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VaR measures have traditionally been categorized according to the transformations procedures they employ. There are four basic forms of transformation in widespread use:
Monte Carlo transformations, and
This article discusses linear VaR measures, which employ linear transformations. Many names have been used to describe linear VaR measures, so you may hear them referred to as parametric, variance-covariance, closed-form, or delta-normal VaR measures. There are shortcomings with most of these names. While linear VaR measures are parametric, so are most VaR measures. While linear VaR measures use variances and covariances, so do all VaR measures, with the exception of historical VaR measures. While some linear VaR measures employ a delta remapping, most do not. Also, while a normal assumption is common with linear VaR measures, it is by no means universal.
I prefer the name "linear" because it describes the one characteristic that is common to all linear transformations: they are applicable to portfolios whose portfolio mapping function is a linear polynomial. Such portfolios include portfolios of equities, physical commodities, or futures. The market value of such portfolios depends linearly upon applicable key factors. Other portfolios are so nearly linear that they can reasonably be approximated (remapped) with a linear polynomial. These include portfolios of forwards (including foreign exchange forwards) and most non-callable debt.
Linear VaR measures are generally not applicable to portfolios that hold options or instruments with embedded options. These include callable bonds, mortgage-backed securities and many structured notes.
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Schematic of
How VaR Measures Work |
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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. |
A transformation procedure accepts two inputs:
a portfolio
mapping function obtained from the
mapping procedure, and
a characterization of the joint distribution of the
key vector obtained from the
inference procedure.
The transformation procedure must combine these to somehow characterize the distribution of the portfolio's value. Based upon that characterization, the transformation procedure then values the desired VaR metric.
Let time 0 correspond to the current time, and let time 1
correspond to the end of the VaR
horizon. Mathematically, a portfolio is defined by a current value
, which is a known constant, and a future value
,
which is a random variable (see the
notation conventions documentation). Typically,
is specified
as a function
of a
random vector 1R, which is the key vector. Its components
, called key factors, represent market variables such as prices, interest
rates, spreads or
implied
volatilities as of time 1. Current values of key factors are indicated
with a constant vector
. The relationship
=
(
)
is called a portfolio mapping,
and the function
is the
portfolio mapping function.
We say that a portfolio is linear
if its portfolio mapping
function
is a linear polynomial. Using matrix notation (see any elementary linear
algebra text, such as Strang (1988)) this means
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[1] |
where b is a row vector and a is a scalar.
Suppose a portfolio comprises 100 shares of Dell stock, 200 shares of IBM stock and a short position of 300 shares of Microsoft stock. In this case, we would define
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[2] |
Assuming none of the stocks goes ex-dividend during the VaR horizon, the mapping procedure would specify:
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[3] |
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If we let b = (100 200 –300), then [3] has form [1].
Linear transformations are based upon an important result
from probability theory related to
linear polynomials of random vectors. Let
and
be the mean vector and covariance matrix of
.
(The superscripts
indicate that both parameters are for time 1,
conditional on information available at time 0.) Let
and
be the mean and
standard
deviation of
conditional on information available at
time 0. Then probability theory tells us that
(see article
linear polynomial of a random vector):
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[4] |
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[5] |
where a prime ' indicates transposition. Note
that these formulas are general. They require that
be a
linear polynomial of
, but
they make no assumptions about
the distribution of
. With these results, it is
possible to value a variety of VaR metrics, including standard deviation
of loss
| 0std(1L) = 0std(0p – 1P) = 0std(1P) | [6] |
and standard deviation of return
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[7] |
On their own, results [4] and [5] are not sufficient to value a quantile of loss VaR metric. Without additional information about a distribution, a mean and standard deviation do not determine the distribution's quantiles.
A standard solution is to assume
is normally
distributed. Because a normal distribution is fully determined by its mean
and standard deviation, this assumption—together with [4]
and [5]—fully specifies the distribution of
.
It should be possible to value any VaR metric.
For quantile of loss VaR metrics, we use the fact that quantiles of a normal distribution occur a fixed number of standard deviations from that distribution's mean. Formulas for some standard quantile of loss VaR metrics are
| 90.0% VaR = 1.282 0std(1P) – [ 0E(1P) – 0p ] | [8] |
| 95.0% VaR = 1.645 0std(1P) – [ 0E(1P) – 0p ] | [9] |
| 97.5% VaR = 1.960 0std(1P) – [ 0E(1P) – 0p ] | [10] |
| 99.0% VaR = 2.326 0std(1P) – [ 0E(1P) – 0p ] | [11] |
These formulas are motivated for the 90% VaR case in Exhibit 2:
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Graphical
Derivation of Formula [8] |
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A graphical derivation of formula [8] is provided. The 90% loss occurs at a portfolio value 1.282 standard deviations below the portfolio's expected value (the mean of the distribution). However, loss is calculated relative to the portfolio's current value as opposed to its expected value, which is why formula [8] includes the [0E(1P) – 0p] term. |
Over a short VaR horizon, such as a day, it is often reasonable to assume the portfolio's expected value equals its current value. In this case, Formulas [8] to [11] simplify to
| 90.0% VaR = 1.282 0std(1P) | [12] |
| 95.0% VaR = 1.645 0std(1P) | [13] |
| 97.5% VaR = 1.960 0std(1P) | [14] |
| 99.0% VaR = 2.326 0std(1P) | [15] |
Because the computations for a linear transformation are so modest, implementations typically run in real time. Simple linear VaR measures can even be implemented on a spreadsheet.
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