ELEMENTARY PORTFOLIO OPTIMIZATION
Kenneth C. Abbott
Vice President
Bankers Trust Company
April, 1993
ELEMENTARY PORTFOLIO OPTIMIZATION
Kenneth C. Abbott*
This paper provides an introduction to portfolio optimization using elementary quadratic
programming techniques. It assumes a rudimentary knowledge of linear algebra, statistics, and
differential calculus. Its purpose is threefold. First, it provides definitions for most of the basic
concepts used in portfolio analysis.
Second, it explains the intuition behind portfolio
risk/expected return frontiers. Third, it describes a simple method of tracing a portfolio's efficient
frontier and calculating the associated portfolio weights.
Section I defines the concepts of expected return and variance for individual assets. Section II
extends the analysis of Section I to define expected return and risk for groups of assets.
Section III develops the concept of the portfolio possibility frontier and shows how to find the
frontier for any set of assets. Appendix I derives the variance of the minimum variance portfolio
for any level of expected return as well as the corresponding asset weights.
Appendix II
provides an example of how portfolio optimization may be implemented in a spreadsheet.
SECTION I: ASSET RETURNS AND VARIANCES
Ignoring taxes and transaction costs, the expected return on a security over one period may be
written as
Rt 
Pt  Pt 1  Dt
Pt 1
where
Rt
= return in period t
Pt-1
= asset price at beginning of period t
Pt
= asset price at the end of period t
*
Thanks to Ken Baron for his help in editing this paper.
(1)
Dt
= cash value of distributions during period t
For example, if a security had prices of 100 and 110 at times t-1 and t, respectively, and paid a
dividend of 5.0 during that period, its expected return would be ((110 - 100)+ 5) )/100, or 15%.
At the beginning of each period, the asset's price is known with certainty. The end of period
price and cash value of distributions are not known, however, making the period expected return
uncertain. Thus the expected return, Rt, may be treated as a random variable.
The value of Rt may vary considerably with t, with each possible value having a certain
probability attached to it.
Calculations involving these probabilities can be quite complex.
However, one may use the average Rt over a period of time as a convenient measure of R t's
central tendency. This measure is usually referred to as its mean. This measure, when used in
portfolio theory, makes the assumption that the average historical return is a good indicator of
future expected returns. It is denoted by  and is defined mathematically as
1
 
T
T
R
(2)
t
t 1
where T is the number of periods for which there are observations. For example, if a security
had returns of 15%, 10%, 12%, 8%, -2% and 1% over 6 periods, its mean return would be (.15
+.10 +.12 +.08 -.02 +.01)/6 or 8.5%.
While this formula provides a measure of central tendency, it gives no insight into how risky the
returns on an individual asset are. For this purpose, a different measure is needed which
examines how dispersed actual returns are about their mean. That measure is variance, which
is denoted by  2 and defined as
2
 2 
1
T 1
T
 ( R   )
t
2
( 3) *
t 1
For instance, the variance of the security in the example above would be 1/5 *((.15-.085) 2 +
(.11-.085)2 +(.12-.085)2 +(.08-.085)2 + (-.02 -.085)2+(.07-.085)2) or .00347.
The variance of Rt is a weighted average of the squared differences of each observation in the
series about their mean. The square root of this measure is called the standard deviation and is
also often referred to as volatility. The use of these measures in portfolio theory also assumes
that the future behavior of the variable can be estimated by examining its past.
*
T-1 is used because when sample data is used to estimate the variance of the whole population,
dividing by T biases the estimate of the variance.
3
SECTION II: PORTFOLIO RETURNS AND VARIANCES
Having defined measures of central tendency and risk for individual assets, one can extend the
concepts to combinations, or portfolios, of assets. Consider a portfolio containing k assets. Let
 i be the expected return on the ith asset and x i be the weight in each asset such that the
weights sum to one. The expected return on this portfolio,  P , is simply the weighted average
of the expected returns of its assets:
k
 p  E ( Rp) 
 x 
i i
( 4)
i 1
For example, consider a portfolio of three assets, x1, x2, and x3, with expected returns of 1%,
4%, and 7% per period. If the portfolio consisted of 24% of asset x 1, 46% asset x2 and 30%
asset x3, the expected return on the portfolio would be (.24 * .01) + (.46 * .04) + (.30 * .07), or
4.18% per period.
Portfolio variance can be derived from the formula for individual asset variance.
1
 p2 
T 1
T
(R
pt
 p ) 2
(5)
t 1
where  p2 is portfolio variance,  p is the portfolio's expected return , and Rpt is the portfolio
return in period t. For example, consider the case of a two asset portfolio with weights x a and xb
associated with the two assets. Let Ra, Rb,  a , and  b , be the returns at time t and the expected
returns for assets a and b, respectively. Here, we can restate the formula for variance as
follows:
4
T
1
 x2R x R =
(T-1)
[(x R
1
(T-1)
[x (R
a a
b b
=
=
1
(T-1)
a at
2
+ xb Rbt ) (xa  a + xb  b )]
(6)
t=1
T
a
at
2
-  a )+ xb (Rbt -  b )]
t=1
T
2
a
[x
(Rat -  a )2 + 2xa xb (Rat -  a )(Rbt -  b ) + xb2 (Rbt -  b )2 ]
t 1
T
2 1

