Para Bellum
Prepare for the worst riskBy Rafael Ballestas  March 26, 2019
"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%:
Knowing the probability distribution,
which tells us probabilities of point values,
we can find probabilities of ranges
with the corresponding cumulative distribution function (CDF
):
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:
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:
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:
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:
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."
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.