An Overview of Value-at-Risk:
Part II - Historical Simulations VaR              

By Romain Berry
J.P. Morgan Investment Analytics and Consulting
romain.p.berry@jpmorgan.com

This article is the third in a series of articles exploring risk management for institutional investors.

In the previous issue, we looked at Analytical Value-at-Risk, whose cornerstone is the Variance-Covariance matrix. In this article, we continue to explore VaR as an indicator to measure the market risk of a portfolio of financial instruments, but we touch on a very different methodology.

We indicated in the previous article that the main benefits of Analytical VaR were that it requires very few parameters, is easy to implement and is quick to run computations (with an appropriate mapping of the risk factors). Its main drawbacks lie in the significant (and inconsistent across asset classes and markets) assumption that price changes in the financial markets follow a normal distribution, and that this methodology may be computer-intensive since we need to calculate the n(n-1)/2 terms of the Variance-Covariance matrix (in the case where we do not proceed to a risk mapping of the various instruments that composed the portfolio). With the increasing power of our computers, the second limitation will barely force you to move away from spreadsheets to programming. But the first assumption in the case of a portfolio containing a non-negligible portion of derivatives (minimum 10%-15% depending on the complexity and exposure or leverage) may result in the Analytical VaR being seriously underestimated because these derivatives have non-linear payoffs.

One solution to circumvent that theoretical constraint is merely to work only with the empirical distribution of the returns to arrive at Historical Simulations VaR. Indeed, is it not more logical to work with the empirical distribution that captures the actual behavior of the portfolio and encompasses all the correlations between the assets composing the portfolio? The answer to this question is not so clear-cut. Computing VaR using Historical Simulations seems more intuitive initially but has its own pitfalls as we will see. But first, how do we compute VaR using Historical Simulations?

Historical Simulations VaR Methodology

The fundamental assumption of the Historical Simulations methodology is that you look back at the past performance of your portfolio and make the assumption – there is no escape from making assumptions with VaR modeling – that the past is a good indicator of the near-future or, in other words, that the recent past will reproduce itself in the near-future. As you might guess, this assumption will reach its limits for instruments trading in very volatile markets or during troubled times as we have experienced this year.

The below algorithm illustrates the straightforwardness of this methodology. It is called Full Valuation because we will re-price the asset or the portfolio after every run. This differs from a Local Valuation method in which we only use the information about the initial price and the exposure at the origin to deduce VaR.

Step 1 – Calculate the returns (or price changes) of all the assets in the portfolio between each time interval.

The first step lies in setting the time interval and then calculating the returns of each asset between two successive periods of time. Generally, we use a daily horizon to calculate the returns, but we could use monthly returns if we were to compute the VaR of a portfolio invested in alternative investments (Hedge Funds, Private Equity, Venture Capital and Real Estate) where the reporting period is either monthly or quarterly. Historical Simulations VaR requires a long history of returns in order to get a meaningful VaR. Indeed, computing a VaR on a portfolio of Hedge Funds with only a year of return history will not provide a good VaR estimate.

Step 2 – Apply the price changes calculated to the current mark-to-market value of the assets and re-value your portfolio.

Once we have calculated the returns of all the assets from today back to the first day of the period of time that is being considered – let us assume one year comprised of 265 days – we now consider that these returns may occur tomorrow with the same likelihood. For instance, we start by looking at the returns of every asset yesterday and apply these returns to the value of these assets today. That gives us new values for all these assets and consequently a new value of the portfolio. Then, we go back in time by one more time interval to two days ago. We take the returns that have been calculated for every asset on that day and assume that those returns may occur tomorrow with the same likelihood as the returns that occurred yesterday. We re-value every asset with these new price changes and then the portfolio itself. And we continue until we have reached the beginning of the period. In this example, we will have had 264 simulations.

Step 3 – Sort the series of the portfolio-simulated P&L from the lowest to the highest value.

After applying these price changes to the assets 264 times, we end up with 264 simulated values for the portfolio and thus P&Ls. Since VaR calculates the worst expected loss over a given horizon at a given confidence level under normal market conditions, we need to sort these 264 values from the lowest to the highest as VaR focuses on the tail of the distribution.

