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This article discusses mapping procedures, which are one of three essential components of a VaR measure. The article assumes familiarity with concepts discussed in the overview article measuring value-at-risk. Exhibit 1 is reproduced from that article.
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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. |
The purpose of a mapping procedure is to characterize a portfolio's exposures. It does so by expressing the portfolio's value as a function of applicable market variables, such as stock prices, exchange rates, commodity prices or interest rates. Let's define some concepts.
Let time 0 be the current time, and 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 (see the
notation conventions documentation). A risk
vector
is a random vector of risk
factors. If a risk vector reflects a future value of some time series, we
may speak of its current value
or historical
values
One particular risk factor and two risk vectors play important roles in VaR measures. We give them special names and notation. These are:
the
portfolio’s future value
;
the
asset vector
; and
the
key vector
.
The portfolio’s future value
represents the market value at time 1 of the portfolio for which VaR is to
be measured. The portfolio is assumed fixed in the sense that it will not
be traded during the period [0,1], and no assets will be added or
withdrawn. We are interested in the portfolio’s current value
only if our VaR metric depends
upon it.
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Asset vector
has asset
values
as components. These represent
accumulated values of specific assets the portfolio may hold.
Realizations may be negative, so our
definition recognizes no accounting distinction between assets and
liabilities. Accumulated value is denominated in the
base currency employed by the
VaR metric. It may reflect such variables as capital gains,
dividends,
coupons, margin payments, reinvestment income, storage costs, insurance,
financing, changes in exchange rates, leasing income, etc.
Every VaR measure must directly characterize a
conditional probability distribution for some vector of risk factors, such
as prices, interest rates, spreads,
or implied
volatilities. Those risk factors
are
called key factors. They are the
components of the key vector
. Occasionally, we use asset values
as key factors. More often, it is convenient to use more basic financial
variables as key factors.
A
portfolio’s holdings is a row vector
indicating the number
of units held by the portfolio of each asset. We use
to define the portfolio's value
in terms of the asset vector
:
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[1] |
Consider a simple example. A portfolio comprises a short straddle. It is short 10 call options on a particular future, and it is short 10 put options on the same future. All options have the same strike and expiration. We define assets with
 |
[2] |
so holdings are
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[3] |
and we define
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[4] |
We represent the mapping schematically as
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[5] |
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At this point in our example, we could be done with the
mapping procedure. We would let
be our key
vector, and [4] would be our portfolio mapping, which we would pass to the
transformation procedure. A problem with doing so is that we would be
using option prices as key factors, which means that the
inference
procedure would need to characterize their joint distribution.
Designing an inference procedure to do this would be difficult. Because of
the limited downside risk of options, their prices have
skewed price distributions. Also, their
standard deviations would be highly dependent upon
whether the options were
in-the-money
or
out-of-the-money. A simpler solution is to not employ options prices
as key factors, but to use more fundamental risk factors such as the
options' underlier prices and implied volatilities.
Consider a portfolio comprising call options with various
strikes on the first nearby Henry Hub natural gas future. Asset values
represent the market values at time 1 of the various strike options. We
define
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[6] |
For simplicity, we are employing a single implied
volatility for all strikes. A more sophisticated model would use multiple
implied volatilities to capture volatility
skew. Using
Black's (1976) pricing formula
for options on futures, we define asset values
as functions of
. Note that, in applying the
options pricing formula, we use day counts as of time 1, not the current time
0. We obtain mapping
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[7] |
Composing
with φ, we obtain portfolio mapping function
.
Our portfolio mapping is
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[8] |
We represent it schematically as
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[9] |
Portfolio mappings of form [5] or [9] are called primary mappings. Primary mappings are portfolio mappings constructed directly from a portfolio's holdings, as indicated in our example. Mathematically, a primary mapping works by valuing each asset held by the portfolio, multiplying the values by the portfolio's holdings in each asset, and summing.
For many VaR measures, output of the mapping procedure is a primary mapping, but not for all. Use of primary mappings can pose certain problems. The most common of these is that primary mappings can be mathematically complicated. This is especially true for large portfolios of instruments such as mortgage-backed securities or exotic derivatives. Applying a transformation procedure to a complicated portfolio mapping can be computationally expensive. For this reason, many mapping procedures replace primary mappings with simpler approximations. Those approximations are called portfolio remappings.
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