portfolio theory finance - pedro barroso1. motivation mean-variance portfolio analysis – developed...

Portfolio Theory

Finance - Pedro Barroso 1

Motivation• Mean-variance portfolio analysis

– Developed by Harry Markowitz in the early 1960’s (1990 Nobel Prize in Economics)

– Foundation of modern finance• Used by all mutual funds, pension plans, wealthy

individuals, banks, insurance companies, ...• There is an industry of advisors (e.g.

Wilshire Associates) and software makers (e.g. BARRA, Quantal) that implement what we will learn in the next few classes


http://www.wilshire.com/

http://www.barra.com/

http://www.quantal.com/

Modern Investment Advice

• The optimal portfolio of risky assets should contain a large number of assets – it should be well diversified – and is the same for all investors

• Investors should control the risk of their portfolio not by re-allocating among risky assets, but through the split between risky assets and the risk-free asset


Individual Securities

• Characteristics of individual securities that are of interest are the:– Expected Return– Variance and Standard Deviation– Covariance and Correlation (to another security or

index)


Expected (mean) return

wherers :return if state s occurs

ps : probability of state s happening

S

sssrpr

1

]E[


Example

• Consider the following two risky asset world. There is a 1/3 chance of each state of the economy, and the only assets are stock A and stock B

Rate of ReturnScenario Probability Stock A Stock BRecession 33.3% -7% 5%Normal 33.3% 12% 25%Boom 33.3% 28% -5%


Stock A Stock B

Rate of Squared Rate of Squared

Scenario Return Deviation Return Deviation Recession -7% 0.0324 5% 0.0011Normal 12% 0.0001 25% 0.0278Boom 28% 0.0289 -5% 0.0178Expected return 11.00% 8.33%Variance 0.0205 0.0156Standard Deviation 14.3% 12.5%

Expected Return

%11%)28(31

%)12(31

%)7(31

)( ArE


Variance and Standard Deviation

• Shortcut

• Standard Deviation is the square root of variance

S

sss rrpr

1

22 ])E[(]Var[

S

sss rrp

1

222 ]E[

]Var[r


Stock A Stock B



Variance

0205.00289.031

0001.031

0324.031

)11.028.0(31

)11.012.0(31

)11.007.0(31 2222

A


Stock A Stock B



Standard Deviation

%3.140205.0 A


Covariance

• Measure of movement in tandem

• Correlation:

S

sBAsBsAs

S

sBsBAsAs

rrrrp

rrrrprr

1,,

1,,BA

]E[]E[

])E[)(]E[(],Cov[

BA

BAAB

rr

],[Cov


Covariance and Correlation

005.0)02267.0(31

00167.031

006.031

1333.017.031

1667.001.031

0333.0)18.0(31

),( ,

BABA rrCov

28.0125.0143.0

005.0),(,

BA

BABA

rrCov

Finance - Pedro Barroso

Stock A Stock B



12

Correlation

• Correlation measures relationship between the return on a stock and the return on another:– Perfect positive correlation: 1– No correlation: 0– Perfect negative correlation: -1


Correlation

Correlation=-0.9

-2

-1

0

1

2

-2 -1 0 1 2

Correlation=+0.9

-2

-1

0

1

2

-2 -1 0 1 2

Correlation=0.0

-2

-1

0

1

2

-2 -1 0 1 2


Estimating Means and Covariances• In real life we do not know probability of each state

of the world and the return that corresponds to it• We need to use historical data to estimate average

returns, variance and covariance of returns

T

titi r

Tr

1

1

T

titi rr

T 1

22 )(1

1̂


)()(1

1ˆ1

ijt

T

tiitij rrrr

T

15

Estimating Means and Covariances

• We can use the functions average(), var(), stdev(), covar(), correl() in Excel

• We are implicitly assuming that the returns came from the same probability distribution in each year of the sample

