Chapter 6

Chapter 6

Efficient Diversification

Relationship Between Risk and Return – Let’s revisit…

Exhibit IPossible Investm ent O utcom es

Taxi & B us Com panies

State of the Econom yPoor Average Good

Bus Com pany 1,300,000 1,210,000 700,000 Taxi Com pany 600,000 1,210,000 1,400,000

Harry Markowitz -- one of the founders of modern finance – contributed greatly to modern financial theory and practice. In his dissertation he argued that investors are 1) risk adverse, and 2) evaluate investment opportunities by comparing expected returns relative to risk which he defined as the standard deviation of the expected returns. This example is based on his seminal work.

Step 1: Calculate the potential return on each investment...

Rit = (Priceit+1 - Priceit)/Priceit

Where:

Rit = The holding period return for investment “i” for time period “t”

Priceit = The price of investment “i” at time period “t”

Priceit+1 = The price of investment “i’ at time period “t+1”

State of the EconomyPoor Average Good

Bus Company 30.00% 21.00% -30.00%Taxi Company -40.00% 21.00% 40.00%

Step 2: Calculate the expected return

(E)Rit = ΣXi Rit

Where: Xi = Probability of a given event

The Expected Return for the Taxi and Bus Companies -

(E)RBus = 1/3(-30%) + 1/3(21%) + 1/3(30%)

= 7%

(E)RTaxi = 7%

Step 3: Measure risk

σi = (ΣXi(Rit - (E)Rit) 2).5

The standard deviation of the Bus Company -

σBus = (1/3(30% - 7%)2 + 1/3(21% - 7%)2 + 1/3(30% - 7%)2).5

= 26.42%

σTaxi = 34.13%

Step 5: Compare the alternatives Expected Return

10% 7% --------------B------T 2% 10% 20% 30% 40% Risk (Standard Deviation)

We have two investment alternatives with the same expected return – which one is preferable?

Bus Company

Taxi Company

Expected Return 7% 7%Standard Deviation 26.42% 34.13%

Conclusion

Based on our analysis the Bus Company represents a superior investment alternative to the Taxi company. Since the Bus company represents a superior return to the Taxi company, why would anyone hold the Taxi company?

Portfolio Analysis of Investment Decision

A s s u m e y o u i n v e s t 5 0 p e r c e n t o f y o u r m o n e y i n t h e B u s c o m p a n ya n d 5 0 p e r c e n t i n t h e T a x i c o m p a n y .

E x h i b i t I VP o r t f o l i o A n a l y s i s o f I n v e s t m e n t D e c i s i o n

S t a t e o f t h e E c o n o m yP o o r A v e r a g e G o o d

B u s C o m p a n y 3 0 . 0 0 % 2 1 . 0 0 % - 3 0 . 0 0 %T a x i C o m p a n y - 4 0 . 0 0 % 2 1 . 0 0 % 4 0 . 0 0 %5 0 p e r c e n t i n e a c h - 5 . 0 0 % 2 1 . 0 0 % 5 . 0 0 %

Portfolio Return...

(E)Rp = Σwj(E)rit

Where: (E)Rp = The expected return on the portfolio wj = The proportion of the portfolio’s total value

= .5(7%) + .5(7%)

= 7%

or,

(E)Rp = -5%(1/3) + 21%(1/3) + 5%(1/3)

= 7%

Portfolio RiskThe standard deviation of the portfolio: σp = 10.68% Note - The expected return of the portfolio is simply a weighted-average of the of the expected returns for each alternative; the standard deviation of the portfolio is not a simple weighted-average. Why?

The formula for the portfolio standard deviation is:

σp = (wa2* σa2 + wb2* σb2 + 2*wa*wb* σa* σb*rab).5

Where:Wa – weight of security AWb – weight of security Bσa = standard deviation of security A’s return σb = standard deviation of security B’s return Corrab = correlation coefficient between security A and B

Risk Reduction

Holding more than one asset in a portfolio (with less than a correlation coefficient of positive 1) reduces the range or spread of possible outcomes; the smaller the range, the lower the total risk.

