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1
On Mean-Variance Analysis
1
Yang Li
Rotman School of Management
University of Toronto
105 St. George Street
Toronto, ON, M5S 3E6
Yang.Li10@rotman.utoronto.ca
Traian A. Pirvu
Dept of Mathematics & Statistics
McMaster University
1280 Main Street West
Hamilton, ON, L8S 4K1
tpirvu@math.mcmaster.ca
Abstract
This paper considers the mean variance portfolio management problem. We exam-
ine portfolios which contain both primary and derivative securities. The challenge in
this context is the well posedness of the optimization problem. We nd examples in
which this problem is well posed. Numerical experiments provide the ecient frontier.
The methodology developed in this paper can be also applied to pricing and hedging
in incomplete markets.
JEL classication: C61; G11
Keywords: Mean-Variance, Portfolios of Options, Quadratic Programming.
1 Introduction
The main objective in portfolio management is the tradeo between risk and return.
Markovitz, [6] and [7] studied the problem of maximizing portfolio expected return for a
given level of risk, or equivalently minimize risk for a given amount of expected return.
One limitation of Markovitzs model is that it considers portfolios of primary assets only.
Recent works consider the optimal management of portfolios containing primary and
derivative assets. Here we mention [8] and [1]. [8] introduces a technique of optimizing
CVaR (conditional value at risk) of a portfolio. [1] notices that the problem of minimizing
CVaR for a portfolio of derivative securities is ill-posed. [1] shows that this predicament
can be overcome by including transaction costs.
There are some works which consider portfolio optimization with non-standard asset
classes; we recall [2], [3], and [5]. In a continuous time model [2] looks at the problem of
maximizing expected exponential utility of terminal wealth, by trading a static position in
derivative securities and a dynamic position in stocks. In a one period model [3] analyses
the optimal investment and equilibrium pricing of primary and derivative instruments. [5]
shows how to approximate a dynamic position in options by a static one and this is done
by minimizing the mean-squared error.
By the best of our knowledge this paper is the rst work to consider the mean vari-
ance Markovitz portfolio management problem in one period model with derivative assets.
1
Work supported by NSERC grants 371653-09 and MITACS grants 5-26761
1
We work in a multivariate normally distributed returns framework. The rst diculty
encountered in such problems stems from the nonlinearity of derivative prices. This can
be overcome by considering delta gamma or related approximations. It is well known in
the industry practice that this approximation performs well for small time intervals. By
performing the delta gamma approximation the portfolio management problem is reduced
to a quadratic program. The second diculty is the lack of convexity for the quadratic
program. We nd two examples in which the quadratic program is convex: a portfolio of
instruments in which every instrument is written on one underlying only; a second example
is a portfolio which contains only two instruments.
The results of our paper can be applied to the problem of pricing and hedging in
incomplete markets. For instance we can consider instruments written on nontradable
factors (e.g. temperature) and they can be hedged with tradable instruments which are
highly correlated (this procedure is called cross hedging). Take as an example weather
derivatives (e.g. HDD or CDD); energy prices are considered as the traded correlated
instrument (in California a highly correlation can be observed between temperature and
energy prices). Perfect hedging is not possible in this paradigm. Minimizing the variance of
the hedging error can be captured as a special case of mean variance optimization problem
for a portfolio of primary and derivative instruments. A survey paper on mean-variance
hedging and mean-variance portfolio selection is [9].
Another possible application of our results is the hedging of long maturity instruments
with short maturities ones. As it is well known, the market for long maturity instruments
is illiquid, thus the issuers use (static) hedging portfolios of the more liquid short maturity
instruments. The interested reader can nd out more about this in [4].
The paper is organized as follows: Section 2 presents the model. Section 3 introduces
the delta gamma approximation. Section 4 presents the reduction to quadratic programs.
Section 5 shows examples in which the quadratic programs are convex. Section 6 presents
the numerical results. Section 7 is an application to pricing and hedging in incomplete
markets. The paper ends with an appendix.
2 The Model
Portfolios returns are derived from the return of individual positions. In practice, it is
not good to model the positions individually because of their correlations. If we have m
instruments in our portfolio we would need m separate volatilities plus data on
m(m1)
2
correlations, so in total
m(m+1)
2
pieces of information. This is hard to get for large m.
The resolution is to map our m instruments onto a smaller number of n risk factors.
The mapping can be nonlinear (e.g. BS for option). Let us assume that the factors
are represented by a stochastic vector process S = (S
1
, S
2
, , S
n
), which at all times
t (0, ) is assumed to be of the form
S
t
= t + W
t
. (2.1)
Here is the vector of returns, is the variance-covariance matrix, and W
t
is a standard
Brownian motion on a canonical probability space (, T
t
, T). The value of portfolio at
time t, denoted V (S, t), is of the form
V (S, t) =
m

