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I have been looking for an illustrative way to quantify the risk of an investor not being able stay above the minimum capital requirement. I would like to find a neat solution for the probability that the investor goes out of business in a very simple set up.

First, denote the solvency capital of an investor with $C$=$1^TX$, where
$X_i$ = value of the portfolio component $i$ ($i=1,2,...,n$) and
$X$ ~ $N(\mu,\Sigma)$

Also, assume that the investor has some obligations in the future and until that the solvency capital has to stay above the minimum solvency capital level denoted as
$C_m=\sqrt{X^TQX}$, where
$Q$ is a positive semi-definite $n$ x $n$ matrix defined by the legislator. The legislator has set $Q$ so that $\sqrt{X^TQX}$ represents their view on the portfolio max loss during the period considered.

I assume that the investor goes out of business whenever the risk buffer is not high enough, i.e. if $C-C_m < 0$. Therefore, in the best case I would find a distribution for $\frac{C}{C_m}$ or $C-C_m$. Do you have an idea of a probability distribution that could be applied here?

I am not very familiar with various probability distributions but,

  • ratios of quadratic forms in normal variables have been discussed here
  • according to these notes I would understand that $X^TQX$ ~ $\chi^2(r,\lambda)$ and so should be the case with $X^TX$ also. Should I rather consider $(X^TX)/(X^TQX)$?
Techonomist
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  • It seems to me that the matrix $Q$ need be symmetric and that $X^\text{T} QX$ is only a non-central $\chi^2$ for specific choices of $Q$, so that $Q^{1/2}\Sigma Q^{1/2}$ has eigenvalues in $\{0,1\}$. Same thing for $X^\text{T} X$ which is not a non-central $\chi^2$ variate, unless $\Sigma$ has eigenvalues in $\{0,1\}$. – Xi'an Dec 20 '20 at 11:21
  • Why is $C_m$ not fixed? I mean it's something like a VaR, so why does it make sense for it to be a random variable too given $Q$? Is it trying to capture the unpredictability of the policy maker? – PaulG Dec 20 '20 at 11:45
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    @Xi'an $Q$ does not need to be symmetric, because it can always be replaced by a symmetric version $\Sigma=(Q+Q^\prime)/2$ without changing the quadratic form $X\to X^\prime Q X = X^\prime \Sigma X.$ Thus, WLG, we may assume this has been done. – whuber Dec 20 '20 at 15:47
  • Thank you for all the comments. @Xi'an despite that the symmetry is not necessary, Q is also symmetric. It is like a covariance matrix which can be decomposed into two diagonal matrices of portfolio component specific risks and a correlation matrix. – Techonomist Dec 20 '20 at 17:39
  • @PaulG, a good question maybe arising from the simplified set up. In reality the investment horizon is infinite and the investor has ever-growing funding obligations. $C_m$ can be considered indeed e.g. as a minimum VaR buffer the investor is required to have for a finite horizon defined by the policy maker. Over time, the capital should have a growth rate at least as high as the growth rate of the value of the future obligations. As the portfolio and the obligations evolve over time so should the minimum risk buffer. Hopefully this clarifies the assumptions. – Techonomist Dec 20 '20 at 17:57
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    See this detailed answer: https://stats.stackexchange.com/a/188650/7224 – Xi'an Dec 20 '20 at 17:58
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    @Xi'an thank you for sharing the detailed answer. I now believe that I can consider a ratio or a sum of two dependent variables that both are non-central $\chi^2$, but I am still unsure is there any joint distribution for them. Any thoughts? – Techonomist Dec 22 '20 at 20:51

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