Using Regression Beta to Estimate an Asset’s Return

Summary

The document clarifies how beta relates the returns of two instruments in a simple linear model. If the goal is to estimate the return of asset Y from the observed return of asset X, beta is the covariance of X and Y returns divided by the variance of X returns. The original question instead places the variance of Y in the denominator, which corresponds to a different regression orientation.

After estimating beta, multiply it by the observed return of X to obtain the model’s fitted return for Y. The residual is Y’s actual return minus that fitted value, representing the portion not explained by X in this model. The answer presents this as a basic regression interpretation, not a complete forecasting procedure. It does not discuss an intercept, estimation uncertainty, changing relationships, or whether the short sample is adequate, so the fitted return should not be treated as a guaranteed outcome.

Key ideas

• For a regression of Y returns on X returns, beta uses the variance of X in its denominator.
• Multiplying beta by the observed return of X gives the fitted return of Y in the stated model.
• The residual is the actual return of Y minus its fitted return.
• The beta formula depends on which asset is treated as the predictor.

Tags

# How to apply derived beta to daily change?


# 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 $y$?

## Answer by Ezy (score 1)

https://quant.stackexchange.com/a/42310

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.