k=1
x
k
(t)V
k
(S, t), (2.2)
2
where V
k
(S, t), k = 1, m, represents the value of the individual instruments (mapped
onto the risk factors), and x
k
(t), k = 1, m stands for the number of shares of instrument
k held in the portfolio at time t. At time t, we choose the portfolio mix x
k
(t), k = 1, m
such that the portfolio return V over time interval [t, t + t]
V = V (S + s, t + t) V (S, t), (2.3)
is optimized in a way which is described below. It turns out to be more convenient to work
with the vector of actual proportions of wealth invested in the dierent assets. Thus, at
time t (0, ), we introduce the portfolio weights w
k
(t), k = 1, m, by
w
k
(t) =
x
k
(t)
V (S, t)
, k = 1, m. (2.4)
In the following we posit the following Markowitz mean-variance type problem; given
some exogenous benchmark return r
e
(t), at time t an investor wants to choose among all
portfolios having the same return r
e
(t), the one that has the minimal variance Var(V ) :
(P1) min
w
Var(V )
such that E(V ) = r
e
(t),
m

k=1
w
k
(t)V
k
(S, t) = 1.
Another possible portfolio management problem is to choose the portfolio with the minimal
variance:
(P2) min
w
Var(V )
m

k=1
w
k
(t)V
k
(S, t) = 1,
There are some diculties in solving (P1) and (P2). First, we might be short of information
of the moments of V . Because V nonlinearly depends on the change of asset prices,
it is not obvious what distribution V would follow even if we perfectly learn the p.d.f of
S. The situation would not get much better if we only require the moment information
of V . The integration for moments might be still hard to calculate explicitly. One way
what of this predicament is to use delta gamma approximation.
3 Delta-Gamma Approximation
This is a second order Taylor expansion
V V =
V
t
t +
T
S +
1
2
S
T
S, (3.1)
where

i
=
V
S
i
,
ij
=

2
V
S
i
S
j
, i = 1, n.
Since
V (S, t) =
m

k=1
x
k
(t)V
k
(S, t),
3
then

i
=
V
S
i
=
m

k=1
x
k
(t)
k
i
,
k
i
:=
V
k
S
i
, i = 1, n, k = 1, m, (3.2)

ij
=

2
V
S
i
S
j
=
m

k=1
x
k
(t)
k
ij
,
k
ij
:=

2
V
k
S
i
S
j
, i = 1, n, j = 1, n, k = 1, m. (3.3)
It is well known that this approximation performs well as long as the time interval t is
not too big. At this point we formulate the approximated versions of (P1) and (P2), as
follows:
(P3) min
w
Var(V )
such that E(V ) = r
e
(t),
m

k=1
w
k
(t)V
k
(S, t) = 1,
(P4) min
w
Var(V )
m

k=1
w
k
(t)V
k
(S, t) = 1.
The next step is to reduce (P3) and (P4) to quadratic programs and this is done in the
next section.
4 Quadratic Programs
Let us rst consider the case of one asset, m = 1. In the light of (2.1), S A(,

t).
For computational convenience we assume is the zero vector and t = 1. Next, replace
the vector of correlated normals, S, with the vector of independent normals Z A(0, I).
This is done by setting
S = CZ with CC
T
= .
In terms of Z, the quadratic approximation of V becomes
V V = a + (C
T
)
T
Z +
1
2
Z
T
(C
T
C)Z,
with
a =
V
t
t. (4.1)
At this point it is convenient to choose the matrix C to diagonalize the quadratic term in
the above expression and this is done as follows. Let

