What is VAR?
Idea behind volatility
Variance - Covariance Method
Monte Carlo Stimulation
• In ﬁnancial mathematics and ﬁnancial risk management, Value at Risk
(VaR) is a widely used risk measure of the risk of loss on a speciﬁc
portfolio of ﬁnancial assets.
• For a given portfolio, probability and time horizon, VaR is deﬁned as a
threshold value such that the probability that the mark-to-market loss on
the portfolio over the given time horizon exceeds this value is the given
• To estimate the probability of the loss, with a conﬁdence interval, we need to
deﬁne the probability distributions of individual risks.
The focus in VaR is clearly on downside risk and potential losses. Its use in
banks reﬂects their fear of a liquidity crisis, where a low-probability catastrophic
occurrence creates a loss that wipes out the capital and creates a client exodus.
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.
Thus, we could compute the VaR for a large investment project for a ﬁrm in
terms of competitive and ﬁrm-speciﬁc risks and the VaR for a gold mining
company in terms of gold price risk.
Idea Behind -Volatility
• A statistical measure of the dispersion of returns for a given security or
• Volatility can either be measured by using the standard deviation or
variance between returns from that same security or market index.
• In other words, volatility refers to the amount of uncertainty or
risk about the size of changes in a security's value.
• Commonly, the higher the volatility, the riskier the security.
• However, is that it does not care about the direction of an investment's
• VAR answers the question, "What is my worst-case scenario?"
• What is the most I can - with a 95% or 99% level of conﬁdence - expect
to lose in dollars over the next month?
• What is the maximum percentage I can - with 95% or 99% conﬁdence -
expect to lose over the next year?
• The historical method simply re-organizes actual historical returns, putting
them in order from worst to best.
• It then assumes that history will repeat itself, from a risk perspective.
• TheWith 95% conﬁdence, we expect that our worst daily loss will not
• If we invest $100, we are 95% conﬁdent that our worst daily loss will not
exceed $4 ($100 x -4%)
• While all three approaches to estimating VaR use historical data, historical
simulations are much more reliant on them than the other two approaches
for the simple reason that the Value at Risk is computed entirely from
historical price changes.
A related argument can be made about the way in which we compute Value
at Risk, using historical data, where all data points are weighted equally. In
other words, the price changes from trading days in 1992 aﬀect the VaR in
exactly the same proportion as price changes from trading days in 1998. To
the extent that there is a trend of increasing volatility even within the
historical time period, we will understate the Value at Risk.
The historical simulation approach has the most diﬃculty dealing with new
risks and assets for an obvious reason: there is no historic data available to
compute the Value at Risk.
Variance - Covariance
• This method assumes that stock returns are normally distributed.
• It requires that we estimate only two factors - an expected (or average)
return and a standard deviation.
• The blue curve above is based on the actual daily standard deviation of
the QQQ, which is 2.64%.
If there are far more outliers in the actual return distribution than would be
expected given the normality assumption, the actual Value at Risk will be much
higher than the computed Value at Risk.
To the extent that these numbers are estimated using historical data, there is a
standard error associated with each of the estimates. In other words, the
variance-covariance matrix that is input to the VaR measure is a collection of
estimates, some of which have very large error terms.
A related problem occurs when the variances and covariances across assets
change over time. This nonstationarity in values is not uncommon because the
fundamentals driving these numbers do change over time.
• The third method involves developing a model for future stock price
returns and running multiple hypothetical trials through the model.
• 100 hypothetical trials of monthly returns for the QQQ. Among them,
two outcomes were between -15% and -20%; and three were between
-20% and 25%.
• That means the worst ﬁve outcomes (that is, the worst 5%) were less than
• Every VaR measure makes assumptions about return distributions, which,
if violated, result in incorrect estimates of the Value at Risk.
• History may not be a good predictory.
• Non Stationary predictions might occur.
Ignored 2,500 years of experience in favor of untested models built by non-
Was charlatanism because it claimed to estimate the risks of rare events,
which is impossible.a
Gave false conﬁdence.
Would be exploited by traders.