Portfolio Construction
Capital Allocation
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Capital Allocation and Security
Selection
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Capital allocation decision is concerned with
how you allocate your funds in broad asset
classes: e.g., the risk free asset and the risky
assets.
Security selection decision is concerned with
how you allocate your money among
individual securities, e.g., IBM stock,
Microsoft stock.
Capital Allocation
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In making the capital allocation decision, I
assume that you have already selected the
(optimal) risky portfolio.
For ease of exposition, I sometimes just call
the risky portfolio as the risky asset.
In constructing their portfolios, Individual
investors or portfolio managers seek to
achieve the best possible trade-off between
risk and return.
To do this, the individual investors or portfolio
managers have to make two decisions:
______________ and _____________
Top-down or bottom-up?
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top-down approach
bottom-up approach
As you will see, the capital allocation decision
and the security selection decision are largely
independent according to the modern
portfolio theory.
Portfolio of A Risk-free Asset
and A Risky Asset
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Assume the expected return of the risky asset
is E(r) and the standard deviation of the
return of the risky asset is σ.
„
Then the expected return of the portfolio is
E(rp) = (1-y) rf + y E(r) = rf + y (E(r)-rf )
y = proportion of funds in the risky asset
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Portfolio of A Risk-free Asset
and A Risky Asset
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By definition, the expected return of the riskfree asset is the ___________ and the
standard deviation of the return of the riskfree asset is _____. The correlation between
the risk free rate and the return on any risky
asset is also _____.
N
Var ( X ) = σ X2 = ∑ pi [ X i − E ( X )]
Portfolio of A Risk-free Asset
and A Risky Asset
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„
„
2
i =1
Cov( X , Y ) = E [( X − E ( X ))(Y − E (Y ))]
=
N
N
∑∑ p ( X
ij
i =1 j =1
i
− E ( X ))(Y j − E (Y ))
Sharpe Ratio
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Example
Reward to risk ratio
Sharpe Ratio =
„
E (RP ) − R f
σP
σP
=
y (E ( R ) − R f )
yσ
=
„
„
„
„
„
σ
„
Suppose
r f = 7%, E (r ) = 15%, σ = 22% and A = 4
„
„
E ( R) − R f
Example
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Suppose
r f = 7%, E (r ) = 15%, σ = 22% and A = 4
E ( RP ) − R f
The portfolio of a risk-free asset and a
risky asset has the same Sharpe Ratio as
the risky asset
Sharpe Ratio =
What about the variance of the portfolio?
σ p2 = w12 σ 12 + w22 σ 22 + 2w1w2 Cov(r1, r2)
In this case,
σp2 = (1-y)2 02 + y2 σ 2 + 2y(1-y ) 0
=y2σ2
And the standard deviation of the return of
the portfolio is simply:
σp = y σ
If you put 20% of your money in the risky
asset and the rest in the risk free asset,
E(R)= ________________________
Stdev(R)= _____________________
Sharpe Ratio= __________________
If you put 80% of your money in the risky
asset and the rest in the risk free asset,
E(R)= ________________________
Stdev(R)= _____________________
Sharpe Ratio= __________________
Capital Allocation Line
E(rp) = rf + y (E(r)-rf )
σp = y σ
Î E(rp) = rf + (σp/σ) (E(r)-rf )
ÎE(rp) = rf + σp (E(r)-rf )/ σ
Expected return is linear in the
standard deviation of the
portfolio. Graphically, it is a
straight line.
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Capital Allocation Line
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Capital Allocation Line
We can represent combinations of a risky
asset and the risk-free asset on a graph (this
is also the investment opportunity set):
Expected
Return
E(Ri)
Capital Allocation Line
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(CAL)
E(Rp)=15%
S=
E ( RP ) − R f
σP
=
0.15 − 0.7
= .36
0.22
• Risky Asset
„
Rf=7%
The slope of the CAL measures the excess
return being earned per unit of volatility.
In our present example, the slope is
• Risk-free Asset
This “reward-to-risk ratio” is commonly
referred to as the Sharpe ratio.
σ
σ = 22%
Indifference Curve
Optimal Capital Allocation
Expected Return
Increasing Utility
Standard Deviation
Utility and Risk Aversion
CAL with Risk Preferences
E(r)
Borrower
Rf
Risky Asset
Lender
σ
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Risk Aversion and Allocation
ƒ Greater levels of risk aversion lead to larger
proportions of the risk free asset
ƒ Lower levels of risk aversion lead to larger
proportions of the portfolio of risky assets
ƒ Willingness to accept high levels of risk for
high levels of returns may result in leveraged
combinations
Optimal Allocation
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Optimal Allocation
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Optimal Allocation
Consider a portfolio with y risky asset and 1-y
risk-free asset. The expected return of this
portfolio is:
„
E (rp ) = y E (r ) + (1 − y )rf
[
= rf + y E (r ) − rf
„
We have shown that different investors will
choose different positions in the risky asset.
In particular, the more risk averse investors
will choose to hold less of the risky asset and
more of the risk-free asset.
How do we quantify this?
We start from the utility function of the
investor:
U = E (r ) − 0.5 Aσ 2
The investor attempts to maximize her utility
level, by choosing the best allocation to the
risky asset, y.
U = E (r ) − 0.5 Aσ 2
]
[
]
= rf + y E (r ) − rf − 0.5 Ay 2σ 2
The variance of the return of the portfolio is:
σ p2 = y 2σ 2
Optimal Allocation
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Optimal Allocation
Taking the first order derivatives of U with
respect to y and set it to zero.
E (r ) −r f − Aσ 2 y = 0
⇒ y* =
E (r ) − rf
Aσ 2
y* =
„
E (r ) − rf
( )
Aσ 2
If r f = 7%, E r = 15%, σ = 22% and A = 4
y* =
E (r ) − rf
Aσ 2
=
4
Optimal Allocation
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Other things being equal, the higher the
expected return of the risky asset, the higher
the optimal allocation in the risky asset.
y* =
y* =
E (r ) − rf
Aσ 2
E (r ) − rf
♦ Other things being equal, the more risk averse the
investor is, the lower the optimal allocation in the
risky asset.
y* =
Aσ 2
E (r ) − rf
Aσ
2
of the return of the risky asset, the lower the
optimal allocation in the risky asset.
y* =
y* =
=
Aσ 2
E (r ) − rf
♦ Other things being equal, the higher the variance
=
Optimal Allocation
y* =
Optimal Allocation
E (r ) − rf
Aσ 2
E (r ) − rf
Aσ 2
=
=
Market Price of Risk
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=
E (r ) − rf
σ2
is called the market price of risk.
Note that the market price of risk is different
from the Sharpe Ratio, which is defined as:
E (r ) − rf
σ
=
„
Both the market price of risk and Sharpe
Ratio captures reward to risk ratio.
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