Value @ risk and its variants

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.