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Originally proposed by HarryMarkowitz in 1950·s
First formal attempt to quantify therisk
Diversification reduces the risk
Round 20 stocks to the portfolio,
unsystematic risk can be reduced tozero
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Wp %
35
20
0
Number of securities in portfolio
10 20 30 40 ...... 100+
Total risk
Systematic Risk
DiversifiableRisk
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Markowitz, the father of modern portfolio theory,
developed the basic principle of portfoliodiversification in a formal way, in quantified
form, that shows why and how portfolio
diversification works to reduce the risk of a
portfolio to an investor.Modern Portfolio theory hypothesizes how
investors should behave.
As the no. of securities in the portfolio increases,
contribution of individual security·s riskdecreases due to offsetting effect of strong
performing and poor performing securities in the
portfolio and the importance of covariance
relationships among securities increases.
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Holding two stocks is less risky than holdingone stock
Assumptions
Investors decisions are on twoparameters-expected return and variance
Investors are risk averse
Investors seek to achieve the highest
expected return at a given level of risk Investors have identical expectations
Common one period investment horizon
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Portfolio Return
Actual Portfolio Return
RP = W1R1+ W2R2 + «WnRn
Expected Return of a Portfolio Return
E(RP) = W1 E(R1) + W2 E(R2) + «Wn E(Rn)
Portfolio Risk
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States that among all investments with a
given return, the one with the least risk is
desirable; or given the same level of risk, the
one with the highest return is mostdesirable.
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Security E(Ri) W
ATW 7% 3%
GAC 7% 4%
YTC 15% 15%FTR 3% 3%
HTC 8% 12%
ATW dominates GAC
ATW dominates FTR
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1. Normal Diversification
This occurs when the investor combines more
than one (1) asset in a portfolio
1 5 10 20 30
Risk
Systematic Risk
Unsystematic
Risk
# of Assets
75% of Co.
Total Risk
25% of Co.
Total Risk
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Unsystematic Risk
... is that portion of an asset·s total risk which
can be eliminated through diversification
Systematic Risk ... is that risk which cannot be eliminated
Inherent in the marketplace
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Superfluous or Naive Diversification
Occurs when the investor diversifies in more
than 20-30 assets. Diversification for
diversification·s sake. a. Results in difficulty in managing such a
large portfolio
b. Increased costs
Search and transaction
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This type of diversification considers the
correlation between individual securities. It is
the combination of assets in a portfolio that are
less then perfectly positively correlated. a. The two asset case:
Stk. A Stk. B
E(R) 5% 15%
W10
%20
%
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Assume that the investor invests 50% of
capital stock in stock A and 50% in B
1. Calculate E(R)
E(Rp) = 7xi E(Ri) E(Rp) =.5(.05) + .5(.15)
E(Rp) = .025 + .075
E(Rp) = .10 or 10%
i=1
n
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2. Graphically
A
B
PortfolioAB
E(R p)
15%
10%
5%
5 10 15 20 25 W
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Portfolio Return of AB will always be on line AB
depending on the relative fractions invested in
assets A and B.
3. Calculating the risk of the portfolio Consider 3 possible relationships between A
and B
Perfect Positive Correlation
Zero Correlation Perfect Negative Correlation
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A and B returns vary in identical pattern.
Hence, there is a linear risk-return
relationship between the two assets.
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A
B
AB
E(R p)
15%
5%
10 15 20 W p
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Therefore, the risk of portfolio AB is simply
the weighted value of the two assets· W.
In this case:
Wp = xA2 WA
2 + xB2 WB
2 + 2 xAxBWAWB VAB
Wp = .25(.10)2+.25(.20)2+2(.5)(.5)(.10)(.20)
Wp = .15 or 15%
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A·s return is completely unrelated to B·s
return. With zero correlation, a substantial
amount of risk reduction can be obtained
through diversification.
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A
B
AB
E(R p)
15%
10%
5%
10 11.2 20 W p
W p = .25(.10)2+.25(.20)2
W p } 11.2%
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A·s and B·s returns vary perfectly inversely.
The portfolio variance is always at the lowest
risk level regardless of proportions in each
asset.
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A
B
AB
E(R p)
15%
5%
5 10 20 W p
10%
W p = .25(.10)+.25(.20)+2(.5)(.10)(.20)(-1)
W p = .05 or 5%
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Although there are no securities with
perfectly negative correlation, almost all
assets are less than perfectly correlated.
Therefore, you can reduce total risk (Wp) through diversification. If we consider many
assets at various weights, we can generate
the efficient frontier.
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E(R p)
W p
M
Efficient
Frontier
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The Efficient Frontier represents all the
dominant portfolios in risk/return space.
There is one portfolio (M) which can be
considered the market portfolio if weanalyze all assets in the market. Hence, M
would be a portfolio made up of assets that
correspond to the real relative weights of
each asset in the market.
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Assume you have 20 assets. With the help of
the computer, you can calculate all possible
portfolio combinations. The Efficient
Frontier will consist of those portfolios withthe highest return given the same level of
risk or minimum risk given the same return
(Dominance Rule)
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4. Borrowing and lending investment funds
at R to expand the Efficient Frontier.
a. We keep part of our funds in a saving
account Lending, OR
b. We can borrow funds for a greater
investment in the market portfolio
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The efficient frontier is
convex as a result of risk
and return characteristics
of the portfolio, which
changes in a non-linearfashion as its components·
weighting are changed.
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The capital asset pricing model defines therelationship between risk and return
E(RA) = Rf + FA(E(RM) ² Rf)
If we know an asset·s systematic risk, we canuse the CAPM to determine its expectedreturn
This is true whether we are talking aboutfinancial assets or physical assets
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Pure time value of money ² measured by the
risk-free rate
Reward for bearing systematic risk ²
measured by the market risk premiumAmount of systematic risk ² measured by
beta
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Consider the betas for each of the assets givenearlier. If the risk-free rate is 2.13% and the marketrisk premium is 8.6%, what is the expected return
for each?
Securit Beta Ex ected etur
DCLK 2.685 2.13 + 2.685(8.6) = 25.22%
KO 0.195 2.13 + 0.195(8.6) = 3.81%
INTC 2.161 2.13 + 2.161(8.6) = 20.71%
KEI 2.434 2.13 + 2.434(8.6) = 23.06%
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