• The estimated mean and variance are themselves random variables since there is estimation error that depends on the particular sample of data used (sampling error)– We can calculate the standard error of our estimates and figure out a

confidence interval for them– This contrasts with the true (but unknown) mean and variance which

are fixed numbers, not random variables


Annualizing Mean and Covariances• Annual return is approximately equal to the sum of the 12

monthly returns; assuming monthly returns are independently distributed (a consequence of market efficiency) and have same variance

• If mean, standard deviation or covariance are estimated from historic monthly returns, estimates will be per month

• To annualize: – mean, variance, covariance: multiply by 12– standard deviation: multiply by sqrt(12)

12321 ... rrrrry

212

23

22

21

22 12... my rrrrr

Microsoft Office Excel 97-2003 Worksheet


Portfolios• Weights: fraction of wealth invested in different assets

– add up to 1.0 – denoted by w

• Example– $100 MSFT, $200 in GE

• Total investment: $100+$200=$300– Portfolio weights

• MSFT: $100/$300 = 1/3• GE: $200/$300 = 2/3

• Can we have negative portfolio weights?


Portfolios• You can have negative weights if you short sell a stock

– Borrow stock from broker– Sell stock and get proceeds (stock price)– Buy stock back later to give it back– Profit/loss = sell price - buy price

• Example– $500 MSFT (buy) , $200 in GE (short sell)

• Total investment: $500-$200=$300– Portfolio weights

• MSFT: $500/$300 = 5/3• GE: -$200/$300 = -2/3


Portfolio Expected Return

• Portfolio return– Average of returns on individual securities

weighted by their portfolio weights

• Then expected return on the portfolio

Remember from stats that E(aX+bY)=aE(X)+bE(Y)

BBAAP rwrwr

)()()( BBAAP rEwrEwrE


Rate of ReturnScenario Stock A Stock B Portfolio squared deviationRecession -7% 5% -2.2% 0.0147Normal 12% 25% 17.2% 0.0053Boom 28% -5% 14.8% 0.0024

Expected return 11.00% 8.33% 9.9%Variance 0.0205 0.0156 0.0075Standard Deviation 14.31% 12.47% 8.64%

Portfolios (60% Stock A, 40% Stock B)

Expected rate of return on the portfolio is a weighted average of the expected returns on stocks in portfolio:

%)3.8(%40%)11(%60%9.9

)()()( BBAAP rEwrEwrE


Portfolio Variance• Variance of a portfolio is

Remember Var(aX+bY)=a2Var(X)+b2Var(Y)+2abCov(X,Y)

ABBA2

BB2

AA2P w2wσwσwσ )()(


ABBABA2

BB2

AA2P w2wσwσwσ )()(

22

Portfolios

Variance of the rate of return on the two stock portfolio:

00746.0)28.0(1247.04.01431.06.02)1247.04.0()1431.06.0(

))(()()(22

ABBBAA2

BB2

AA2P ρσwσw2σwσwσ




Portfolios

Observe the decrease in risk that diversification offers

Portfolio with 60% in stock A and 40% in stock B has less risk than either stock in isolation




% Stock A Risk Return0% 12.47% 8.33%5% 11.67% 8.47%

10% 10.91% 8.60%15% 10.21% 8.73%20% 9.58% 8.87%25% 9.03% 9.00%30% 8.58% 9.13%35% 8.25% 9.27%40% 8.05% 9.40%45% 7.98% 9.53%50% 8.07% 9.67%55% 8.29% 9.80%60% 8.64% 9.93%65% 9.10% 10.07%70% 9.66% 10.20%75% 10.30% 10.33%80% 11.01% 10.47%85% 11.77% 10.60%90% 12.58% 10.73%95% 13.43% 10.87%

100% 14.31% 11.00%

Efficient Frontier - Two Stocks

We can consider other portfolio weights besides 60% in stock A and 40% in stock B …

Portfolio Risk and Return Combinations

7%

8%

9%

10%

11%

12%

7% 9% 11% 13% 15%Portfolio Risk (standard deviation)

