Diversification and Portfolio Risk

Asset Allocation With Two Risky Assets

6-1

Combinations of risky assetsWhen we put stocks in a portfolio, p <

Why?

When Stock 1 has a return E[r1] it is likely that Stock 2 has a return E[r2] so that rp that contains stocks 1 and 2 remains close to

What statistics measure the tendency for r1 to be above expected when r2 is below expected?

Covariance and Correlation

(Wii)

E[rp]

<

>

n = # securities in the portfolio

Averaging principle

6-2

Portfolio Variance and Standard Deviation

Q

1I

Q

1JJIJI

2p )]r,Cov(r W[Wσ

portfolio the in stocks of number total The Q

lyrespective J and I stock in invested portfolio total the of PercentageW,W JI

J Stock and I Stock of returns the of Covariance)r,Cov(r JI

)r,r(Cov)r,Cov(r & σ )r,(r Cov then J I If IJJI2

IJI

22

222121

21

21

2 2 W)r,r(CovWWWp

Variance of a Two Stock Portfolio:

6-3

Covariance Calculation• Ex ante. Using scenario analysis with probabilities the

covariance can be calculated with the following formula:

• Ex post. Using a time series of returns, the covariance can be calculated with the following formula:

1

( , ) ( ) ( ) ( )S

S B S S B Bi

Cov r r p i r i r r i r

6-4

N

1T21 n

)r(r)r(r

1n

n)r,Cov(r

2T2,1T1,

Covariance and correlation

• The problem with covariance

Covariance does not tell us the intensity of the comovement of the stock returns, only the direction.

We can standardize the covariance however and calculate the correlation coefficient which will tell us not only the direction but provides a scale to estimate the degree to which the stocks move together.

6-5

Measuring the correlation coefficient

• Standardized covariance is called the _____________________

For Stock 1 and Stock 2

21

21(1,2) σσ

)r,Cov(rρ

correlation coefficient or

6-6

Correlation Coefficients: Possible Values

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

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

The closer The closer is to -1, the better diversification.is to -1, the better diversification.

Range of values for correlation coefficients:

-1.0 < < 1.0

7

and diversification in a 2 stock portfolio

•

•

•

•

•

Typically is greater than ____________________

(1,2) = (2,1) and the same is true for the COV

The covariance between any stock such as Stock 1 and itself is simply the variance of Stock 1,

(1,1) = +1.0 by definition

We have no measure for how three or more stocks move together.

zero and less than 1.0

6-8

The effects of correlation & covariance on diversification

Asset A

Asset B

Portfolio AB

6-9

6-10

The effects of correlation & covariance on diversification

Asset C

Asset D

Portfolio CD

= W1 + W2 W1 = Proportion of funds in Security 1W2 = Proportion of funds in Security 2

= Expected return on Security 1= Expected return on Security 2

Two-Security Portfolio: Return

r1

E( )rp

r2

r1 r2

portfolio the in securities # n ;rW)rE(n

1i

iip

Wii=1

n

= 1Wii=1

n

WiWii=1i=1

n

= 1

6-11

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: Risk

6-12

Example 1: Calculating portfolio risk using a time series of returns

– next 4 slides

6-14

Returns ABC XYZ 1 0.2515 -0.2255 2 0.4322 0.3144 3 -0.2845 -0.0645 4 -0.1433 -0.5114 5 0.5534 0.3378 6 0.6843 0.3295 7 -0.1514 0.7019 8 0.2533 0.2763 9 -0.4432 -0.4879

10 -0.2245 0.5263 AAR 0.09278 0.11969

Squared deviations

from average ABC XYZ

0.025192 0.119156 0.115206 0.037912 0.14234 0.033926 0.055734 0.398275 0.212171 0.047572 0.349896 0.04402 0.059624 0.338968 0.025767 0.024527 0.287275 0.369166 0.100667 0.165332