C be a square matrix such that

C

C
T
= (4.2)
(e.g., the one given by the Cholesky factorization). The matrix
1
2

C
T

C is symmetric and
thus admits the representation (more about this can be found in the appendix)
1
2

C
T

C = UU
T
, (4.3)
4
where = diag(
1
, ,
n
), and U is an orthogonal matrix such that UU
T
= I. Next, set
C =

CU and observe that
CC
T
=

CUU
T

C
T
= , (4.4)
1
2
C
T
C =
1
2
U
T
(

C
T

C)U = U
T
(UU
T
)U = .
Thus, with
b = C
T
, (4.5)
we get
V V = a +b
T
Z +Z
T
Z := Y.
4.1 Moment Generating Function
In this subsection, we explore the moment generating function of Y and further derive the
mean and variance of Y . Since the random variable Y is student distributed, it is well
known that
E(Y ) = exp(()), (4.6)
where
() = a +
n

j=1

j
() = a +
n

j=1
1
2

2
b
2
j
1 2
j
log (1 2
j
)

, (4.7)
for all satisfying max
j

j
<
1
2
. Direct computations lead to
d

e
()

d
= exp(())
d
d
= exp(())

a +
1
2
n

j=1

2b
2
j
(1 2
j
)
2
(2
j
)
(1 2
j
)
2

2
j
1 2
j

and
d
2

e
()

d
2
= exp(())

a +
1
2
n

j=1

2b
2
j
(1 2
j
)
2
(2
j
)
(1 2
j
)
2

2
j
1 2
j

2
+exp(())

1
2
n

j=1

2
j
2
j
(1 2
j
)
2
+
(2b
2
j
8b
2
j

j
+ 4b
2
j

j
)(1 2
j
)
2
(2b
2
j
(1 2
j
) +
2
b
2
j
2
j
)((2
j
)2(1 2
j
))
(1 2
j
)
4

Thus, the rst and second moments of Y are

e
()

=0
= a +
n

j=1

j
and
E(Y
2
) =
d
2

e
()

d
2 =0
= (a +
n

j=1

j
)
2
+
n

j=1
(b
2
j
+ 2
2
j
).
5
Hence,
Var(Y ) = E(Y
2
) E
2
(Y ) =
n

j=1
(b
2
j
+ 2
2
j
).
In order to ease the notations we assume that V (S, t) = 1, so the vector of shares x equals
the vector of proportions w (also notice that for simplicity we dropped the t dependence
of w). Next, we would like to express the mean and variance of Y in terms of x. In the
light of (4.3), (4.2), (3.3) and (8.1) it follows that
E(Y ) = a +
n

j=1

j
= a + tr

UU
T

(4.8)
= a +
1
2
tr

C
T

= a +
1
2
tr

j=1
x
j

= a +x
T
p,
where the vector p is dened by
p :=
1
2

tr

, tr

, . . . , tr (
n
)

T
.
As for the variance, recall that with b of (4.5) it follows that (see (3.2) and (4.4))
m

k=1
b
2
k
= b
T
b = (C
T
)
T
C
T
=
T
C
T
C =
1
2
x
T

x, (4.9)
where

= 2(
1
, ,
n
) (
1
, ,
n
)
T
.

is positive semidenite by Lemma 8.1 (see appendix). Next, in the light of (4.4) and
(8.1) it follows that
m

k=1

2
j
=
1
4
tr

(C
T
C)
T
(C
T
C)

=
1
4
tr

CC
T
CC
T

=
1
4
tr ()
=
1
4
tr

j=1
x
j

=
1
4

j=1
x
2
j
tr

(
j
)
2

+ 2

i=j
x
i
x
j
tr

=
1
4
x
T
Qx,
6
where the matrix Q is dened by
Q
ij
= tr

, i = 1, n, j = 1, n. (4.10)
Therefore, we end up with
Var(Y ) =
m

j=k
(b
2
k
+ 2
2
k
) =
1
2
x
T
(

+Q)x. (4.11)
Thus, from (4.8) and (4.11), the portfolio problem (P3) (recall that x = w) becomes
(P5) min
x
1
2
x
T
(