 1
=xa 
(Rat -  a )2  + 2xa xb 

(T-1) t=1

(T-1)
T
2 1

(Rat -  a )(Rbt -  b ) + xb 

t=1

(T-1)
T
(R
bt
t 1

-  b )2 

The first and last terms are the variances of the individual assets times the square of their
respective weights. The term in the middle represents the covariance of the assets multiplied
by two times the product of their weights.
Covariance is important because it measures the extent to which two variables offset each
other. Although it has little practical meaning outside of statistics, it is used implicitly in many of
its financial applications. For two random variables, covariance represents the average value of
the product of the variables less their means:
Cov( Ra , Rb ) 
1
T 1
T
 (R
at
  a )( Rbt   b )
( 7)
t 1
If there is a positive relationship between the two series, with small values of X associated with
small values of Y and large values of X associated with large values of Y, then the covariance
will be positive. If the reverse is true, i.e. small values of X associated with large values of Y
and large values of X associated with small values of Y, then the covariance will be negative.
Notice that the covariance of a variable with itself is simply its variance. Thus, a table showing
the covariances of k assets would have as its diagonal the variances of the individual assets.
Such a table is called a covariance matrix or a variance/covariance matrix.
The combination of one unit each of any two assets, a and b would have a return with variance
as follows:
5
 2Ra , Rb   2a   2b  2Cov( Ra , Rb )
(8)
Otherwise, the calculation is as follows:
 2xa Ra  xb Rb  xa2  2a  xb2  2b  2 xaxbCov( Ra , Rb )
( 9)
where the x's are the weights of the assets and sum to 1.
For a portfolio of k equally weighted assets with one unit of each asset, then, the portfolio
variance would simply be the sum of each entry in the covariance matrix. This is so because
the diagonal contains all of the k individual variances, while the upper and lower triangulars both
contain a complete set of the
(k 2 k )
2
pairwise covariances.
Portfolio variance for non-equally weighted portfolios may be generalized as:
n
n
  x x Cov ( R , R )
i j
i
j
(10 )
i 1 j1
where the x's are the respective weights attached to assets in the portfolio and the R's the
returns.
Portfolio variance is often estimated using correlation instead of covariance.
Correlation
expresses the relationship specified by covariance in a more descriptive manner, re-scaling
 and defined as follows:
covariance to a number between -1 and +1. It is usually denoted as 
Corr( Ra , Rb )   ab 
Cov( Ra , Rb )
 a  b
(11)
Thus, from equation (9), the variance of the return of a two asset portfolio is:
 2xa Ra  xb Rb  xa2  2a  xb2  2b  2 xaxb ab  a  b
6
(12 )
Note that using this formula does not change the calculation, since  a,b is multiplied by the
product of the standard deviations. Correlation describes the extent to which a scatterplot of the
two variables forms a straight line. The concept is important because it provides a unit-free
means of describing the relationship between any two assets in a portfolio. When the returns of
two assets have a correlation of +1 or -1 (a perfect positive or negative correlation), the return
of one is simply a constant times the return of the other. If the returns of two assets have a
correlation of zero, there is no linear relationship between those returns.
A Note on Matrix Notation
Matrix notation greatly simplifies the expressions for portfolio
return and variance by removing the summation notation (S) from the equations. If x is a
column vector containing portfolio weights, xt the transpose of that vector, R a column vector
containing the expected returns for each asset in the portfolio, and V the variance/covariance
matrix of the asset returns, the following equations may be used for portfolio return and
variance:
 p  x t R
(13)
 2p  x t Vx
(14 )
Note that these equations are simply restatements of equations 4 and 10.
7
