Decomposition of Index Returns

In this section we will get a deeper insight into the return structure of commodity indices. As already mentioned introductory to Section 4.1 there are three different index types calculated by the index issuer: the total return index, representing the return development of a fully collateralized commodity investment, the excess return index, representing the return development of a leveraged commodity investment, and the spot return index, representing the simple commodity price changes over time. But how are the three types connected to each other? Figure 4.8 shall give a first overview of their team play.¹⁰¹

Total Return = Excess Return + Interest Rate Return

XXXX

Figure 4.8: Decomposition of Commodity Index Return

A single asset’s index is nothing else but a time series of the prices realized by the underlying asset. In stock markets this equals a buy and hold trading strategy. In commodity markets it is not that easy because commodities are traded with futures contracts, i.e. the underlying has a maturity and therefore, investments have to be rolled over different positions by and by resulting in the so-called futures or excess return as the pure return produced by commodity investments. It depends of the actual price changes of the underlying commodity covered in the spot return and the roll return realized by rolling futures positions forward under the current term structure. Later in this section we will see the mathematical derivation of this dependence structure in Theorem 4.1.

The most common way to construct a single commodity index is to roll someone’s position from the first to the nearest longer term contract because the nearby con-tracts have generally the highest liquidity. Futures investments need minimal cash requirements that are only used to serve margin calls. But to actually add com-modities as part of an investment portfolio someone has actually to invest a certain amount reserved for commodity investment. Because this is not possible with

fu-101Figure 4.8 and the following calculations are based on log returns as of Definition C.2. Com-pare [Kat Oomen 2006]. For a commodity return decomposition based on simple returns see [Geer 2000].

tures contracts, someone has to invest the reserved amount into a reference asset called collateral. The issuers of the main indices usually use T-Bills producing his-torically an annualized return of 3-4%. Because log returns are additive,¹⁰² the first decomposition of Figure 4.8 of total return into excess and interest rate return is quite intuitive. But what about the second decomposition of excess return into spot and roll return?

To answer this question we will first give an example calculation by constructing the futures return time series by rolling the maturing contract into the next nearby con-tract for the crude oil and copper futures concon-tract already known from in Figure 3.2 in Section 3.1 and second derive the mathematical illustration in Theorem 4.1. For it, Table 4.3 and Table 4.5 summarize the price movements of the respective con-tracts. The column header give the maturity T of the respective contract and the raw header the respective date t at which the price of the contract is measured.

The respective spot return time series is constructed by using the price of the front month futures contract as a proxy.¹⁰³ The respective values are highlighted by bold letters.

Crude Oil (US dollar) Jan 06 Feb 06 Mar 06 Apr 06 May 06 Jun 06 Jul 06 30. Dec 2005 57.98

→

31. Jan 2006 68.35

→

Table 4.3: Construction of a Futures Return Series for Crude Oil

First, we will examine the construction of a futures return series exemplified by the crude oil price series. The construction follows the arrows in Table 4.3 and is based on the following thought: From the end of November 2005 to the end of December 2005 the investor holds the January 2006 contract. Before the contract expires in January 2006 he closes his position and at the same time he opens a new position in the February 2006 contract which he holds until the end of January 2006.

Following Definition C.2 the futures return is given as:

r_F(t) ≡ ln F (t, T ) F (s, T )



, 0 ≤ s < t ≤ T

102See Theorem 4.1.

103The procedure is inspired by [Markert 2005] and [Gorton Rouwenhorst 2004].

Implicating, the investor realizes a crude oil futures return of:

Again, before the contract expires he closes his February 2006 position and opens a position in the March 2006 contract. The crude oil futures return time series is continued with the following value:

Running the described construction methodology over the reported times a whole futures return time series evolves. The results are reported in the first column of Table 4.4 and also known as excess return as of Figure 4.8.

