What Is Value At Risk.The Value at Risk also called VaR is a method for quantifying exposure to market risk using statistical techniques
CALCULATION OF VAR
The simplest way is to obtain it through simulations with the historical data of the returns of an asset (or portfolio), for which we must determine what are the possible values that the asset / portfolio may have under different scenarios, therefore we are assuming that what happened in the past can happen again in the future.
Excel or any statistical package (for example crystall ball ), facilitate the calculation, the historical data can be obtained from information sources such as yahoo finance .
Suppose we have 251 price data for the only asset that makes up our portfolio, and that today that portfolio is worth: Vo = $ 100 Million
The first thing we have to do is calculate the daily logarithmic returns: Rt = ln (Pt + 1 / Pt), then we calculate the value of the portfolio for each day according to a probable scenario calculated as follows:
Vt = Vo * (1 + Rt)
Since Vo = 100:
If what we have is a portfolio that has X1, X2. X3 ,,, Xn amount of assets, what we have to do is multiply the shares% of each asset (q1, q2, q3, qn) by the yield of each asset (Rt), add them and then multiply that result by Vo
Vt = (q1 * (1 + Rt1)) + (q2 * (1 + Rt2)) + (q3 * (1 + Rt3))….) * Vo
As a next step, in the Excel Menu bar: Tools, Data analysis , we select histogram , which classifies the possible values of the portfolio according to ranges / classes that Excel determines (or you determine), versus the frequency / distribution with that these data are accumulated in that certain value of the portfolio like this:
If we look at this graph we can see that it gives us an idea of how the expected value of my portfolio may behave: that is, 76% of the data has a value between -∞ and $ 101.80, which is concentrated in a range between $ 99,301 and $ 100.56 with a total of 68 data out of the 250.
As we are interested in the minimum value that the portfolio may have ( Vc) , with 95% confidence (1-α), we must calculate that portfolio Vc with an α = 5% which is statistically called the percentile ; Vc will then be that portfolio in which 5% of the worst possible scenarios are below it.
Using the Excel formula PERCENTILE (matrix, K), where matrix corresponds to the Vt data series and k corresponds to α = 0.05 we have:
Vc = 95.938 and as VAR = Vo-Vc.
Then VAR = 100- 95.938 = 4.06
Therefore, there is a 5% probability (5 times out of 100) of obtaining a daily loss greater than $ 4.06M if the market is in normal conditions and we do not change the composition of the portfolio.
Another way to calculate VaR is by directly assuming a normal probability distribution for returns. Therefore, if with this assumption we calculate the standard deviation σ and the average yield Rp, we can directly know where the worst 5% values of the portfolio are.
Since we need to know how many standard deviations the asset has to move to obtain the maximum loss with 95% confidence, under the assumptions of a normal distribution, it will approximately be:
95% = -1.65 * σ or 99% = -2.33 * σ
Suppose the average daily returns of an asset and its standard deviation over 250 pieces of data are:
| Average | 0.337% |
| Desv are | 0.02480449 |
Therefore, with a 95% confidence level, we can expect that the maximum daily loss is = -1.65 * 0.0248 = -4.09%, that is, if the value of my portfolio today is worth $ 100M, there are 5 chances out of 100 that it can lose $ 4.09M in one day.
The average and Dev data correspond to a single asset, however, when there are several assets in a portfolio, we not only have to take into account the expected returns of each asset or their deviations, but also the relationships that exist between each one. of the assets among themselves, that is, their covariances, using Excel we can calculate them using the formula COVAR (matrix1; matrix2), where matrixX = range of the data of the daily returns Rt of asset x. If we have 3 assets we will obtain a matrix like the following:
According to the participation of each asset in% (q) in the portfolio, we can then now determine the performance of the portfolio and its variance:
Rp = q1 * (1+ R 1)) + (q2 * (1+ R 2)) + (q3 * (1+ R 3) where R = average return on asset i
The standard deviation of the portfolio will then be: √ (Portfolio variance)
and by approximation 95% = -1.65 * σ or with 99% = -2.33 * σ we can find the VAR of the portfolio.
Conclusions
It must be taken into account that in these VaR calculation methods, we are assuming two major assumptions: what happened in the past may happen in the future, we are implicitly giving equal weight of importance to all the data, assuming then also that the variance will remain constant over time , which is why it is not entirely true, since there are non-stationary variables over time, which modify the behavior and variance / volatility of an asset, as well as we ignore the incorporation of new risks in assets and the same market.
Furthermore, if we assume a normal probability distribution , not all assets necessarily behave in this way, since if there are many extreme values in one of the tails of the distribution, the calculated VaR would be underestimating these values.
