Suppose
you are going to initiate a trading position of $ 100 million involving market
risk. What is the important question in your mind as you initiate this trading
position?
Of
course, you will be interested to know, how much you may lose on this
investment.
This
is a question that every investor asks when considering investing in a risky
asset. Value at Risk tries to provide an answer to this question.
We
will examine the following in this paper:
• An overview of VaR
• Pros & Cons of three methods used
to estimate
WHAT IS VALUE AT RISK?
Value
at Risk measures the potential loss in value of a risky asset or portfolio over
a defined period for a given confidence interval – say 95%.
What
does this 95% signify?
Say
for example you have calculated VaR on an asset of $100 million. Daily Var at 95% confidence level is $ 1
million. What does it mean?
It
can be interpreted in two ways:
·
There is 95% chance that the value of the asset
will NOT drop more than $1 million over any day.
·
Or, there is 5% chance that the value of the
asset will drop more than $ 1 million over any day.
While
Value at Risk can be used by any entity to measure its risk exposure, it is
used most often by commercial and investment banks to capture the potential
loss in value of their traded portfolios from adverse market movements over a
specified period. The focus in VaR is
clearly on downside risk and potential losses.
There
are three key elements of VaR – a specified level of loss in value, a fixed
time period over which risk is assessed and a confidence interval. The VaR can
be specified for an individual asset, a portfolio of assets or for an entire
firm.
MEASURING VALUE AT RISK
There
are three basic approaches that are used to compute Value at Risk, though there
are numerous variations within each approach.
Approach
I - Variance-Covariance Method (Risk Metrics fame)
Value
at Risk and the usage of the measure can be traced back to the RiskMetrics
service offered by J.P. Morgan in 1995. Publications by J.P. Morgan in 1996
describe that ther eturns on individual risk factors are assumed to follow
normal distributions.
Advantages & Weaknesses
The
strength of the Variance-Covariance approach is that the Value at Risk is
simple to compute, once you have made an assumption about the distribution of
returns and inputted the means, variances and covariances of returns. However,
there are three key weaknesses:
•
Wrong distributional assumption: If conditional
returns are not normally distributed, the computed VaR will understate the true
VaR. There could be far more outliers in
the actual return distribution than would expect.
•
Wrong Inputs: Even if the normal distribution
assumption holds up, the VaR can still be wrong if the variances and
covariances estimates are incorrect.
•
Dynamic variables: A related problem occurs when
the variances and covariances across assets and securities change over time. This
is not uncommon because the fundamentals driving these numbers do change over
time.
Approach
II – Historical Simulation Method
To
run a historical simulation, we begin with time series data – for example daily
change in the price of the underlying. The
key aspects of the historical simulation approach are:
•
Assumption of normality is NOT needed.
•
The second is that each day in the time series
carries an equal weight when it comes to measuring the VaR
•
Basically it is assumed that the history may
repeating itself
Advantages & Weaknesses
Historical
simulations are relatively easy to run, however they do have its
weaknesses.
a)
Past may not repeat: While all three approaches
to estimating VaR use historical data, historical simulations are much more
reliant on them than the other two approaches. For example, a portfolio manager
of Oil Corporation that determined its oil price VaR, based upon past data
would have been exposed to much larger losses than expected over during 2008
oil volatility. During July 2008, oil prices touched record high of $147/- per
barrel but dropped below $ 40 during Dec 2008.
b)
Ignores Trends in the data: The approach takes
all data points with equal weight. In other words, continuing the oil price
example, it is assumed that the price changes from trading days in 2007 affect
the VaR in exactly the same proportion as price changes from trading days in
2008. If there is a trend of increasing volatility, we will understate the VaR.
c)
New assets: The historical simulation approach is
not suitable for new risks and assets because there is no historic data
available to compute the Value at Risk.
Approach
III - Monte Carlo Simulation
Monte
Carlo simulations rely on simulations to build up distributions. Once the
distributions are specified, the VaR process starts. In each run, the market
risk variables take on different outcomes and the value of the portfolio
reflects the outcomes.
After
a repeated series of runs, numbering usually in the thousands, you will have a
distribution of portfolio values that can be used to assess Value at Risk. For
instance, assume that you run a series of 10,000 simulations and derive
corresponding values for the portfolio. These values can be ranked from highest
to lowest, and the 95% percentile Value at Risk will correspond to the 500th
lowest value and the 99th percentile to the 100th lowest value.
The
strengths and weaknesses of the simulation approach apply to its use in
computing Value at Risk. Quickly reviewing the criticism, a simulation is only
as good as the probability distribution for the inputs that are fed into it.
While Monte Carlo simulations are often touted as more sophisticated than
historical simulations, many users directly draw on historical data to make
their distributional assumptions.
Monte
Carlo simulations become more difficult to run for two reasons. First, you now
have to understand the probability for several (running into hundreds) market
risk variables. Second, the number of simulations that you need to run to
obtain reasonable estimate of Value at Risk will have to increase substantially
CONCLUSION
Are the estimates of Value at Risk same under the
three approaches?
Historical
simulation and variance-covariance methods will yield the same Value at Risk if
the historical returns data is normally distributed. Similarly, the
variance-covariance approach and Monte Carlo simulations will yield roughly the
same values if all of the inputs in the latter are assumed to be normally
distributed with consistent means and variances. As the assumptions diverge, so
will the Var.
Which approach is the best to estimate of VaR?
The
decision depends upon the risk manager, based on the task at hand.
-
If you are assessing the Value at Risk for
portfolios, that do not include options, over very short time periods (a day or
a week), the variance-covariance approach does a reasonably good job,
notwithstanding its heroic assumptions of normality.
-
If the Value at Risk is being computed for a risk
source that is stable and where there is substantial historical data (commodity
prices, for instance), historical method is suited.
-
If the historical data is volatile and dynamic and
the normality assumption is questionable, Monte Carlo simulations do best.
