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
Finance - Pedro Barroso 2
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
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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)
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Expected (mean) return
wherers :return if state s occurs
ps : probability of state s happening
S
sssrpr
1
]E[
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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%
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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
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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
Finance - Pedro Barroso 8
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%
Variance
0205.00289.031
0001.031
0324.031
)11.028.0(31
)11.012.0(31
)11.007.0(31 2222
A
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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%
Standard Deviation
%3.140205.0 A
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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
Finance - Pedro Barroso 11
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
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%
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
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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
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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
Finance - Pedro Barroso
)()(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
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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
Finance - Pedro Barroso
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?
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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
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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
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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
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Portfolio Variance• Variance of a portfolio is
Remember Var(aX+bY)=a2Var(X)+b2Var(Y)+2abCov(X,Y)
ABBA2
BB2
AA2P w2wσwσwσ )()(
Finance - Pedro Barroso
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
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%
Finance - Pedro Barroso 23
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
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%
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% 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
Finance - Pedro Barroso 27
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
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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[()(
Finance - Pedro Barroso
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
Finance - Pedro Barroso 30
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
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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
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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
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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
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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.
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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
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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
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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)(
)(
Finance - Pedro Barroso
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
Finance - Pedro Barroso
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
Finance - Pedro Barroso 41
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
Finance - Pedro Barroso 42
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%
Microsoft Office Excel 97-2003 Worksheet
Finance - Pedro Barroso 43
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
Finance - Pedro Barroso
Optimal Portfolio• To find optimal portfolio choice
2f2 )(
0)(:T
TTTTfT
rrEwwrrEFOC
2)1()()(max
22TT
fTTTPw
wrwrEwrU
T
Finance - Pedro Barroso 45
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?
Finance - Pedro Barroso
Microsoft Office Excel 97-2003 Worksheet
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
Microsoft Office Excel 97-2003 Worksheet
Finance - Pedro Barroso 47