I tried to prove some of the beta adjustment formulas this afternoon and I was in particular curious of the “(1+r)^t” factors that you have with equity futures. Then I thought, this is probably linked to the beta of the equity futures and I did the following calculations. Beta (futures) = Covar (future, market) / Var(market)
In the calc for number of contracts needed, the futures beta is on the denominator (ie number of contracts = (Target B - Current B)/ Futures B * value/contract size. Futures beta = market beta discounted by RFR ie 1/(1+RFR)^T So dividing by 1/(1+RFR)^T is the same as multiplying by (1+RFR)^T so when the aim is to convert equity to synthetic cash (ie get out of the market altogether) or to convert cash to synthetic equity (ie get 100% market exposure) the numerator is -1 (sell futures) or +1 (buy futures) respectively. Rather than say “divide by 1/(1+RFR)^T” (on the denominator), they use the shortcut and say “multiply by (1+RFR)^T” (ie on the numerator). personally I always stick to the (T-C)/F method in all cases. It is a general rule that works for changing Beta, Duration, changing stock/bond allocation, and even swaps - for swaps it is (T-C)/S instead of (T-C)/F. Just remember that if you are getting out of the market altogether (stock to synthetic cash) or using cash to create synthetic stock, calculate your own Futures Beta (discount by RFR over the term) instead of using the futures contracts that may be provided. Works for me.
> Futures beta = market beta discounted by RFR ie > 1/(1+RFR)^T How do you get this formula? I just get the inverse i.e. Futures beta = (1+RFR)^T Then with your formula, yes it works, no issue. > Just remember that if you are getting out of the > market altogether (stock to synthetic cash) or > using cash to create synthetic stock, calculate > your own Futures Beta (discount by RFR over the > term) instead of using the futures contracts that > may be provided. Works for me. Yes, I would say: - when using equity futures, use the (1+RFR)^T factor to adjust for beta; - when using bond futures, use the conversion factor and yield beta to adjust for beta. That could help me if you find why the beta of a Futures is the one you give. MH
i just learned years ago that futures beta is always less then the physical (market) beta - due to discounting at the RFR. Seemed to tie in with the examples in reading 42 so I haven’t questioned it…