|
|||
 |
|||
|
This article discusses portfolio remappings, which are widely used in production VaR measures. The article assumes familiarity with concepts discussed in the overview article measuring value-at-risk and the article mapping procedures. Exhibit 1 is reproduced from the first of those articles.
|
Schematic of
How VaR Measures Work |
|
|
 |
|
All practical VaR measures accept portfolio data and historical market data as inputs. They process these with a mapping procedure, inference procedure, and transformation procedure. Output comprises the value of a VaR metric. That value is the VaR measurement. |
One of the essential tasks a VaR measure must perform in order to quantify the market risk of a trading portfolio is to characterize the exposures of that portfolio. This is the purpose of the measure's mapping procedure. Output of the mapping procedure is a portfolio mapping, which becomes an input for the measure's transformation procedure.
The most direct way to construct a portfolio mapping is to construct a primary mapping. Based upon the portfolio's holdings, the portfolio's value is expressed as a weighted sum of the values of the assets it holds. Asset values may not be directly observable in the market, but these can be expressed in terms of more fundamental market variables, such as relevant exchange rates, interest rates, commodity prices, etc—what are called key factors.
Let's express this mathematically (see the
notation conventions documentation). We let time 0 be the current time, and
we let time 1 be the
end of the VaR horizon. A risk factor is any random variable
whose value will be realized during the interval (0,1] and will affect the
market value of a portfolio at time 1. A risk
vector
is a random vector of risk
factors.
One particular risk factor and two risk vectors are of particular interest. These are:
the
portfolio’s value
at the end of the VaR horizon;
the
asset vector
, whose components are values
of assets held by the portfolio; and
the
key vector
,
whose components are key factors
.
The portfolio's
holdings
is
a row vector indicating the number of units of each asset held by the
portfolio. Using vector multiplication, we indicate the first step of
constructing a primary mapping as
 |
[1] |
We represent this mapping schematically as
 |
[2] |
|
|
On its own, formula [1] defines a
simple primary mapping. We can leave it in this form, in which case
will play the dual role of both asset vector and key vector. More
commonly, we express
in terms of more
fundamental market variables, and these become our key factors. We define
mapping
|
|
[7] |
The function φ may be quite complicated. Essentially, it must value each asset based upon the key factors. If the assets are exotic derivatives or mortgage-backed securities, φ will need to incorporate sophisticated techniques from financial engineering.
Composing
with φ, we obtain portfolio mapping function
.
Our portfolio mapping is
 |
[8] |
We represent it schematically as
 |
[9] |
This is the most general form of primary mapping.
Explicitly or implicitly, every mapping procedure constructs a portfolio mapping by first constructing a primary mapping. Some mapping procedures stop at this point. The primary mapping is their output, which is passed to the transformation procedure. A drawback of this approach is the fact that primary mappings can be extremely complicated. If a portfolio holds several thousand exotic derivatives, the function φ could take hours of processing time to value. Many transformation procedures—especially Monte Carlo transformation procedures—must value a portfolio mapping function numerous times. They need a portfolio mapping function that is relatively easy to value.
Accordingly, many mapping procedures apply certain approximations to a primary mapping to obtain what is known as a portfolio remapping. For these mapping procedures, output comprises the simpler portfolio remapping, and this is what is passed to the transformation procedure..
Formally, a remapping is an
approximation of a risk vector 1Q with some other
risk vector
.
We are primarily interested in portfolio
remappings, which approximate a portfolio's value
with some other random variable
.
If we have a portfolio mapping
=
(),
portfolio remappings may take three forms:
-
A function remapping approximates
=
()
by replacing
with
an approximate mapping function
,
so
.
-
A variables remapping approximates
=
()
by replacing
with alternative key vector
,
so
.
-
A dual remapping approximates
=
()
by replacing both
and
, so
.
The first and third forms are most common.
Many function remappings approximate a portfolio mapping function
with a
linear or
quadratic polynomial to facilitate use of a
linear transformation or
quadratic
transformation. These are called global
remappings. They also have other names, depending upon the
specific nature of the remapping. They may be called
linear remappings or
quadratic remappings, depending
upon the type of polynomial constructed. If a quadratic approximation is
constructed based upon the portfolio's
deltas and
gammas, the result may be
called a delta-gamma remapping.
Because deltas and gammas are highly localized exposure metrics, better
results are generally obtained by using
interpolation or the
method of
least squares. With these approaches, the portfolio mapping function
is
valued at several points and interpolation or the method of least squares
is used to fit a quadratic polynomial
to the results. These are called
interpolation remappings or
least squares remappings.
We represent global remappings schematically as
 |
[10] |
Schematics such as [10] take a bit of getting used
to, but they are extremely helpful for understanding specific remappings.
They generalize more simple schematics such as [2] and
[9]. In them, horizontal arrows indicate mappings
(exact relationships). Vertical arrows indicate remappings
(approximations). In schematic [10], we see that
mapping function
replaces mapping function
, but
key vector 1R remains unchanged. The result is an
approximation
for .
Another type of function remapping is a
holdings remapping. These
remappings replace the assets held by a portfolio with just a
handful of assets that, together, exhibit similar exposures. A simple
example of a holdings mapping is to replace a large number of fixed cash
flows with just a handful maturing on specific (perhaps annual)
maturities. A more sophisticated holdings remapping might replace several
thousand derivatives positions with just a handful that have the same
combined market value, delta and
vega. With a holdings remapping, the only thing
that changes is the portfolio's holdings
.
This is illustrate in schematic [11]:
 |
[11] |
Dual remappings may be used to reduce the dimensionality of the key
vector
.
Principal component analysis can be used for
this purpose, in which case the remapping may be called a
principal component remapping.
Consider a portfolio mapping
=
(),
where is n-dimensional with
mean vector
and multicollinear covariance matrix
(the superscripts 1|0 indicate that these are parameters for time 1
conditional on information available at the current time 0). We represent
in terms of its principal
components:
 |
[12] |
where
is
the matrix whose columns are orthonormal eigenvectors of
,
and 
is an n-dimensional column vector
of the principal components of
. We convert [12]
to an approximate relationship by discarding principal components
that have variances close to 0. Suppose we retain m principal
components and discard n – m. Let
be
the m-dimensional vector of retained principal components. Let
be
the matrix whose columns are the eigenvectors corresponding to those
principal components. We obtain
 |
[13] |
Schematically, the remapping is
 |
[14] |
The above examples indicate just a few of the many forms portfolio remappings take in practice.
|
Related Internal Links |
||||||||||||||||
|
|
||||||||||||||||
|
||||||||||||||||
|
Sponsored Links |
||||||||||||||||
|
|
||||||||||||||||
 |
|
|||||||||||||||
|
Related Books |
||||||||||||||||
|
|
||||||||||||||||
|
||||||||||||||||
|
Sponsored Links |
||||||||||||||||
|
|
||||||||||||||||
 |
|
|||||||||||||||
|
Related Forum Discussions |
||||||||||||||||
|
|
||||||||||||||||
|
||||||||||||||||
|
||||||||||||||||
_125.jpg)
