# Historical VaR

Historical VaR or Historical Simulation VaR is a method to estimate VaR (Value at Risk) that uses historical data.

One of the ways to calculate VaR by the historical method is by accumulating past returns and ordering them from highest to lowest. Next, we identify the 5% of data with the lowest returns, and the highest of that 5% lowest returns will be the VaR.

The data to calculate the historical VaR are the historical prices of the securities. Therefore, a larger time series (say 5 or 10 years) will result in higher simulated results and therefore will be more accurate than a 3-month time series.

The main disadvantage of the historical model to calculate the VaR is that it is assumed that the returns obtained in the past will be repeated in the future.

The VAR by historical simulation is one of the ways of calculating the VaR, always a little more laborious than the parametric VaR and less precise than the VaR by Monte Carlo simulation. It is about applying to the portfolio of financial assets, historical variations in the price of the securities to generate contrastable scenarios with the initial position (known as spot in English), generating different possible simulated results from which the VAR will be obtained.

## Example of historical VaR at 95% confidence

Although hundreds of data are usually used to calculate the VaR to simplify its understanding we will use only 40 data. Imagine an asset that had the following results over the past few years:

2015 | 2016 | 2017 | 2018 | |
---|---|---|---|---|

January | 2,00% | 3,06% | 0,00% | 8,15% |

February | 4,05% | -3,56% | -2,14% | -2,95% |

March | -2,85% | 7,81% | 4,69% | 1,69% |

April | 6,25% | 2,75% | 2,25% | -7,35% |

May | 3,00% | 1,13% | 1,88% | |

June | 2,50% | -8,75% | -5,25% | |

July | -7,00% | 4,81% | 1,09% | |

August | 1,45% | 15,81% | 9,49% | |

September | 12,65% | -10,19% | -6,11% | |

October | -8,15% | 3,88% | 2,33% | |

November | 3,10% | 3,13% | 1,88% | |

December | 2,50% | 5,25% | 1,88% |

If we want to calculate the VaR at 95% confidence, we must choose the 5% of worst results, which in this case are 2 (5% of 40 data). We then choose the second worst result of the entire period, which is -8.75%. If we assume that the investment in this asset is 1 million euros, the 5% VaR will be 87,500 euros, that is, there is a 5% probability of losing at least 87,500 euros and a 95% probability that this loss is less. Therefore, the company will have to take into account that five out of every 100 months will lose at least 87,500 euros, or that one out of every 20 months will lose at least 87,500 euros.

The more historical data we have, the more accurate the VaR measurement will be.

## Steps to calculate the VAR by historical simulation of a portfolio

The steps to follow are those:

1. Selection of the series of historical prices of our portfolio and calculation of the weight of each one of them in the portfolio.

2. Calculation of the continuous field variation rates:

3. The variation rates obtained are applied to the market price of each of the securities (we are using continuous capitalization, but compound capitalization can also be used).

4. The possible values of the sub-portfolio are calculated based on the position of each security within the portfolio and the simulated price.

5. Calculation of equity in each of the simulated scenarios. To do this, we will add the results obtained from each title.

6. Calculation of the variation rate of the simulated portfolio with respect to the initial portfolio (market value of the initial or spot portfolio).

7. Calculation of VaR. For this we have to choose the level of trust.

Monte Carlo simulation
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