The Portfolio Choice Problem

1
The Portfolio Choice Problem
2
The Classical Problem

• An investor wishes to save for some fixed horizon
  using a combination of risky and risk-free
  investments.
• How can his/her wealth be optimally divided among
  these assets.

3
Mean-Variance Analysis

• In the special case where the investor cares only
  about mean and variances (or standard deviation)
  we can characterize the set of all possible
  outcomes as a function of their portfolio choice.
• This results in the efficient frontier.

4
The Efficient Frontier
Efficient investments
?
All possible (?, ?) pairs that an investor can
attain by holding risky securities.
rf
?
All possible (?, ?) pairs that an investor can
attain by holding any security lie in this cone.
5
The Practical Problem

• We wish to characterize efficient portfolios in
  order to quantify an investment decision.
• We may also want to enforce constraints on the
  problem. For example
• What if short selling is not allowed?
• What if borrowing and lending rates differ?

6
Formally Stating the Problem

• The efficient frontier can be found by solving an
  optimization problem.
• In words, we wish to minimize portfolio variance
  (equivalently standard deviation) given any level
  of desired mean return.

7
Basic Inputs

• In order to undertake the optimization we need to
  be able to calculate portfolio means and
  variances.
• We can calculate these if we know
• The mean return of every asset in the portfolio,
  and
• The covariance matrix for the returns of the
  assets.

8
The Return Data Matrix
9
Calculating Means and the Covariance Matrix

• Means
• Covariances

T rows
10
The Covariance Matrix
11
Portfolio Return Properties

• Once individual asset mean returns and variances
  are known, it is possible to characterize
  portfolio means and variances.
• This is the first step to determine the set of
  efficient portfolios.

12
Calculating Portfolio Means and Returns

• Given any set of portfolio weights (?) the
  portfolio mean and variance can be easily
  calculated

13
A Formal Description of the Efficient Frontier

• The portfolio on the efficient frontier with mean
  return E solves the following constrained
  optimization problem

Subject to
14
Calculating Efficient Risky Portfolios Without
Constraints

• Any two efficient risky portfolios (mutual funds)
  can be combined to generate the entire efficient
  frontier.
• Formulas for the minimum variance and tangency
  portfolios are given by

15
The Efficient Frontier
Tangency portfolio
?
All efficient risky portfolios are combinations
of the minimum variance and tangency portfolio
rf
?
Minimum variance portfolio
16
Efficient Portfolios with Constraints

• In general, the efficient set will contain short
  positions in some assets.
• In many applications, no short-sale constraints
  may be enforced.
• To find efficient portfolios in this case a
  numerical solution technique is required.

17
Efficient Portfolios with Short Sales Constraints

• The portfolio on the efficient frontier with mean
  return E solves the following constrained
  optimization problem

Subject to
18
Using Excel to find Efficient Portfolios

• Excel can be used to determine the set of
  efficient portfolios.
• Two approaches are possible
• Given the mean and covariance matrices, use
  solver to calculate optimal weights.
• Use excels matrix manipulation functions
• transpose()
• mmult()
• minverse()