I've taken three months of price return data for two instruments and calculated a $\beta$ between the two using the formula $\beta = \frac{Cov(x,y}{Var(y)}$ with the goal of estimating what the percentage change in instrument $y$ should be based on what the percentage change in instrument $x$ is.

I have been applying this by multiplying $\beta$ by the percentage change of $x$ to determine a beta-adjusted percentage change for $y$, but I am wondering if I should actually multiply the derived $\beta$ by the current percentage change of $y$ instead. Could anyone shed some light on whether or not I am properly applying this $\beta$ to arrive at an expected percentage change for $y$?

  • $\begingroup$ It appears that you're looking at the Capital Asset Pricing Model (CAPM), It is important to note that the CAPM can be used for prediction, however, I feel that you're slighlt confused. Risk premium is the price of risk, $[E(R_m)-r_f]$ multiplied by a measure of risk, often called $\beta$. The definition of $\beta$ that you have is correct, but are you trying to find out how your two securities impact each other, if so I would suggested a granger causaility model. $\endgroup$ – user22485 Oct 18 '18 at 12:43
  • $\begingroup$ Sort of, I am attempting to derive an estimate for the change in a back month commodity future against how much the front month has changed. So in this case I am treating my front month as a 'spot' and looking at expected changes in the back. I know the instruments are related, and cointegrated over the time interval I am looking at so I'm not sure if granger causality is going to tell me anything new. I am just looking to estimate a coefficient that shows me expected change in the back when applied to what has happened in the front. $\endgroup$ – jod51 Oct 18 '18 at 13:17
  • $\begingroup$ So are you using lagged co-integrated techniques? $\endgroup$ – user22485 Oct 19 '18 at 6:58
  • $\begingroup$ I still think a regression model would help you, but your variables are I(1), (if they are cointegrated), this makes GC using OLS pretty useless. There is a great paper by Campbell and Yogo (2006) and they look at these predictive regression with persistent explanatory variables, its a little bit complicated but worth a read. $\endgroup$ – user22485 Oct 19 '18 at 7:01

First of all if your model is $r_Y\sim~\beta r_X$ (with $r_X$ and $r_Y$ the returns of assets X and Y) then $\beta = Cov(r_X,r_Y)/Var(r_X)$ not $Cov(r_X,r_Y)/Var(r_Y)$ as stated by OP

Then assuming this model, if you observe $r_X$ the model would associate $\hat{r}_Y = \beta r_X$ to explain the actual return $r_Y$. The unexplained return would be $\epsilon_Y = r_Y - \hat{r}_Y$

  • $\begingroup$ Yep, thx. That's not the issue here. $\endgroup$ – jod51 Oct 24 '18 at 13:04
  • $\begingroup$ i think that's exactly the issue at hand, i responded to your question and corrected your typo on the definition of $beta$ $\endgroup$ – Ezy Oct 25 '18 at 12:20

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