Optimal Risky Portfolio
Optimal Risky Portfolios
„
„
„
Portfolio of Two Risky Assets
„
„
Suppose you hold a proportion w in asset A
and (1-w) in asset B
The portfolio expected return and risk is given
by
E(RP ) = wE(RA ) + (1 − w) E (RB )
σ p 2 = w2σ A 2 + (1 − w) 2 σ B 2 + 2w(1 − w)σ A ⋅ σ B ⋅ ρ AB
An Example
„
„
Suppose you hold two assets in your
portfolio, GE and IBM.
Let the portfolio weight of GE be w, and
then the portfolio weight of IBM be (1-w)
„ If w = 1, you hold only GE,
„ If w = 0, you hold only IBM,
„ If w = 0.5, you have an equally
weighted or naively diversified portfolio.
When choosing the optimal allocation between a
risk-free asset and a risky portfolio, we have
assumed that we have already selected the optimal
risky portfolio.
In this section, we learn how to determine the
optimal risky portfolio.
We start from two risky assets. Most of the intuition
carries to the case of more than two risky assets.
Return and Risk of A Portfolio
„
„
Given the expected returns of A and B,
variances of A and B, and the covariance
(correlation) between A and B, we compute
the expected return and variance of the
portfolio for a series of portfolio weights.
Then we plot the expected returns against
variances.
An Example
„
Based on data for 1982-2001, we find that
„ The average monthly return is 1.68%
for GE, and 1.22% for IBM
„ The standard deviation is 6.49% for GE
and 8.10% for IBM
„ The correlation between GE and IBM is
0.377
1
Equally weighted portfolio
„
„
Equally weighted portfolio
σ p 2 = w2σ A 2 + (1 − w) 2 σ B 2 + 2w(1 − w)σ A ⋅ σ B ⋅ ρ AB
The expected return of the equally weighted
portfolio is:
„ 0.5*1.68%+0.5*1.22%=1.45%
The standard deviation of this portfolio is
more complicated
= 0.52 × 6.49% 2 + 0.52 × 8.10% 2 +
2 × 0.5 × 0.5 × 6.49% × 8.10% × 0.377
= 0.00368
⇒ σ p = 6.07%
„
Another Portfolio
„
Diversification Benefit
Calculate the expected return and standard
deviation of the portfolio consisting of 80%
GE and 20% IBM
„
„
„
„
0.018
Variance
0.017
$1,000 GE, $0 IBM
0.016
Standard Deviation
The portfolio risk is lower than either of the
individual stocks
This is called the benefit of diversification
I repeat the above steps for other portfolio
weights, w=0, 0.1, 0.2, …, 0.9, 1.0
I plot the expected return against standard
deviation
Diversification and Risk
Portfolios of GE and IBM
$800 GE, $200 IBM
0.015
NonSystematic
$600 GE, $400 IBM
$400 GE, $600 IBM
0.014
$200 GE, $800 IBM
0.013
0.012
This portfolio is less risky than either of GE
and IBM!!!
$0 GE, $1000 IBM
Investment Opportunity Set
Systematic
0.011
49
47
45
43
41
39
37
35
33
31
29
23
27
25
17
0.095
21
0.09
15
0.085
19
0.08
11
0.075
13
0.07
9
0.065
7
0.06
5
0.055
3
1
0.01
0.05
Number of Assets
Expected Return
2
Feasible Portfolios
With N Risky Assets
Efficient Frontier
Investment Opportunity Set
or Feasible Set
Expected
Return E(R)
„
You can construct the efficient frontier using
one of two equivalent approaches.
B
C
„
A
„
Efficient
frontier
Std dev σ
„
You can do this using the Solver in Excel.
What if a risk-free asset is
available?
Utility Maximization
Higher Utility
Indifference Curves
„
C
Expected
Return E(R)
For given expected return, find the
minimum variance
For given variance, find the maximum
expected return
.A
„
B
We have covered the capital allocation
problem between a risk-free asset and a risky
asset.
Recall that the capital allocation line is the
straight line through the risk-free asset and
the risky asset.
A is the Utility maximizing
risky-asset portfolio
Std dev σ
The Optimal Risky Portfolio
has the Highest Sharpe Ratio
Capital Market Line
Expected
Return E(R)
Optimal Risky
Portfolio
D
•
C
•
E
Riskless
Asset •
•
•
CALB
B
What if a risk-free asset is
available?
„
„
CALA
•A
„
The feasible set of portfolios becomes more
attractive
We can identify an optimal risky portfolio
which dominates all other risky portfolios
(irrespective of risk preferences)
The optimal (tangency) portfolio has the
highest Sharpe ratio among all feasible
portfolios
Std dev σ
3
Optimal Risky Portfolio is the
Market Portfolio
Utility Maximization with
a Risk-free Asset
„
Expected
Return E(Ri)
Optimal Risky
Portfolio
.
Riskless
Asset
.
.E
Capital
Market Line
„
D
.
Efficient
frontier
„
„
Everybody holds a combination of risk-free
asset and the optimal risky portfolio
This optimal risky portfolio is the same for
everybody regardless of how risk averse you
are
It must be the market portfolio
If not, then there must be some assets that no
one holds, which cannot be true
Std dev σi
Passive Strategy is Efficient
„
„
„
The optimal risky portfolio is the same for
every investor, and is the market portfolio
No need for stock selection
Investors need only to adjust the mix of riskfree asset and the market portfolio based on
risk aversion
The Optimal Risky Portfolio
„
„
Key Question: How do we find the optimal
risky portfolio?
By choosing asset weights wi that maximize
the Sharpe Ratio:
Max S p =
wi
The Optimal Risky Portfolio
(with 2 risky assets)
„
For two risky assets, we know that the
portfolio return and standard deviation are
given by
E (R p ) − R f
σp
The Optimal Risky Portfolio
(with 2 risky assets)
„
Therefore, we need to maximize the ratio
Sp =
E(RP ) = wE(RA ) + (1 − w) E(RB )
wE(RA ) + (1 − w) E(RB ) − R f
w σ A + 2w(1 − w)σ AB + (1 − w) 2 σ B
2
2
2
σ p = w2σ A 2 + 2w(1 − w)σ AB + (1 − w)2 σ B2
„
by choosing w appropriately. This can be
done using Solver in Microsoft Excel.
4
The Optimal Risky Portfolio
(with 2 risky assets)
The Optimal Risky Portfolio
(with N risky assets)
♦ Or, by the following formula which gives the
„
weights for the optimal portfolio comprised of
only two assets:
WA =
(E (R ) − R )σ − (E (R ) − R )σ
]σ + [E (R ) − R ]σ − [E (R ) − R + E (R ) − R ] σ
A
[E (R ) − R
A
f
2
B
2
B
f
B
WB
f
=
B
2
A
f
A
AB
f
B
f
(1 − WA )
The Optimal Risky Portfolio
(with N risky assets)
„
„
„
„
AB
Steps to solve for the optimal risky portfolio
weights using Solver in Microsoft Excel:
„ Identify all of the risky assets to be
included in the investment universe
„ Compute return series (from prices) for
each risky asset and the risk-free asset
and determine the average return for each
Solver
Compute the covariance matrix of the risky
assets
Enter the formula for the Sharpe ratio into
a cell
Set up a column of cells for the portfolio
weights
Use Solver to maximize the Sharpe ratio
by changing the weights, subject to the
constraint that the weights sum to one
Solver
Solver
5
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Solver
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Solver
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