tail dependency for portfolio optimization

Question

This question pops up in my head every few weeks and I'm struggling to really understand the concept / theory behind it.

We all know there are different kind of measures of dependencies out there. From Pearson's $\rho$ to Kendall's $\tau$ to more sophisticated tail dependency measures from copula:

$$\lim_{q\downarrow0}\frac{C(q,q)}{q}$$

where $C$ is the copula of two random variables $X, Y$.

In portfolio theory we often asses risk with the standard deviation in optimization, i.e. $\sqrt{w\Sigma w}$. The nice thing about pearson correlation is that we know $Var(w^TX)= w\Sigma w$ if $\Sigma$ is the covariance matrix of the random variables $X$. This simple fact allows us to use the dependency measure (pearson correlation) to define a risk measure which has a simple structure.

If we think deeper about it. We have a dependency matrix $\Sigma$ containing pairwise dependency measures. Using $f_\Sigma (w):= w\Sigma w$ maps then these pairwise dependency to a single risk number for the total portfolio.

First question Can we come up with such meaningful $f$ for other pairwise dependency measures as well to obtain an overall single risk number for the portfolio? Meaningful in this context simply means that we really capturing a sort of risk of the overall portfolio.

I often see that the same quadratic $f$ is used but one just replaces the pairwise dependency matrix. For example using Kendall's tau, correlation or tail dependencies as entries in $\Sigma$.

Second question Does this replacing even make sense from a mathematical and risk perspective? What are the pitfalls then of just blindly using such a $\Sigma$ with a different dependency measure? For example the package FRAPO calculates a minimal tail dependency portfolio by quoting from this link

Akin to the optimisation of a global minimum-variance portfolio, the minimum tail dependennt portfolio is determined by replacing the variance-covariance matrix with the matrix of the lower tail dependence coefficients as returned by tdc.

At least to me it's not clear why this should work and represent a number which tells you something about the risk of the portfolio.

---

Example for answer Wintermute

I'm not sure if I understand / agree. I did the following example in R. Suppose we have 4 assets with the following correlation matrix:

> cor <- matrix(c(1, 0.9, -0.95, -0.96, 0.9 , 1, -0.98, -0.92, -0.95, -0.98, 1, 0.97, -0.96, -0.92, 0.97, 1), nrow=4, byrow=T)
> cor
      [,1]  [,2]  [,3]  [,4]
[1,]  1.00  0.90 -0.95 -0.96
[2,]  0.90  1.00 -0.98 -0.92
[3,] -0.95 -0.98  1.00  0.97
[4,] -0.96 -0.92  0.97  1.00

i.e. two asset are highly correlated two are highly uncorrelated. Assume we have an equally weighted portfolio and volatilities (standard deviation of the assets are $(0.12,0.08,0.15,0.02)$. Then the covariance matrix is

> cov <- matrix(c(cor[1,1]*0.12*0.12, cor[1,2]*0.12*0.08, cor[1,3]*0.12*0.15, cor[1,4]*0.12*0.02, cor[2,1]*0.12*0.08, cor[2,2]*0.08*0.08, cor[2,3]*0.08*0.15, cor[2,4]*0.08*0.02, cor[3,1]*0.12*0.15, cor[3,2]*0.15*0.08, cor[3,3]*0.15*0.15, cor[3,4]*0.15*0.02, cor[4,1]*0.12*0.02, cor[4,2]*0.02*0.08, cor[4,3]*0.02*0.15, cor[4,4]*0.02*0.02),nrow=4,byrow=T)
> cov
          [,1]      [,2]     [,3]      [,4]
[1,]  0.014400  0.008640 -0.01710 -0.002304
[2,]  0.008640  0.006400 -0.01176 -0.001472
[3,] -0.017100 -0.011760  0.02250  0.002910
[4,] -0.002304 -0.001472  0.00291  0.000400

checking quickly if I didn't mess around and if the matrices are positive semi-definite:

> cov2cor(cov)
      [,1]  [,2]  [,3]  [,4]
[1,]  1.00  0.90 -0.95 -0.96
[2,]  0.90  1.00 -0.98 -0.92
[3,] -0.95 -0.98  1.00  0.97
[4,] -0.96 -0.92  0.97  1.00
> eigen(cov)
$values
[1] 4.237828e-02 1.181651e-03 1.278374e-04 1.223631e-05