The next step to encode the different futures return elements is to construct the spot return. We use the bold highlighted prices in Table 4.3 because the front month futures contract serves as proxy. Following Definition C.2 the spot return is given as:

r_P(t) ≡ ln P (t) P (s)



, 0 ≤ s < t ≤ T Implicating, the first crude oil spot return value is given by:

r_P(J an) = ln P (Jan)

The second value is gives by:

r_P(F eb) = ln P (F eb)

Again, running the described calculation rule over the reported times a whole spot return time series evolves. All values are listed in the second column of Table 4.4.

Although crude oil went up in price over the last months and could realize a high spot return the positive slope of the term structure as shown in Figure 3.2 disembogue into a negative difference between futures and spot returns over the whole period as documented in the last column of Table 4.4. This gap is caused by rolling a maturing futures contract into the next nearby month futures contract. Because the market is in contango the next nearby month futures contract is more expensive than the maturing futures contract and the investor realizes a loss amounting to -17.4% by

Future Return Spot Return Difference = Roll Return

Table 4.4: Spot, Future and Roll Return Time Series for Crude Oil

rolling his position forward. The so-called roll return first introduced in Figure 4.8 is mathematically derived in Theorem 4.1:

Theorem 4.1 Roll Return

Let F (t, T ) denote the commodity futures price at time t ∈ [0, T ] and let P (t) be the commodity spot price at time t ∈ [0, T ]. Moreover, we have 0 ≤ s < t ≤ T . Then the roll return is given by:

rr(t) = ln F (t, T )

Proof: Recall, the spot price, denoted by P (t), is approximated by the front month futures price, denoted by F (t, T ), i.e. we have: P (t) = F (t, T ). Therewith, As shown in Figure 3.2 the copper market is in backwardation, e.g. the negative slope of the term structure disembogues into a positive roll return what we will show in the following example. The price data of the respective futures contract are given in Table 4.5.

Calculating the return series with the same methodology described for the crude oil

Copper (US dollar) Jan 06 Feb 06 Mar 06 Apr 06 May 06 Jun 06 Jul 06 30. Dec 2005 4,538

→

31. Jan 2006 4,912

→

↓ 4,853 4,815 4,768 4,721

28. Feb 2006 4,881

→

31. Mar 2006 5,440

→

↓ 5,400 5,375

28. Apr 2006 7,118

→

31. May 2006 8,001

→

30. Jun 2006 7,425↓

Table 4.5: Construction of a Futures Return Series for Copper

example we end up with the values given in Table 4.6. Recall, the futures return series is calculated by following the arrows and the spot return series by following the bold letters. The ”backwarded” term structure produced a positive roll return amounting to 3.9% as shown in the last column of Table 4.6.

Future Return Spot Return Difference = Roll Return

Jan 2006 9.0% 7.9% 1.1%

Feb 2006 -0.1% -0.7% 0.5%

Mar 2006 11.7% 10.8% 0.8%

Apr 2006 27.2% 26.9% 0.3%

May 2006 12.4% 11.7% 0.7%

Jun 2006 -7.1% -7.5% 0.4%

Total 53.0% 49.2% 3.9%

Table 4.6: Spot, Future and Roll Return Time Series for Copper

The examples above have shown the impact of the term structure to the investors return. If a market is in contango the negative roll return will diminish the final return in spite of price increases yielding to positive spot returns. To push back the negative rolling impact in contangoed markets, someone could think about ex-tending the rolling periods. For instance, if the investor of the crude oil example had avoided rolling forward the positions monthly, and instead would have invested in January 2006 directly into the July 2006 contract he would have realized a fu-tures return of ln ^68.94_63.25
= 8.6% because the roll return would have decreased to ln ^63.25_57.98
= −8.7%. This conclusion is used by Merrill Lynch. In May 2006 they introduced the ML Oil Return and Income Index that rolls forward its oil futures positions every third month.¹⁰⁴ Backtesting has shown that in fact they could realize an excess return in comparison to one month rolling, long only oil futures indices

104See [Merrill Lynch 2006]. To be precise, Merrill Lynch employ a short option trading facility as well to minimize the negative influence of contango to the roll return.

over the last 2 years. Given the current term structure as of July 2006 of NYMEX crude oil shown in Figure 4.9 this strategy is expected to work over the next nine months namely until April 2007 properly. From this point on, the market is ex-pected to be in backwardation again yielding to positive roll returns. Implicating, monthly rolling will be more attractive again.

