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1.
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Definition
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In
economics and finance, Value at Risk (VaR) is the maximum loss not
exceeded with a given probability defined as the confidence level,
over a given period of time. It is commonly used by security houses
or investment banks to measure the market risk of their asset
portfolios (market value at risk), however VaR is a very general
concept that has broad applications. VaR is widely applied in
finance for quantitative risk management for many types of risks.
VaR does not give any information about the severity of loss by
which it is exceeded. Other measures of risk include
volatility/standard deviation, semi-variance (or downside risk) and
expected shortfall.
Other Resources:
-
MC2 Management Consulting:
Often
abbreviated as VAR, these are a class of Models used by
financial institutions to measure the risk in complex derivative
portfolio positions.
More…
-
FHLB Dallas:
Measures the worst expected loss over a given time interval
under normal market conditions at a given confidence level.
More…
-
Optimization Partner:
The Value-at-risk (VaR) at p% is the amount one risks to lose
with probability p%.
More…
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2.
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Examples, Types, or
Variations
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A
variety of models exist for estimating VaR. Each model has its own
set of assumptions, but the most common assumption is that
historical market data is our best estimator for future changes.
Common models include:
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(a) variance-covariance (VCV), assuming that risk factor returns
are always (jointly) normally distributed and that the change in
portfolio value is linearly dependent on all risk factor
returns,
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(b) the historical simulation, assuming that asset returns in
the future will have the same distribution as they had in the
past (historical market data),
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(c) Monte Carlo simulation, where future asset returns are more
or less randomly simulated
The
variance-covariance, or delta-normal, model was
popularized by J.P Morgan (now J.P. Morgan Chase) in the early 1990s
when they published the RiskMetrics Technical Document. In
the following, we will take the simple case, where the only risk
factor for the portfolio is the value of the assets themselves.
Example -
Consider a
trading portfolio. Its market value in US dollars today is
known, but its market value tomorrow is not known. The
investment bank holding that portfolio might report that its
portfolio has a 1-day VaR of $5 million at the 95% confidence
level. This implies that (provided usual conditions will prevail
over the 1 day) the bank can expect that, with a probability of
95%, the value of its portfolio will decrease by 5 million or
less during 1 day, or in other words: it can expect that with a
probability of 5% (i. e. 100%-95%) the value of its portfolio
will decrease by more than 5 million during 1 day. Stated yet
differently, the bank can expect that the value of its portfolio
will decrease by 5 million or less on 95 out of 100 usual
trading days, in other words by more than 5 million on 5 out of
every 100 usual trading days.
Historical simulation
is the simplest and most transparent method of calculation. This
involves running the current portfolio across a set of
historical price changes to yield a distribution of changes in
portfolio value, and computing a percentile (the VaR). The
benefits of this method are its simplicity to implement, and the
fact that it does not assume a
normal
distribution of asset returns. Drawbacks are the
requirement for a large market database, and the computationally
intensive calculation.
Monte Carlo simulation
usually involve
principal
components analysis of the VCV matrix, followed by
random simulation of the components. Benefits are the ability to
handle any underlying distribution, plus a more accurate
assessment when non-linear risk factors are present in the
portfolio (e.g. options). Drawbacks include the inherently
opaque nature of Monte Carlo calculations, and the
computationally intensive process.
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3.
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Formula
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In the following, return means percentage change in value.
A variety of models exist for estimating VaR. Each model has its
own set of assumptions, but the most common assumption is that
historical market data is our best estimator for future changes.
Common models include:
-
(1)
variance-covariance (VCV), assuming that risk factor returns
are always (jointly) normally distributed and that the
change in portfolio value is linearly dependent on all risk
factor returns,
-
(2) the
historical simulation, assuming that asset returns in the
future will have the same distribution as they had in the
past (historical market data),
-
(3)
Monte
Carlo simulation, where future asset returns are
more or less randomly simulated
The variance-covariance, or
delta-normal, model was
popularized by J.P. Morgan Chase (formerly J.P. Morgan) in the
early 1990's. In the following, we will take the simple case,
where the only risk factor for the portfolio is the value of the
assets themselves. The following two assumptions enable to
translate the VaR estimation problem into a linear algebraic
problem:
(1) The
portfolio is composed of assets whose deltas are linear, more
exactly: the change in the value of the portfolio is linearly
dependent on (i.e. is a linear combination of) all the changes
in the values of the assets, so that also the portfolio return
is linearly dependent on all the asset returns.