SECTION III: PORTFOLIO POSSIBILITY SETS
The locus of all possible risk/expected return combinations for a set of assets is called the
portfolio possibility set. It is usually represented on a graph with the expected return on the Y
axis and either the standard deviation or variance on the X axis. Generally it is represented with
risk along the X axis and expected return along the Y axis.
Its shape and location are
determined not only by the absolute riskiness of the individual assets, but also by the way asset
risks offset each other. An example of a portfolio possibility set is seen in Figure 1.
Figure 1: Portfolio Possibility Set
E
X
P
E
C
T
E
D
R
E
T
U
R
N
D
A
C
B
E
RISK (STANDARD DEVIATION)
Since each point on the graph can be denoted by a pair of coordinates representing mean and
standard deviation, the graph is said to be in mean-standard deviation space.
Among this set one can find many portfolios which are relatively undesirable. For instance, the
portfolio associated with point A in Figure 1 is clearly preferable to that of point B since it
represents a portfolio with both higher expected return and lower risk. It is also preferable to
that of point C, because point A has the same level of expected return, but at a lower risk level.
At the same time, the portfolio associated with point D is superior to that of point A, since it has
the same level of risk, but a higher expected return. In this sense, portfolio represented by point
D is said to dominate the other three portfolios, since it shows a level of riskiness less than or
8
equal to those of the others and a expected return which is superior. Similarly, portfolio A
dominates portfolios B and C, since its expected return is as good and its level of risk superior.
The portfolio associated with point E is the point on the frontier furthest to the left. It is called
the minimum variance portfolio. and represents the lowest risk portfolio attainable with a set of
assets.
Efficient portfolios are those portfolios that are not dominated by any other available portfolios.
These portfolios lie at the edge of the portfolio possibility set and have the greatest expected
return among available portfolios with the same level of risk. Equivalently, they have the lowest
risk among available portfolios with the same expected return. The locus of points representing
efficient portfolios is said to be the efficient frontier of the assets. It is represented on Figure 1
with a solid black line. Note that the efficient frontier begins at point E and does not include the
entire edge of the portfolio possibility set. This is because many of the points at the lower edge
of the portfolio possibility set are dominated by points on the efficient frontier. The points on the
portfolio possibility frontier that are not efficient are marked with a dotted black line.
The shape of the portfolio possibility frontier (and thus the efficient frontier) is a function of the
correlation between its asset's returns. The shape of the portfolio possibility frontier for two
asset portfolios with different correlations provides a useful example of how correlation affects
that shape. Figure 2 provides graphs of these frontiers. For the purposes of this example, it is
assumed that the weights of both assets are positive.
9
Figure 2: The Sha pe of the Portfolio Possibility Frontier
0.55
R
E
T
U
R
N
A
C
c o rrela tio n = 0
c o rrela tio n = -1
c o rrela tio n = +1
B
0.1
0.1
0.38
RISK (STANDARD DEVIATION)
If the returns of two assets are perfectly positively correlated, their portfolio possibility frontier is
a straight line, like that connecting points A and B. This is so because a portfolio of perfectly
correlated assets has a standard deviation which is a linear combination of the standard
deviations of the individual assets, and thus a straight line dependent upon the relative weights
of the assets in the portfolio:
 2 xa Ra  xb Rb  xa 2  a 2 + xb 2 b 2 + 2 xa xb ab a  b

xa 2 a 2
+
xb 2 b 2
 ( xa  a  xb  b )

