# cointegration applied to Portfolio Construction & Risk management

There are all sorts of applications of cointegration to generating alpha on mean-reverting timeseries: comparing spot vs. futures, bond spreads, identifying mean-reverting residuals, etc.

But there is not much literature on applying co-integration to portfolio risk. Overwhelmingly, the variance-covariance matrix is used to measure and minimize portfolio risk. Cointegration imposes stricter requirements on the relationship between two time-series than mere correlation. There are also fewer false-but-useful assumptions in the variance-covariance method such as the i.i.d, homoskedasticity, and normal nature of returns.

So are there any citations in the literature that describe portfolio construction procedures for risk minimization using co-integration (or a combination of cointegration and correlation)?

I found this paper - Optimal Hedging Using Cointegration (1999) but it's more of an empirical case-study rather than a framework for thinking about risk thru the lens of cointegration.

• There is a generic page about cointegration, some answers to your question can be found on it: quant.stackexchange.com/tags/cointegration/info – lehalle May 8 '12 at 7:56
• @Thanks -- unfortunately none of them cover portfolio construction via cointegration (as opposed to alpha generation which is well covered) – Ram Ahluwalia May 8 '12 at 13:55

The following paper (Identifying Small Mean Reverting) is not directly related to portfolio risk minimization but it provides a method to build tradable mean reverting portfolios based on a multivariate co-integration approach. It has the advantages of providing a theoretical framework along with two algorithms. It also takes into account financial strict financial problems such as transactions costs.

http://www.cmap.polytechnique.fr/~aspremon/PDF/MeanRevVec.pdf

As far as your portfolio is a linear combination of instruments (for non linear portfolios, this is more difficult to explain), the associated risk you would like to monitor is the variation through time of this combination. You stress test your portfolio when you measure its variations when the market conditions are unexpected (in the sense that you can decide for instance to move a market factor) and measure its Value at Risk when you estimate its value-level assciated to a given quantile given usual conditions.

Usual market conditions are difficult to define, but it is often about stationarity of the market context. If you identify a transformation of the market that is stationary, it is reasonnable to test the risk level of your portfolio for any realization of the market in this stationary state space. It is there that you can find a link with cointegration.

Mainly because when two variable are cointegrated, their combination is somehow more stationary than each of them take alone. Of course links between stationarity and cointegration are in practice not easy to deal with (see for instance Kunst, R. M., 2002. Testing for stationarity in a cointegrated system. Tech. rep., Institut für Höhere Studien (IHS)) but from a theoretical viewpoint it should be better to compute VaR using assumptions in a space in which no variable have remaining cointegrated relations.

I have not found any literature directly handling this. But you could pose the question in terms of risk management of stationary series. Here is extreme values in stationary series. You can try this Quantile Cointegrating Regression. This Sensitivity of portfolio VaR and CVaR to portfolio return characteristics is an interesting twist. Here is an another version of the paper you citedCointegration and Asset Allocation or cointegration.It will be interesting to go through them in detail. Of course, MC would be a sure way to do VaR or CVaR. You may find that the VaR may vary for the sub samples of the in-sample dataset. To me, that variability may be more important in predicting or monitoring the out-of-sample VaR on day to day basis. I do not have any literature showing whether the stationarity broke down or improved during the credit crisis of 2008, considering you are short one vs another or how sensitive the cointegrated series are wrt correlation changes. Would be nice to have a regime switching model to detect when the changes are occurring. Normally the portfolio is already experiencing drawdown by the time the model realizes the change. Need to give some more thought to EVT and drawdown.

Cointegration really shouldn't be a risk relationship. It's a sort of equilibrium pricing relationship.

If you take a bunch of prices/yields and regress them in levels, you are looking for a 'fair value' (or rich/cheap) analysis for pricing. If you look in differences you are looking for the hedges. The analysis is very different.

In effect, cointegration is like an infinite-horizon correlation. It only tells you what will happen in the long-run. Its impact on the short-run is limited. It does have an impact, but this is for forecasting (some famous analysis showed that cointegration relations do not really help that much with long-run forecasts, but do in fact help with adjustments to short-run forecasts).

To use cointegration for risk just does not make sense. First, as a risk manager, you want to look at returns. Returns are $I(0)$. You can't have cointegration relationships between $I(0)$ time-series. They're already stable. You would never look at risk using price levels themselves, either, right? That would not make sense. But that's where you'd look for a cointegration relation.

Another way of looking at it is, it's valid to find cointegration using PCA, but PCA in levels. IF a set of $N$ assets (in prices, $I(1)$ variables) has $k$ cointegrating vectors, then it has $N-k$ common trends, which are $I(1)$. In other words, our $N$ assets are linear combinations of $k$ cointegration relations and $N-k$ non-stationary time-series. PCA always searches out the highest contributions to variance, but $I(1)$ variables must always be higher contributors to variance than $I(0)$ (in the long-run, of course, since I think we can imagine cases where this is not true in the short run, i.e., comparing a high variance Ornstein-Uhlenbeck process to a very low-vol Brownian motion. In the long-run the BM will have higher variance, while the OU will have bounded variance).

PCA in levels is sometimes used as an (inefficient) means of extracting cointegration relations.

I think it's best to leave the cointegration to traders. Yes, if they find a relationship that seems stable with nice carry, then they should try to take advantage of it. As a risk manager, you must poke holes in their theory and show them that the cointegration relationship sometimes breaks quite severely! (To do so, you can find changepoint detectors in cointegration relations and try them out, or just look at the cointegration relationship over different time periods and see if the coefficients drift or break).