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Portfolio Theory and Capital Asset Pricing Model
Prof. Ashok Thampy
IIMB
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Markowitz Portfolio Theory
• Combining stocks into portfolios can reduce standard deviation, below the level obtained from a simple weighted average calculation.
• Correlation coefficients make this possible.
• The various weighted combinations of stocks that create this standard deviations constitute the set of efficient portfoliosefficient portfolios.
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Markowitz Portfolio Theory
Coca Cola
Reebok
Standard Deviation
Expected Return (%)
35% in Reebok
Expected Returns and Standard Deviations vary given different weighted combinations of the stocks
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Efficient Frontier
Standard Deviation
Expected Return (%)
•Each half egg shell represents the possible weighted combinations for two stocks.
•The composite of all stock sets constitutes the efficient frontier
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Efficient Frontier
Standard Deviation
Expected Return (%)
•Lending or Borrowing at the risk free rate (rf) allows us to exist outside the
efficient frontier.
rf
Lending
BorrowingT
S
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Efficient Frontier
A
B
Return
Risk (measured as )
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Efficient Frontier
A
B
Return
Risk
AB
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Efficient Frontier
A
BN
Return
Risk
AB
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Efficient Frontier
A
BN
Return
Risk
AB
Goal is to move up and left.
WHY?
ABN
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Efficient Frontier
Return
Risk
Low Risk
High Return
High Risk
High Return
Low Risk
Low Return
High Risk
Low Return
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Efficient Frontier
Return
Risk
Low Risk
High Return
High Risk
High Return
Low Risk
Low Return
High Risk
Low Return
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Portfolio Risk
)rx()r(x Return PortfolioExpected 2211
)σσρxx(2σxσxVariance Portfolio 21122122
22
21
21
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Portfolio Risk
n
1iii )r(x Return Portfolio Expected
n
ji 1,ij)σ( jixxVariance Portfolio
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Portfolio RiskThe shaded boxes contain variance terms; the remainder contain covariance terms.
1
2
3
4
5
6
N
1 2 3 4 5 6 N
STOCK
STOCKTo calculate portfolio variance add up the boxes
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Limits of Diversification
covariance average x 1/N) - (1 variance average x (1/N) variance Portfolio
)covariance .(average (N variance average Variance Portfolio 2
221
).1
NNx
NN
As the number of stocks in the portfolio becomes very large, the portfolio variance tends towards the average covariance.
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Portfolio Diversification
Suppose you make a portfolio constructed by taking equalProportions of n assets; that is xi = 1/n for each i. then The corresponding portfolio return and variance is :
n
1ii)(r
n1
Return Portfolio Expected
n
σ)σ(
1 2
1,ij2
n
jinVariance Portfolio
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Question : Find the minimum variance portfolio
abba
abab
abba
abba
ababaaaa
abaabaaa
abbabbaa
xx
Solving
xxxx
xxxx
xxxx
p
p
p
22
0)21(2)1(22
)1(2)1(
2
22
2
22
2
22
2
22222
22222
and
:get we this
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Return
Risk
Risk Free
Return, = rrf
Efficient Portfolio
Market Return = rm
The one-fund theorem: There is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund F and the risk free asset.
F
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Capital Market Line
Return
Risk
.
rf
Risk Free
Return =
Efficient Portfolio
Market Return = rp
pσ
Slope = (rp-rf)/ pσ The portfolio that maximizes theSlope gives the efficient portfolio.
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The capital market line is mathematically expressed asFollows:
asset. efficient arbitrary anof returnof rate theof deviation standard the and
value expected the are and and return,of rate market theof deviation standard and
values expected the are and where
r
r
rrrr
MM
M
fMf
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Capital Asset Pricing Model
2)(
M
iMifMifi
i
rrrr
r
where
:satisfies i asset anyof return,of rateexpected the efficient, is M portfolio market theIf :CAPM The
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Portfolio Beta
Beta of a portfolio is the weighted average beta of individualAssets in the portfolio.
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Security Market Line
Return
.
rf
Risk Free
Return =
Efficient Portfolio
Market Return = rm
BETA1.0
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Security Market LineReturn
BETA
rf
1.0
SML
SML Equation = rf + B ( rm - rf )
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Systematic and Unsystematic Risk
risk icunsystemat risk systematic
Variance
VarrVarrVar
Cov
E
rrrr
iMiii
Mi
i
ifMifi
)()()(
0),(
0)(
)(
22
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Testing the CAPM
Avg Risk Premium 1931-65
Portfolio Beta1.0
SML
30
20
10
0
Investors
Market Portfolio
Beta vs. Average Risk Premium
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Testing the CAPM
Avg Risk Premium 1966-91
Portfolio Beta1.0
SML
30
20
10
0
Investors
Market Portfolio
Beta vs. Average Risk PremiumIn the period 1966-91, returnhas not been proportionate to betaas predicted by the CAPM-SML.