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I heard that we can use, say, Eurostoxx Futures as a benchmark to compute the beta of the index's components. Is this relevant? If so, how do we deal with the futures' expiry?

Thanks

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You can use either the spot value of the index (STXE) or the Futures Price for the current contract (FX*0) to compute the Market Return ($R_M$) on any day. The advantages and disadvantages are:

  • The Spot price (which is computed from the prices of the underlying stocks) is easily available and computing returns is easy ($\frac{S_{t}-S_{t-1}}{S_{t-1}}$). The quality of the Closing spot price is very good, but the Open price may contain some stale data from the previous day and therefore should not be relied on, the intraday prices may also be somewhat questionable for the same reason (if some underlying stocks have not traded in a while).

  • The Futures price has the advantage that it always represents a tradeable price and therefore always incorporates current information. There is no problem of data quality in this case, however as you point out contracts sometimes expire and are replaced by another contract. To deal with this is easy enough: on a normal day you use $\frac{F_{t}-F_{t-1}}{F_{t-1}}$ where $F$ is the current contract (for example Sep 2020). On the first day when you no longer wish to use the Sep contract, because it is about to expire, you compute the return as $\frac{G_{t}-G_{t-1}}{G_{t-1}}$ where $G$ is the next contract, in this case the Dec 2020 contract. The error introduced by this procedure is negligible for most purposes ($R_F \approx R_G$).

If you only need daily data at the close the Spot price method is probably best.

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  • $\begingroup$ Thanks! And what about contango or normal backwardation? How does it affect rolling returns? $\endgroup$ – mbz0 Aug 19 '20 at 19:06
  • $\begingroup$ For computing Beta we use daily (or weekly) returns on the market. It is true that the return on the future includes two components: the market return and a gradual movement from convergence of future to spot. But the market return is 1 or 2% per day up or down, the convergence return is a steady 1 or 2% per year, so it is completely negligible (260 times smaller) compared the market return on a daily basis. So for computing Beta futures work fine. $\endgroup$ – noob2 Aug 19 '20 at 23:40
  • $\begingroup$ Thanks a lot! Do you have any clue about how to test the stationarity of rolling futures? $\endgroup$ – mbz0 Aug 20 '20 at 7:48
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You may use EuroStoxx 50 futures as a market proxy for stocks in the index as well as other stocks. You could also use the STOXX Europe 600 (a better index, but with much lower futures liquidity) or build your own index from the DAX, CAC 40, FTSE 100, IBEX, MIB, AEX, BEL20, SMI, ATX, OMX, WIG, HEX, etc.

Handling futures expiry is usually done by switching to the next contract (like rolling a contract except rolling involves trading and thus fees). This can be done a few days before expiry or you can use other rules for doing the switch:when volume of one crosses the other and when open interest crosses are common rules. The difference between these rules will not likely impact your regression.

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