The risk or volatility is c>Σc = Var (c>X).
2
,→ The portfolio choice, i.e., the selection of c, is such that the return is
maximized for a given risk bound.
17.2
Efficient Portfolio
A variance efficient portfolio is one that keeps the risk (17.2) minimal under the constraint
that the weights sum to 1, i.e., c>1p = 1. For a variance efficient portfolio, we therefore try
to find the value of c that minimizes the Lagrangian
1
L =
c>Σc − λ(c>1p − 1).
(17.3)
2
A mean-variance efficient portfolio is defined as one that has minimal variance among all
portfolios with the same mean. More formally, we have to find a vector of weights c such
that the variance of the portfolio is minimal subject to two constraints:
1. a certain, pre-specified mean return µ has to be achieved,
2. the weights have to sum to one.
Mathematically speaking, we are dealing with an optimization problem under two con-
straints.
The Lagrangian function for this problem is given by
L = c>Σc + λ1(µ − c>µ) + λ2(1 − c>1p).
17.2
Efficient Portfolio
409
IBM
Consolidated Edison
0.4
0.4
0.2
0.2
y
0
y
0
-0.2
-0.2
-0.4
-0.4
PanAm
Gerber
0.4
0.4
0.2
0.2
y
0
y
0
-0.2
-0.2
-0.4
-0.4
Delta Airlines
Texaco
0.4
0.4
0.2
0.2
y
0
y
0
-0.2
-0.2
-0.4
-0.4
Figure 17.1.
Returns of six firms from January 1978 to December 1987.
MVAreturns.xpl
With tools presented in Section 2.4 we can calculate the first order condition for a minimum:
∂L = 2Σc − λ1µ − λ21p = 0.
(17.4)
∂c
EXAMPLE 17.1 Figure 17.1 shows the returns from January 1978 to December 1987 of six
stocks traded on the New York stock exchange (Berndt, 1990). For each stock we have chosen
the same scale on the vertical axis (which gives the return of the stock). Note how the return
of some stocks, such as Pan American Airways and Delta Airlines, are much more volatile
than the returns of other stocks, such as IBM or Consolidated Edison (Electric utilities).
As a very simple example consider two differently weighted portfolios containing only two
assets, IBM and PanAm. Figure 17.2 displays the monthly returns of the two portfolios.
The portfolio in the upper panel consists of approximately 10% PanAm assets and 90% IBM
assets. The portfolio in the lower panel contains an equal proportion of each of the assets.
The text windows on the right of Figure 17.2 show the exact weights which were used. We
410
17
Applications in Finance
weights
equally weighted portfolio
0.4
0.500 IBM
0.500 Pan Am
0.2
Y
0
-0.2
0
50
100
X
weights
optimal weighted portfolio
0.4
0.896 IBM
0.104 Pan Am
0.2
Y
0
-0.2
0
50
100
X
Figure 17.2.
Portfolio of IBM and PanAm assets, equal and efficient
weights.
MVAportfol.xpl
can clearly see that the returns of the portfolio with a higher share of the IBM assets (which
have a low variance) are much less volatile.
For an exact analysis of the optimization problem (17.4) we distinguish between two cases:
the existence and nonexistence of a riskless asset. A riskless asset is an asset such as a
zero bond, i.e., a financial instrument with a fixed nonrandom return (Franke, Härdle and
Hafner, 2001).
Nonexistence of a riskless asset
Assume that the covariance matrix Σ is invertible (which implies positive definiteness). This
is equivalent to the nonexistence of a portfolio c with variance c>Σc = 0. If all assets are
uncorrelated, Σ is invertible if all of the asset returns have positive variances. A riskless asset
17.2
Efficient Portfolio
411
(uncorrelated with all other assets) would have zero variance since it has fixed, nonrandom
returns. In this case Σ would not be positive definite.
The optimal weights can be derived from the first order condition (17.4) as
1
c =
Σ−1(λ1µ + λ21p).
(17.5)
2
Multiplying this by a (p × 1) vector 1p of ones, we obtain
1
1 = 1>c =
1>Σ−1(λ
),
p
1µ + λ21>
2 p
p
which can be solved for λ2 to get:
2 − λ11>Σ−1µ
λ
p
2 =
.
1>Σ−11
p
p
Plugging this expression into (17.5) yields
!
1
1>Σ−1µ
Σ−11
c =
λ
p
p
1
Σ−1µ −
Σ−11p
+
.
(17.6)
2
1>Σ−11
1>Σ−11
p
p
p
p
For the case of a variance efficient portfolio there is no restriction on the mean of the portfolio
(λ1 = 0). The optimal weights are therefore:
Σ−11
c =
p
.
(17.7)
1>Σ−11
p
p
This formula is identical to the solution of (17.3). Indeed, differentiation with respect to c
gives
Σc = λ1p
c = λΣ−11p.
If we plug this into (17.3), we obtain
1
L =
λ21pΣ−11p − λ(λ1pΣ−11p − 1)
2
1
= λ − λ21pΣ−11p.
2
This quantity is a function of λ and is minimal for
λ = (1pΣ−11p)−1
since
∂2L = Σ > 0.
∂c>∂c
412
17
Applications in Finance
THEOREM 17.1 The variance efficient portfolio weights for returns X ∼ (µ, Σ) are
Σ−11
c
p
opt =
.
(17.8)
1>Σ−11
p
p
Existence of a riskless asset
If an asset exists with variance equal to zero, then the covariance matrix Σ is not invertible.
The notation can be adjusted for this case as follows: denote the return of the riskless asset
by r (under the absence of arbitrage this is the interest rate), and partition the vector and
the covariance matrix of returns such that the last component is the riskless asset. Thus,
the last equation of the system (17.4) becomes