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The Optimal Risky Portfolio
Lecture No.3² SAPM
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Portfolio construction Process
Capital allocation between risky portfolio and risk-free
assets
± Depends upon risk aversion and risk-return trade-off
Asset allocation among asset classes
± Broad outlines of portfolio established
Security selection
± Specific securities selected for the portfolio
Steps 2 and 3 lead to optimal risky portfolio
Optimal risky portfolio is the combination of risky assets that
provides the best risk-return trade-off
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Diversification and portfolio risk
Two sources of risk ²
± firm-specific risk or unique risk
± Market risk or systematic risk (inflation, business cycles,
exchange rates etc )
Diversification can reduce firm-specific risk to zero if
specific risk is independent ( known as the insurance principle)
Diversification cannot eliminate the systematic risk, i.e. risk
attributable to market-wide sources
Hence investors only care about systematic risks
Return on assets compensates only for systematic risks
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Portfolio expected return and risk
Consider 2 risky assets, a debt mutual fund D and an equity
mutual fund E
If the weights of the 2 assets are Wd and We then portfolio
expected return E(rp) is
and portfolio risk (p) is given as
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Correlation and Portfolio Risk Expected return of the portfolio is the weighted average of
expected returns of component assets with their proportions as
weights
Portfolio variance is driven by the covariance between
component assets If correlation between assets is 1 (perfect positive correlation)
then portfolio standard deviation = weighted average of
component standard deviations
If correlation less than 1, portfolio standard deviation islessthan weighted average of component standard deviations
If correlation is -1 (perfect negative correlation), portfolio
variance is lowest ² we can construct a zero-variance portfolio
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Case of perfect positive correlation
When rho=1, equation for portfolio variance
becomes
Or
Hence standard deviation of portfolio = weighted
average of component standard deviations
Thus no benefit from diversification
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Correlation > 0 < 1
When assets are less than perfectly positively correlated
we can construct the minimum variance portfolio
The Minimum Variance Portfolio has a standard
deviation less than that of the component assets Equation for obtaining weights for Minimum Variance
Portfolio for portfolio consisting of 2 assets D and E
Example
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When correlation is zero
The equation for portfolio variance becomes
The weights for the minimum variance portfolio
are
and
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Case of perfect negative correlation
For assets with perfect negative correlation
The weights for the zero variance portfolio are
and
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Example-Portfolio return and risk
The expected return and risk of the two assets are E(rd)= 0.08,
E(re)=0.13, (d) =0.12 and (e)=0.20
Wd We E( rp) Portfolio Std Deviation for the gi ven correl ation
-1 0 0.3 1
0.00 1.00 0.1300 0.2000 0.2000 0.2000 0.2000
0.10 0.90 0.1250 0.1680 0.1804 0.1840 0.1920
0.20 0.80 0.1200 0.1360 0.1618 0.1688 0.1840
0.30 0.70 0.1150 0.1040 0.1446 0.1547 0.1760
0.40 0.60 0.1100 0.0720 0.1292 0.1420 0.1680
0.50 0.50 0.1050 0.0400 0.1166 0.1311 0.1600
0.60 0.40 0.1000 0.0080 0.1076 0.1226 0.1520
0.70 0.30 0.0950 0.0240 0.1032 0.1170 0.1440
0.80 0.20 0.0900 0.0560 0.1040 0.1145 0.1360
0.90 0.10 0.0850 0.0880 0.1098 0.1156 0.1280
1.00 0.00 0.0800 0.1200 0.1200 0.1200 0.1200
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Observations
For =1, portfolio standard deviation is simply weighted average
of asset standard deviation (no benefit of diversification)
For =0.30 and =0, portfolio standard deviation
± decreases initially as equity component increases indicating
diversification benefit and
± increases as portfolio becomes concentrated in equity
± we can find the minimum-variance portfolio which has
standard deviation less than that of individual assets
For =-1, diversification is most effective due to ´perfect hedgeµ
± For =-1, we can construct a zero-variance portfolio
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Portfolio expected return and SD
Exhibit 4.5: Mean Standard Deviation Diagram: Portfolios of Two Risky Securities with
Arbitrary Correlation, V
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The Minimum Variance Frontier
We plot the set of portfolios with the lowest variance at a given
level of expected return ( recall the mean variance criterion)
Above set known as the Mean Standard Deviation frontier or
Minimum Variance frontier Portion of mean-SD frontier below the global minimum variance
portfolio is inefficient
For each frontier portfolio on the lower portion there exists
another frontier portfolio on the upper portion with same but a
higher E(r)
Portion above the global minimum variance portfolio is known as
the efficient frontier in the absence of a risk-free asset
Investors will only choose a portfolio on the efficient frontier
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The Efficient Frontier
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Risky Portfolio with a risk-free asset Given a risky portfolio of 2 risky assets, a debt mutual fund and
an equity mutual fund and a risk-free asset with return rf
How do we find the optimal risky portfolio?
Plot CALs starting from the risk-free rate and passing through
the opportunity set of risky assets ² debt and equity funds
The highest CAL will have highest slope, i.e.Max S = (E(rp)-
rf)/p
Optimal risky portfolio is the tangency point of the highest CAL
to the portfolio opportunity set Tangency portfolio consists of risky assets only
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Risky portfolio + risk-free asset
We draw
CALs
from the
risk-freerate to
various
portfolios
on the
efficient
frontier
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The Optimal risky portfolio
We find theCAL from the
risk-free rate to
the point of
tangency to the
efficientfrontier
Portfolio at
tangency point
will have
highestReward-to-risk
ratio
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Optimal Risky Portfolio ² 2 assets
Equation for determining weights of optimal risky portfolio with
2 risky assets
And
where RD and RE are excess returns on debt and equity funds
i.e. Expected return less the risk-free rate
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Optimal Complete Portfolio
Calculate the weights of the optimal risky portfolio as
above
Compute the E(r ) and of the optimal risky portfolio
We have the risk-free rate and the investor·s degree of
risk aversion
Proportion to be invested in risky portfolio is
Balance is the proportion invested in risk-free asset
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Optimal Complete Portfolio
Point P where the
CAL is tangent to
the efficient frontier
depicts the optimal
risky portfolio At Points 1 and 2
the indifference
curves of 2 different
investors are tangent
to the CAL
Points 1 and 2depict the
O ptimum
com plete portfolio
for those investors
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Summary
Note that optimal risky portfolio is the same for all investors
± F ormula for computation of weights of optimal risky portfolio does not
include the investor·s degree of risk aversion
Hence the fund manager will offer the same optimal risky portfolio
to all his investors- his job becomes easier !
The optimal com plete portfolio for each investor (the allocation
of funds between the risky portion and risk-free portion) will be
different
It will depend on investor·s preferences, i.e. his degree of risk aversion and indifference curve
More risk averse investors will have lower proportion of the
optimal risky portfolio in their complete portfolio than less risk-
averse investors