"Si vis pacem, para bellum", goes the old adage. If you want peace,
prepare for war. In our case, the worst possible risky scenario our
information assets could go into. While probability distributions, loss
exceedance curves, simulated scenarios, etc, are all great for the
quants in the office, at the end of the day big, important decisions
need to be supported by single numbers that can be easily compared to
one another. In risk management, this number is the Value at Risk or
VaR
. Fortunately, once you have one you have the other.
VaR
measures this scenario by telling us beyond how much our losses
will not go, with a certain degree of confidence, over a definite
period of time. Thus a daily 1% VaR
of $10 million means that the
probability that you will lose more than ten million is 1%, i.e., are
99% confident that the losses will not exceed that.
So we need to define over what time period our VaR
will be taken and
how extreme the worstcase scenario. Typical periods and confidences
used in the industry are a single day or week, and confidence levels of
95% or 99%.
There are at least three workable ways to compute the value at risk:

Examining the distribution of the returns,

Using the loss exceedance curve (
LEC
)
The normal distribution is perhaps the most popular one for modeling realword situations and natural phenomena, and with good reason. It could be used, to model the value of a portfolio over a oneyear period, with mean return 10%, and standard deviation (volatility) 30%:
Figure 1. Normal distribution of value
Knowing the probability distribution, which tells us probabilities of
point values, we can find probabilities of ranges with the corresponding
cumulative distribution function (CDF
):
Figure 2. Cumulative distribution function of value
Looks like a vertically reflected LEC
. In a cumulative probability
plot the VaR
is just the xvalue corresponding to the confidence.
We can use a spreadsheet,
for this, with the
NORM.DIST
function. The probability that the loss exceeds 20% is
=NORM.DIST(20,10,30,1)
0.158655253931457
i.e., around 15.8%. The 10 and 30 above are the distribution parameters, and the 20 is the value whose probability we need. Notice that it is negative, meaning a loss. The 1 means to make the computations cumulative.
We can also use the inverse function so that, given a probability, we get the point at which this probabilty is attained. It is the same process as above, but backwards.
At what point is the 1% probability? More exactly, for which value V
is it true that the probability that the final value is less than or
equal to V is 1%? That’s just the 1% VaR
:
=NORM.INV(0.01,10,30)
59.7904362212252
This is the 1% quantile, or
the first percentile of the distribution, the point under which the
remaining 1% of points are, weighing by the probability. Thus the Value
at Risk in this example will be 59.8% of what we invested. Had we
invested $100 million, then we know the VaR
is $59.8 million, and
hence that the losses will not exceed that amount in 99% of the cases,
only in that rare 1%. Notice that the VaR
, being a single figure, does
not tell us exactly or otherwise what the losses might be in that
catastrophic 1%. But if we are ready to lose that much, we are halfway
prepared for the metaphoric war.
The tail (or conditional) value at risk, or TVaR
(CVaR
) for
short, tries to fill that void by giving us the expected value or mean
in the catastrophe region, i.e., in case of a VaR
breach. Much like
the actual mean of a distribution is a center of gravity of sorts, where
we could "hold" the PDF
in balance, besides being the value with more
repetitions if we repeatedly draw numbers from such a distribution:
Figure 3. Expected value of a beta distribution. Via Wikimedia.
The TVaR
is thus the expected value of the loss, given that the VaR
has been surpassed. In terms of the above analogy, it is the center of
gravity of the "catastrophe" region of the distribution plot:
Figure 4. Illustration of VaR
and TVaR
. Via
Nematrian.
In our case, since we are mainly interested in cybersecurity risk, which we quantify via simulations, we can always rerun them and aggregate the results differently in order to obtain the density function and recreate the example above. But given that the main result of our simulations was a loss exceedance curve:
Figure 5. Loss exceedance curve
We can just use this to obtain the VaR
, just like we did with the
distribution CDF
. This graph is already cumulative, so there is no
need to compute areas under the curve behind the scenes. We simply
obtain the value in millions corresponding to the percentage of the
scenario in which we are interested. In this particular graph, the 5%
yearly VaR
appears to be $500 million (recall that this graph has a
logarithmic scale in the xaxis). The 1% is not even visible here, but
at least that tells us that it must be beyond $1000 million.
Monitoring a shorttermed VaR
can be usefulto evaluate the performance
of risk management or to understand events from the past:
Figure 6. Artificial VaR monitoring (via MathWorks) and real example from Bankers Trust, via [3].
In the first we see a steady, if slow, decline in VaR
over the years.
Also notice how the returns are almost always above their corresponding
valuesatrisk, save for a few rare breaches, which is to be expected.
In the image to the right there is an interesting moment around February
1994, where there is a sharp decrease in the VaR
, after which it
pretty much stays stable under the risk appetite line (dashed). This
phenomenon is explained in Jorion’s book ^{[3]} as a
response to a rise in interest rates at that moment, which was just as
sharp as the decrease in the VaR
.
However, a decreasing VaR
is not all. Shying away from investments to
keep the VaR
low will, by symmetry, mean lower chance of great
returns:
"A risk manager has two jobs: make people take more risk the 99% of the time it is safe to do so, and survive the other 1% of the time.
VaR
is the border."— Aaron Brown
So, the VaR tells us in a single number what can happen with an
investment or any risky situation the worst that might happen. However
its greatest strength is also where it falls short. This particular
number, while it gives an upper bound for the losses, is also unable to
tell us anything else about what happens in that 1% of the cases. The
TVar
tries to fill this void, but it is still just a number, meaning
that it inherits this same weakness.
References

S. Benninga and Z. Wiener (1998). ValueatRisk (
VaR
). Mathematica in Education and Research 7(4) 
P. Jorion (2006). Value at Risk: The New Benchmark for Managing Financial Risk. McGrawHill.

N. Pearson (2002). Risk Budgeting: portfolio problem solving with valueatrisk. Wiley.
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