(15)
+ 2 xa xb  a  b
2
 xa Ra  xb Rb  xa  a  xb b
The expected return of the portfolio is a linear combination of the expected returns of the
individual assets. Thus, if the weights change by a certain amount, the expected return and the
standard deviation will move in fixed proportions to each other, creating a straight line.
The thin black lines in Figure 2 illustrate the portfolio possibility frontier for two assets with a
correlation of -1. Here the frontier assumes the shape of two line segments emanating from the
same point on the Y axis. Again, the standard deviation of the portfolio is a linear combination
10
of those of its component assets. Here, however, the perfect negative correlation permits the
risk of one asset to be completely offset by the other at some point, creating a combination with
zero risk. This can be represented as follows:
 2 xa Ra  xb Rb  xa 2 a 2 + xb 2  b 2 + 2 xa xb ab  a b

xa 2  a 2
+
xb 2  b 2
(16)
- 2 xa xb  a  b
 ( xa  a  xb  b )2

 xa Ra  xb Rb  xa  a  xb b
The existence of an absolute value sign in the formula for standard deviation in this case
indicates that there are two sets of weights which will bring about the same level of risk.
Because these two sets of weights correspond to different portfolio expected returns, the
portfolio possibility frontier consists of two line segments which meet at a point on the Y axis.
This point, the point of zero risk, is is represented by point C on the graph. Here, the ratio of
weights equals the ratio of the standard deviations:
xa  a

