Table of contents:

- 1 What does the basis development look like in a price index?
- 2 But how is arbitrage possible?
- 3 The interest rate risk
- 4 The “roll risk
- 5 The dividend risk
- 6 Performance Index
- 7 Price Index
- 8 Why can’t an arbitrageur close out his position again and again?
- 9 How is the basis in a future calculated on the bond market?

It is only index arbitrage that makes a functioning interaction between a cash and a derivatives market possible.

Index arbitrage is a market-determining and price-influencing element in daily exchange trading and thus a partial aspect that every market participant should understand, at least in its basic structure. Strictly speaking, it is the index arbitrage which makes an attractive derivatives market possible in the first place due to its liquidity provision and its balancing function. In the following we will take a look at the individual elements of this business area. The focus here is on the presentation of the basic principle.

The spread (the basis) between cash and future …

… plays the central role in this. So let us first address the importance of the spread (basis), the difference between the fair futures price and the price of the underlying asset (in our example this is a stock index). In technical terminology, the “basis” is used here to distinguish it from the spread between the bid and ask price. We will therefore refer to the technical vocabulary and will refer to the corresponding difference between futures and spot (underlying) as the “basis”.

Let us first look at how the fair value (price) of a future is calculated using the example of a stock index. The basis is an underlying asset (index), which is composed of several shares. Each share has holding costs in the form of interest costs, which are incurred for financing the purchase of the share. This is usually offset by so-called holding income, as most stock companies pay out dividends. Sometimes other income also flows, e.g. from subscription rights or bonus payments. Holding costs and holding income are clearly calculable, provided that the amount of interest over the term can be correctly estimated, as well as if upcoming dividend payments, subscription rights and any upcoming bonus payments are known. If the holding costs exceed the returns, the prices of the futures quote above the price of the underlying asset (stock or index) – in this case the basis would be positive, if the returns are higher, the prices of the futures quote below the price of the underlying asset – in this case the basis would be negative.

If the basis is determined between a futures and a price index (e.g. Dow Jones, IBEX 35, CAC-40), dividends of the stocks contained in the indices must be taken into account. Thus, the basis can be negative (future quoted under cash) if holding income is generated during the term that exceeds the costs. When calculating the basis between a futures and a performance index (e.g. DAX 30 as a performance index – namely the index to which the FDAX refers), the basis is usually positive (provided that negative interest must not be assumed), since in this case, income (such as dividends) has no effect on the price performance of the underlying instrument. In a performance index, dividends are immediately “reinvested” when the price is calculated, whereas in price indices, dividend payments from the stock companies included in the index lead to price markdowns in the index itself.

Insertion: This fact also makes it nonsensical to compare the price of the Dow Jones with the price of the DAX 30 index (as is often done in the long term). Rather, the comparison of the Dow Jones with the DAX price index would be possible, although here too the share weighting is different, which also makes the two starting values difficult to compare. But the main difference is the different view of the dividend. While this is deducted in the Dow Jones, it is reinvested in the DAX 30. In fact, the DAX price index is only at 6,175.31 points (as of 20 March 2015), while the DAX 30 closed at 12,039.37 on the same day. This makes it clear that almost 50 percent of the price in the DAX-30 represents reinvested dividends.

This means that earnings are “under the table” when calculating the basis between the DAX-30 and its future, and we only take into account the holding and financing costs. Additional factors that are taken into account when calculating the basis are individual tax rates and transaction costs. These last two factors in particular are different for each arbitrageur, which means that there is not ONE generally valid basis, but the basis per arbitrageur is individual.

How do we have to imagine this in concrete terms? Let’s assume that you want to invest 300,000 euros. You decide to invest in the DAX index, so you would have to use this money to buy all 30 DAX stocks according to their weighting, but then you would have the DAX exactly in your portfolio. What is the situation in your portfolio now? You own 30 different DAX shares, in the exact DAX index weighting in the total amount of 300,000 euros. Alternatively, you could also put a DAX future into your portfolio. All you have to do is advance the margin required by Eurex, which is assumed to be 21,000 euros for the sake of simplicity. This means that you would now have 269,000 euros “left over”, which you could take to the bank and receive interest for the term. If we now leave out transaction fees and taxes for the sake of clarity, this interest income gives you a financial advantage, which, converted into points, represents the basis between the cash and the future. Since income from dividends is not taken into account in the DAX-30 Performance Index (because it is immediately and completely reinvested), the basis is positive here, because now it only reflects the interest advantage.

