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Value-at-Risk (VaR) is a powerful tool for assessing market risk, but it also poses a challenge. Its power is its generality. Unlike market risk metrics such as the Greeks, duration and convexity, or beta, which are applicable to only certain asset categories or certain sources of market risk, VaR is general. It is based on the probability distribution for a portfolio's market value. All liquid assets have uncertain market values, which can be characterized with probability distributions. All sources of market risk contribute to those probability distributions. Being applicable to all liquid assets and encompassing, at least in theory, all sources of market risk, VaR is an all-encompassing measure of market risk.
As with its power, the challenge of VaR also stems from its generality. In order to measure market risk in a portfolio using VaR, some means must be found for determining the probability distribution of that portfolio's market value. Obviously, the more complex a portfolio is—the more asset categories and sources of market risk it is exposed to—the more challenging that task becomes.
It is worth distinguishing between three concepts:
A VaR measure is an
algorithm with which we calculate a portfolio's VaR.
A VaR model is the
financial theory, mathematics, and logic that motivate a VaR measure. It is the
intellectual justification for the computations that are the VaR measure.
A VaR metric is our
interpretation for the output of the VaR measure.
Examples of VaR metrics are one-day 95% USD VaR or one-week standard deviation of return EUR VaR. A VaR measure is just a bunch of computations. What justifies our interpreting the output of those computations as, say, two-week 99% EUR VaR? The answer is the VaR model. The VaR model is the intellectual link between the computations of a VaR measure and the interpretation of the output of those computations, which is the VaR metric.
This article focuses on VaR measures and VaR models. Conveniently, these can be discussed without regard for specific VaR metrics. The reason is that valuation of a VaR metric is the final step of any VaR measure. The real work for a VaR measure is to somehow characterize a probability distribution for a portfolio's market value. Valuing a specific VaR metric based on that characterization is a final step—it is almost an afterthought. By changing that final step of a VaR measure, we can alter the VaR measure to support a different VaR metric. Accordingly, to a large extent, any VaR measure can support any VaR metric, and we can discuss VaR measures without considering the specific VaR metrics they are to support.
Measure time in trading days. Let 0 be the current time.
We know a portfolio's current market value
. Its
market value
in one trading
day is unknown (see the
notation
conventions documentation). It is a random variable. Out notation uses preceding superscripts
to denote time. We find it convenient to indicate random quantities with capital letters and known constants with lower case letters.
Our task is to ascribe to
a probability
distribution. One way that we might simplify this task is to assume some
standard distribution. Doing so reduces the problem from one of estimating an
entire distribution to that of estimating the handful of parameters necessary to
specify that standard distribution. Depending upon the standard distribution
which is assumed, this simple approach may yield a
closed-form solution for the
portfolio's VaR.
For example, a normal distribution is fully described with
two parameters, its mean and
standard
deviation. If we assume
is normally
distributed, then all we need do in order to measure VaR is estimate the mean
and standard deviation
of that
distribution. (The preceding superscripts in our notation indicate that
parameters are for the portfolio's time 1 value conditional on information
available at time 0.) Together with the normality assumption, these two parameters provide all the information necessary to value any
other parameter—VaR
metric—related to the
distribution of .
For example, if our VaR metric is one-day 95% USD VaR, we can calculate VaR
as
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1.645
+ ( –
) |
[1] |
This formula is based on the fact that the 5%-quantile of a normal distribution always occurs 1.645 standard deviations below its mean. See Exhibit 1 to understand [1] or see the article linear value-at-risk for a more detailed discussion.
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Example: One-Day 95% USD VaR |
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If we assume a portfolio's value 1P
is normally distributed, then we can calculate its 95% VaR with
formula [1]. The 5%-quantile of a normal
distribution occurs 1.645 standard deviations below its mean. Of
course, losses are measured relative to a portfolio's current value
0p and not its expected value
|
In practice, a portfolio's expected value
will often be
close to its current value
. This is especially
true over short VaR horizons, such as the one
trading day horizon of our example. In this circumstance, it may be reasonable
to set =
. With this
simplification, our formula [1] for 95% VaR
becomes
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1.645
|
[2] |
Based upon similar assumptions, formulas for 90%, 97.5% and 99% VaR are
90% VaR ~ 1.282
97.5% VaR ~ 1.960
99% VaR ~ 2.326
Estimating the standard
deviation
of the portfolio's market value is analogous to the task of estimating the standard
deviation of portfolio return, a task you may be familiar with from
portfolio theory. Except for the
fact that VaR deals with market values instead of returns, we may adopt this
familiar mathematics
of portfolio theory for estimating VaR.
We use a general result from
probability. Suppose
are random
variables having standard deviations
and correlations
. Suppose another random variable
Y is defined as a linear polynomial of the
:
| Y = b1X1 + b2X2 + + bmXm + a | [3] |
Then the standard deviation of Y is given by (see article linear polynomial of a random vector)
 |
[4] |
Formula [4] is completely general. So long as
Y is a linear polynomial of the
,
we can use [4]. We need no other assumptions or information about the random variables
.