Sum 1.37387 1.578853 Average 0.137387 0.157885

2ABC =

ABC =

2XYZ =

XYZ =

1.37387 / (10-1) = 0.15265

39.07%

1.57885 / (10-1) = 0.17543

41.88%

Calculating Variance and CovarianceEx post

6-15

Returns ABC XYZ 1 0.2515 -0.2255 2 0.4322 0.3144 3 -0.2845 -0.0645 4 -0.1433 -0.5114 5 0.5534 0.3378 6 0.6843 0.3295 7 -0.1514 0.7019 8 0.2533 0.2763 9 -0.4432 -0.4879

10 -0.2245 0.5263 AAR 0.09278 0.11969

COV(ABC,XYZ) =

ABC,XYZ =

ABC,XYZ =

Deviation from average

ABC XYZ 0.15872 -0.34519 0.33942 0.19471

-0.37728 -0.18419 -0.23608 -0.63109 0.46062 0.21811 0.59152 0.20981

-0.24418 0.58221 0.16052 0.15661

-0.53598 -0.60759 -0.31728 0.40661

Product of

deviations -0.05479 0.066088 0.069491 0.148988 0.100466 0.124107 -0.14216 0.025139 0.325656 -0.12901

Sum 0.533973 Average 0.053397

0.533973 / (10-1) = 0.059330

COV / (ABCXYZ) =

0.3626ABC = 39.07%

XYZ = 41.88%

0.059330 / (0.3907 x 0.4188)

N

1T21 n

)r(r)r(r

1n

n)r,Cov(r

2T2,1T1,

E(rp) = W1r1 + W2r2

W1 =

W2 =

=

=

Two-Security Portfolio Return

E(rp) = 0.6(9.28%) + 0.4(11.97%) = 10.36%

Wi = % of total money invested in security i

0.6

0.4

9.28%

11.97%

r1

r2

6-16

p2 =

p2

=

p2

=

p =

p <

Two-Security Portfolio Risk

Q

1I

Q

1JJI

2p J)]Cov(I, W[Wσ

W121

2 + 2W1W2 Cov(r1r2) + W222

2

0.36(0.15265) +

0.1115019 = variance of the portfolio

33.39%

Let W1 = 60% and W2 = 40% Stock 1 = ABC; Stock 2 = XYZ

40.20%

ABC = 39.07%

XYZ = 41.88%

2(.6)(.4)(0.05933) + 0.16(0.17543)

33.39% < [0.60(0.3907) + 0.40(0.4188)] =

W11 + W22

6-17

2ABC = 0.15265

2XYZ = 0.17543

COV(ABC,XYZ) = 0.05933

ABC,XYZ = 0.3626

Example 2: Calculating portfolio risk using scenario analysis with probabilities

– next 5 slides

Scenario Probability Stock Fund Return Bond Fund ReturnRecession 0.3 - 11% 16%Normal 0.4 13% 6%Boom 0.3 27% - 4%

Step 1: Calculate the expected return for the each fund using our formula from Chap.5 for discrete random variables:

Spreadsheet #1

)s(r)s(p)r(Es

1i

Column B Column C Column E Stock Fund Bond Fund

Scenario Probability Rate of Return Col B x Col C

Rate of Return Col B x 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 Expected or Mean Return: SUM: 10.0 SUM: 6.0

19

Scenario Probability Stock Fund Return Bond Fund ReturnRecession 0.3 - 11% 16%Normal 0.4 13% 6%Boom 0.3 27% - 4%

Step 2: Calculate the risk (i.e., variance and standard deviation) for the each fund using formulas for discrete random variables:

Spreadsheet #2

)r(Var and )]r(E)s(r)[(s(p)r(Vars

1i

22

12345678910

A B C D E F G H I J Stock Fund Bond Fund Deviation Deviation

Rate from Column B Rate from Column Bof Expected Squared x of Expected Squared x

Scenario Prob. Return Return Deviation Column E Return Return Deviation Column IRecession 0.3 -11 -21 441 132.3 16 10 100 30Normal 0.4 13 3 9 3.6 6 0 0 0Boom 0.3 27 17 289 86.7 -4 -10 100 30

