Thus far we have
dealt with portfolios of at most two assets, with only one involving any risk.
It is time to turn to the general relationship between the characteristics of a
portfolio and the characteristics of its components.
Let there be n
assets and s states of the world, with R an {n*s} matrix in which the
element in row i and column j is the return (or value) of asset i in state of
the world j. Here is an example with n=3 and s=4:
Good Fair
Poor Bad
Asset1
5 5 5
5
Asset2
10 8 6
-5
Asset3
25 12 2
-20
Let x be an {n*1}
vector of asset holdings in a portfolio. For example:
x
Asset1
0.20
Asset2
0.30
Asset3
0.50
What will be the
return of the portfolio in each of the states? This is easily computed. The
{1*s} vector of portfolio returns in the states (rp) will be:
rp = x'*R
Here:
Good
Fair Poor Bad
rp
16.50 9.40 3.80
-10.50
Now, let p be an
{s*1} vector of the probabilities of the various states of the world. In this
case:
p
Good
0.40
Fair
0.30
Poor
0.20
Bad
0.10
The expected
return (or value) of the portfolio will be:
ep = rp*p
In this case:
ep = 9.13
It is useful to
write the expression for expected return in terms of its fundamental
components:
ep = x'*R*p
The product of the
three terms can be computed in either of two ways. Above, we computed x'*R,
then multiplied the result by p. Alternatively, we could have multiplied x' by
the result obtained by multiplying R times p:
ep = x'*(R*p)
The parenthesized
expression is an {n*1} vector in which each element is the expected return (or
value ) of one of the n securities. Let e be this vector:
e = R*p
Here:
e
Asset1
5.00
Asset2
7.10
Asset3
12.00
Using these
results we may write:
ep = x'*e
That is, the
expected return (or value) of a portfolio is equal to the product of the vector
of its asset holdings and the vector of asset expected returns (or values).
This is the case whether the returns are discrete, as in this derivation, or
continuous (that is, drawn from continuous distributions).
The units utilized
for the values in vectors x and e will depend on the application. In some
cases, physical units (e.g. shares) may be appropriate for x; in others, values
(e.g. dollars); and in yet others, proportions of total value. Whatever the
units selected, to find the end-of-period value of a portfolio, the
end-of-period values per unit of exposure should be placed in vector e and the
number of units of each asset held placed in vector x. To find the expected
return (or value-relative) for a portfolio, multiply the expected returns (or
value-relatives) in vector e by the exposures to the assets in vector x.
Whatever the
application, the relationship between the expected outcome of a portfolio and
the expected outcomes for its components is relatively simple and intuitive.
For example, the expected return on a portfolio is a weighted average of the
expected returns on its components, with the proportionate values used as
weights. Since the relationship is linear, the marginal effect on
portfolio expected return of a small change in the exposure to a single
component will equal its expected outcome:
d(ep)/d(x(j)) = e(j)
If the expected
outcome were the only relevant characteristic of a portfolio, it would be easy
to make investment decisions. But risk is also relevant. And, as we will see,
its determination presents a more substantial challenge.
For present
purposes we will use as a measure of portfolio risk the standard deviation of
the distribution of its one-period return or the square of this value, the
variance of returns.
By definition, the
variance of a portfolio's return is the expected value of the squared deviation
of the actual return from the portfolio's expected return. It depends, in turn,
on the possible asset returns (R), the probability distribution across states
of the world (p) and the portfolio's composition (x). The relationship is,
however, somewhat complex.
To begin it is
useful to create a matrix of deviations of security returns from their expectations.
This can be accomplished by subtracting from each security return the
corresponding expectation:
d = R - e*ones(1,s)
The result (d)
shows the deviation (surprise) associated with each security in each of the
states of the world. Here:
Good
Fair Poor Bad
Asset1
0.00 0.00 0.00
0.00
Asset2
2.90 0.90 -1.10
-12.10
Asset3
13.00 0.00 -10.00
-32.00
The deviation
(surprise) associated with the portfolio in each of the states of the world can
be obtained by multiplying the transpose of the composition vector times the
asset deviation matrix:
dp = x'*d
In this case:
Good
Fair Poor Bad
dp
7.37 0.27 -5.33
-19.63
To determine the
variance of the portfolio, we wish to take a probability-weighted sum of the
squared deviations. A simple way to do so uses the dot-product operation, in
which elements are treated one by one:
vp = sum(p'.*(dp.^2))
However, there is
a more elegant and (as will be seen) far more useful way to do the computation.
