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
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
