Creating synthetic cash v creating synthetic T-bill

Schweser QBank has one question asking you to convert an equity portfolio to a synthetic t-bill position and uses [B(target)-B(existing)]/B(futures) where B(target)=0 then has question asking you to convert an equity position to a synthetic cash position and uses [V(equity)*(1+Rf)]/(V(futures) how do you know when to use which…is it literally down to the wording of synthetic cash (use 1+Rf formula) or synthetic t-bill (use Beta formula)? Pls explain the logic if possible? Surely there is no diff btwn the 2 as both have B=0 and earn Rf?? TVM

Textbook, Vol 5, 363 has a discussion on this. Look like both formula will result in the same number of futures if the portfolio to hedge = index underlying the future. Here is the exact extraction from the book. “In Section 3.2, we gave a different formula to reduce the portfolio beta to zero. These formulas do not appear to be the same. Would they give the same value of Nf? In the example here, we sell the precise number of futures to completely hedge the stock portfolio. The stock portfolio, however, has to be identical to the index. It cannot have a different beta. The other formula, which reduces the beta to zero, is more general and can be used to eliminate the systematic risk on any portfolio. Note, however, that only systematic risk is eliminated. If the portfolio is not fully diversified, some risk will remain, but that risk is diversifiable, and the expected return on that portfolio would still be the risk-free rate. If we apply that formula to a portfolio that is identical to the index on which the futures is based, the two formulas are the same and the number of futures contracts to sell is the same in both cases.” If someone have interpretation different from me, please let me know.

In the first case, you are reducing the sensitivity of the equity by reducing beta to 0. So its basically like you have no equity, so there is no question of the value growing at any rate. In the second scenario, you are converting your equity into cash, and this cash will earn the risk free rate just like in a normal situation. In conclusion, if you are trying to “reduce” your equity position, which in this case is completely reducing it, hence the 0 beta, then you use the first formula. We see this case many times in problems where they want to switch allocations from equity to debt without actually unwinding their positions. You are completely hedged against any movement in the equity markets.

Does anyone else have a clearer (no offense to the previous two answers) explanation for this? Ran across this in last year’s mock exam and had never seen the V(1+r)^t/contract formula before, had only seen the Schweser (target-current)/contract. In the question(s) I’m referencing, the first was someone who had 100MM in Eurodollar deposits and wanted to “reallocate to a synthetic index fund of European equities”, and the question asked…aww hell, I’ll just start a new thread with the question and see what people come up with.