(2) The asset
returns are jointly normally distributed.
The
implication of (1) and (2) is that the portfolio return is
normally distributed because it always holds that a linear
combination of jointly normally distributed variables is itself
normally distributed.
We will
use the following notation:
-
means
“of the return on asset i“ (for σ and
μ) and "of asset i" (otherwise)
-
means
“of the return on the portfolio” (for σ and
μ) and "of the portfolio"
(otherwise)
-
all
returns are returns over the holding period
-
there are
N assets
-
μ=
expected value, i. e. mean
-
σ =
standard deviation
-
V =
initial value (in currency units)
-
-
=
vector of all ωi
(T means transposed)
-
=
covariance matrix = matrix of
covariances between all N asset returns, i. e. an
NxN matrix
The
calculation goes as follows.
(i)
(ii)
The
normality assumption allows us to z-scale the calculated
portfolio standard deviation to the appropriate confidence
level. So for the 95% confidence level VaR we get:
(iii)
The benefits
of the variance-covariance model are the use of a more compact
and maintainable data set which can often be bought from third
parties, and the speed of calculation using optimized linear
algebra libraries. Drawbacks include the assumption that the
portfolio is composed of assets whose delta is linear, and the
assumption of a normal distribution of asset returns (i. e.
market price returns).
VaR has three parameters; these are...
-
The time
horizon (period) we are going to analyze (i. e. the length
of time over which we plan to hold the assets in the
portfolio - the "holding period"). The typical holding
period is 1 day, although 10 days are used, for example, to
compute capital requirements under the European Capital
Adequacy Directive (CAD). For some problems, even a holding
period of 1 year is appropriate.
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The
confidence level at which we plan to make the
estimate. Popular confidence levels usually are 99% and 95%.
-
The unit
of the currency which will be used to denominate the value
at risk.
VaR, with
the parameters: holding period x days; confidence level
y%, measures what will be the maximum loss (i. e.
decrease in portfolio market value) over x days, if one
assumes that the x-days period will not be one of the
(100 − y)% x-days periods that are the worst under
normal conditions.
Note that VaR
cannot anticipate changes in the composition of the portfolio
during the day. Instead, it reflects the riskiness of the
portfolio based on the portfolio's current composition.
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4.
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Related Terms
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-
Market to Market (MTM)
- Covariance
- Monte Carlo Simulation
-
Market Risk
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5.
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As
Used in the Hedge Fund World
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Value-at-Risk is a very widely used measure of
market risk. It is a statistical measure which evaluates the market
risk of a portfolio using probabilities based on a percentage of
common confidence and time horizon.
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6.
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Applications
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7.
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Misused & Abused
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Unfortunately, VaR is not the panacea of risk
measurement methodologies. A subtle technical problem is that VaR is
not sub-additive. That is, it's possible to construct two
portfolios, A and B, in such a way that VaR (A + B) > VaR(A) + VaR(B).
This is unexpected because we'd hope that portfolio diversification
would reduce risk.
The theory of coherent risk measures outlines the properties we'd
want any measure of risk to possess. Artzner, et al wrote the
canonical paper on the subject. In this paper they outline, in
axiomatic fashion, the properties a risk measure should possess to
be considered coherent. An example of a coherent risk measure is
Expected Tail Loss (ETL) (also known as Conditional Value-at-Risk (CVaR)).
Other names are Expected shortfall and worst conditional
expectation.
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8.
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Additional Sources of Information
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-
Books
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News
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Scholarly Papers
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Back to Terms
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