First, we create a {s*s} matrix with the state probabilities on the main
diagonal and zeros elsewhere. This can be done in one statement:
P = diag(p);
In this case, P
will be:
Good Fair
Poor Bad
Good
0.40 0.00 0.00
0.00
Fair
0.00 0.30 0.00
0.00
Poor
0.00 0.00 0.20
0.00
Bad
0.00 0.00 0.00
0.10
The variance of
the portfolio is then given by a more conventional matrix expression:
vp = dp*P*dp'
For our portfolio:
vp =
65.9641
and
sdp = sqrt(vp)
= 8.1218
To see why the
latter procedure for computing variance is more useful, we substitute the
vectors used to compute dp:
vp = (x'*d)*P*(x'*d)'
There is an easier
way to write the last portion. Remember that the transpose operation turns a
matrix on its side. From this it follows that:
(a*b)' = b'*a'
For example, let a
be a {ra*c} matrix and b a {c*rb} matrix. Then (a*b) is {ra*rb} and (a*b)' is
{rb*ra}. Now consider the expression to the right of the equal sign. The first
term (b') is of dimension {rb*c}, while the second is of dimension {c*ra}.
Their product will thus be of dimension {rb*ra}. Since each element will
represent the sum of the same set of products as in the result produced by the
expression on the left, the resulting matrices will in fact be the same.
We can use this
result to note that:
(x'*d)' = d'*x''
But two transpose
operations will simply turn a matrix on its side, then turn it back, giving the
original matrix. Therefore:
(x'*d)' = d'*x
And the expression
for portfolio variance can be written as:
vp = (x'*d)*P*(d'*x)
Of course the
multiplications can be performed in any desired order. For example:
vp = x'*(d*P*d')*x
The parenthesized
term is of great importance in portfolio analysis - - enough to warrant its own
section in this exposition.
The matrix
described in the previous section is termed the covariance matrix for
the assets in question. Each of its elements is said to measure the covariance
between the corresponding assets. Using C to represent the covariance matrix:
C = d*P*d'
In this example:
Asset1 Asset2
Asset3
Asset1
0.00 0.00 0.00
Asset2
0.00 18.49 56.00
Asset3
0.00 56.00 190.00
The variance of a
portfolio depends on the portfolio's composition (x) and the covariance matrix
for the assets in question:
vp = x'*C*x
which of course
gives the same value found earlier (65.9641).
Well and good. But
what do the covariance numbers mean? How are we to interpret the fact that the
covariance of Asset2 with Asset3 is 56.00, while that of Asset3 with itself is
190.00, and so on?
Examination of the
matrices involved in the computation of C provides the answer. Recall that
C=d*P*d'. Consider the covariance of Asset2 and Asset3. It uses the information
in row 2 of matrix d and that in column 3 of matrix d' (the latter is, of
course, also in row 3 of matrix d). It also uses the vector of probabilities
along the diagonal of matrix P. The net result, written in a slightly casual
notation is that:
C(2,3) = sum(d(2,s)*p'(s)*d(3,s))
where the sum is
taken over the states of the world.
As this expression
shows, the covariance between two assets is a probability-weighted sum of the
product of their deviations. To verify this we can adapt the expression above
to make it legal in MATLAB:
c23 =
sum(d(2,:).*p'.*d(3,:))
The answer is
56.00, precisely equal to the value in the second row and third column of the
covariance matrix.
Put in terms of
prospective results: the covariance between two assets is the expected value of
the product of their deviations from their respective expected values. It
immediately follows that the covariance of asset i with asset j is the same as
the covariance of asset j with asset i. Thus the matrix is symmetric around its
main diagonal -- note that the value in row 2, column 3 is the same as that in
row 3, column 2. It also follows from the expression for covariance that the
covariance of an asset with itself is its variance. The asset variances thus
lie on the main diagonal of the covariance matrix. In this case:
va = diag(C)
Here:
va
Asset1
0.00
Asset2
18.49
Asset3
190.00
The asset standard
deviations are of course the square roots of these numbers:
sda = sqrt(diag(C))
In this case:
sda
Asset1
0.00
Asset2
4.30
Asset3
13.78
Note that the
first asset's return is certain. Hence its variance and standard deviation are
zero. The second asset is risky, with a standard deviation of 4.30. The third
is considerably more risky, with a standard deviation of 13.78.
Since the
covariance matrix includes asset variances along the main diagonal, the entire
matrix is sometimes termed a variance-covariance matrix. For brevity we
will use the simpler term covariance matrix, but it should be remembered
that the diagonal elements are both covariances and variances.