xb  b
(17)
When the two assets' returns are uncorrelated (have a correlation coefficient of zero), the
frontier is a hyperbola between the two assets' individual risk/expected return points. The
curved line in Figure 2 shows the portfolio possibility frontier in the case when the correlation of
the two assets' returns is zero. In this case, the term containing correlation drops out, making
the locus of points at the frontier a hyperbola:
 2xa Ra + xb Rb  xa 2 a 2 + xb 2 b 2 + 2 xa xb ab a  b
 xa 2  a 2 + xb 2  b 2
 xa Ra + xb Rb  xa 2  a 2  xb 2  b 2
11
(18)
For two asset portfolios where the correlation is between zero and -1, the frontier lies between
the hyperbola of r = 0 and the kinked line of r = -1. For portfolios where the correlation is
between zero and +1, the frontier lies between the hyperbola of r = 0 and the straight line of r
= +1. Note that the frontier shifts to the left (less risk for a given level of expected return) as the
correlation decreases.
This shows that diversification into non-perfectly correlated assets
reduces portfolio risk.
For a portfolio of k assets whose weights sum to 1, the formula for portfolio possibility frontier as
a function of the desired expected return will be:
2 
1
[cr *2 2br *  a ]
( ac  b 2 )
(19 )
where
r*
= desired level of expected return
i
= vector of ones of length k
V
= kxk variance/covariance matrix
R
= vector of mean returns of length k
a = Rt V-1 R
b = Rt V-1 i
c = it V-1 i
and the superscript "-1" denotes the matrix inverse.
The efficient part of this frontier is represented by all points for which r* ³ b/c, which is the
expected return of the minimum variance portfolio. The derivation of these formulae can be
seen in Appendix I.
SUMMARY
There are 4 steps to finding the efficient frontier given historical asset returns:
1) calculating the mean expected return of each series,
2) creating the variance/covariance matrix,
12
3) finding the minimum variance portfolio, and
4) tracing the efficient frontier using the scalars a, b, and c defined above
13
APPENDIX I: OPTIMIZATION
Once the data for portfolio assets' means and variances have been assembled, it is possible to
solve for the minimum variance portfolios consisting of those assets at any desired expected
return levels. The risk/expected return combinations thus obtained form the efficient frontier, or
optimal set of risk/expected return combinations. The solution involves the application of
differential calculus to the equation for portfolio variance, subject to the constraints that the
weights must sum to 1 and that the portfolio must achieve some expected return.
The
techniques used below represent a simple application of quadratic programming. Note that the
following assumes an outright investment of capital, i.e. the size of the positions are based upon
some amount of capital allocated to the portfolio. In this case, we assume captial equals one.
The objective function states the problem as variance minimization subject to two constraints:
min  x t Vx
subject to
( 20 )
xtR  r *
x ti  1
The Lagrangian is thus:
min L  x t Vx  1 ( x t R  r *)   2 ( x t i  1)
( 21)
where
x
= weights vector of length k
r*
= desired level of expected return
i
= vector of ones of length k
V
= k x k variance/covariance matrix
R
= vector of mean returns of length kl1,l2
= Lagrange multipliers
14
The variance of the portfolio is xtVx . The term l1(xtR-r*) equals zero only when the expected
return equals r*. Similarly, the term l2(xti-1) equals zero only when the weights sum to one.
Taking the partial derivative of the Lagrangian with respect to xt, we get
L
 2 Vx  1 R   2 i
x t
( 22 )
To find the minimum* , we must set the derivative in equation (22) equal to zero and solve for x.
Doing this, we get:
x
1
(  1 V 1 R   2 V 1 i )
2
( 23)
Solving for the values of l1 and l2 allows one to solve for the weights. Premultiplying both sides
of (23) by , we see that:
r*  R t x 
1
( 1R t V 1 R   2 R t V 1 i )
2
( 24 )
(Note that r*=Rt x from the constraint in equation 20)
Similarly, premultiplying both sides of (23) by it:
1  itx 
1
( 1i t V 1 R   2 i t V 1 i )
2
( 25)
Therefore, both equations can be put in matrix form:
*
Since the function is convex with respect to the origin, setting the first derivative equal to zero finds a
minimum
15
r * a
 1   b
  
1 
b  2 
c   2 
2
(26)
with a = Rt V-1 R , b = Rt V-1 i, and c = it V-1 i as in equation (19).
Here, it is possible to solve for l1 and l2 since we have two equations and two unknowns.
1 
 2  a
   
 2  b
2
b r *
c  1 
a b 
1
b c   (ac  b 2 )


 c b 
b a 


1
(27)
Inverting the matrix,
1
(28)
Substituting, we can now easily solve for l1 and l2.
  1   cr * b 
 2  ac  b2  ac  b2 
    br * a  (29)
 2  2  2
 2  ac  b ac  b 
From (22), we see that 2Vx = l1R + l2i. Dividing both sides by 2 and premultiplying both sides
by xt, we can now calculate the variance of the portfolio:
 2  x t Vx 
1

r * 2
2
2
( 30 )
Substituting (29) into (30) variance can be restated as a function of the scalars a, b, and c
defined above:
16
2 
1
[cr *2 2br *  a ]
( ac  b 2 )
( 31)
Using this formula and substituting in different values of r*, one can easily trace the locus of
points comprising the portfolio possibility frontier.
To identify the efficient part of the frontier, one must first find the minimum variance portfolio.
This is done by differentiating (31) with respect to r*:
d 2
1

[2cr * 2b]
dr * ( ac  b2 )
( 32 )
Setting this equal to zero we get:
2cr*  2b
2b b
r* 