We can apply [4] to estimate the standard deviation
of
our portfolio's market value. Suppose the portfolio has holdings
in m
assets. The assets' accumulated market values at time 1 are random variables,
which we denote
.
Then
 |
[5] |
Based on [5], we can apply [4]
to obtain .
All we need as inputs are standard deviations and correlations for the
. These might
be inferred by applying methods of
time series analysis to historical price data for the assets. In some cases,
this is feasible. In others, it is not. Collecting historical price data for
every asset held by a portfolio may be a daunting task.
A more manageable approach may be to model the portfolio's
behavior, not in terms of individual assets, but in terms of specific risk
factors. Depending upon the composition of the portfolio, risk factors
might include exchange rates, interest rates, commodity prices, spreads, implied
volatilities,
etc. We call the n modeled risk factors key factors.
We denote their values at time 1 as
. The key
factors comprise an ordered set (or "vector"), which we call the
key vector. We denote it
:
 |
[6] |
In all likelihood, the number n or key factors we need to model will be substantially less than the number m of assets held by the portfolio.
Selecting which key factors to model is as simple—or
complex!—as choosing a set of market variables such that a pricing formula
for
each asset
held by the portfolio can be expressed in terms of those variables. That is, for
each asset, there must exist a valuation function
such that
|
=
(
) |
[7] |
Because the value of the portfolio
is a linear polynomial of the asset values
, we can now express
in terms of the key
factors:
 |
[8] |
This is a functional relationship that specifies the portfolio's market
value in terms
of the key factors
. Shorthand
notation for the relationship is
 |
[9] |
Relationship [9] is called a
portfolio mapping. The function
is called the
portfolio mapping function.
As an example, 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
 |
[10] |
Assuming none of the stocks goes ex-dividend during the VaR horizon, the portfolio mapping is
 |
[11] |
This is a very simple portfolio mapping. A slightly more complex example is a portfolio comprising a call option on a futures contract. In this case, we define
 |
[12] |
and the portfolio mapping function
is simply
Black's (1976) pricing formula
for options on futures. Obviously, if a portfolio holds many complicated
instruments, the portfolio mapping function will be equally complicated.
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|
The portfolio mapping function
maps the n-dimensional
space of the key factors to the one-dimensional space of the portfolio's market
value.
Given a realization for
,
gives us the
corresponding value of
. That doesn't
solve our problem. We're not interested in one possible realization of
. We need to
characterize the entire distribution of
. Somehow, we
need to apply the portfolio mapping function
to the entire
joint distribution of
to obtain
the entire distribution of
. The question is: how?
After all,
beyond purporting its existence, we know very little about the portfolio mapping
function . It could be some complicated function with discontinuities and other
inconvenient properties.
A simple solution exists if
is a linear
polynomial, as is the case in the above example of
a portfolio with holdings in Dell, IBM, and Microsoft stock. As indicated by [11],
is a linear
polynomial for that example. If we assume that
is normally
distributed and that
=
, then all we need to
calculate is .
Given standard deviations
and correlations
for the
, we can
apply [4] to obtain
.
But what if
is not a linear
polynomial? In our example of an option portfolio,
is given by
Black's (1976) option pricing formula. That is decidedly non-linear, so we
cannot use [4] to obtain
. Furthermore,
we cannot reasonably assume that
is normally
distributed. Because options limit downside risk, they
skew the probability distribution of
. Normal
distributions aren't skewed.
These issues can be understood graphically.
Consider Exhibit 2. It illustrates with two graphs the situation
if a portfolio mapping function
is a linear
polynomial. The graph on the left is of
.
It shows how the price of the portfolio responds linearly to changes in a
single key factor
. In that graph,
evenly spaced values for
have been mapped into corresponding
values for . The resulting values of
are also evenly spaced, indicating
that the mapping causes no distortions. If
is normally distributed, so will be
.
That normal distribution for
is depicted in the graph on the right.
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Example: Linear Portolio |
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A linear mapping function
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If the portfolio price function is non-linear,
may
not be normally distributed. This is illustrated in Exhibit 3 with a portfolio
consisting of a single call option in an underlier
.
|
Example: Long Call Option |
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 |
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A non-linear mapping function
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The left graph of Exhibit 3 depicts the familiar "hockey
stick" price function for a call option. Evenly spaced values for
do not map into evenly spaced values for
. If
is normally distributed, the resulting
distribution of will not be normal. As shown on the right, it will be skewed.
That skewness reflects the call option's limited downside risk.
Portfolios can have more complex price distributions. For example,
a range forward is a long-short options position
which, when applied to a short position in an underlier
, behaves as
illustrated in Exhibit 4.
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Example: Range Forward Hedging a Short Position |
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A long-short options position can result in a bimodal distribution for 1P. |
In the left graph of Exhibit 4, we see that values of
cluster in two regions, resulting in the dramatically non-normal price
distribution shown on the right.
These three examples illustrate how linearity of can
simplify the task of calculating a portfolio's value-at-risk. Non-linear
portfolios often exhibit unusual price distributions. These can differ markedly
and in unpredictable ways from normal distributions. Such portfolios require
more sophisticated modeling techniques.