Step 4 – Read the simulated value that corresponds to the desired confidence level.

The last step is to determine the confidence level we are interested in – let us choose 99% for this example. One can read the corresponding value in the series of the sorted simulated P&Ls of the portfolio at the desired confidence level and then take it away from the mean of the series of simulated P&Ls. In other words, the VaR at 99% confidence level is the mean of the simulated P&Ls minus the 1% lowest value in the series of the simulated values. This can be formulated as follows:

(1)

where:

• VaR_1-α is the estimated VaR at the confidence level 100 × (1 - α)%
• μ(R) is the mean of the series of simulated returns or P&Ls of the portfolio
• R_α is the αth worst return of the series of simulated P&Ls of the portfolio or, in other words, the return of the series of simulated P&Ls that corresponds to the level of significance α

We may need to proceed to some interpolation since there will be no chance to get a value at 99% in our example. Indeed, if we use 265 days, each return calculated at every time interval will have a weight of 1/264 = 0.00379. If we want to look at the value that has a cumulative weight of 99%, we will see that there is no value that matches exactly 1% (since we have divided the series into 264 time intervals and not a multiple of 100). Considering that there is very little chance that the tail of the empirical distribution is linear, proceeding to a linear interpolation to get the 99% VaR between the two successive time intervals that surround the 99th percentile will result in an estimation of the actual VaR. This would be a pity considering we did all that we could to use the empirical distribution of returns, wouldn’t it? Nevertheless, even a linear interpolation may give you a good estimate of your VaR. For those who are more eager to obtain the exact VaR, the Extreme Value Theory (EVT) could be the right tool for you. We will explain in another article how to use EVT when computing VaR. It is rather mathematically demanding and would require us to spend more time to explain this method.

Applications of Historical Simulations VaR

Let us compute VaR using historical simulations for one asset and then for a portfolio of assets to illustrate the algorithm.

Example 1 – Historical Simulations VaR for one asset

The first step is to calculate the return of the asset price between each time interval. This is done in column D in Table 1. Then we create a column of simulated prices based on the current market value of the asset (1,000,000 as shown in cell C3) and each return which this asset has experienced over the period under consideration. Thus, we have 100 x (-1.93%) =
-19,313.95. In Step 3, we simply sort all the simulated values of the asset (based on the past returns). Finally, in Step 4, we read the simulated value in column G which corresponds to the 1% worst loss. As there is no value that corresponds to 99%, we interpolate the surrounding values around 99.24% and 98.86%. That gives us -54,711.55.

This number does not take into account the mean, which is 1,033.21. As the 99% VaR is the distance from the mean of the first percentile (1% worst loss), we need to subtract the number we just calculated from the mean to obtain the actual 99% VaR. In this example, the VaR of this asset is thus 1,033.21 – (-54,711.55) = 55,744.76. In order to express VaR in percentage, we can divide the 99% VaR amount by the current value of the asset (1,000,000), which yields 5.57%.

Table 1 - Calculating Historical Simulations VaR for one asset
Asset Price	Histogram of Returns

Example 2 - Historical Simulations VaR for one portfolio

Computing VaR on one asset is relatively easy, but how do the historical simulations account for any correlations between assets if the portfolio holds more than one asset? The answer is also simple: correlations are already embedded in the price changes of the assets. Therefore, there is no need to calculate a Variance-Covariance matrix when running historical simulations. Let us look at another example with a portfolio composed of two assets.

Table 2 - Calculating Historical Simulations VaR for a portfolio of two assets
Portfolio Unit Price	Histogram of Returns

As you can see, we simply add a couple of columns to replicate the intermediary steps for the second asset. In this example, each asset represents 50% of the portfolio. After each run, we re-value the portfolio by simply adding up the simulated P&L of each asset. This gives us the simulated P&Ls for the portfolio (column J).