Po

rtfol

io R

etur

n

100% A

100% B

% Stock A Risk Return0% 12.47% 8.33%5% 11.67% 8.47%

10% 10.91% 8.60%15% 10.21% 8.73%20% 9.58% 8.87%25% 9.03% 9.00%30% 8.58% 9.13%35% 8.25% 9.27%40% 8.05% 9.40%45% 7.98% 9.53%50% 8.07% 9.67%55% 8.29% 9.80%60% 8.64% 9.93%65% 9.10% 10.07%70% 9.66% 10.20%75% 10.30% 10.33%80% 11.01% 10.47%85% 11.77% 10.60%90% 12.58% 10.73%95% 13.43% 10.87%

100% 14.31% 11.00%

Efficient Frontier - Two Stocks

Note that some portfolios are “better” than others; they have higher returns for the same level of risk or less

Portfolios in the frontier above the MVP are efficient

7%

8%

9%

10%

11%

12%

7% 9% 11% 13% 15%

Por

tfolio

Ret

urn

Portfolio Risk (standard deviation)

Portfolio Risk and Return Combinations

100% A

100% B

MVP

Minimum Variance Portfolio (MVP)• Portfolio with lowest possible variance

%3.55

%7.44)005.0(20156.00205.0

)005.0(0156.0

2

0)21(2)1(22:

)1()1(()(

22

2

22

B

A

ABBA

ABBA

AABBAAAA

2P

ABAA2

BA2

AA2P

w

w

σσσ

w

wσwσwwσ

FOC

w2w)σwσwσMin


Portfolios with Different Correlations

100% B

E(re

turn

)

100% A

= 0.2 = 1.0

= -1.0

• Relationship depends on correlation coefficient– If= +1.0 no diversification effect– If< 1.0 some diversification effect– If= –1.0 diversification can eliminate all the risk


Portfolios with Different Correlations • Standard deviation of a portfolio is

– If= +1.0

– If< 1

– If= -1.0

BAAA2

BAAA

BAAA2

BA2

AAP

σwσwσwσw

σσw2wσwσwσ

)1(])1([

)1(])1[()(


BAAABAAA

2BAAA

2BAAA

BAAA2

BA2

AAP

σwσwσwσw

σwσwσwσw

σσw2wσwσwσ

)1( )1(

])1([ ])1([

)1(])1[()(

BAAA

ABBAAA2

BA2

AAP

σwσw

σσw2wσwσwσ

)1(

)1(])1[()(

29

Portfolio with Many Stocks• For portfolio with N stocks, we need:

– N expected returns (one for each asset)– N variances (one for each asset)– N(N-1)/2 covariances (for each pair of assets)

N

iiiP rEwrE

1

)()(

ijj

N

i

N

jiji

N

jiiijj

N

i

N

ji

2P σwwwσwwσ

1 ,11

22

1 1

2


Opportunity Set for Many Stocks

Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios (it is an area rather than a line)

E(re

turn

)

P

Individual Assets


Efficient Frontier for Many Stocks

Section of opportunity set above minimum variance portfolio is efficient frontier (north-west edge): -offers minimum risk for a given expected return-offers maximum expected return for a given risk

E(re

turn

)

P

Minimum Variance Portfolio

Efficient frontier

Individual Assets


Diversification and Portfolio Risk

• Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns

• This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another

• However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion


Portfolio Risk and Number of Stocks

Systematic Risk

Market Risk, Non-diversifiable risk

Idiosyncratic Risk

Diversifiable Risk, Nonsystematic Risk Firm Specific Risk

Number of stocks

In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not

Portfolio risk

Limits to Diversification• Consider equal-weighted portfolio wi=1/N

portfolio variance depends on average stock variance and average correlation among stocks

• When N grows:

• Average correlation is 0.20, average standard deviation is 50%, so minimum portfolio volatility about 22%

222

2

22

22

22

11)1(

112/)1(2/)1(1

211

12

1

iijiiji

i jiij

ii

i jiij

iip

NN

NNN

NN

NNNN

NNN

NN

ijiijpn

22lim


Systematic Risk

• Risk factors that affect a large number of assets• Also known as non-diversifiable risk or market risk• Includes such things as changes in GDP, inflation,

interest rates, etc.