$vectors
            [,1]        [,2]       [,3]        [,4]
[1,] -0.56773237  0.78783396 -0.2372159  0.02694845
[2,] -0.37734597 -0.49612636 -0.7636994 -0.16802342
[3,]  0.72548256  0.36316206 -0.5520474 -0.19243722
[4,]  0.09468381 -0.03591096 -0.2360838  0.96644184

> eigen(cor)
$values
[1] 3.840405106 0.115703083 0.038349747 0.005542064

$vectors
           [,1]       [,2]        [,3]        [,4]
[1,] -0.4960354  0.5839966 -0.63706112 -0.08396417
[2,] -0.4947472 -0.7033709 -0.19753124 -0.47061241
[3,]  0.5078001  0.2206994 -0.08387845 -0.82848975
[4,]  0.5013114 -0.3398664 -0.74033705  0.29168255

Looks all about right. Now lets calculate $w^T\Sigma w$ and $w^TCw$. According to you we should see a difference. However, they are pretty inline with each other:

> sum(w*(cov%*%w))
[1] 9.55e-05
> sum(w*(cor%*%w))
[1] 0.0075
> w
[1] 0.25 0.25 0.25 0.25

the risk (using the covariance matrix) is zero as we would expect.

Answer 1

Since your standard for meaningful risk measurement is pretty low ("Meaningful in this context simply means that we really capturing a sort of risk of the overall portfolio.") There are limited cases where such an approach might make sense. From a broader perspective I think it makes very little sense.

If your matrix of pairwise risk numbers is symmetric and positive definite, it defines a metric (actually an inner product) in the vector space of all possible portfolios. With this metric you can do all those things you can do with any metric: Distinguish between large ("risky") and small ("safe") portfolios, optimise easily (due to convexity of the distance funtion), perform risk/return optimisation and so on. I think this is exactly the point quintuple has made.

Will this give you the full picture? Without strong additional assumptions on the underlying multivariate distribution of asset returns definitely not. The best example for such an assumption is of course being multivariate normal. Actually the reason why the covariance matrix is so important in this case is not that it provides a simple way to calculate the variance. It is important since covariance (together with the means) determines the distribution! This means if you know the covariance (and the means) you know everything about the distribution there is to know, including the risk of total portfolio return, for whichever way you may define your risk.

To drive this point of adequate assumptions down further: Even linear correlations may no longer be "meaningful" if you leave multivariate normality. See this paper on pitfalls of correlation.

Furthermore the whole idea of measuring only pairwise dependence is highly problematic. Pairwise dependence is just one aspect of total dependence (hence of total risk). To see a true shocker have a look at this presentation. It is in German sadly but on page 9 you find a nice graphic. It is an example of three uniform(0,1) random variables. The left hand scatters show that they are pairwise independent (not just uncorrelated, really independent!) while the righthand side gives you the full picture. In this example the third variable $U_3$ is actually a deterministic function of the two other variables $(U_1,U_2)$, i.e. it has arguably the strongest possible dependence on the other two.

If you would like to understand how this guy has constructed the example have a look at his paper on shaping tail dependencies.

Answer 2

Let $X$ be an $n \times 1$ random vector and $w$ a vector of coefficients. Then we know that $$Var(w^TX)=w^T \;Var(X) \;w=w^T\Sigma w$$ where $\Sigma$ is the covariance matrix of $X$. Now suppose that $C$ is the correlation matrix of $X$ and $e$ is the $n \times 1$ vector of all ones. We see that $$w^T C w = w^Te +\sum_{i \neq j}w_iw_jc_{ij}$$ where $c_{ij}$ is the correlation between the $i^{th}$ and $j^{th}$ components of $X$. Now $c_{ij}$ tells you the strength of any possible linear relationship between the $i^{th}$ and $j^{th}$ components of $X$ (1 or -1 means a perfect linear relationship). What does it mean to sum correlations? In the context of multiple regression it can be used to compute $R^2$ to see how much of the variance is described by your model, however in the context of portfolio risk I'm not sure it tells us anything meaningful. For example if half the correlations are 1 and the other half -1 $w^T C w$ would be small and you might think you have low risk. However you should have a strong linear dependance between all of your stocks. I would say the same for Kendall's tau. In general I would say it is dangerous to use any measure which is not first and foremost rational. In addition it is dangerous to use any code which you do not completely understand. It is not even remotely clear to me how the code in the link you provided works so I would stay away. If you want to measure the tail risk of your portfolio I would recommend fitting a power law to your time series of portfolio returns. This will give you a good idea of how kurtotic your portfolio returns are.