Inferring a New Risk Factor’s Correlations from a Known Asset
Summary
The document considers how to add an unobserved risk factor to a correlation model when its correlation with one existing asset is known. The proposed method writes the new factor as a linear regression on that asset plus a residual. Choosing the regression coefficient from the known correlation and the two volatilities preserves the required covariance with the anchor asset.
If the residual is assumed uncorrelated with every other asset, the method implies covariances with those assets through the anchor asset’s relationships. If that assumption is not appropriate, residual covariances must also be included. This gives a way to construct a covariance extension, but it depends on assumptions about the residual and concerns covariances rather than fully specifying distributions or a simulation model. The document does not establish that this construction is uniquely correct for every application.
Key ideas
- Represent the new factor as a linear component tied to a known asset plus a residual.
- Set the linear coefficient using the known correlation and the volatilities to preserve covariance with the anchor asset.
- Assuming the residual is uncorrelated with other assets determines the new factor’s covariances from the anchor asset’s covariances.
- If the residual has additional correlations, those covariances must be modeled explicitly.
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# How to infer correlation?
# How to infer correlation?
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?
## Answer by Richi Wa (score 2, accepted)
https://quant.stackexchange.com/a/17664
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.