This straightforward step of simply re-composing the portfolio after every run is one of the reasons behind the popularity of this methodology. Indeed, we do not need to handle sizeable Variance-Covariance matrices. We apply the calculated returns of every asset to their current price and re-value the portfolio.

As we have noted, correlations are embedded in the price changes. In this example, the 99% VaR of the first asset is 55,744.76 (or 5.57%) and the 99% VaR of the second asset is 54,209.71 (or 5.42%). We know that VaR is a sub-additive risk measure – if we add the VaR of two assets, we will not get the VaR of the portfolio. In this case, the 99% VaR of the portfolio only represents 3.67% of the current marked-to-market value of the portfolio. That difference represents the diversification effect. Having a portfolio invested in these two assets makes the risk lower than investing in any of these two assets solely. The reason is that the gains on one asset sometimes offset the losses on the other asset (rows 10, 12, 13, 17-20, 23, 26-28, 30, 32 in Table 2). Over the 265 days, this happened 127 times with different magnitude. But in the end, this benefited the overall risk profile of the portfolio as the 99% VaR of the portfolio is only 3.67%.

Advantages of Historical Simulations VaR

Computing VaR using the Historical Simulations methodology has several advantages. First, there is no need to formulate any assumption about the return distribution of the assets in the portfolio. Second, there is also no need to estimate the volatilities and correlations between the various assets. Indeed, as we showed with these two simple examples, they are implicitly captured by the actual daily realizations of the assets. Third, the fat tails of the distribution and other extreme events are captured as long as they are contained in the dataset. Fourth, the aggregation across markets is straightforward.

Disadvantages of Historical Simulations VaR

The Historical Simulations VaR methodology may be very intuitive and easy to understand, but it still has a few drawbacks. First, it relies completely on a particular historical dataset and its idiosyncrasies. For instance, if we run a Historical Simulations VaR in a bull market, VaR may be underestimated. Similarly, if we run a Historical Simulations VaR just after a crash, the falling returns which the portfolio has experienced recently may distort VaR. Second, it cannot accommodate changes in the market structure, such as the introduction of the Euro in January 1999. Third, this methodology may not always be computationally efficient when the portfolio contains complex securities or a very large number of instruments. Mapping the instruments to fundamental risk factors is the most efficient way to reduce the computational time to calculate VaR by preserving the behavior of the portfolio almost intact. Fourth, Historical Simulations VaR cannot handle sensitivity analyses easily.

Lastly, a minimum of history is required to use this methodology. Using a period of time that is too short (less than 3-6 months of daily returns) may lead to a biased and inaccurate estimation of VaR. As a rule of thumb, we should utilize at least four years of data in order to run 1,000 historical simulations. That said, round numbers like 1,000 may have absolutely no relevance whatsoever to your exact portfolio. Security prices, like commodities, move through economic cycles; for example, natural gas prices are usually more volatile in the winter than in the summer. Depending on the composition of the portfolio and on the objectives you are attempting to achieve when computing VaR, you may need to think like an economist in addition to a risk manager in order to take into account the various idiosyncrasies of each instrument and market. Also, bear in mind that VaR estimates need to rely on a stable set of assumptions in order to keep a consistent and comparable meaning when they are monitored over a certain period of time.

In order to increase the accuracy of Historical Simulations VaR, one can also decide to weight more heavily the recent observations compared to the furthest since the latter may not give much information about where the prices would go today. We will cover these more advanced VaR models in another article.

Conclusion

Despite these disadvantages, many financial institutions have chosen historical simulations as their favored methodology to compute VaR. To many, working with the actual empirical distribution is the “real deal.”

However, obtaining an accurate and reliable VaR estimate has little value without a proper back testing and stress testing program. VaR is simply a number whose value relies on a sound methodology, a set of realistic assumptions and a rigorous discipline when conducting the exercise. The real benefit of VaR lies in its essential property of capturing with one single number the risk profile of a complex or diversified portfolio. VaR remains a tool that should be validated through successive reconciliation with realized P&Ls (back testing) and used to gain insight into what would happen to the portfolio if one or more assets would move adversely to the investment strategy (stress testing).