For the special
case in which the probability of each state is the same, it is possible to
compute the covariance matrix more simply using the standard MATLAB function cov.
However, the function assumes that the inputs represent a sample of
observations drawn from a larger population and hence adjusts the values in the
matrix upwards to offset the bias associated with measuring deviations from a
fitted mean. In effect, each value produced by the MATLAB function cov will
equal the one given by our formulas times (s/(s-1)), where is the number of
states (observations).
To use the cov
function, simply provide the matrix of observations, with each row representing
a different observation (state) and each column a different asset class. For
example, if the returns in our {n*s} matrix R were historic observations and we
were willing to assume that they were equally probable we could compute:
C = cov(R')
which would give:
Asset1 Asset2
Asset3
Asset1
0.00 0.00 0.00
Asset2
0.00 44.92 122.58
Asset3
0.00 122.58 360.92
These values are,
of course, quite different from those found earlier, due to both the assumption
of equal probabilities and the correction for bias.
With this aside
completed, we return to our forward-looking example.
It is relatively
easy to find a meaning for the elements on the main diagonal of the covariance
matrix. But what of the remaining ones? How can one interpret the fact that the
covariance of Asset2 with Asset3 is 56.00?
The solution is to
scale each covariance by the product of the standard deviations of the associated
assets. The result is the correlation coefficient for the two assets,
usually denoted by the Greek letter rho:
rho(i,j) = C(i,j)/(sda(i)*sda(j))
The matrix of
correlation coefficients is termed (unimaginatively) the correlation matrix.
We denote it Corr. To compute it, we compute a matrix containing the products
of the asset standard deviations:
sda*sda':
Asset1 Asset2
Asset3
Asset1
0.00 0.00 0.00
Asset2
0.00 18.49 59.27
Asset3
0.00 59.27 190.00
We need to divide
each element in the covariance matrix by the corresponding element in this
matrix. This can be done in one equation:
Corr = C./(sda*sda')
Giving:
Asset1 Asset2
Asset3
Asset1
NaN NaN NaN
Asset2
NaN 1.00 0.94
Asset3
NaN 0.94 1.00
Notice that the
elements associated with asset pairs in which one of the assets is riskless are
NaN (not a number), since they involve an attempt to divide zero (the
covariance) by zero(the product of two standard deviations, one of which is
zero).
While the
correlation of two assets, one of which is riskless, is not really a number, it
sometimes proves helpful to set it to zero. This can be accomplished by
adjusting the matrix of the cross-products of the standard deviations to have
ones in the cells for which the true value is zero. A simple way to do this is
to add to the original matrix a matrix with 1.0 in such positions. Since
"true" is represented in MATLAB as 1.0, a single matrix expression
does the job. Here is a set of statements that accomplishes the objective:
z = sda*sda';
z = z + (z==0);
CC = C./z;
where CC is the
desired correlation matrix. In this case:
Asset1 Asset2
Asset3
Asset1
0.00 0.00 0.00
Asset2
0.00 1.00 0.94
Asset3
0.00 0.94 1.00
In most cases, the
covariance matrix is known, and the correlation matrix derived from it as an
aid in interpretation. However, there are cases in which standard deviations
and correlations are estimated first, and the covariance matrix derived from
those estimates. To do this, we simply reverse the terms in the definition of
correlation. For the element in row i, column j:
C(i,j) = rho(i,j)*sda(i)*sda(j)
And, for the
entire matrix:
C =CC.*(sda*sda')
Note that the
adjusted matrix CC was used in the latter computation to avoid NaN values in
the cells associated with the riskless asset.
Asset covariances
are the main ingredients for computing portfolio risks. But we have shown that
standard deviations are much easier to interpret than are asset variances.
Similarly, correlations often prove more useful for communicating relationships
than do covariances.
Correlation
coefficients measure the extent of the association between two variables. Each
such coefficient must lie between -1 and +1, inclusive. A positive coefficient
indicates a positive association: a greater-than-expected outcome for one
variable is likely to be associated with a greater- than-expected outcome for
the other while a smaller-than-expected outcome for one is likely to be
associated with a smaller-than-expected outcome for the other. A negative
coefficient indicates a negative association: a greater-than-expected outcome
for one variable is likely to be associated with a smaller-than-expected
outcome for the other while a smaller-than- expected outcome for one is likely
to be associated with a greater-than-expected outcome for the other.