+Q)x
s.t. a +x
T
p = r
e
,
n

k=1
V
k
(t, S)x
k
= 1.
and (P4) turns into
(P6) min
x
1
2
x
T
(

+Q)x
s.t.
n

k=1
V
k
(t, S)x
k
= 1.
It turns out that the problem (P5) has a similar form with the classical mean variance
portfolio problem. However, the essential dierence is that (P5) is not always tractable.
Depending on the data, (P5) could be either a convex quadratic programming or NP-hard.
The following is a well known result in quadratic programming.
Theorem 4.1. (P5) is a convex quadratic program (thus solvable in polynomial time) as
long as the matrix

+Q is positive semidenite. Else, (P5) is NP-hard.
In the following we want to nd examples in which (P5) is convex, and this is done in the
next section.
5 Convex Quadratic Programs
As Theorem 4.1 states, (P5) is not always tractable. We found two instances in which
(P5) is convex.
5.1 Case 1
Let us consider a portfolio in which every instrument is a map of one factor (possible
dierent from instrument to instrument) only. Take for example a portfolio of options
written on one (possible dierent) underlying. Thus,
V (S, t) =
m

k=1
x
k
V
k
(S
k
, t).
7
Since V
k
only depends on the factor S
k
, the matrix
k
(see (3.3)) has a nonzero element
(
k
kk
) only. We prove that in this case (P5) is a convex quadratic program. Given this
special structure it follows that Q
ij
of (4.10) becomes
Q
ij
= tr

=
i
ii

j
jj

ij

ji
. (5.1)
After some algebra manipulation one gets
Q = D
T
( )D, (5.2)
where D = Diag(
1
,
2
, . . . ,
m
) and denotes the Hadamard product. Recall that for
two matrices A R
mn
and B R
mn
, the Hadamard product is the matrix A B with
the entries
(A B)
ij
:= A
ij
B
ij
.
According to Schur Product Theorem (see appendix) it follows that is positive
semidenite. Next, by Lemma 8.1 it follows that D
T
()D is also positive semidenite.
At this point we can state the result
Theorem 5.1. (P5) is a convex quadratic program.
Proof. So far we proved that the matrices

and Q are positive semidenite. Hence (P5)
is a convex quadratic program, since

+Q is positive semidenite.
Let us move to the second case in which we can establish convexity of (P5).
Case 2
Let us consider a portfolio containing two instruments only. That is,
V (S, t) =
2

k=1
x
k
V
k
(S, t).
Thus
Q =

tr

tr

tr

tr

tr

(C
T

1
C)
T
C
T

1
C

tr

(C
T

1
C)
T
C
T

2
C

tr

(C
T

1
C)
T
C
T

2
C

tr

(C
T

2
C)
T
C
T

2
C

.
We claim that the matrix Q is positive semidenite. First, tr

(C
T

1
C)
T
C
T

1
C

is
nonnegative. For the determinant condition we apply the Cauchy-Schwarz inequality
['X
1
, X
2
`[ |X
1
| |X
2
|,
with X
1
= C
T

