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Let's say a have a correlation matrix $\Omega$ for 25 assets which I use to generate a Monte-Carlo simulation. Let's assume that $\Omega$ is valid (i.e positive-semi-definite, etc...) and estimated empirically with market data.

Now assume that I want to add a risk factor $r$, but I only know the correlation $\rho$ of that risk factor to a given asset $m$. As $r$ is not easily observable, I can't include it in the empirical estimation. Plus, traders can't mark the correlation with the remaining 24 asset without making the correlation matrix invalid (i.e not positive-definite anymore).

I was actually thinking about:

  • generating $N$ correlated standard normal numbers for the 25 assets for which correlation is known
    • yielding a set $Z$ of size $N \times 25$.
  • Taking from $Z$ the random numbers corresponding to asset $m$, let's denote them $Z_m$, which is a vector of $N$ standard normal random numbers
  • Correlating a new set of indepentand standard normal numbers $X$ with $Z_m$
    • yielding $Y$, a vector of size $N$
  • "Appending" $Y$ to $Z$.

Somebody suggested me to do something different:

  • Extend $\Omega$ adding $r$
  • Setting $\Omega_{r,m}=\rho$
  • For each asset $i$ in $\Omega$ such that $i \neq r,m$:
    • Set $\Omega_{r,i}=\Omega_{m,i} \cdot \Omega_{r,m}$
  • Correlate $Z$ as mentioned in the first point above.

Theoretically, is one of these methods "more right than the other"?

Is there another common approach to solving this kind of problem?

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You have the risk factor $F$ and the asset that it is correlated to $r_m$. You can calculate the variances of each of these, say $\sigma^2_F$ and $\sigma^2_m$. If you do not care about the distribution but just work with variances and correlations then can look at an OLS setting: $$ F = \beta r_m + \epsilon $$ with $\beta = \rho \frac{\sigma_F}{\sigma_m}$ and $\epsilon$ uncorrelated. Then the covariane is preserved: $$ Cov(F,r_m) = Cov(\beta r_m + \epsilon, r_m) = Cov(\beta r_m,r_m) = \beta \sigma^2_m = \rho \sigma_m \sigma_F. $$

If we assume that $\epsilon$ is uncorrelated with all other $r_i$ then for any other asset $r_i$ you have $$ Cov(F,r_i) = Cov(\beta r_m + \epsilon, r_i) = \beta Cov(r_m,r_i), $$ and you get a full covariance matrix. In the case $\epsilon$ is correlated to $r_i$ you add $Cov(\epsilon,r_i)$.

This could be a way to go if you are just interested in co/variances.

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  • $\begingroup$ If I assume $\epsilon$ is uncorrelated, then following your equation $ \rho_{F,i} = \frac{Cov(F,r_i)}{\sigma_F \sigma_i} = \beta \frac{ Cov(r_m,r_i)}{\sigma_F \sigma_i}$. Replacing $\beta = \rho_{F,m} \frac{\sigma_F}{\sigma_m}$, you get $ \rho_{F,i} = \rho_{F,m} \frac{ Cov(r_m,r_i)}{ \sigma_m \sigma_i} = \rho_{F,m} \rho_{i,m} $ right? $\endgroup$ – SRKX May 5 '15 at 7:42
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    $\begingroup$ Looks correct to me ... so thus would be approach 2) but with an OLS story behind it. $\endgroup$ – Ric May 5 '15 at 9:04
  • $\begingroup$ But is there any advantage compared to version 1)? $\endgroup$ – SRKX May 5 '15 at 9:06
  • $\begingroup$ if you do MC sample you usually don't have the exact covariance (see e.g. Sampling with exact covaraince). So if you can have the exact covariance - why not use it? $\endgroup$ – Ric May 5 '15 at 9:29
  • $\begingroup$ Since I know nothing about the relation of $F$ and the other assets so how can I know the exact covariance? $\endgroup$ – SRKX May 5 '15 at 9:36

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