# How do you calculate value at risk? Two ways of calculating VaR

##### In todays video we learn how to calculate VaR or Value at Risk.

Hi my name is patrick boyle welcome back to my youtube channel where we learn all about derivatives and quantitative finance in today’s video we’re going to learn two methods for calculating var or value at risk this video is part of a short series i’m doing on risk management in finance if you click on my channel button and then on playlists you’ll see the playlists

I’ve put together collecting all of the videos on this topic if this is the first video you’re watching make sure you click the subscribe button so that you can see more content like this going forward by the end of this video you’re going to know how to calculate var by two different methods and the pros and cons of the two methods that we’re covering okay so there

Are two basic ways to calculate value at risk the historical approach and the model approach using the historical approach we look at the historical time series of a portfolios return and then we take the bottom x percentile microsoft excel does this with the formula equals percent and then you put in the array of returns that you’re looking at comma 5% to calculate

95 var using the model approach we look at the expected standard deviation of the portfolio and use the normal distribution tables to find the x percent are so anyhow let’s go into each of those in a little bit of more detail so the historical approach to quickly understand var what we’ll do is we’ll use an example that i work through in my book trading and pricing

Financial derivatives which is linked to in the description below and we’ll calculate the var on the returns of an equity mutual fund vanguards vf aix which is based on the financials index the last 15-years daily percentage returns can be seen on the screen right now daily returns are taken and put in order and converted into a histogram of daily returns you can

See that histogram on screen right now in statistics the 68 95 99 rule is shorthand to remember the percentage of values around the mean in a normal distribution with a width of one two and three standard deviations more accurately sixty-eight point two seven percent ninety-five point four five percent and ninety nine point seven three percent of the values lie

Within one two and three standard deviations of the mean respectively var assumes normally distributed returns implying that 68% of the time we expect daily returns to fall between minus one point eight percent and plus one point eight percent for this index fund now that’s actually surprisingly symmetrical it’s not always a plus or minus at the same number in the

Worst five percent of days this portfolio will lose six point six percent or more as shown in the left tail of the histogram that you can see on screen right now as you can also see the worst one day return for this fund was a nine percent loss and the biggest update for the fund was a twelve point six percent gain so that’s data that you get from looking at past

Returns and of course then the question is how representative is the past of the expected future so that’s really all there is to the historic approach to calculating far as you can see all we did was download a representative sample of the daily returns and look for the five percent worth days and that’s really it it’s not very complicated now there are though

A few things we should think about when looking at this data when we look at the daily returns data which you see on screen right now you can see that the worst days in this example were not evenly spread throughout the fifteen years of data occurring once every twenty days for example you can instead see a lot of clustering and i mentioned this idea in the last

Video often in the real world we see the five or ten worst days of the year occurring right next to each other we’ll talk more about this in my next video on time varying volatility i know that a video would a title like that would be a huge hit here on youtube so i just had to make it it’s also worth but the 15 years of data that we use they’re included the credit

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Crunch 2007 and 2008 huffing that data in there will give you a good feel for what markets look like when they get quite wild if instead you use this historic approach with only the 5 last years of data you might be making the mistake of thinking that markets are a lot calmer than they actually are var like all of our financial models suffers from the garbage in

Garbage out problem if you’re a risk management professional you really have to pay attention to how the models you are using work when they’re helpful and when they’re useless anyhow that was the historic approach let’s next look at the model-based approach with the model-based approach instead of using historic data we’re going to use the standard deviation of

The portfolio in question if a portfolio standard deviation or sigma is expected to be one and a quarter percent and we want to know it’s one day 99 var we use the normal distribution tables which you can see on screen right now the closest number we can find to one percent is 0.99 percent and this is two point three three standard deviations to the left of the

Mean this means that the worst one percent of days we expect to have a two point three three standard deviation loss we assume that the expected daily return for the portfolio is around zero percent which is reasonable on daily returns on an equity index hence two point three three times the standard deviation of one and a quarter percent gives us minus two

Point nine one percent as the 99 var the worst one percent of day should thus involve losses of two point nine one percent or greater according to this calculation if we multiply this return by the size of the portfolio we will get 99 var in dollar terms for a 1 million dollar portfolio one would expect to lose twenty nine thousand one hundred and twenty-five

Dollars or more on the war 1% of days in normal markets so that’s how the calculation is done now the more astute financial students out there may be asking where do we get the standard deviation from we could of course just calculate the standard deviation from historic data but that would be a historic standard deviation and what we’re looking for here is some

Sort of forward-looking estimate of standard deviation the likely standard deviation over the coming days and weeks so if we’re looking for a forward-looking or predicted standard deviation where do we get that well we could take it from implied volatility of the stocks in our portfolio now if you don’t know what implied volatility is above there’s a link to my

Video explaining that idea but it is a forward-looking estimate of volatility for a given underline as you can see these two models are quite different from each other and will give you quite different numbers so which one should you use that’s really up to you but it might be worthwhile looking at both as you can see with the model-based approach when the market

Gets more volatile implied volatility will rise and without you making any changes to your portfolio your var will change there is a bit of a built-in assumption with this approach that the current level of volatility will be maintained in the market in the next video we’ll look at the idea of time varying volatility and how that might affect our approach to risk

Management don’t forget to subscribe and hit the bell button so that you can be notified when each new video is released i upload one new video each week hit the like button if you found this video helpful all of these videos are based on my book if you are interested there’s a link to that in the description below anyhow see you later bye

Transcribed from video
How do you calculate value at risk? Two ways of calculating VaR By Patrick Boyle

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