What is special about the basis is that it is reduced daily, as the interest advantage is melting away daily. Consequently, on the expiration date, the basis is at zero, as there is no more interest income on the last trading day. Any small arithmetical differences can be explained by different transaction costs. As a result, the basis strives towards zero at expiration, not only in this example, but for all futures.

### What does the basis development look like in a price index?

As mentioned above, a price index takes into account the distribution of dividends in the price trend. Therefore, the distribution of dividends must also be taken into account when calculating the basis. If the income from the dividends exceeds the financing costs, the basis of the future at the cash desk must be negative. However, the basis decreases towards the expiration date, so that here too the price of the future and the price of the underlying instrument (cash) are almost identical on the expiration date. Here too, the only possible deviations are the different transaction costs.

### But how is arbitrage possible?

Both the future and the cash register are two independent, freely tradable securities on the exchange. Their only mathematical connection to find a fair price relationship to each other is the above described basis. However, this basis is not a fixed band that guarantees that the cash and the future always have a fair relationship to each other, which means that during trading, the future and the cash can move closer to each other or further away from each other, whereby there are differences to the calculated basis. This situation results in rewarding arbitrage opportunities, and once they are in place, it is usually not long before they are exploited.

If the processing of a large buy order in the future leads to an increase in the base compared to the cash desk, an arbitrageur becomes active by selling the future and buying the cash desk. This has the following effect on the market: demand in the spot stocks increases, and the pressure to supply in the future also increases. As a result, the price of the cash register rises while the price of the future falls, reducing the basis between the cash register and the future. If, on the other hand, a stronger sell order in the future or a stronger buy order in the cash register reduces the basis between the two values, the opposite arbitrage effect sets in and the arbitrator sells cash and in return increases demand in the future. This causes the basis to expand again, the future rises and the cash desk falls.

Buying the cash and selling the future at the same time is called “cash and carry”, the reverse transaction, i.e. selling the cash and buying the future, is called “reverse cash and carry”.

The ideal case for any arbitrageur would be if he could carry out both cash and carry and reverse cash and carry transactions on every trading day or at least at short intervals, and thus realise the arbitrage proceeds in a reasonable period of time. The fact is that once an arbitrage profit has been made, it can only be collected without risk when the position once built up is also closed again. As long as an arbitrage position is open, there are residual risks, which we will look at below.

### The interest rate risk

Probably the most obvious risk of an arbitrageur in a performance index is the interest rate risk. As we have seen in advance, the interest rate for the term of a contract is a decisive valuation component. If the interest rate rises unexpectedly during the term of a contract, the basis between the future and the cash position expands mathematically. This effect occurs because the financing advantage resulting from the lower capital expenditure when purchasing a future is increased. If, on the other hand, the interest rate unexpectedly falls during the term of the respective contract, the basis between the cash desk and the future naturally also shrinks. If such a development occurs unexpectedly, a change in interest rates is directly reflected in the arbitrageur’s portfolio through unexpected gains or losses.

Example (1): the arbitrageur has a cash and carry position of 10,000 futures short and the equivalent in cash long with an assumed fair underlying of ten points. If an unexpected increase in interest rates for the remaining term of the position leads to an expansion of the basis by only one point, the increase in the price of the basis relative to 10,000 futures short results in an immediate loss of EUR 250,000 (EUR 10,000 * EUR 25). In contrast, an unexpected interest rate reduction would have positive effects, because the unexpected merging of the basis by one point would result in a profit of 250,000 €.

Example (2): the arbitrageur holds a reverse cash and carry position of 10,000 futures long and the equivalent in cash short for an assumed fair underlying of ten points. An unexpected interest rate increase would now yield a profit of 250,000 euros per point for the remaining term of the position, as the basis rises, while an unexpected interest rate decrease would lead to a corresponding loss.

In order to avoid these risks, it is very important to feed in a correct interest rate expectation. This risk increases the longer the position remains open in the arbitrator’s book.

However, the interest rate risk is also reflected in a reverse cash and carry position in a special way. Here the arbitrageur holds short cash against long future. The spot values are usually borrowed, so that stock lending costs are due and must be taken into account in the base calculation. If there is now an unexpected change in the interest rate, this effect can also affect the borrowing costs, which means that the arbitrageur’s individual basis can develop to his disadvantage.

### The “roll risk

We differentiate between futures on whose expiration there is effective exercise combined with effective delivery of the underlying asset (e.g. commodity futures and bond futures) and forward contracts based on cash settlement (e.g. equity indices). In the case of the latter, there is no physical delivery of the underlying on the expiration date, but the value is settled on the basis of a cash settlement. These futures also include the FDAX. An additional problem arises in these futures on this settlement basis.

