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This article discusses VaR metrics. It assumes familiarity with concepts described in the articles value-at-risk and measuring VaR.
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.
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Let's introduce some notation. We measure time in units equal to the length of the VaR horizon. The present time is time 0. The end of the VaR horizon is time 1.
To distinguish between known quantities and random quantities, we denote the former with lowercase letters and the latter with capital letters. With this convention, we denote the portfolio's current market value as 0p and its market value at the end of the VaR horizon as 1P. The preceding superscripts 0 and 1 denote time (see the notation conventions documentation).
Formally, a VaR metric is a real function of:
the distribution of 1P, conditional
on information available at time 0; and
the portfolio’s current value 0p.
Standard deviation of portfolio simple return 1Z, conditional on information available at time 0, is a VaR metric:
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[1] |
Quantiles of portfolio loss, 1L = 0p – 1P, make intuitively appealing VaR metrics. If a portfolio's conditional .95-quantile of 1L is USD 12.5MM, then such a portfolio can be expected to lose less than USD 12.5MM on 19 days out of 20.
An example of a risk metric that is not a VaR metric is standard deviation of cash flow. Because this generally cannot be expressed as a function of 0p and the conditional distribution of 1P, it is not a VaR metric.
VaR metrics can be quite elaborate. Semi-variance of portfolio return 1Z is one example. Define
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[2] |
Then the semi-variance of 1Z is simply the variance of 1Z–.
Another VaR metric is expected tail loss (ETL), which is sometimes called expected shortfall. This is the average portfolio loss, assuming that the loss exceeds some quantile of loss. For example, a 90% ETL VaR metric indicates the expected loss conditional on that loss exceeding its own .90-quantile.
To fully specify a VaR metric, we must indicate three things:
the
period of time—1 day, 2 weeks, 1 month, etc.—between time 0 and time 1;
this is the VaR horizon;
the function
of 0p and the conditional distribution of 1P;
the currency
in which 0p and 1P are denominated;
this is the base currency.
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We adopt a convention for naming VaR metrics:
The
metric’s name is given as the horizon, function and currency, in that
order, followed by “VaR.”
If the
horizon is expressed in days without qualification, these are understood
to be trading days.
If the
function is a quantile of loss, it is indicated simply as a percentage.
For example, we may speak of a portfolio’s
1-day standard
deviation of simple return USD VaR,
2-week
95% JPY VaR, or
1-week 90% ETL
GBP VaR, etc.
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