## Historical Simulation VaR

The Historical Simulation VaR approach is the third approach to value at risk and quite a popular approach in banking institutions along with Monte Carlo. The main point here to remember is that the historical simulation does not make any assumptions about the distributions of returns. In effect the historical simulation uses an empirical return distribution to calculate VaR. The simple assumption here is that the past returns give some indication of what the future will be like and therefore an idea of possible loss distributions. This can of course be a dangerous assumption for various reasons which we will discuss later.

### Historical Simulation VaR Implementation

In order to calculate historical VaR we would create a theoretical P&L of the portfolio that we hold. In order to do this we would calculate the P&L for each asset in the portfolio over a specified time horizon (1 day). Let us break down the steps :

– Calculate the price returns of all the assets in the portfolio. This will require deciding on the time interval – usually taken as day , but can be up to 30 days for investments in hedge funds etc. where more frequent data is not available.

– Apply the price changes that have been calculated to the current portfolio. So lets say we have a years worth of returns (252 days), we would apply these returns to every asset in the current portfolio with the assumption that these returns may occur between now and tomorrow. We can then also calculate the returns for the portfolio itself. So we would end up with 252 possible values for the portfolio tomorrow.

Where

rit is returns of asset I over time t

N is the notional amount invested in each asset i.

– Sort the portfolio simulated P&L from the lowest to the highest value. So we will very simply sort our 252 portfolio values. We can also plot this on a histogram to get a visual representation.

– Read off the required percentile value from the simulated vector of P&L’s.

### Weighted Historical Simulation VaR

We have so far weighted all data regardless of how old it is with the same relevance. However this could be a suspect decision particularly when dealing with seasonal variations – e.g. natural gas which will have different price behavior in different seasons. We can also have a scenario where a recent price shock will significantly distort the VaR and may not be indicative of a broader market regime. This can significantly overestimate VaR. Conversely during a unduly low period of volatility which is unusual this would cause VaR to be underestimated.

These problems can be countered by weighting observations appropriately. For seasonally affected prices for example we would give the winter prices of natural gas more weight than the summer price. The more common implantation is to weight more recent data points more heavily. This is done by using a exponentially weighted moving average (EWMA) method.

A nice way to look at this is by saying if current volatility was 10% , however volatility a month ago was 5%, then any returns from a month ago would be somewhat unrepresentative of the current market conditions. We could quite simply adjust this as follows:

where

r*it is the adjusted rate of return

σi,C is the Current volatility

σi,t is the volatility over the return period.

So actual returns may rise or fall depending on the ratio of current volatility to previous volatility.

### Pros and Cons of the Historical Simulation VaR Method

Pros

– Easy to understand at an intuitive level and easy to communicate

– East to implement , with various systems available commercially or even possible to implement on a spreadsheet (data sources like Bloomberg , Reuters etc)

– Shock events or stress scenarios can be incorporated quite easily to see the effect of a major shock on the portfolio.

– As there are no distribution assumptions the model can deal with fat tails, skewness, etc.

– Historical Simulation VaR allows for weighting of observations.

Cons

– It can be quite hard to get the length of time series of observations right. If the time series if too long it becomes potentially irrelevant (aged data) and may have events not relevant to present market conditions.

– If the time series is too short then the data might not capture rich enough information to give a true indication of market conditions. This can lead to some level of subjectivity on the decision of the length of the historical VaR period.

– Regulators generally impose 1 business years worth of data as a minimum.

– A long time period can also cause problems of data collection – particularly when there are new issues or illiquid isssues.

– Historical VaR can take time to work through regime shifts – for example if there is a significant change in the way a FX pair behaves for some underlying structural reason, this can take a while to work through the system.

– There can be distortions caused by ghost effects which are the result of updates to the historical sample.

– Historical VaR can never really account for any event that might occur but has not happened in the sample period.

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