2c c
( 33)
Substituting this value back into equation 31:
2 

2
1   b
 b 
c
 2b   a 



2
 c  
(ac  b )   c 
(34)
1  ac b 2 

(ac  b 2 )  c c 
( ac  b 2 )
(ac  b 2 ) c
1

c

 1 b
Thus, the coordinates of the minimum variance portfolio are  ,  in mean-variance space
 c c
 1 b 
and 
,  in mean-standard deviation space.
 c c 
17
18
APPENDIX II: SPREADSHEET OPTIMIZATION
Generating a portfolio's efficient frontier in a spreadsheet involves three steps: 1) creating the
expected returns vector (R) and covariance matrix (V), 2) generating the scalars a, b, and c
from equation 19, and 3) using these scalars to get the coordinates of the minimum variance
portfolio and any other portfolios of interest on the frontier. The method suggested below can
be used with Lotus 123 or Microsoft Excel. When the commands differ significantly, a separate
paragraph will be used to distinguish between the two.
This description assumes some
experience with either spreadsheet. Figure 3 at the end of this paper shows a suggested layout
for the analysis using a five asset portfolio. It is suggested that the reader use this as a guide.
Creating the returns vector and covariance matrix Since individual asset returns will usually
be recorded in columns, averages taken beneath these cells will form a row vector of length n,
where n is the number of different assets in the portfolio.
This vector can be used as Rt, the
transpose of the returns vector. To create R, Rt can be Range Valued and Range Transposed
(Cut, Paste Special, Values and Copy, Paste Special, Transpose in Excel) or created cell by cell
using the + key. It is very important that the orientation of these vectors be correct. If it is not,
conformability errors will result. Next to these vectors, it is usually convenient to store the i and
it vectors,
The covariance matrix can be created in 123 using one of the many third party statistics add-ins
available. The diagonal on the covariance matrix should be checked to make sure that its
values match the individual asset variances, since some packages provide biased estimators
(dividing by n instead of n-1) of sample variance. Immediately below this matrix, its inverse
should be created using Data Matrix Invert.
In Excel, the covariance matrix can be created using Options Analysis Tools Covariance. This
is not recommended, however, since Excel only calculates population covariance and thus
19
gives biased estimates for the sample. It is suggested that the correlation matrix be created
using Options Analysis Tools Correlation, the upper triangular filled in, and this each item
multiplied by the relevant sample standard deviations.
This matrix can then be inverted with
the minverse() function. When doing this, it is important to identify the range that will hold the
matrix first, enter the minverse() function in its upper left hand corner, and then hit [Control]
[Shift][Enter] to perform the calculation.
Finding the minimum variance portfolio. To find the minimum variance portfolio, the values
of a, b, and c from equation (19) must first be calculated. This can be done two ways. The first
method can be used in 123 or Excel and requires several intermediate calculations. First, the 1
x n row vector RtV-1 must be created. This vector is postmultiplied by R to create a, and by i to
create b. To create c, the 1 x n row vector itV-1 must be calculated and postmultiplied by i.
Note that in all three cases, the row vector (which is itself the product of a square matrix
postmultiplied by a column vector) is multiplied by a column vector to create a scalar.
Alternately, these values can be created in Excel using a compound MMULT function. This
method does all of the vector/matrix multiplication in one cell for each scalar. These Excel
