VaR with monte carlo
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bob
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Congratulations, you have just identified problem #1 with MC VaR. Problem #2 is that you want to represent some sort of correlation structure among the assets.
The most straightforward method is to choose some historical data for your n assets, generate the covariance matrix on the excess returns (perhaps by using exponentially decaying weights as you go further back into the past), then use Cholesky decomposition on the resulting matrix. You can then simulate returns by multiplying the decomposed matrix by a random vector of independent standard normals.
This solves problem #1 by assuming that excess asset returns are normal and problem #2 by essentially regressing each asset, in sequence, against the previous ones and then capturing the residual variance in a new factor.
In practice it is common to impose correlation structure via a factor model, where each asset is envisioned as a linear combination of m << n market factors, plus an idiosyncratic risk component. Such an approach may lend some understanding of the interrelation of the assets, but to do it, you need a set of factors; determining these is at least as much art as science. If you're interested in learning about these kinds of approaches, check out the Fama/French factors; French has historical data available on his site.
In the end, many if not most MC approaches tend to assume that excess returns are normally distributed. At the microscale, this is mostly because of the normal distribution's many convenient properties. However, there is ample empirical evidence that, particularly over short time horizons, excess asset returns are anything but normal. This is a shortcoming that everyone claims to be aware of, yet in practice I do not believe there is any single generally accepted method of doing anything about it.
At the macroscale, in my view it is mainly economic considerations that not only permit but encourage the use of a measure that is apt to understate risk. With this in mind, it becomes at least doubtful that a method with a better fit to empirical data (i.e., bigger risk, implying higher costs to the holders of these assets) would gain wide acceptance while normal distributions are still considered an acceptable choice.
The most straightforward method is to choose some historical data for your n assets, generate the covariance matrix on the excess returns (perhaps by using exponentially decaying weights as you go further back into the past), then use Cholesky decomposition on the resulting matrix. You can then simulate returns by multiplying the decomposed matrix by a random vector of independent standard normals.
This solves problem #1 by assuming that excess asset returns are normal and problem #2 by essentially regressing each asset, in sequence, against the previous ones and then capturing the residual variance in a new factor.
In practice it is common to impose correlation structure via a factor model, where each asset is envisioned as a linear combination of m << n market factors, plus an idiosyncratic risk component. Such an approach may lend some understanding of the interrelation of the assets, but to do it, you need a set of factors; determining these is at least as much art as science. If you're interested in learning about these kinds of approaches, check out the Fama/French factors; French has historical data available on his site.
In the end, many if not most MC approaches tend to assume that excess returns are normally distributed. At the microscale, this is mostly because of the normal distribution's many convenient properties. However, there is ample empirical evidence that, particularly over short time horizons, excess asset returns are anything but normal. This is a shortcoming that everyone claims to be aware of, yet in practice I do not believe there is any single generally accepted method of doing anything about it.
At the macroscale, in my view it is mainly economic considerations that not only permit but encourage the use of a measure that is apt to understate risk. With this in mind, it becomes at least doubtful that a method with a better fit to empirical data (i.e., bigger risk, implying higher costs to the holders of these assets) would gain wide acceptance while normal distributions are still considered an acceptable choice.
Thanks for your reply. It seems that it's more complicated than I expect.
When you say “over short time horizons”, what do you mean by “short”, one month can be regarded as short ?
When I ask the question, I suppose there is no correlation between these 3 stocks, say they are IBM, HSBC and Boeing. I mean, if we regard the distribution as normal, and assume no correlation at all, what we need is the vol of each stock (find from the historical data). I mean, the difference between these 3 stocks are the vol only, otherwise they are identical. Am I right ?
For the case with correlation, I didn't consider. However, I'm now interested in knowing more. I mean, I want to ask a question. Is it necessary for us to consider the correlation ? I mean, when I use MC, I need consider correlation ? Can it be so simple as “IBM, HSBC and Boeing” ? It seems that I haven't figured out the problem #2.
