Deriving the single factor model

Consider the following regressions, with the common factor $x$:

$y_1 = \beta_1 \cdot x + \gamma_1 \cdot \epsilon_1$

$y_2 = \beta_2 \cdot x + \gamma_2 \cdot \epsilon_2$

With $\epsilon_1$, $\epsilon_2 \tilde{} N(0,1)$ as usual, and $Cov(x, \epsilon_j) = 0$, for all $i$.

How do you derive the single factor model below?

$y_1 = \sqrt{\rho} \cdot x + \sqrt{1 - \rho} \cdot \epsilon_1$

$y_2 = \sqrt{\rho} \cdot x + \sqrt{1 - \rho} \cdot \epsilon_2$

where $\rho$ is the correlation between $y_1$ and $y_2$.

I've looked for a proof of this, but haven't been able to find it anywhere...

• I do not think this is correct. Do you want to model the default times based on a single factor? Feb 22 '16 at 20:45
• who said anything about default times? but sure, why not. Let's assume they are default times.
– J4y
Feb 23 '16 at 15:18
• You need to provide more background for your question. For assets or stocks, there is no such thing as for the same coefficient for a single factor for different assets. A single factor or multiple factors are usually used to model default times. But this all depend on what you want to achieve. Feb 23 '16 at 15:53
• What if all the assets are the same type (e.g. jet-fuel and diesel) or stocks are in the same industry (e.g. pairs like Pepsi and Coca-Cola)? Wouldn't it be plausible to assign the same coefficients then?
– J4y
Feb 23 '16 at 17:35
• In that case, you will also need a constant term to capture the value difference. For example, one is Microsoft and another is Apple. Feb 23 '16 at 18:03

After some digging arround, I found that you can derive the single factor model under the following assumptions:

$\beta_1 = \beta_2$

$\gamma_1 = \gamma_2$.

In addition, we require $Cov(\epsilon_1, \epsilon_2) = 0$ and $Cov(\epsilon_i, x) = 0$ for $i = 1, 2$.

Finally, we would require $y_1$ and $y_2$ to be normalized $\tilde{} N(0, 1)$ for the correlation to be equal to the co-variance.

This way:

$\sigma_i^2 = \beta^2 + \gamma^2$

$\sigma_{i,j} = \beta^2$

and we can derive the above.