

BS2B

Min

34.0)12.1592(1156676

12.159676

wMin(B) = 0.66

The minimum-variance portfolio proportions are:

21

BSBS2B

2B

2S

2SMin )r,r(Covww2ww

The mean and standard deviation of the minimum variance portfolio are:

= [(0.342 1156) + (0.662 529) + (2 0.34 0.66 159.12)]1/2= 22.34%

E(rMin) = (0.34 18%) + (0.66 12%) = 14.04%

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Example 9-AnswerExample 9-Answer-d-dInvestment opportunity set

for stocks and bonds

min varB

SCAL

0

2

4

6

8

10

12

14

16

18

0 10 20 30 40

Standard Deviation (%)

Exp

ected

Retu

rn

(%

)

E(rt)= (15.6 %+ 16.8%)/2= 16.20

σt= (24.5 %+ 28.59%)/2= 26.54

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Example 9-AnswerExample 9-Answer-e and f-e and f

442.054.26

5.420.16)(

p

fp rrE

The reward-to-variability ratio of the optimal CAL is:

The equation for the CAL is:

CCp

fpfC

rrErrE

442.05.4

)()(

Setting E(rC) equal to 15% yields a standard deviation of 23.75%.

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Example 9-AnswerExample 9-Answer-g-g

•The mean of the complete portfolio as a function of the proportion invested in the risky portfolio (y) is:

E(rC) = (l - y)rf + yE(rP) = rf + y[E(rP) - rf] = 4.5 + y(16.20- 4.5)

Setting E(rC) = 15% y = 0.89 (89% in the risky portfolio)1 - y = 0.11 (11.00% in T-bills)

From the composition of the optimal risky portfolio:Proportion of stocks in complete portfolio = 0.89 0.7080 = 0.63Proportion of bonds in complete portfolio = 0.89 0.2920 = 0.25

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The EndThe End

Thanks

Department of Banking and Finance SPRING 2007-08 Efficient Diversification Efficient Diversification by Asst. Prof. Sami Fethi.

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