Another interesting question is, if it is not a stock, it's a plain option, what distribution should be used ? Assume the stock is normal, and use BS model to calculate the price instead of using any distribution ?
When you say “over short time horizons”, what do you mean by “short”, one month can be regarded as short ?
When I ask the question, I suppose there is no correlation between these 3 stocks, say they are IBM, HSBC and Boeing. I mean, if we regard the distribution as normal, and assume no correlation at all, what we need is the vol of each stock (find from the historical data). I mean, the difference between these 3 stocks are the vol only, otherwise they are identical. Am I right ?
For the case with correlation, I didn't consider. However, I'm now interested in knowing more. I mean, I want to ask a question. Is it necessary for us to consider the correlation ? I mean, when I use MC, I need consider correlation ? Can it be so simple as “IBM, HSBC and Boeing” ? It seems that I haven't figured out the problem #2.
Another interesting question is, if it is not a stock, it's a plain option, what distribution should be used ? Assume the stock is normal, and use BS model to calculate the price instead of using any distribution ?
bob
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Monthly equity returns are much closer to normal than, for example, daily returns are. Monthly returns may not, however, be a particularly useful data set for an actively managed portfolio if holding times are generally less than a month.
Doing risk without considering the joint behavior of assets can be disastrous. For example, all three equities you just mentioned have pairwise positively correlated daily returns, including a correlation of almost 0.35 for IBM and BA. Assuming independent assets will generally understate the variance of any long-only portfolio.
Adding options to the mix makes things even more interesting. You need a way of modeling the distribution of your option's implied vol and associating that somehow with the equity returns. They are definitely associated, since equity options basically all exhibit a negative vol skew. How you do that is a question that involves many technical issues, not all of which have clear best solutions.
Since it sounds like you don't have a lot of background in this area, I'd encourage you to pick up a book on the subject for further insight.
Doing risk without considering the joint behavior of assets can be disastrous. For example, all three equities you just mentioned have pairwise positively correlated daily returns, including a correlation of almost 0.35 for IBM and BA. Assuming independent assets will generally understate the variance of any long-only portfolio.
Adding options to the mix makes things even more interesting. You need a way of modeling the distribution of your option's implied vol and associating that somehow with the equity returns. They are definitely associated, since equity options basically all exhibit a negative vol skew. How you do that is a question that involves many technical issues, not all of which have clear best solutions.
Since it sounds like you don't have a lot of background in this area, I'd encourage you to pick up a book on the subject for further insight.
Sanket Patel
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Have one more question in mind, if we really construct the correlation structure. As far as I understand, to put it simple, it relies on historical data to build some statistics. Then, to put it further simple, the VaR is just a statistical value derived from historical data, is it ?
If so, why don't I use historical simulation to calculate the VaR directly ?
As you know, the drawback of MC is the computational consumption.
For every method, there must be pros & cons. How is the practice in industry ?
Sanket Patel
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I'm no expert on VaR but the times I have calculated VaR (for equity portfolios) I have used historical simulation. The nice thing about historical simulation is that it is a non-parametric approach; thus, unlike the MC method, you're not forced to impose any particular distribution on your data.
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hist sim - hidden assumption about distribution
Hist sim is often said to be non-parametric. This is true because the distribution is not stated explicitly by the modeler with parameters; you just take whatever the distribution was in the past. But this history still has parameters - they are just not set by the modeler. Your forecast relies on the future distribution being like the past data you are sampling. This is pretty well known now to be false - doing this in December 2007 gave no clue about what 2008 held. It's well known that you can get very different hist sim VaR numbers depending on whether you take 1, 3, 5, or 10 year historical data samples to run your portfolio over.