The figures below
provide examples. In each case the probabilities of the points shown are
assumed to be equal.
In the above
examples the variables are roughly jointly normally distributed with means of
zero and standard deviations of 1.0 -- roughly, because each of the 100 points
is drawn from such a joint distribution so the (sample) distribution of the
actual results departs somewhat from the underlying (population) distribution.
Note that in the
case of perfect positive correlation (+1.0), the points fall precisely along an
upward- sloping straight line. In this case it has a slope of approximately 45
degrees due to the nature of the variables. In general, the line may have a
greater or smaller slope. Nonetheless, a necessary and sufficient condition for
perfect positive correlation is that all possible outcomes plot on an
upward-sloping straight line.
In the case of
perfect negative correlation the plot has the opposite characteristic. All
points will plot on a downward-sloping straight line. Here too, the slope will
depend on the magnitudes of the variables, but the line will be
downward-sloping in any event.
As the figures
show, in the case of less-than-perfect positive correlation (between 0 and
+1.0), the points will tend to follow an upward-sloping line, but will deviate
from it. The closer the correlation coefficient is to zero, the greater will be
such deviations and the more difficult it will be to see any positive
relationship. In the case of less-than-perfect negative correlation (between 0
and -1), the points will tend to follow a downward-sloping line. Here too, the
closer the correlation coefficient is to zero, the greater will be the
deviations and the more obscure the relationship.
If the correlation
coefficient is zero, the best linear approximation of the relationship will be
a flat line. This does not preclude the possibility that there is a non-linear
relationship between the variables. The figure below shows a case in which the
correlation coefficient is zero, but knowledge of the value of the variable on
the horizontal axis would help a great deal if one wished to predict the value
of the variable on the vertical axis. In this case the variables are uncorrelated,
but they are not independent.
In the special
case in which probabilities are equal, one can use the MATLAB function corrcoef
to compute a correlation matrix directly from an {n*s} matrix of values of n
assets in s different states of the world, with each row representing a
different state (observation) and each column a different asset. For example:
corrcoef(R')
would give:
Asset1 Asset2
Asset3
Asset1
NaN NaN NaN
Asset2
NaN 1.00 0.96
Asset3
NaN 0.96 1.00
In this case the
only source of the differences from our forward- looking estimates is the use
of equal probabilities rather than the predicted probabilities. Since the
correlation coefficient is the ratio of estimated variance to the product of
two estimated standard deviations, any adjustment of the covariance matrix for
sample bias cancels out, leaving the correlation coefficients unaffected.
Analysts
frequently utilize historic returns to estimate the covariances among future
returns. If all returns, past and future, were drawn from a stable joint
distribution, it would be desirable to use as many observations from the past
as possible in order to maximize the accuracy of the resultant estimates of the
true underlying process that will generate future returns. However, if the
parameters of the distribution are likely to have changed over time, the
situation is more difficult. One can utilize a great deal of data, much of
which may be of limited relevance for the future. Alternatively, a small amount
of recent data can be employed, with the attendant danger of substantial
estimation errors. Which is better -- a great deal of possibly irrelevant data
or too little relevant data?
There is no easy
answer to the question. The optimal procedure ultimately will depend on the
manner in which covariances evolve through time. Some Analysts approach the
problem by limiting the historic data to, say, 60 monthly observations, with
each observation assigned the same weight (probability). Others select only
periods in which underlying conditions are assumed to have been similar to
those existing at the present time (e.g. periods following recessions if a
recession has recently been experienced). Yet others employ complex procedures
in which covariances are assumed to be positively correlated but with a
tendency to eventually revert to a long-run mean value. Here we focus on a
simple procedure utilized in a number of asset allocation models that assumes
that the future is more likely to be like the recent past than the distant
past.
In an exponential
weighting scheme each historic observation is assigned a multiple of the
weight assigned to its predecessor. For example, observation t could be
assigned a weight equal to 2^(t/h) divided by a constant (k), with the latter
set so that the sum of all the weights equaled 1.0. In such a scheme h
can be interpreted as an assumed half-life. To see why, consider the
weights assigned to months t and t-h:
w(t)
= (2^(t/h))/k
w(t-h) = (2^((t-h)/h))/k
Thus:
w(t)/w(t-h) = (2^(t/h))/((2^t/h)/2) = 2
Thus if month t is
the most recent month and h=60, the observation 60 months ago will be assigned
half as much weight as the most recent month.
