13' Optimal Portfolios

1
13. Optimal Portfolios

• Effect of Diversification
• Expected return and standard deviation of a
  portfolio
• Variance/covariance matrix using the matrix
  algebra
• Efficient Frontier
• Risk free asset and CML and CAPM

2
Notes

• The textbook used the population standard
  deviation STDEVP ( ) for the exhibits
• However usually we will deal with the sample, so
  the sample standard deviation STDEV ( ) is
  used.
• Reading on Creating a Variance Matrix is
  optional (P 400 403).

3
Risk Reduction with Diversification
St. Deviation
Unique Risk
Market Risk
Number of Securities
4
Two-Security PortfolioReturn and Risk
5
Covariance
?1,2 Correlation coefficient of
returns
?1 Standard deviation of returns for
Security 1 ?2 Standard deviation of
returns for Security 2
6
Simple Question

• Suppose there are two stocks in your portfolio.
• 50 - 50 investment into these two stocks
  produce an optimal portfolio?
• Probably not. It depends on the return-risk
  nature of the two individual stocks. We use
  Solver to solve this problem (See Exhibit 13-1
  thru 13-4). What is the optimal W1 and W2?

7
Excel functions for Basic Statistics

• Expected Return Average ( )
• Variance VAR ( )
• Standard Deviation STDEV ( )
• Covariance COVAR ( , )
• Correlation (Rho) CORREL ( , )

8
Back to the Optimal Portfolio Problem

• Choose the investment weight (W1, W2) so that
  portfolio risk is minimized, given a portfolio
  return.
• Solver Function
• Min Portfolio Risk
• Choosing weights
• Constraints portfolio return x sum of
  weights ??
• and others
  ..

9
In General, For An N-Security Portfolio
10
Basic Concepts - Markowitz Model

• The minimum variance portfolio What
  combinations result in lowest level of risk for a
  given return?
• The optimal trade-off between return and risk is
  described as the efficient frontier upper-half
  of the MVP set.
• Any combination of MVPs results in a MVP.

11
The Minimum-Variance Frontier of Risky Assets
12
The Efficient Portfolio Set
13
Extending to Include Riskless Asset

• The optimal combination becomes linear.
• A combination of a single risky and riskless
  assets will dominate.

14
Capital Allocation Lines with Various Portfolios
from the Efficient Set
15
Efficient Frontier with Lending Borrowing
CAL
E(r)
B
Q
P
A
S
rf
F
St. Dev
16
The Separation Theorem

• A portfolio manager only needs the same risky
  portfolio, P (tangent), to all clients regardless
  of their degree of risk aversion
• The portfolio choice problem may be separated
  into two independent tasks
• First determine the optimal risky portfolio (P)
• Then choose the allocation of the complete
  portfolio between the risk-free asset (F) and the
  risky asset (P)