One half-and-half way to generate VaR that is a slightly more sophisticated than hist sim: Take a 5 or 10 year data set across all of the time series you will need. Figure the % change to each asset on each day. Number each day. Then draw random integers from 1 to 1260, or 1 to 2520, and just pull a slice of the historical data, one of the days' changes to all the assets out of the list and change all of the assets' value that much. How much did each of the assets change on the 5th day in the list? The 972nd day? Etc. Draw as long a horizon as you want, 5 draws for 5 days, 20 draws for 20 days, etc. That's 1 path. Do this for 10,000 paths, then find the 5% marker and call it VaR. It's sort of like 10,000 fake hist sims. When we used this at the bank I worked at, we used to call this "hist sim with shuffle-up", or something like that. It's like shuffling cards. It's relatively fast to more complicated VaR methods (full blown Monte Carlo), you just have to draw the random numbers, not alot of calculations (just change the value of the assets), you get away from having to specify a daily correlation structure (just assume the historical), but you also get away from only having one sample path, which is a drawback of hist sim VaR. And it worked pretty well for VaR (until it didn't, and the bank almost died, of course).
Hist sim is often said to be non-parametric. This is true because the distribution is not stated explicitly by the modeler with parameters; you just take whatever the distribution was in the past. But this history still has parameters - they are just not set by the modeler. Your forecast relies on the future distribution being like the past data you are sampling. This is pretty well known now to be false - doing this in December 2007 gave no clue about what 2008 held. It's well known that you can get very different hist sim VaR numbers depending on whether you take 1, 3, 5, or 10 year historical data samples to run your portfolio over.
One half-and-half way to generate VaR that is a slightly more sophisticated than hist sim: Take a 5 or 10 year data set across all of the time series you will need. Figure the % change to each asset on each day. Number each day. Then draw random integers from 1 to 1260, or 1 to 2520, and just pull a slice of the historical data, one of the days' changes to all the assets out of the list and change all of the assets' value that much. How much did each of the assets change on the 5th day in the list? The 972nd day? Etc. Draw as long a horizon as you want, 5 draws for 5 days, 20 draws for 20 days, etc. That's 1 path. Do this for 10,000 paths, then find the 5% marker and call it VaR. It's sort of like 10,000 fake hist sims. When we used this at the bank I worked at, we used to call this "hist sim with shuffle-up", or something like that. It's like shuffling cards. It's relatively fast to more complicated VaR methods (full blown Monte Carlo), you just have to draw the random numbers, not alot of calculations (just change the value of the assets), you get away from having to specify a daily correlation structure (just assume the historical), but you also get away from only having one sample path, which is a drawback of hist sim VaR. And it worked pretty well for VaR (until it didn't, and the bank almost died, of course).
bob
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Good question. There are a number of issues relating to historical data, some of which Pat has touched on. One major one is the scarcity of data--or relevant data, anyway.
Typically, for historical VaR you generate historical transitions in the needed market data, treat these as samples from their "real" distributions, shock today's market data and reprice each time to get P&L's. Once you have P&L's, you look for the mth worst realization, with m chosen according to the number of samples (mainly a function of your lookback period) and your chosen "significance level," which is sort of a terrible misnomer but is nevertheless the standard term.
What you're looking at is the mth order statistic of a sample from some distribution, and like any statistic it is itself random. The farther out in the tail you go, the greater its variance. Not only that, but the particular shape of the distribution--which is generally not known--has a lot to do with how (potentially) bad this statistic is as a measure of the "actual" P&L in question, and it gets worse when distributions have fatter lower tails. 95% VaR estimated on a normal distribution with a typical year's worth of info--250 data points or so--is pretty good as these things go: You have decent confidence that your estimate lies within half a standard deviation of the source distribution (!) of the "actual" value.
Moreover, such point estimators are biased; again, the amount of the bias depends upon the source distribution. (Funnily, in the literature this estimate is most often described as "asymptotically unbiased," meaning that as your number of samples approaches infinity, the bias goes away. Omniscience is nice, no?)
By comparison, an estimate of a distribution's mean and variance based on a sample of size 250 seems quite solid. Any error in this estimate or in your choice of distribution will of course be reflected in the MC VaR you end up calculating as a result, but assuming your choice of distribution is correct or reasonably so (a big if) you can actually do better than the raw point estimate via MC.
