Forward Pricing for Futures-Based ETFs and Their Roll Policies

# Why is the forward price of an ETF that holds futures equal to the current price of the ETF?


# Why is the forward price of an ETF that holds futures equal to the current price of the ETF?












I've seen some ETFs that only hold futures (and some cash) whose options are being priced with a forward price equal to the current ETF price, even for 1 year out options (at least according to Bloomberg).

Let's say that oil spot is trading at 46/barrel. The 1 year out future is trading at 50 (10% anual interest rate). If this ETF can only hold futures, and has assets for 1M, it will buy 20.000 of them. We also assume that there are 1M shares, which comes out to 1 dollar per share. 1 year later and assuming 0% drift (i.e. oil is still trading at 46/barrel), the future should converge to 46 too, making the ETF be worth 20.000 * $46 = 920.000, which comes out to 0.92 dollars per share. That means that if I'm pricing 1 year out options, the 1 year out forward price should be lower than the current ETF price.

That is, I should be pricing my options using 0.92 as the forward price and not 1 dollar, correct?

## Answer by will (score 2, accepted)

https://quant.stackexchange.com/a/68768

The two answers above have covered how the value of the ETF is calculated (i.e. based on the NAV, or Net Assets Value) but what they haven't covered is the forward of the ETF.

Shares are open ended - that is, they don't have a preset expiry date. When you buy shares of an ETF, these too are open ended*. This creates an issue if the ETF holds futures though, as these expire (either into cash, or are physically delivered, but again this doesn't matter here for reasons that will be obvious below). In order to make the ETF open ended, i.e. remove the issue of an expiry date, the ETF will have some predefinied methodoology for rolling the current futures position to the subsequent futures as those it's holding approach their expiry date. This methodology is typically defined in the ETF prospectus, though in some cases the ETF will instead track an index which itself prescribes how it rolls through the futures.

Given your example of oil futures, here are a couple of WTI crude oil futures ETFs which behave differently:

United States Oil Fund: This used to hold the front month futures and roll on a monthly basis, as documented in the prospectus as the benchmark futures contract, but last year in April they used their right according to that same prospectus to invest in different futures to the benchmark in order to increase liquidity. They would roll over 3 business days near the start of the month (something like this). They've changed it now and it's over a longer window, and also it holds more than just the front month as mentioned before, but it's still rolling futures that it holds itself.

CMCI Oil ETF: This ETF works a little differently, in that it tracks the level of an index which itself rolls oil futures (albeit it holds a broader range of futures and rolls according to a different methodology), and the ETF recieves that expsoure via a swap rather than directly investing in the futures itself.

In both of the above cases, the concept of rolling is mentioned. What this means is that the ETF will at some point unwind the current futures it is holding, and trade into new ones with a longer maturity. Let's start with a simple example where the futures are fully funded (i.e. they behave like shares and you need to hand over the total cash value when you enter into the futures position), and the roll happens on the first of each month (where we conveniently ignore weekends):

- 1 jan 2020: WTI front future = \$100, ETF nav = \$100. ETF holds 1 future.

- 2 jan 2020: WTI front future = \$99, ETF nav = \$99, ETF still holds 1 future as there has been no roll.

- ...

- 31 jan 2020: WTI front future = \$110, ETF nav = \$110. ETF still holds 1 future.

- 1 feb 2020: WTI front future = \$110 still, but 2$^\mathrm{nd}$ future is worth \$100. The fund rolls out of it's exposure in front month futures, reciving \$110 cash, and invests this full exposure into the 2$^\mathrm{nd}$ future, meanign the fund now holds 1.1$\times 2^\mathrm{nd}$ month futures, and the nav is still the same \$110 it was before the roll.

This same thing happens each roll, you can think of it as fully unwinding all positions the ETF holds at the settlement levels on the roll date, and then taking all of the cash that creates and investing it into the new position it needs to be holding after the roll on that date, as sepcified in the ETF prospectus.

So this explains the roll methodology. Why does this mean there is no forward curve if the ETF only holds (and will only hold in the future) futures? This is because the value of a future is the expected value of the underlying of the future on the expiry date of said future. This means that when you buy a future, you are striking it at the expected value on the expiry date, which is the same as today, and the same as any day in the future - the forward curve is flat. The expected PnL of buying a future with the plan to sell it at a future date is always zero**. For this reason, a strategy which is predefined to buy and sell futures at their market values must also have a flat forward curve.

Now, the reality is a bit more complicated than this. For a couple of reasons:

- ETFs have management fees, such that if there is a 1% p.a. fee on the ETF, and it's at \$100 now, then it's reasonable to assume that the expected value in 1Y is \$99, due to the management fees being extracted from the fund assets.

- Futures based ETFs do not need to deploy all of their cash to enter into the futures positions, since the margin requirements are typically <20%. This means that they can utilise the remaining cash to invest in other interest bearing instruments. In the case of USO, it invests in a number of interest bearing cash instruments (see here). This means that USO can enhance the returns of the ETF to make it look more competetive, by not wasting the unused cash in the fund. Another alternative to this is to invest the cash in a portfolio of stocks and shares, and then swap the exposure of that out with a counterparty who requires it in excahnge for them paying you a funding rate which may be higher than govt bonds. The impact of this is normally fed through to the funding cost, effectively reducing it. If the fund holds bonds with an aim of extracting some extra performance from the cash component, then this also needs to be considered in the forward curve.

There will be other reasons too, but they become more niche and i don'ot think are important here. The above two reasons though mean that even though the ETF NAV is mainly based on futures, there may be other reasons that will pull the forward away from being flat. These typically being:

- ETF management fees

- floating rate components to the underlyings - be it bonds included in the portfolio, or a floating rate included in the underlying index of the ETF.

*In the case of ETNs, or Exchange Traded Notes, they can have an expiry, which is the date in the future when the issuer will reedeem the shares in the ETN, but other than this they function in the same way. For the sake of the above, you can consider ETFs, ETCs, and ETNs in similar ways, as we're ignoring credit implications.

**Ignore things like cost of funding the margin, cost of VaR, etc.

## Answer by Mild_Thornberry (score 1)

https://quant.stackexchange.com/a/68706

The ETF price you describe is a Martingale. You’ve set up a scenario where your oil price growth (0%) is less than your discount (10%), which is how you are forcing the forward price lower than 1. If your growth and discount expectations are equal, which is the only appropriate way to price this or any other financial instrument, then your forward price of oil is 50, and your forward price of your ETF is 1, which is the current price and consistent with Bloomberg.

## Answer by JoshK (score 1)

https://quant.stackexchange.com/a/68708

You are misunderstanding how ETFs work. The ETF is worth the value of the underlying trust. That's it.

Simple example. The trust on day one (pre-launch) has \$100mm and 10mm shares. That means the NAV of each share is $10. Let's say this is a futures only ETF.

Now with futures you can buy up to your margin limit. BUT in general most trusts will only buy fully collateralized futures, so the fund will buy \$100mm worth of the futures. So, say the future contract price * multiplier is \$1mm. So that would mean the trust buys 100 of these futures.

That's it. It's that easy. You now value the ETf based on the liquidation value, which would involve selling those futures and getting your cash back.

You might think that the futures aren't discounted right, but that's not the ETF's problem. The ETF just reflects the market value of the assets that are inside its trust.