Figure 4.9: Term Structure of NYMEX Crude Oil as per July 2006

Generally, the big public commodity indices described in Section 4.2 roll every month over a five day period each with 20% of the total futures investment caused by liquidity reasons. Trading volume is clustered around the front month contracts.

For instance, the most traded commodity futures contract worldwide, the NYMEX crude oil future, has in July 2006 approximately 230.000 open interests in the con-tract maturing in August 2006 less than half of this about 130.000 open interests in the contract maturing one month later namely in September 2006 and the contract maturing one year later namely in July 2007 has just 10.000 open interests. The example is supported by different issuer’s studies proofing that liquidity is clustered around the nearby contracts. For instance, following [Merrill Lynch 2006] the second nearby futures contract has only a trading volume of two thirds of the trading vol-ume of the first month futures contract. Nevertheless, Deutsche Bank has changed its trading strategy. They implemented the so-called optimum yield rolling strategy.

Depending on the shape of the forward curve, they roll the contracts forward into contracts that under liquidity requirements maximize the roll return.

Investor’s attention is generally attracted by asset classes that are on an upwards move. Legends like Jim Rogers have helped to establish commodities as an asset class and to convince many investors that the only way for commodities is up. The economic boom of emerging market countries and the long lasting expansion of pro-duction capacities will push prices further over the next years. But investors first have to understand that there is not the ”average commodity”. Therefore, the first part of this section, i.e. Section 5.1, will concentrate on single commodity returns and their interactions. Introductory, Section 5.1.1 shall give a first inside into their different risk and return profiles. We will use the conclusions from Section 3 and Section 4.4 to decompose excess returns, i.e. the pure commodity return, aiming to identify whether commodities offer a risk premium or not and how much risk an investor has to bear when investing into selected commodities. An interesting observation will be, that in contrast to traditional asset classes, the risk measure volatility goes up in bullish markets. Commodity price surges come in line with low inventories and the fear of supply interruptions yields into nervous market move-ments.

Although, the different types of commodities are influenced by their own specific risk factors, technological progress allows new substitution possibilities. So, com-modities that are on the first view totaly different among each other, might be more and more driven by the same risk factors and demand sources. But in which ex-tent can similar price movements be observed? Section 5.1.2 will show that only commodities of the same group show high overlapping among their price movement characteristics while combining different commodity groups will yield into balanced risk and return profiles. Section 5.1.3 will finally give the mathematical explanation of diversification and therewith will state, why commodity indices are suitable to get balanced commodity exposure.

The second part of this section, i.e. Section 5.2, will further focus on the statis-tical properties of such a balanced commodity exposure’s return. While different research focused on the construction and analysis of artificial commodity indices including e.g. [Gorton Rouwenhorst 2004] and [Erb Harvey 2006], little is done in analyzing actual market indices, e.g. [Kat Oomen 2006]. We will close the gap by analyzing the DJ-AIGCI total return index and its pure commodity return compo-nents.¹⁰⁵ We will uncover roll returns and show their impact on total returns in

105The switch from excess return in Section 5.1 to total return in the Section 5.2 is motivated as follows: The first part of Section 5 concentrates on the characteristics of single commodity

Section 5.2.1 and 5.2.2. Our findings in Section 5.2.3 stand in contrast to findings of [Gorton Rouwenhorst 2004] and [PIMCO 2006]. While we report negative skewness, they published positive. We reason this with two facts: First, they both construct artificial indices that are not investable and second, they consider a period from 1970 until 2005. Therewith, the value development considers the two major price surges over the last 100 years.

To close this section, we will report two major time series characteristics: stationarity in Section 5.2.4 and autocorrelation in Section 5.2.5. Our findings are in line with [Kat Oomen 2006]. They’ve already reported that the facility of autocorrelation in selected commodity returns, including among others corn, soybeans, live cattle, oil and gold, got lost in index returns.¹⁰⁶