# How to apply derived beta to daily change?

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 $$y$$?

• 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. Oct 18 '18 at 12:43
• 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. Oct 18 '18 at 13:17
• So are you using lagged co-integrated techniques? Oct 19 '18 at 6:58
• 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. Oct 19 '18 at 7:01

## 1 Answer

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 $$\epsilon_Y = r_Y - \hat{r}_Y$$

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