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Derivatives

# COMPUTING VAR USING EXPONENTIALLY WEIGHTED HISTORICAL SIMULATION

## Exponentially weighted historical simulation (EWHS) puts a portfolio through a series of historical scenarios with heavier weightings given to more recent events.

Exponentially weighted historical simulation (EWHS) puts a portfolio through a series of historical scenarios with heavier weightings given to more recent events. It is based on the assumption that what happened recently is more likely to occur again than what happened long ago.

This approach can overcome some of the weaknesses found in some of the more-commonly-known VaR methods. Unlike the Variance-Covariance (VC) method, EWHS can handle non-linear risks and is thus suitable for portfolios containing options, does not make any assumptions about the returns distribution--VC assumes normality--and does not need to forecast any parameters. In a 100-asset portfolio, the VC method would require 100 volatility and 4950 [= n(n-1)/2] correlation forecasts!

Compared to Monte Carlo simulation, EWHS is less computationally intensive, intuitive, easy to implement, and easy to present to management.

IMPLEMENTING EWHS

With some knowledge of basic statistics, anyone can implement EWHS. Let's go through the steps for computing a 95% one day VaR.

1. Decay Factor

First, we select a decay factor, *, which will determine the weight distribution of the return series. The weight given to a scenario as at T days ago is

The smaller the *, the heavier the weighting given to recent data as observed in the graph below. The recommended value for * is between 0.94 and 0.99, depending on the financial instruments concerned. I find 0.96 works well in most cases.

From the same figure, we can also see that the line corresponding to *=0.96 dies off after about 125 days which means that we need data only up to a certain point in the past. To determine the exact number of data points required, we first set the tolerance level, *L. If *L =1% and *=0.96, we need data up to T=K where

2. Revaluation Rates

Daily revaluation rates are collected and stored as ratios to their previous-day values. For example, if the SGD 3M rates are 3.125% and 3.25% as at 10 and 11 days ago respectively, the ratio will be

After collecting the required number of data sets, we go on to generate the necessary historical scenarios by multiplying the latest revaluation rates with each ratio. If the latest revaluation rate for the SGD 3M rate is 2.5%, then the scenario corresponding to T=10 is 2.0438% (=2.5*0.9165).

3. Portfolio Revaluation

Finally, the portfolio is revalued against each set of historical scenarios. Following that, the daily profit and loss, the difference between two consecutive revaluations, is computed and ranked from the worst loss to the highest profit. Remember that each scenario has a weight attached. These weights are accumulated as in the example in the table below. The 95% VaR is the loss with cumulative weight closest to 5%. In this example, VaR=USD0.8m.

"ARTIFICAL INTELLIGENCE"

The beauty of EWHS, besides simplicity, lies in its ability to regulate the VaR number according to the relationship between VaR and realized P&L. This "artificial intelligence" feature is desirable to banks that post capital based on their internal VaR model. When a portfolio's realized loss exceeds its VaR, subsequent VaR will increase and thus reduce the probability of breaching the VaR again and ensure satisfactory backtesting results. Likewise, when realized losses are constantly lower than the VaR numbers, the VaR that follows will be lowered. Banks are happier when they can post lower capital.

To illustrate, let us look at a portfolio consisting of 10 million shares of Microsoft Corp. We compute the 95% 1-day VaR using both EWHS with a decay factor of 0.96 and Unweighted Historical Simulation (UWHS) with 250 data points. For a one-year period we expect, out of the 250 working days, 12.5 (=0.05*250) breaches. A breach occurs when a loss is greater than VaR. EWHS has 14 breaches which is satisfactory while UWHS has 17, 4.5 more than expected.

From the figure below, we can see that after the breach on February 9, EWHS VaR increased substantially and avoided the breaches UWHS had. On April 3, although the loss was huge, EWHS VaR did not increase immediately. This is because its objective is not to avoid breaches completely, but to create the expected number of breaches--12.5 in this example.

Notice that during the stable period in 1999, EWHS VaR is slightly lower than that of UWHS, thus incurring a lower capital charge.

Like every other VaR method, the performance of EWHS hinges heavily on the accuracy of the data collected, which can be a major problem when it comes to illiquid products and currencies. This weakness, however, is more than offset by the many benefits that EWHS brings.

This week's Learning Curve was written by Eric Sim, senior risk analyst at Standard Chartered Bankin Singapore.

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