State of the EconomyPoor Average Good

Bus Company 30.00% 21.00% -30.00%Taxi Company -40.00% 21.00% 40.00%50 percent in each -5.00% 21.00% 5.00%

Correlation coefficient = CovarianceAB /σAσB

Covariance = ΣpAB(A – E(A))*(B – E(B))

= 1/3(30% - 7%)(-40% - 7%) + 1/3(21% - 7%)(21% - 7%) + 1/3 (-30% - 7%)(40% - 7%)= -.0702

Correlation coefficient = -.0702/((.2642)*(.3413)) = -.78

Standard Deviation of a Two-Asset Portfolio

σp = (wa2* σa2 + wb2* σb2 + 2*wa*wb* σa* σb*rab).5

Where: Wa – weight of security A (.5) Wb – weight of security B (.5) σa = standard deviation of security A’s return (26.42%) σb = standard deviation of security B’s return (34.13%) rab = correlation coefficient between security A and B (-.78)

σp = ((.5)2 (26.42)2 + (.5)2 (34.13)2 + 2(.5)(.5)(26.42)(34.13)(-.78)).5

σp = (174.50 +291.21 - 351.67).5

σp = 10.68%

Risk Reduction Expected Return 10% 7% ----P--------B------T 2% 10% 20% 30% 40% Risk (Standard Deviation)

The net effect is that an investor can reduce their overall risk by holding assets with less than a perfect positive correlation in a portfolio relative to the expected return of the portfolio.

Extending the example to numerous securities...

ExpectedReturn

Risk (Standard Deviation)

Each point represents the expected return/standard deviation relationship for some number of individual investment opportunities.

Extending the example to numerous securities...

Expected Return Risk (Standard Deviation)

This point represents a new possible risk - return combination

More on Correlation & the Risk-Return Trade-Off

Efficient Frontier

ExpectedReturn

Risk (Standard Deviation)

Each point represents the highest potential return for a given level of risk

Breakdown of Risk

Total Risk = Diversifiable Risk + Non Diversifiable Risk Diversifiable Risk = Company specific risk Nondiversifiable Risk = Market risk

Total Risk () Number of Securities

Company specific or diversifiable risk

Market risk or Non-diversifiable risk

Total Risk

Diversification and Risk

Why Diversification Works, I.

Correlation: The tendency of the returns on two assets to move together. Imperfect correlation is the key reason why diversification reduces portfolio risk as measured by the portfolio standard deviation.

Positively correlated assets tend to move up and down together.

Negatively correlated assets tend to move in opposite directions.

Imperfect correlation, positive or negative, is why diversification reduces portfolio risk.

Why Diversification Works, II.

The correlation coefficient is denoted by Corr(RA, RB) or simply, A,B.

The correlation coefficient measures correlation and ranges from:

From: -1 (perfect negative correlation)

Through: 0 (uncorrelated)

To: +1 (perfect positive correlation)

Why Diversification Works, III.

Why Diversification Works, IV.

Why Diversification Works, V.

Correlation and Diversification

Minimum Variance Combinations -1< r < +1

11 22

- Cov(r1r2) - Cov(r1r2)

W1W1==

++ - 2Cov(r1r2) - 2Cov(r1r2)

22

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

s2

s2

s 2 s 2 s2

s2

Choosing weights to minimize the portfolio variance

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11

Minimum Variance Combinations -1< r < +1

22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 21212 = .2= .2

E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 ssss rr

11 22

- Cov(r1r2)- Cov(r1r2)

W1W1==

++ - 2Cov(r1r2)- 2Cov(r1r2)

22

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

2 2

2 2 2 211 22

- Cov(r1r2)- Cov(r1r2)

W1W1==

++ - 2Cov(r1r2)- 2Cov(r1r2)

22

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

2 2

2 2 2 2WW11

==(.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 = (1 -- .6733) = .3267.6733) = .3267

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 = (1 -- .6733) = .3267.6733) = .3267

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 = (1 -- .6733) = .3267.6733) = .3267

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 = (1 -- .6733) = .3267.6733) = .3267Cov(r1r2) = r1,2s1s2

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E[rp] =

Minimum Variance: Return and Risk with r = .2

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

22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2

E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1

1/22222p (0.2) (0.15) (0.2) (0.3267) (0.6733) 2 )(0.2 )(0.3267 )(0.15 )(0.6733σ

sp2 =sp

2 =

%.. /p 081301710 21

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 = (1 -- .6733) = .3267.6733) = .3267

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 = (1 -- .6733) = .3267.6733) = .3267

1

.6733(.10) + .3267(.14) = .1131 or 11.31%

W12s1

2 + W22s2

2 + 2W1W2 r1,2s1s2

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WW11==

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

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

WW11 = .6087= .6087

WW22 = (1 = (1 -- .6087) = .3913.6087) = .3913

WW11==

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

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

WW11 = .6087= .6087

WW22 = (1 = (1 -- .6087) = .3913.6087) = .3913

Minimum Variance Combination with r = -.3

11 22

- Cov(r1r2)- Cov(r1r2)