1
C and X
2
= C
T

2
C. Here the inner product 'X
1
, X
2
` is dened by
'X
1
, X
2
` = tr

X
1
X
T
2

.
Consequently,

+Q is positive denite, which lead to the following Theorem.
Theorem 5.2. (P5) is a convex quadratic program.
8
6 Numerical Results
Assume that there is one riskless asset, SC 91 Day T-Bills, and four risky assets, SC Uni-
verse Bonds, S&P/TSX, S&P 500, and MSCI EAFE. Further, assume that the arithmetic
returns are multivariate normally distributed. The expected return, standard deviation
and the covariance matrix is obtained from the historical data (this is taken from Jan.
1993 to Dec. 2002).
2
Next, consider a vanilla call option on SC Universe Bonds, a vanilla
Table 1: Expected Return and Standard Deviation
SC91TBIL SCUNOVER SP/TSX SP500 MSEAFEC
Return 4.85% 8.91% 9.07% 11.68% 6.54%
STDEV 0.44% 5.08% 16.80% 13.90% 14.35%
Table 2: Covariance Matrix
SC91TBIL SCUNOVER SP/TSX SP500 MSEAFEC
SC91TBIL 0.00% 0.01% 0.01% 0.01% 0.00%
SCUNOVER 0.01% 0.26% 0.24% 0.10% 0.02%
SP/TSX 0.01% 0.24% 2.82% 1.64% 1.49%
SP500 0.01% 0.10% 1.64% 1.93% 1.30%
MSEAFEC 0.00% 0.02% 1.49% 1.30% 2.06%
call option on S&P/TSX, a binary put option on BiS&P 500, and a binary put option on
MSCI EAFE. The initial asset prices, strike prices and maturity dates are given in Table
3. The derivatives are priced using BS type formulas.
Table 3: Initial prices, strike prices, and expire date
SCUNOVER SP/TSX SP500 MSEAFEC
Initial Price 100 50 80 100
Strike Price 80 51.25 100 150
Expire Date (days) 20 40 60 80
7 Quadratic Hedging
The results we established so far can also be applied to hedging. The motivation comes from
incomplete markets. Indeed, nancial markets are fundamentally incomplete. It is well known
that in incomplete markets perfect hedging is not possible. One way to solve this problem is to
consider quadratic hedging; that is, minimize the variance of the hedging error. Let F be a payo
of the form F = V
1
(S
t+t
, t + t), for some map V
1
. We would like to hedge this payo by some
instruments which are of the form V
k
(S, t), k = 2, , l (with l possible less than n, whence the
incompleteness). For simplicity assume that in this market borrowing and lending of cash is done
2
Data from http://www.math.mcmaster.ca/grasselli/john.pdf. For the covariance matrix, we only
take 4 decimals.
9
at zero interest rate (this can be easily achieved if one takes the zero coupon bonds as numeraire).
Given the number of shares (x
1
, x
2
, , x
l
) in the hedging portfolio, the hedging error is

l+1

k=1
x
k
V
k
(S, t),
with x
1
= 1, and V
l+1
(S, t) = 1. Therefore, the problem of minimizing the variance of hedging
error is of the form (P2). The initial amount needed to nance the hedging portfolio is
x
l+1
+V
1
(S
t
, t).
8 Appendix
8.1 Trace and its Properties
In linear algebra, the trace of a square matrix A is dened to be the sum of the elements on the
main diagonal (the diagonal from the upper left to the lower right) of A, i.e.,
tr(A) = a
11
+a
22
+ +a
nn
=
n

i=1
a
ii
.
Property 8.1. If A {
mn
and B {
nm
, then
tr(AB) = tr(BA). (8.1)
Property 8.2. If P is an invertible matrix, then
tr(P
1
AP) = tr((AP)P
1
) = tr(A). (8.2)
Property 8.3. If A {
nn
with real or complex entries and if
1
,
2
, . . . ,
n
are the (complex
and distinct) eigenvalues of A (listed according to their algebraic multiplicities), then
tr(A) =
n

i=1

i
. (8.3)
8.2 Symmetric Matrices and Semidenite Positive Matrices
Property 8.4 (Eigenvalue Decomposition). Let M be a symmetric real matrix. Then, there
exists an Eigenvalue Decomposition such that
M = QQ
T
=
n

i=1

i
q
i
q
T
i
, (8.4)
where is a diagonal matrix with all the eigenvalues of M along its diagonal, Q is an orthonormal
matrix, i.e., QQ
T
= I, and each column q
i
of Q is an eigenvector of M corresponding to the
eigenvalue
i
.
Lemma 8.1. Let P R
nn
and A R
nn
be a semidenite positive matrix. Then the matrix
PAP
T
is also semidenite positive..
Proof. According to the denition, we have
x
T
PAP
T
x = (P
T
x)A(P
T
x) 0,
for any x = 0.
10
8.3 Schur Product Theorem
Property 8.5 (Schur Product Theorem). Suppose A, B {
nn
are positive semidenite ma-
trices. Then A B is also positive semidenite.

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