PORTFOLIO THEORY
 When we think about investments, 2 things matter:
1. the expected return
2. the dispersion of possible returns
The Mean - Standard Deviation Rule:
 It is usually assumed that security returns will be described
adequately by the mean, and standard deviation. This is
reasonable if security returns are normally distributed.
 The mean standard deviation rule is based upon the following two
assumptions:
1) Individuals prefer more return to less
2) Individuals prefer less risk to more
 When choosing a portfolio of assets, investors will seek out the
portfolio that provides the highest return for a given level of risk,
or alternatively, the lowest level of risk for a given level of return.
Portfolios with this characteristic are said to be efficient.
THE RISK OF A PORTFOLIO OF SECURITIES
 A portfolio is a collection of assets such as stocks, bonds, real
estate, etc.
 The return to a portfolio of assets is an average return of all of the
assets, weighted by the relative amount invested in each asset.
However, the risk of a portfolio of assets is fundamentally
different from the average risk of all assets in the portfolio.
 The process of diversification will reduce the risk of assets when
held in portfolios. Think of the following picture of two asset’s
returns:
25
20
15
10
5
0
-5
1
2
3
4
5
6
7
-10
-15
-20
-25
 The riskiness of a portfolio depends not only on the riskiness of
each asset in the portfolio, but also on the relation between the
returns from the two assets.
CORRELATION
 Correlation measures the tendency for the returns of two assets to
move together. In technical terms, it measures the strength and
direction of the linear relationship between two variables.
 For example, if the correlation between assets A and B is equal to
0.30, we can say that the amount of change in ‘A’ is predictable
about 30% of the time for each unit change in ‘B’.
The mathematical formula for portfolio risk
Two Assets: a & b
wa and wb are the fractions of total funds invested in each asset.
a and b are the standard deviations of returns for each asset.
ρab is the correlation between the two assets’ returns. It must lie
between –1 and +1.
The definition of expected return and standard deviation for a 2asset portfolio:
r p  wa ra  wb rb
 P  wa2 a2  wb2 b2  2 wa wb a b  ab
What about n assets?
n
Portfolio return: r p   wi ri
i 1
Portfolio risk:  p 
n 2 2
n n
w


 i i
  wi w j ij
i 1
i 1 j 1
i j
Note: [ ij  ij i j ]
where wi = the percent of the value of the portfolio in asset i
ri = the expected return for asset i
n = number of assets in the portfolio
i2 = the variance of rates of return for asset i
ij = the covariance between the rates of return for assets i
and j
ij =the correlation coefficient between the rates of return
for assets i and j
.
MATRIX THEORY
 A matrix is a rectangular array of numbers. Matrix
multiplication allows you to easily compute the expected return
and standard deviation of a multi-asset portfolio.
Ex: Two-asset case:
 r1 
rp  [w1 w 2 ]  
r2 
 w1r1  w 2r2
 The first matrix is a row vector (1 x 2), and the second matrix
is a column vector (2 x 1).
 In order to multiply matrices, the number of columns of the
first matrix must be equal to the number of rows of the second.
The resulting matrix will have the number of rows equal to
that of the first matrix and the number of columns equal to
that of the second matrix.
Example: A 2 x 3 matrix multiplied by a 3 x 4 matrix will result in a
2 x 4 matrix.
Example of Matrix Multiplication for the Two-asset Portfolio Variance:
 11 12   w 1 


 p 2  w1 w 2  

 w 

 21 22   2 
 w1212  w 2 2 2 2  2w1w 212
Example:
.30 .02 .50
 p  .50 .50 
 
.02 .40 .50
2
 .185
CALCULATING EFFICIENT N-ASSET PORTFOLIOS WHEN
THERE ARE SHORT-SALE RESTRICTIONS
Use Excel’s Solver to optimize the portfolio model by identifying the
best proportion of the portfolio to invest in each asset without allowing
short positions.
 One objective is to minimize the risk of the portfolio σp2
By changing the decision variables wi of the proportion to invest
in each asset i
Subject to the constraints:
Σ wi = 100% and that wi > 0
rp = desired return level
This calculates the Minimum Variance Portfolio (MVP+)
which is the lowest risk portfolio on the efficient frontier
o
Efficient Frontier with Short Sales
Portfolio
Return rp
Efficient Frontier without Short Sales
MVP+
Standard Deviation σp
 A second objective is to maximize the return of the portfolio rp
By changing the decision variables wi of the proportion to invest
in each asset i
Subject to the constraints:
Σ wi = 100% and that wi > 0
σp2 ≤ Target variance
This calculates a constrained maximized return portfolio
which is the point on the efficient frontier when the x-axis
value is the square root of the target variance

A third objective is to maximize the Sharpe Index (rp - rf)/ σp
By changing the decision variables wi of the proportion to invest
in each asset i
Subject to the constraints:
Σ wi = 100% and that wi > 0
This calculates the tangency portfolio which assumes that
investors can borrow and lend at a risk-free rate. This
assumption results in a linear efficient frontier that begins at
rf on the y intercept and is tangent to the efficient frontier of
risky assets.