functions
would
be
input
as
=mmult(mmult(r_transpose,v_inverse),r)
for
a,
=mmult(mmult(r_transpose,v_inverse),i) for b and =mmult(mmult(i_transpose,v_inverse),i) for c.
It is recommended that only readers familiar with linear algebra use this method.
Given the values of these three scalars, the formulas in equations (33) and (34) can be used to
get the minimum variance portfolio.. To generate the rest of the frontier, one need only plug in
different values of r* into equation (31). Note that this equation finds the minimum variance.
The square root of this value is the standard deviation of the minimum variance portfolio at a
given level of r*. To graph the frontier, an XY graph should be used with the standard deviation
(or variance) on the x axis and the return on the y axis. The points should form a parabola if
variance is used and a hyperbola if standard deviation is used
20
To generate the x vector (the portfolio weights), the values of l1 and l2, can be used in
conjunction with two other intermediate column vectors, V-1R and V-1i which must be
calculated first Once this is done, x can be calculated by multiplying l1/2 and l2/2 by V-1R and
V-1i, respectively, and adding them. To generate the weights at different levels of expected
return, the Data Table commands in 123 and Excel should be used.
Once x is calculated, one can check the calculations by calculating portfolio three ways. First,
apply the portfolio weights to the individual asset returns on a month to month basis and
calculate the variance of the column. Second, use equation (30) and multiply l1/2 by r* and
add it to l2/2. Third, calculate xtVx itself by multiplying xt by V to form a row vector, and then
multiplying this vector by x. If there are no mistakes in the calculations, these three values
should be exactly alike.
21
Figure 3: Spreadsheet Optimization
DESCRIPTIVE
STATISTICS
ASSET1
ASSET2
INTERMEDIATE VECTORS
ASSET3
ASSET4
ASSET5
AVG
0.0095
0.0091
0.0065
0.0074
0.0119
VAR
0.0057
0.0129
0.0022
0.0029
0.0027
STD
0.0757
0.1134
0.0471
0.0537
0.0517
RTR*VINV
2.68669
-0.90546
-1.15368
1.08396
5.05007
ITR*VINV
253.18
-69.88
272.24
211.87
138.01
INTERMEDIATE SCALARS
COVAR (V)
a
0.07801
ASSET1
ASSET1
0.0057
0.0063
0.0000
0.0000
0.0000
b
6.76159
0.078
6.762
ASSET2
0.0063
0.0129
0.0005
0.0002
0.0008
c
805.41863
6.762
805.419
ASSET3
0.0000
0.0005
0.0022
0.0009
0.0017
ASSET4
0.0000
0.0002
0.0009
0.0029
0.0011
r*
0.00834
ASSET5
0.0000
0.0008
0.0017
0.0011
0.0027
1
1.00000
CORREL
ASSET2
ASSET1
ASSET3
ASSET2
ASSET4
ASSET3
ASSET5
ASSET4
ASSET5
1.0000
0.7382
0.0000
-0.0046
-0.0082
/2
-0.00255
ASSET2
0.7382
1.0000
0.0964
0.0326
0.1381
/2
0.00126
ASSET3
0.0000
0.0964
1.0000
0.3701
0.6950
ASSET4
-0.0046
0.0326
0.3701
1.0000
0.3980
ASSET5
-0.0082
0.1381
0.6950
0.3980
1.0000
RETURNS
47.05818
-0.39506
-0.39506
0.00456
WEIGHTS
/2* vinv*r
V INVERSE
I TRANSPOSE
abc inverse
LAMBDAS
ASSET1
R TRANSPOSE
abc matrix
/2*vinv*i
x
-0.00684
0.31976
0.31292
0.00231
-0.08826
-0.08595
394.3343
-198.5558
-4.5306
-9.8435
71.7752
0.00294
0.34383
0.34677
-198.5558
179.3238
1.1874
9.5258
-61.3607
-0.00276
0.26758
0.26482
-4.5306
1.1874
889.8718
-86.5697
-527.7217
-0.01286
0.17431
0.16145
-9.8435
9.5258
-86.5697
420.8945
-122.1403
71.7752
-61.3607
-527.7217
-122.1403
777.4620
0.0095
0.0091
0.0065
0.0074
0.0119
1
1
1
1
1
i
xtr
0.31292
-0.08595
0.34677
0.26482
0.16145
xtr*v
0.00124
0.00124
0.00125
0.00124
0.00123
VARIANCE
METHOD 1
METHOD2
xtr*v*x
/2 xr*
0.009480
1
0.009097
1
0.006541
1
RISK/RETURN
TABLE
0.007402
1
return
std
0.011941
1
0.003250
0.0498731
0.004925
0.0425236
0.006600
0.0373261
0.008275
0.0352458
0.009950
0.0368152
0.011625
0.0416234
0.013300
0.0487207
0.014975
0.0572622
0.016650
0.0666954
0.00124
22
0.00124
error
+/2
0.00000