Variance = SUM 222.6 Variance: 60Standard deviation = SQRT(Variance) 14.92 Std. Dev.: 7.75

20

Scenario Probability Stock Fund Return Bond Fund ReturnRecession 0.3 - 11% 16%Normal 0.4 13% 6%Boom 0.3 27% - 4%

Step 3: Calculate the covariance and correlation coefficient of the 2 funds’ returns. These are formulas for discrete random variables that we haven’t seen before:

Spreadsheet #3

s

1iBBSSBS )]r)i(r)][(r)i(r)[(i(p)r,r(CovCovariance

12345678

A B C D E F GDeviation from Mean Return Covariance

Scenario Probability Stock Fund Bond Fund Product of Dev Col B x Col ERecession 0.3 -21 10 -210 -63Normal 0.4 3 0 0 0Boom 0.3 17 -10 -170 -51

Covariance = SUM: -114Correlation coefficient = Covariance/(StdDev(stocks)*StdDev(bonds)) = -0.99

BS

BSSB

)r,r(CovtCoefficien nCorrelatio

21

Step 4: Calculate the expected return of a PORTFOLIO that invests in the stock and bond funds:

rp = wB rB + wS rS

For example, let’s calculate the return for a portfolio that has 60% of its money invested in the stock fund and 40% of the portfolio invested in the bond fund:

rp = wB rB + wS rS

= (0.4)(6.0%) + (0.6)(10.0%)

= (2.4%) + (6.0%)

= 8.4%

22

Step 5: Calculate the portfolio the risk (i.e., variance and standard deviation) of a PORTFOLIO that invests in the stock and bond funds:

For a portfolio that has 60% of its money invested in the stock fund and 40% of the portfolio invested in the bond fund:

s 2P = (0.4) 2(0.0775) 2 + (0.6) 2(0.1492) 2 + (2)(0.4)(0.6)(0.0775)(0.1492) (-0.99)

= 0.008014 + 0.000961 - 0.00549

= 0.00348, or 0.348%

Standard deviation ( s P ) = √ 0.00348 = 0.059, or 5.9%

Deviation Standard Portfolio and VariancePortfolio 22pp

SBSBSB2

S2

S2

B2

B2

p ww2ww

23

= +1

= .3

E(r)

13%

8%

12% 20% St. Dev

TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS

Stock A Stock B

WA = 0%

WB = 100%

WA = 100%

WB = 0%

= 0

= -150%A

50%B

6-24

Summary: Portfolio Risk/Return Two Security Portfolio

• Amount of risk reduction depends critically on _________________________.

• Adding securities with correlations _____ will result in risk reduction.

• If risk is reduced by more than expected return, what happens to the return per unit of risk (the Sharpe ratio)?

correlations or covariances

< 1

6-25

2p = W1

2122

p = W121

2 + W22

+ W22

+ W323

2+ W323

2

+ 2W1W2+ 2W1W2 Cov(r1r2) Cov(r1r2)

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

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

Three-Security Portfolio n or Q = 3

Q

1I

Q

1JJIJI

2p )]r,Cov(r W[Wσ

6-26

For an n security portfolio there would be _ variances and _____ covariance terms.