The weight
assigned to any month relative to that assigned to its predecessor will be:
(2^(t/h))/(2^((t-1)/h))
which will equal 2^(1/h).
Thus if a 60-month half life is utilized, each month's observation will be
given a weight equal to 2^(1/60) or 1.0116 times that given the prior month
(1.16% higher).
It is relatively
straightforward to compute a set of such weights using MATLAB. Assume that
there are T observations. The vector of dates (1,2,...T) is given by:
d = 1:1:T;
The vector of
2^(t/h) values will be:
w = 2.^(d/h)
where h is the
desired half-life.
The weights can
easily be normalized so that they sum to 1.0:
p = w/sum(w)
We denote the
result p since the weights will serve as probabilities. In a sense, the
assumption is made that the probability is p(t) that next month's returns will
equal those that occurred in month t.
Having selected a
set of probabilities, we proceed as before to estimate expected returns,
deviations and the covariance matrix:
e = R*p;
d = R - e*ones(1,T);
C = d*diag(p)*d';
The library function wcov
obviates the need to remember all these formulas. It takes as inputs a matrix
of returns for n assets in s states (or from s historic time
periods). For convenience, the return matrix can have assets in the rows and
states (observations) in the columns or vice-versa. The function assumes
(reasonably) that the number of states (observations) exceeds the number of
assets and proceeds accordingly.
To cover more
cases, the half-life parameter can be specified as zero, in which event the
states (observations) are given equal weights.
The simplest way
to utilize the function is as follows:
C = wcov(R,h)
where R is the
matrix of returns, h is the half-life and C is the resultant covariance matrix.
If more
information is desired, one or more additional variables may be indicated. For
example:
[C,sda]
= wcov(R,h)
will also return
sda as the vector of standard deviations.
The statement:
[C,sda,CC]
= wcov(R,h)
will also return
CC as the matrix of correlation coefficients, following the convention that the
correlation coefficient is zero if the corresponding covariance is zero.
Finally, the
statement:
[C,sda,CC,e] = wcov(R,h)
will also return
the expected returns, based on the assumption that future probabilities equal
the weights computed from the assumed half-life.
The Weighted Statistics
Worksheet allows you to compute both equal-weighted and
exponentially-weighted means, standard deviations and correlation coefficients.
An example using returns for a set of
Vanguard mutual funds from 1991 through 1995 provides a chance for you to
experiment with different weighting schemes. Try a half-life of zero for equal
weights, then compare the results with those obtained with other values (for
example, 12, 24, .. 60). You might even wish to try a negative half-life to
weight earlier observations more heavily than later ones.
You may also wish
to copy and paste other return series into the weighted statistics worksheet so
that you can calculate the resulting historic statistics.
It is remarkably
easy to determine the covariances between two portfolios. Recall the formula
for computing the covariance of portfolio x:
vp = x'*C*x
where x is the
vector with the portfolio composition and C is the covariance matrix for asset
returns.
Now, let xa
represent one portfolio and xb another. For example:
xa
Asset1
0.10
Asset2
0.50
Asset3
0.40
xb
Asset1
0.40
Asset2
0.10
Asset3
0.50
Assume that the
covariance matrix (C) is:
Asset1 Asset2
Asset3
Asset1
0.00 0.00 0.00
Asset2
0.00 18.49 56.00
Asset3
0.00 56.00 190.00
The covariance
between the two portfolios is given by:
cab = xa'*C*xb
Which in this case
equals 55.16.
The relationship
can be extended to cover a case in which there are multiple portfolios. Let X
be an {n*p} matrix containing information on the composition of p
portfolios of n assets. For example:
xa xb
Asset1
0.10 0.40
Asset2
0.50 0.10
Asset3
0.40 0.50
Then the
covariance matrix for the portfolios is given by:
Cp = X'*C*X
Which gives:
xa xb
xa
57.42 55.16
xb
55.16 53.28
Note that the elements
on the main diagonal indicate the variances of the two portfolios, while the
other elements equal their covariance.
It is
straightforward to compute the covariance of each asset with a given portfolio.
Recall the statement for the covariance of portfolio xa with portfolio xb:
cab = xa'*C*xb
This can be
computed in two operations:
cab = xa'*(C*xb)
For example, with
xb:
Asset1
0.40
Asset2
0.10
Asset3
0.50
and C:
Asset1 Asset2
Asset3
Asset1
0.00 0.00 0.00
Asset2
0.00 18.49 56.00
Asset3
0.00 56.00 190.00
then
cab = xa'*cp
where cp = C*xb,
or:
cp
Asset1
0.00
Asset2
29.85
Asset3
100.60
Now, assume that
xa contains only the first asset:
xa
Asset1
1.00
Asset2
0.00
Asset3
0.00
Clearly, the
covariance of xa with xb will equal the first value in vector cp (0.00).