Here is the general problem we face in calculating value-at-risk.
To calculate VaR, we need to characterize the distribution of
conditional on
information available at time 0. Our
puzzle has two pieces:
-
The first piece of the puzzle is the key factors
. Because they are observable financial variables,
historical data should be available for them. Based on this, we can characterize
the joint distribution of
. We may do
so with
standard deviations
and correlations
for the
, or we may do
so in some other manner. Our problem, then, is to convert that
characterization of the distribution of
into a
characterization of the distribution of
. On its own, our
characterization of the distribution of
is not enough to do
this. Because it is independent of the portfolio's composition, it cannot,
on its own, tell us how risky the portfolio is. -
The second piece of the puzzle is the portfolio mapping [9] that relates
to
. That
formula will change over time, evolving to reflect the portfolio's changing
composition. Formula [9] contributes to our analysis what the
characterization of the distribution of
does not. It reflects
the portfolio's composition. On its own, however, it cannot tell us how risky
the portfolio is, for it contains no information relating to market volatility.
We need to combine these two pieces of the puzzle in order
to estimate .
Somehow we must filter the market information contained in the characterization
of the distribution of
through the
portfolio information contained in the portfolio mapping [9].
Every VaR measure must address this
problem. Accordingly, all VaR measures share certain common components related
to solving this problem. All must specify a portfolio mapping. All must somehow
characterize the joint distribution of
.
All must somehow combine these two pieces to characterize the distribution of
.
Exhibit 5 is a schematic summarizing these three processes that are common to all
practical VaR measures.
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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. |
Any practical VaR measure must include three procedures:
2. inference procedure, and
Recall that risk has two components:
exposure, and
uncertainty.
By specifying a portfolio mapping, a mapping procedure
describes exposure. By characterizing the joint distribution for
, an
inference procedure describes uncertainty. A transformation
procedure combines exposure and uncertainty to describe the distribution
of
, which it then
summarizes with the value of some VaR metric. In so doing, the
transformation procedure describes risk.
 |
A mapping procedure accepts a portfolio's composition as
an input. Its output is a portfolio mapping function
that defines
as a function of
. Specifying
is largely
an exercise in financial engineering.
Since must
value an entire portfolio, it can be complicated. For example, if a
portfolio holds 1000 exotic
derivatives,
will be
extremely complicated—and may take hours to value, even on a computer. For
this reason, a mapping procedure may employ certain approximations, called
remappings, to simplify
.
The purpose of an inference procedure is to characterize
the joint probability distribution of the key vector
conditional
on information available at time 0.
It generally accepts historical market data as an input and applies
techniques of
time series analysis to characterize the joint distribution
conditional on information available at time 0. Techniques currently
employed tend to be crude. The most common are those of uniformly-weighted
moving averages (UWMA) and exponentially-weighted moving averages (EWMA).
What is needed is time-series methods that can address
conditional
heteroskedasticity in high dimensions. While research is ongoing, such
methods are not yet perfected.
A transformation procedure combines the outputs from the mapping and inference procedures and uses them to characterize the distribution of 1P, conditional on information available at time 0. Based on that characterization, and perhaps the portfolio's current value 0p, the transformation procedure (or "transformation") determines the value of the desired VaR metric. The result is the VaR measurement.
Much research has focused on transformation procedures. Four basic forms of transformations are used:
Monte Carlo transformations, and
Linear transformations are simple and run in real time.
Based on [4], they apply only if a portfolio mapping function
is a
linear polynomial. Quadratic transformations are slightly more
complicated, but also run in real time (or near-real time). They apply
only if a portfolio mapping function is a quadratic polynomial and 1R
is joint-normal. Monte Carlo and historical transformations are widely
applicable, but tend to run slowly (run times of an hour or more are not
uncommon). Both employ the
Monte Carlo method.
They both generate a large number of realizations 1r[k]
for 1R and value 1P for each.
The histogram of realizations 1p[k]
for 1P provides a discrete approximation for the
conditional distribution of 1P. From this, any VaR
metric can be valued. Monte Carlo and historical transformations differ
only in how they generate the realizations 1r[k].
Monte Carlo transformations generate them with pseudorandom number
generators. Historical transformations draw them from historical market
data.
Traditionally, VaR measures have been categorized according to the transformation procedures they employ. There are:
linear VaR measures (other names include: parametric,
variance-covariance, closed form, or delta normal VaR measures)
quadratic VaR measures (also called delta-gamma VaR
measures)
This naming convention may have had unfortunate consequences. By focusing attention on the role of transformation procedures, the convention tends to downplay the important roles of mapping and inference procedures. Over the past 10 years, most VaR-related research has focused on transformations. Important research on mapping and inference procedures has lagged.
To apply a VaR measure, it must be implemented in some manner. For a very simple portfolio—perhaps one comprising a single asset—a VaR measure might be implemented with pencil and paper. In actual trading environments, they are coded as software and run on computers. An implemented VaR measure is a VaR implementation.
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