W1W1==

++ - 2Cov(r1r2)- 2Cov(r1r2)

22

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

2 2

2 2 2 211 22

- Cov(r1r2)- Cov(r1r2)

W1W1==

++ - 2Cov(r1r2)- 2Cov(r1r2)

22

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

2 2

2 2 2 2

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

22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2

E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 -.31

Cov(r1r2) = r1,2s1s2

WW11==

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

(.15)(.15)22 + (.2)+ (.2)22 -- 2(2(--.3)(.15)(.2).3)(.15)(.2)WW11

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

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

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WW11==

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

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

WW11 = .6087= .6087

WW22 = (1 = (1 -- .6087) = .3913.6087) = .3913

WW11==

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

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

WW11 = .6087= .6087

WW22 = (1 = (1 -- .6087) = .3913.6087) = .3913

Minimum Variance Combination with r = -.3

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

22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2

E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 -.3

E[rp] =

1/22222p (0.2) (0.15) (-0.3) (0.3913) (0.6087) 2 )(0.2 )(0.3913 )(0.15 )(0.6087σ

sp2 = sp

2 =

%.. /p 091001020 21

0.6087(.10) + 0.3913(.14) = .1157 = 11.57%

W12s1

2 + W22s2

2 + 2W1W2 r1,2s1s2

1

Notice lower portfolio standard deviation but higher expected return with smaller

12 = .2

E(rp) = 11.31%

p = 13.08%

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Individual securities

We have learned that investors should diversify.

Individual securities will be held in a portfolio.

We call the risk that cannot be diversified away, i.e., the risk that remains when the stock is put into a portfolio – the systematic risk

Major question -- How do we measure a stock’s systematic risk?

Consequently, the relevant risk of an individual security is the risk that remains when the security is placed in a portfolio.

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Systematic risk

Systematic risk arises from events that effect the entire economy such as a change in interest rates or GDP or a financial crisis such as occurred in 2007and 2008.

If a well diversified portfolio has no unsystematic risk then any risk that remains must be systematic.

That is, the variation in returns of a well diversified portfolio must be due to changes in systematic factors.

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Single Index Model Parameter Estimation

Risk Prem Market Risk Prem

or Index Risk Prem

= the stock’s expected excess return if the market’s excess return is zero, i.e., (rm - rf) = 0

ßi(rm - rf) = the component of excess return due to movements in the market index

ei = firm specific component of excess return that is not due to market movements

αi

errrr ifmiifi

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Estimating the Index ModelExcess Returns (i)

SecurityCharacteristicLine.. ....

.. ....

.. ..

.. ....

.. ....

.. ..

.. ....

......

.. ..

.. ....

.. ....

.. ..

.. ....

.. ....

.. ..

..

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

Excess returnson market index

Ri = a i + ßiRm + ei

Slope of SCL = betay-intercept = alpha

Scatter Plot

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Estimating the Index ModelExcess Returns (i)

SecurityCharacteristicLine.. ....

.. ....

.. ..

.. ....

.. ....

.. ..

.. ....

......

.. ..

.. ....

.. ....

.. ..

.. ....

.. ....

.. ..

..

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

Excess returnson market index

Variation in Ri explained by the line is the stock’s systematic risk

Variation in Ri unrelated to the market (the line) is unsystematic risk

Scatter Plot

Ri = a i + ßiRm + ei

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Components of Risk

Market or systematic risk:

Unsystematic or firm specific risk:

Total risk = Systematic + Unsystematic

risk related to the systematic or macro economic factor in this case the market index

risk not related to the macro factor or market index

ßiM + ei

i2 = Systematic risk + Unsystematic Risk

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Comparing Security Characteristic Lines

Describe e

for each

6-37

Measuring Components of Risk

si2 =

where;

bi2 sm

2 + s2(ei)

si2 = total variance

bi2 sm

2 = systematic variance

s2(ei) = unsystematic variance

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The total risk of security i, is the risk associated with the market + the risk associated with any firm specific shocks.

(its this simple because the market variance and the variance of the residuals are uncorrelated.)

Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk / Total Risk

Examining Percentage of Variance

ßi2 s

m2 / si

2 = r2

bi2 sm

2 / (bi2 sm

2 + s2(ei)) = r2

6-39

The ratio of the systematic risk to total risk is actually the square of the correlation coefficient between the asset and the market.

Sharpe Ratios and alphas

When ranking portfolios and security performancewe must consider both return & risk

“Well performing” diversified portfolios provide high Sharpe ratios:

Sharpe = (rp – rf) / p

The Sharpe ratio can also be used to evaluate an individual stock if the investor does not diversify

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