# No-arbitrage arguments: how do additional fees affect futures on an index?

I am considering a fund that replicates the returns of an index minus a fee, using the following case-study my lecturer used regarding SPY:

In practice, futures and forwards can be written on assets which are not directly tradeable. Consider the E-mini futures contract example on the S&P 500 index, where due to transaction costs it is not possible to invest directly in the index. Now, SPY is a managed fund which replicates the returns on the index minus an $$k\approx1\%$$ per annum management fee. So, if you denote $$X_t$$ as the value of the S&P 500, and $$Y_t$$ as the value of one unit in SPY, then you have $$Y_t=X_t\mathrm{e}^{-kt}$$. You can then derive the E-mini futures price in terms of the price of a unit of SPY.

Is the no-arbitrage futures price on $$Y_t$$ just naively $$F_{t,T}=Y_t\mathrm{e}^{r(T-t)}=X_t\mathrm{e}^{(r-k)(T-t)}?$$

If so, what is the justification behind it? Furthemore, I'm thinking that just as how dividend payouts cause a fall in stock price, transaction costs should cause a rise in this index price, which really doesn't make sense.

It feels like I am missing something painfully obvious, but can’t figure it out. I know this is a silly question, so if it has been asked before/it’s common sense please let me know.

## 1 Answer

I might have misunderstood the question; but the index is just a weighted-average of its constituent stocks. As such, it does not trade, and thus does not incur any transaction costs. The forward price on said index is just the spot, adjusted for interest-rate versus (expected) dividend basis. Lest there be arbitrage.

Trading all of the stocks to replicate the index might well generate transactions costs (given inflows and outflows), assuming full physical replication. But the transaction costs of rolling quarterly futures are de-minimus. Hence the attraction of futures.

The futures themselves don't need any adjustment for transaction costs, because the index itself doesn't trade, and thus generate costs. The problem is with anyone wishing to replicate the index. So SPX futures have no transaction costs; but SPY (the ETF) does. But nobody trades futures on SPY...

Hopefully, this makes sense. Shout if it doesn't. DEM

• Hi, thanks for the answer. I think I have misinterpreted my notes and problem sheet as well. The problem was meant to be an exercise on applying no arbitrage principles, which may have led to my confusion. My lecturer also used the SPY example and I have updated my question with what she wrote, was wondering if you could see what conceptual flaws I have too? I’ll reread more into detail and try to understand. Commented Jan 7, 2021 at 15:43