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
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
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:
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
⇒ σ p = 6.07%
Calculate the expected return and standard deviation of the portfolio consisting of 80%
GE and 20% IBM
$1,000 GE, $0 IBM
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
$600 GE, $400 IBM
$400 GE, $600 IBM
$200 GE, $800 IBM
This portfolio is less risky than either of GE and IBM!!!
$0 GE, $1000 IBM
Investment Opportunity Set
Number of Assets
With N Risky Assets
Investment Opportunity Set or Feasible Set
You can construct the efficient frontier using one of two equivalent approaches.
Efficient frontier Std dev σ
You can do this using the Solver in Excel.
What if a risk-free asset is available? Utility Maximization
For given expected return, find the minimum variance
For given variance, find the maximum expected return
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
What if a risk-free asset is available?
The feasible set of portfolios becomes more attractive We can identify an optimal risky portfolio which dominates all other risky…