VALUE AT RISKVAR

ContentWhat is VAR?

Idea behind volatility

VAR questions

Historical Method

Variance - Covariance Method

Monte Carlo Stimulation

Limitations

Criticisms

VAR

• 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 probability level.

VAR

• 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

market index.

• 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 movement.

• VAR answers the question, "What is my worst-case scenario?"

VAR Questions

• 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?

Historical Method• 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.

Historical Method

• TheWith 95% confidence, we expect that our worst daily loss will not exceed 4%.

• If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%)

Historical Method

Weaknesses• 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 Method

• 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%.

Variance - Covariance Method

Variance - Covariance Method

Weaknesses

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.

Monte Carlo Stimulation

• 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 -15%.

Monte Carlo Stimulation

Limitations

• 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.

Criticism

Ignored 2,500 years of experience in favor of untested models built by non-traders.

Was charlatanism because it claimed to estimate the risks of rare events, which is impossible.a

Gave false confidence.

Would be exploited by traders.

Q/A