we have to go back to basics. remember how futures work.
buying futures now would not need any cash exchange upon the start but upon expiration, you can either choose 1) to get the underlying assets (stocks/etfs/shares) delivered to you from the seller or 2) just receive the difference of future price that you bought at the start and the spot price at expiration (cash settlement)
_ the reverse also applies to selling futures. i.e. you don’t get any cash NOW but will do so during expiration, depending on method of settlement and if your short position is profitable. _
there are also features such as margin deposit , marking to market or posting collateral of futures (so you do need cash reserves between the start and expiration) but right now those things (calculating margin deposits, margin call levels, collateral needed or calculating daily mark to market values) are not scope of the curriculum. the curriculum only deals with synthetically rebalancing your portfolio.
in a sense, the curriculum ignores them and so it’d be more correct to say that these are actually forwards and not futures, but that’s not we are here for.
the 1st equation seemingly assumes that you choose the first settlement method (having the underlying asset delivered to you) upon expiration. so let’s say you now have 1,000,000 USD in cash, the futures price is 1000 USD and the multiplier is 10. the short-term risk free rate is 1% and expiration of the future is 1 year. using the formula, we get number of futures equal to 101.
so you buy 101 futures now (this act of buying assumes no initial outlay) and 1 year later, you have to pay 1,010,000 USD and get the 101x10(1,010) underlying assets delivered to you. how do we get the 1,010,000 USD needed? by investing the 1,000,000 USD at the risk-free rate of 1% per year.
_ if you choose the 2nd settlement method _ (only deliver the net gains/losses from the future position from both counterparties, i.e. compare the future price and spot price of the underlying at expiration), then you don’t really need all of that 1,000,000 USD invested in rfr (1,000,000 x 1+rfr basically says here’s how many future contracts you can buy now if you invest that money at rfr), although it’s prudent to do so in case your long exposure nets a significant loss and you really need that liquidity to pay the counterparty.
this is what creating a synthetic long position in equity out of cash is all about.
the 2nd formula is a different scenario, you’re already invested in 2 different asset classes and would like to rebalance, but you don’t want to change your portfolio straight away due to rebalancing costs, taxes, illiquidity etc etc. so, a 60/40 in stocks/bonds portfolio and you want to rebalance to 50/50 stocks/bonds? short stock futures and long bond futures.
here’s the amazing part, we have already established that starting a future position is “free” and there’s no cash exchanged (although in reality, this is definitely not the case). and since you DON’T HAVE ANY CASH in the portfolio to begin with, and entering these two future positions do not give or costs you any cash, there’s NO CASH at all. the risk-free rate becomes irrelevant in this practice and so is future value of cash. YOUR PORTFOLIO is still 60/40 stocks/bonds but because of your futures positions, your total exposure is 50/50 stocks/bonds. THIS IS DIFFERENT IF YOU DO SELL some stocks in your portfolio and then buy bond futures or just the bonds themselves, because now, we have cash from the sale of stocks, but we didn’t do that. shorting stock futures will not give us any cash NOW, but only at expiration (assuming our short position is profitable).
i’m sorry if I’m explaining everything from the basics, but remembering the basics really help in understanding more complex concepts.