Why is the CAPM Beta defined this way - Beta hedging

Question

Let's say I have two equity indices X and Y. Assume they are negatively correlated with some leverage. I want to hedge X with Y.

I have seen many ways of computing a beta to describes the relationship

$(X_t)$ and $(Y_t)$ are strictly positive. $t\in\{0,1,..,n\}$ daily,

$D^X_t = X_t - X_{t-1}$ (difference series)

$R^X_t = \frac{X_t-X_{t-1}}{X_{t-1}}$ (return series)

Let's say today is t=n, I observe all the past values up to t=0. The goal of my daily hedging is to find at time n the beta, the quantity of Y to buy for each dollar of X, such that $E[D^X_{n+1}-\beta_{n+1} D^Y_{n+1} ] =0 $

The CAPM Beta could be defined as: $\beta_t=\frac{{cov}(R_t^X,R_t^Y)}{ V[R_t^Y]}$ (which I could estimate in many ways let's ignore that here unless someone has a better method than a Kalman filter).

My questions are:

1. could I define $\beta_t=\bigg (\frac{{cov}(R_t^X,R_t^Y)}{ V[R_t^X]} \bigg )^{-1} $

   $\frac{{cov}(R_t^X,R_t^Y)}{ V[R_t^X]}$ makes more sense to me because I make the market of X, I think in terms of move of X.

2. could I define $\beta_t = \frac{{cov}(D_t^X,D_t^Y)}{ V[D_t^Y]}$? If I estimate it dynamically with a kalman filter for example I understand that the series $D$ and therefore this beta strongly depends on the level of X and Y whereas $R$ is rescaled.

I could further expand on my studies but I do not want to loose 90% of the potential audience nor skew the answer.

Answer 1

In a CAPM framework, the Beta of Y to X = Correlation * Y Volatility/X Volatility.

Which alternatively is Covariance / X Volatility

So to (1) above, it's your latter formulation not the reciprocal.

To (2) above, you could use prices rather than returns, but this can be problematic depending on the precise thing you want to measure. It's generally assumed that asset price returns are normally/lognormally distributed (or some distribution that at least resembles Gauss!), which is why asset betas default to returns rather than prices.

You simply have to ask yourself which of "a buck on the Apple share price" or "5% on the Apple share price" is the appropriate metric in the case you're looking at.

Suppose you were interested in the effects of oil prices on Japan's trade balance. Japan is a huge energy importer. If she imports X million barrels per day, then the import effect will be proportional to the absolute price change. It doesn't matter whether oil goes from 30->40 or 130-140 bucks a barrel. Oil is a small but significant fraction of Japan's international trade, so the +10 move isn't going to change the total value of Japan's trade very much. Ergo the +10 rather than the 33%-vs-13% is the more appropriate.

And sometimes a mix of your "D" and "R" makes most sense. Suppose you were interested in the impact of the bond market on the stock market. There a +/-X basis point move in bond yields is associated with a Y% change in stock prices. With $13tr of negative yielding bonds in the world, every time they move through zero, using a % change in yields would imply an infinite impact on stocks. Conversely a historic decline from 6% to 5% would imply very little impact, when in fact it was significant. Conversely, these impacts don't add or subtract a point to the S&P index; but they do move it in a logarithmic fashion.

At its heart, CAPM is just a pompous (economists have this sin in spades as a tribe, sorry) term for regression ;-) So all the usual caveats about stationary, heteroskedactic etc. etc. apply here too. As with any regression problem, you just need a rational and hopefully intuitive reason to pick any set of inputs over related alternatives.