Idiosyncratic Risk

• Risk factors that affect a limited number of assets• Includes such things as labor strikes, part shortages,

earnings announcements• Risk that can be eliminated by combining assets into

a portfolio (need about 60 stocks)• If we hold only one asset, or assets in the same

industry, then we are exposing ourselves to risk that we could diversify away


Total Risk

• Total risk = systematic risk + idiosyncratic risk• Standard deviation of returns is a measure of total

risk• For well-diversified portfolios, idiosyncratic risk is

very small• Consequently, the total risk for a diversified portfolio

is essentially equivalent to the systematic risk


Riskless Borrowing and Lending

• Allow for lending and borrowing at risk-free rate (T-bill) – zero risk and zero covariance with stock returns• Capital allocation line (CAL): feasible combinations of stock(s) and riskless asset• CAL Slope is Sharpe ratio : excess return per unit of risk

E(re

turn

)

P

Efficient Frontier

rfp

i

iP

rrErrE

f

f)(

)(


f)( rrE

39

iiP

iiiP

wrwrEwrE

f)1()()(

Efficient Frontier with Riskless Asset

• Efficient frontier is the capital allocation line (CAL) with the steepest slope • Investors allocate their wealth between riskless asset and tangency portfolio

rf

E(re

turn

)Tangency portfolio T

P


pT

TP

rrErrE

f

f)(

)(

40

TTP

fTTTP

w

rwrEwrE

)1()()(

Efficient Frontier with Riskless Asset

• If E(rP)< E(rT): riskless lending, wT < 100%

• If E(rP) > E(rT): riskless borrowing, wT > 100%

rf

E(re

turn

)Tangency portfolio T

P


Tangency Portfolio – Two Stocks• Portfolio weights of tangency portfolio (max Sharpe)

AB

ABBAABBA

ABBBAA

ABAABAAA

BAAA

w

P

w

wwrrErrErrErrE

rrErrEw

wwww

rrEwrEwrrEAA

1)()()()(

)()(

)1(2)1(

)()1()(max

)(max

ff2

f2

f

f2

f

2222f

P

f


Tangency Portfolio – Two Stocks• Using our previous example of two stocks (A and B) and a

riskless rate of 8%:– stock A weight 0.69, stock B weight 0.31– expected return 10.17%, std.dev. 9.54%, Sharpe 0.228

• Investor wants to form portfolio with an expected return of 9%:– Combine tangency portfolio with riskless asset– E(rP) = 9% = 10.17% x wT + 8% x (1 – wT)

wT = 46% (wA = 0.46 x 0.69 = 32%, wB =0.46 x 0.31 = 14%)

wf = 54%

– P = 0.46 x 9.54% = 4.4%



Optimal Portfolio• Investor with quadratic utility function

– Only cares about mean and variance of returns– How do indifference curves plot?– Where is coefficient of risk aversion: e.g. = 4– How do we determine ? With questionnaires

• How much would you pay to avoid a 50-50 chance of doubling or losing x dollars?

2)()(

2P

PP rErU


Optimal Portfolio• To find optimal portfolio choice

2f2 )(

0)(:T

TTTTfT

rrEwwrrEFOC

2)1()()(max

22TT

fTTTPw

wrwrEwrU

T


Optimal Portfolio: Example• Optimal combination of tangency and risk-free asset for an

investor with risk aversion γ = 4• Optimal weight on the tangency portfolio: 0.6• So the weight on the risk-free asset is 0.4• To find the weights on stocks, multiply the weight on T by the

weights that stocks have in T– Weight on stock A: 0.6 × 0.69 = 0.41– Weight on stock B: 0.6 × 0.31 = 0.19

• Expected return 9.30%, standard deviation 5.69%

• What if the risk aversion coefficient was 2?



46

Tangency portfolio – Many Stocks• Portfolio weights of tangency portfolio must be

solving the optimization problem (max Sharpe):

• We can obtain weights using Solver in Excel

2/1

f

p

f maxmax

i jijji

iii

w

p

w

ww

rwrkk



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