The ___________ are the dominant effect on

nn(n-1)

covariances

2p

Possible Risky Investments

Using data from example 2, we calculate the return and risk (standard deviation) of portfolios that invests in different weights of stock and bond funds:

12345

67891011121314151617181920212223

A B C D EInput data

E(rS) E(rB) S

B

SB

10 6 14.92 7.75 -0.99 Portfolio Weights Expected Return

wS wB = 1 - wS E(rP) = Col A x A3 + Col B x B3 Std Deviation*

0.0 1.0 6.00 7.750.1 0.9 6.40 5.500.2 0.8 6.80 3.270.3 0.7 7.20 1.180.4 0.6 7.60 1.510.5 0.5 8.00 3.660.6 0.4 8.40 5.900.7 0.3 8.80 8.150.8 0.2 9.20 10.400.9 0.1 9.60 12.661.0 0.0 10.00 14.92

* The formula for portfolio standard deviation is: SQRT[ (Col A*$C$3)^2 + (Col B*$D$3)^2 + 2*$E$3*Col A*$C$3*Col B*$D$3 ]

27

Possible Risky Investments (continued) Graph the return and risk (standard deviation) of portfolios that invests in

different weights of stock and bond funds:

Investment Opportunity Set for Stocks and Bonds Funds Calculated in Class

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00

Standard Deviation (%)

Exp

ecte

d R

etu

rn (

%)

100% in Bond Fund

100% in Stock Fund

The minimum variance portfolio

28

Possible Risky Investments (continued) Question: Would you ever want to invest in a portfolio that had a higher % of $ invested

in the bond fund than that of the “minimum variance portfolio?Answer: No. You would expect a lower return for risk than you expect in other

combinations!

Investment Opportunity Set for Stocks and Bonds Funds Calculated in Class

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00

Standard Deviation (%)

Exp

ecte

d R

etu

rn (

%)

100% in Bond Fund

100% in Stock Fund

The minimum variance portfolio

29

Minimum Variance Combinations -1< < +1

11 22

- Cov(r1r2) - Cov(r1r2)

W1W1==

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

22

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

2

2

22 2

2

Choosing weights to minimize the portfolio variance

6-30

11

Minimum Variance Combinations -1< < +1

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

11 22

- Cov(r1r2)- Cov(r1r2)

W1W1==

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

22

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

22

22 2211 22

- Cov(r1r2)- Cov(r1r2)

W1W1==

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

22

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

22

22 22WW11

==(.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) = 1122

6-31

E[rp] =

Minimum Variance: Return and Risk with = .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

p2

=p2

=

%.. /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%

W121

2 + W222

2 + 2W1W2 1,212

6-32

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 = -.3

11 22

- Cov(r1r2)- Cov(r1r2)

W1W1==

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

22

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

22

22 2211 22

- Cov(r1r2)- Cov(r1r2)

W1W1==

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

22

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

22

22 22

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) = 1122

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)

6-33

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 = -.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

p2

= p2

=

%.. /p 091001020 21

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

W121

2 + W222

2 + 2W1W2 1,212

1

Notice lower portfolio standard deviation but higher expected return with smaller

12 = .2

E(rp) = 11.31%

p = 13.08%

6-34

Extending Concepts to All Securities

Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result.

The set of portfolios that provide the optimal trade-offs are described as the efficient frontier.

The efficient frontier portfolios are dominant or the best diversified possible combinations.

All investors should want a portfolio on the efficient frontier. … Until we add the

riskless asset6-35

•

•

•

E(r)E(r)

The minimum-variance frontier of The minimum-variance frontier of risky assetsrisky assets Efficient Frontier is the best diversified set of

investments with the highest returns

6-36

GlobalGlobalminimumminimumvariancevarianceportfolioportfolio

EfficientEfficientfrontierfrontier

IndividualIndividualassetsassets

MinimumMinimumvariancevariancefrontierfrontier

St. Dev.

Found by forming portfolios of securities with the lowest covariances at a given E(r) level.

E(r)E(r)The EF and asset allocation

EfficientEfficientfrontierfrontier

St. Dev.

20% Stocks80% Bonds

100% Stocks

EF including international & alternative investments

Ex-Post 2000-2002

80% Stocks20% Bonds

60% Stocks40% Bonds40%

Stocks60% Bonds

100% Stocks

6-37