If xa contained
only the second asset, its covariance with xb would equal the second value in
vector cp (29.85). And so on.
The conclusion is
not hard to reach. Vector cp contains the covariances of the asset classes with
portfolio xb.
More generally,
if:
cp = C*x
then cp(i) is the
covariance of asset i with portfolio x.
Note that the
covariance of an asset with a portfolio will be a weighted average of its
covariances with all the assets (including itself), with the composition of the
current portfolio used as weights.
The risk of a
portfolio is not a linear function of the vector of its components. Rather, the
variance of a portfolio is a quadratic function of its composition. This
thwarts the intuition of most Analysts and Investors. Indeed, the nature of
risk may be the single most important argument for the use of quantitative
analysis in investment management. Neither Investors nor Analysts can be blamed
for this fact. Nor can Harry Markowitz. Nature made risk a quadratic function.
Markowitz only discovered it.
Given this central
fact of investment life, it follows that the impact on the risk of a portfolio
of a small change in the amount invested in a particular asset is not simply a
function of the risk of that asset. The impact will depend on the covariances
of the asset with all the other assets currently in the portfolio and on the
composition of the portfolio.
Consider a
portfolio x and a "difference vector" d. We wish to
determine the effect on portfolio variance of a switch from portfolio x
to portfolio x+d.
The variance of x
will be:
vx = x'*C*x
While that of
(x+d) will be:
vv = (x+d)'*C*(x+d)
The latter can be
expanded by noting first that (x+d)'=x'+d', giving:
vv = (x'+d')*C*(x+d)
then by
multiplying out all the terms:
vv = x'*C*x + x'*C*d + d'*C*x + d'*C*d
Since the first term
of the latter expression equals the variance of x, the change in variance is
given by the sum of the last three terms:
dvp = x'*C*d + d'*C*x + d'*C*d
The first two
terms are the same. This follows from the facts that: (1) the transpose of a
scalar is the same scalar, (2) the transpose of a product of matrices is the
product of their transposes, taken in reverse order and (3) the covariance
matrix C is symmetric, so that its transpose equals the original matrix. Given
this, we may write:
dvp = 2*d'*C*x + d'*C*d
We are interested
here in the effect on variance of a small change in the holding of one asset.
Thus d will contain only one small non-zero element. For example,if we wished
to know the effect of a small change in the holdings of asset 2 we could set:
d
Asset1
0.00
Asset2
0.01
Asset3
0.00
Since the elements
in d will be either zero or very small (very much less than 1.0), the final
term in the earlier expression (d'*C*d) will be even smaller. Indeed, as d
approaches zero, the one element in d'*C*d will approach zero considerably
faster, since it involves the square of the non-zero element in d. For purposes
of computing the marginal effect of a change we may ignore the final term,
giving:
dvp = 2*d'*C*x
or
dvp = d'*2*C*x
From this it
follows that d(vp)/d(x(j)) will equal the j'th row of 2*C*x. More generally,
2*C*x is the vector of marginal risks of the asset classes:
mr = 2*C*x
with mr(j)
indicating the change in portfolio variance per unit change in the amount
invested in asset j.
Note finally, that
C*x is the vector of the covariances of the assets with portfolio x, which we
have denoted cp. Thus:
mr = 2*cp
and two times the
covariance of an asset with a portfolio indicates the marginal risk of that
asset, given the composition of the portfolio. In our case:
mr
Asset1
0.00
Asset2
59.70
Asset3
201.20
Thus variance
would not be affected by a small change in the holding of asset 1. It would
increase at a rate of 59.70 per unit change in Asset 2 and at a rate of 201.20
per unit change in Asset 3. Of course these figures hold only approximately for
finite changes in the assets, with the error greater, the larger the underlying
change. Moreover, they assume that only one element in the portfolio is
changed. If the assets represent zero investment strategies this may be
feasible. If, however, they are true investments, at least one holding will
have to be decreased for another to be increased. We will take these aspects into
account in later discussions. For now it suffices to have determined the vector
of derivatives of variance with respect to asset holdings.
https://drive.google.com/file/d/0Bx3mfFH5R-y3dzZadEFJektVRUE/edit?usp=sharing
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