While the cash position in a stock index could be held practically indefinitely, the counter position in futures is limited to the respective term. If an arbitrageur were not to become active until the expiration of his futures position, his spot position would be unhedged after the expiration of the futures. A delivery of the shares is excluded, the sale of the entire position in the market without negative effects is not possible. Consequently, if an arbitrator is unable to close his position at the expiration of the futures contracts, he must roll the open futures position into the next expiration month. There is a further spread risk here. The difference between the front month and the following month is the difference between the respective values of their respective bases. However, as in practice these two futures can also be traded independently of each other, there can also be differences between the two prices in reality, so that the difference between the two futures can be higher or lower than it should be in arithmetical terms. This in turn creates arbitrage opportunities, but increases the risk of an arbitrage position being held.

### The dividend risk

The dividend risk differs depending on the performance index or price index.

### Performance Index

In a performance index, the dividends are mathematically reinvested in full, so that a dividend distribution of a share has no influence on the price level of the index. In a price index, on the other hand, the price of the underlying instrument (index) falls when a share included in the index pays a dividend and its price is reduced by the amount of the dividend. Nevertheless, dividend payments also have their pitfalls for an arbitrageur in a performance index. The arbitrageur must reflect his cash position in a form that corresponds absolutely to the underlying asset (performance index). The distribution of a dividend and its immediate reinvestment changes the weighting of the share in the index in a first step. The arbitrageur must also carry out this weighting in his portfolio. This means that in the case of a cash and carry position, the dividend received must also be immediately reinvested by the arbitrageur in the purchase of the corresponding share. In the case of a reverse cash-and-carry position, the arbitrageur must adjust the short position of the corresponding stock by the respective factor that changes due to the dividend payment.

### Price Index

With a price index, a dividend distribution has a direct influence on the price performance of the underlying instrument (index). This means that the arbitrageur must have a clear idea in advance which company will pay a dividend and how much. If the dividend payment falls within the term of his positioning, the amount of the dividend has a direct influence on the level of the basis between the index and the futures. The more dividends that are paid out during the term of the contract, the higher the income from the cash positions will be and thus possibly exceed the interest advantage. As a result, we have a negative basis. If trading now takes place on the basis of this calculated basis, but it turns out in retrospect that an included dividend is either not paid or not paid in full, this leads to a change in the basis, comparable to an unexpected interest rate increase or decrease. An incorrectly calculated dividend therefore results in an incorrectly calculated basis and can therefore quickly convert an alleged arbitrage gain into an arbitrage loss.

### Why can’t an arbitrageur close out his position again and again?

In theory, it should be possible to take advantage of the fluctuation of a future around its fair basis on both sides, so that an arbitrageur should have several opportunities to enter into both cash-and-carry and reverse cash-and-carry positions in a short period of time and thus keep his overall arbitrage positions small or be able to close them again and again in a short time. The practice usually looks different. In most cases, one can estimate which of the two possible positions are most likely to be held by the respective arbitrageurs on the basis of the price movements of the underlying instrument. In dominant uptrends, futures tend to outperform their underlying asset (index). Statistically speaking, in phases of uptrends, a too high valuation of the future outweighs a too low one. Thus, in most cases, uptrends lead to the establishment of a cash and carry position with the arbitrageurs. Conversely, it is in phases of steadily falling markets. In these phases, the price falls below the base, so that reverse cash and carry positions dominate.

As the risks described above mean that even the largest banks do not provide arbitrageurs with unlimited funds, persistent trend movements can lead to the respective arbitrageur exhausting the available funds. If such a case occurs, it is hardly possible to build up further arbitrage positions. In smaller markets, such as Spain, France, Italy and similar places, such a development can lead to the future being over- or undervalued for a long period of time without significant arbitrage being carried out. In order to be able to act again, counter positions are sometimes taken, which in themselves generate losses, but which in turn release funds to be able to exploit the higher arbitrage opportunities.

### How is the basis in a future calculated on the bond market?

The basic principle of the base calculation in the bond market is similar to the base calculation in the stock market. We calculate the forward price (price of the futures) of a bond by adding up the costs and revenues of holding a bond. The holding costs are the interest or financing costs. In contrast, we receive income from the bond in the form of coupon payments (interest payments). If we take the spot price of the bond and add the basis, we get the fair value of the futures.

In part 2 we will discuss how short term traders can benefit from the fact that arbitrage is taking place in the market.