## $\beta$ and $\gamma$

As Xi'an noted in his answer the $\beta$ and $\gamma$ are related to each other by relating to the conditional means $X|Y$ and $Y|X$ (which in their turn relate to a *single* joint distribution) these are not symmetric in the sense that $\beta = 1/\gamma$. This is neither the case if you would 'know' the true $\sigma$ and $\rho$ instead of using estimates. You have $$\beta = \rho_{XY} \frac{\sigma_Y}{\sigma_X}$$ and $$\gamma = \rho_{XY} \frac{\sigma_X}{\sigma_Y}$$

See also [simple linear regression on wikipedia][1] for computation of the $\beta$ and $\gamma$. 

It is this correlation term which sort of disturbs the symmetry. When the $\beta$ and $\gamma$ would be simply the ratio of the standard deviation $\sigma_Y/\sigma_X$ and $\sigma_X/\sigma_Y$ then they would indeed be each others inverse. The $\rho_{XY}$ term can be seen as modifying this as a sort of [regression to the mean][2]. With perfect correlation $\rho_{XY} = 1$ then you can fully predict $X$ based on $Y$ or vice versa. But with $\rho_{XY} < 1$ you can not make those perfect predictions and the conditional mean will be somewhat closer to the unconditional mean, in comparison to a simple scaling by $\sigma_Y/\sigma_X$ or $\sigma_X/\sigma_Y$.

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## Is a regression line the right method?

You may wonder whether these conditional probabilities and regression lines is what you need to determine your ratios of $X$ and $Y$. It is unclear to me how you would wish to use a regression line in the computation of an optimal ratio.

Below is an alternative way to compute the ratio. This method does have symmetry (ie if you switch X and Y then you will get the same ratio).


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## Alternative

Say, the yields of bonds $X$ and $Y$ are distributed according to a multivariate normal distribution$^\dagger$ with correlation $\rho_{XY}$ and standard deviations $\sigma_X$ and $\sigma_Y$ then the yield of a hedge that is sum of $X$ and $Y$ will be normal distributed:

$$H = \alpha X + (1-\alpha) Y \sim N(\mu_H,\sigma_H^2)$$

were $0 \leq \alpha \leq 1$ and with 

$$\begin{array}{rcl}
\mu_H &=& \alpha \mu_X+(1-\alpha) \mu_Y \\
\sigma_H^2 &=& \alpha^2 \sigma_X^2 + (1-\alpha)^2 \sigma_Y^2 + 2 \alpha (1-\alpha) \rho_{XY} \sigma_X \sigma_Y \\ 
& =& \alpha^2(\sigma_X^2+\sigma_Y^2 -2 \rho_{XY} \sigma_X\sigma_Y) + \alpha (-2 \sigma_Y^2+2\rho_{XY}\sigma_X\sigma_Y) +\sigma_Y^2
\end{array} $$

The maximum of the mean $\mu_H$ will be at $$\alpha = 0 \text{ or } \alpha=1$$ or not existing when $\mu_X=\mu_Y$. 

The minimum of the variance $\sigma_H^2$ will be at $$\alpha = 1 - \frac{\sigma_X^2 -\rho_{XY}\sigma_X\sigma_Y}{\sigma_X^2 +\sigma_Y^2 -2 \rho_{XY} \sigma_X\sigma_Y} = \frac{\sigma_Y^2-\rho_{XY}\sigma_X\sigma_Y}{\sigma_X^2+\sigma_Y^2 -2 \rho_{XY} \sigma_X\sigma_Y} $$

The optimum will be somewhere in between those two extremes and depends on how you wish to compare losses and gains

Note that now there is a symmetry between $\alpha$ and $1-\alpha$. It does not matter whether you use the hedge $H=\alpha_1 X+(1-\alpha_1)Y$ or the hedge $H=\alpha_2 Y + (1-\alpha_2) X$. You will get the same ratios in terms of $\alpha_1 = 1-\alpha_2$.

### Minimal variance case

In the minimal variance case (here you actually do not need to assume a multivariate Normal distribution) you get the following hedge ratio as optimum $$\frac{\alpha}{1-\alpha}  = \frac{var(Y) - cov(X,Y)}{var(X)-cov(X,Y)}$$ which is in terms of the regression coefficients $\beta = cov(X,Y)/var(X)$ and $\gamma = cov(X,Y)/var(Y)$ can also be expressed as following $$\frac{\alpha}{1-\alpha}  = \frac{1-\beta}{1-\gamma}$$

**In a situation with more than two variables/stocks/bonds you might generalize this to the last (smallest eigenvalue) principle component.**

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## Variants

Improvements of the model can be made by using different distributions than multivariate normal. Also you could incorporate the time in a more sophisticated model to make better predictions of future values for the pair $X,Y$.


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<sup>$\dagger$ This is a simplification but it suits the purpose of explaining how one can, and should, perform the analysis to find an optimal ratio *without* a regression line.</sup>


  [1]: https://en.wikipedia.org/wiki/Simple_linear_regression#Fitting_the_regression_line
  [2]: https://en.wikipedia.org/wiki/Regression_toward_the_mean