Futures are financial instruments created to avoid the uncertainty of future price evolution of all kind of underlying assets, whether it be stocks/indexes, commodities, bonds, currencies, etc.

The future contract is an agreement to buy or sell a specified asset on a certain future date at an agreed price.

## What is the fair price of the future?

We can view it by imagining that we are actually** buying or selling the underlying asset** in the moment we want to calculate the future price:

Let’s imagine that today we’d like to buy an ETF S&P 500 Index tracker. We would follow these steps:

- Borrow the money from the bank
- Buy the ETF at time t
_{0}(spot price p_{0}) - Sell the ETF at time t
_{1}(spot price p_{1}) - Return the loan to the bank

Our profit would have been the performance of the index, the dividends received plus the interests given for them during the contract (D), minus the interests we have to pay back (I).

So, the future price at t_{0} should be such that our future profit be equal to the shown before, that is:

Or, writing the formula in terms of time to the expiration date:

###### * For educational purposes I preferred not to present this formula in continuous compounding terms, which is the most common manner.

Where r is the interest rate, T-t_{0} is the time left to the expiration date T and d_{i} represent the income amounts provided by the asset in t_{i} (in the case of a stock or an index, that would be the dividends). Following this last formula**, the future price and the spot price would theoretially be equal at the delivery date**.

If the price were different to this, there would be an arbitrage opportunity. To clarify this point, here’s an example.

### Arbitrage opportunity example

Imagine that at a certain moment, the 3-month future of a stock has a value of $32 (we assume it does not deliver dividends), while the spot price is $30, and the annual interest rate is 7%. In this situation, you could:

- Borrow $30 at an interest rate 7% to buy the stock.
- At the same time, you sell a 3-month future at $32.

3 months later:

- You have to deliver the stock (since you have sold a future) and collect $32 (the agreed price in the future contract).
- You must repay the loan: 30x(1+0.07)
^{(3/12)})=30.51 - You obtain a gain of 32-30.51 = $1.49
**with no risk**!

## Discussion

Moving on to the practical use of the futures, one of the many uses is **hedging the risk exposure of an index**. If an investor had a percentage of its portfolio invested in an index, he may decide at a certain moment to hedge the risk of a sudden fall by selling a future contract of the index.

**What result can we expect from the hedging opperation?**

To answer this question, let’s work with the series of the Euro Stoxx 50 Index and its future price series (constructing the future series doing roll-overs 4 or 5 days previous to each maturity date). Let’s assume that there are no costs derived from the roll-overs.

According to the annual returns shown next, we would think that the future follows approximately the series of the Euro Stoxx 50 price, which does not consider the dividends delivered by the components of the index.

However, what if we looked at the annual returns from 2009? Would not it be the gross return Index, with the dividends reinvested, more similar to the future annual returns?

Well, if we had correctly understood the future price formation, we would have realised that none of the series would be the same as the future evolution, not even theoretically.

As I explained before, the future price must **include the cost of borrowing money and the “gain”** due to the dividends. Then, what is happening from 2002 to 2008 is that the financing cost is compensating the dividend contribution, giving an annual return similar to the Euro Stoxx 50 price. Nevertheless, when the financing costs fall, as it is the case from 2009 on, the future return is much more similar to the gross index.

In the following graph, where the difference between the Euro Stoxx 50 Gross and Price Index series has been taken as an approximation of the dividend contribution, we can verify that when the risk free rate cost is higher than our approximation of the dividends contribution, the future return is lesser than the price index return, and the opposite when the dividends are higher.

Only in 2008, despite the fact that the financing cost is apparently much higher than the dividends contribution, the future return is very similar to the price index return, instead of being lesser. The reason is that in this case, the dividends contribution to the future return cannot be estimated in the way we attempted: the Gross Index suppose reinvestment of the dividends (this was dealt in Approach to Dividend Adjustment Factor Calculation), while the future contracts assume no reinvestment. So, in years with such a draw down as in 2008, the actual dividends contribution in the futures would be higher than what we show in the graph, counteracting the financing cost.

## What can you expect from hedging an index selling futures?

Based on the previous discussion, we can say that if you are exposed to an index and you follow a policy of not reinvesting the dividends received, a hedging position by selling futures of the index should give a net return similar to the risk-free interest rate.

You must notice that this is quite obvious, considering that to hedge an index position you could just **sell your whole position and put the money at the risk-free interest rate**, obtaining the same result as the future operation, except for the costs of selling and buying your investments.