6.18    Annual chain volume indexes in the ASNA are derived by compounding successive year-to-year Laspeyres indexes. A Laspeyres volume index from year \(y-1\) to year \(y\) is derived by dividing the value of the aggregate in year y at year \(y-1\) prices (i.e. using the volumes in year \(y\) but the prices of year \(y-1\)) with the current price value in year \(y-1\); that is:

\(\large{L_Q} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }},\)

where \(P_i^y\) and \(Q_i^y\) are prices and quantities of the \(i^{th}\) product in year \(y\) and there are \(n\) products.

6.19    Annual chain Laspeyres volume indexes can be formed by multiplying consecutive year-to-year indexes; that is:

\(\large L_Q^y = \frac{{\sum\limits_{i = 1}^n {P_i^0Q_i^1} }}{{\sum\limits_{i = 1}^n {P_i^0Q_i^0} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^1Q_i^2} }}{{\sum\limits_{i = 1}^n {P_i^1Q_i^1} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^2Q_i^3} }}{{\sum\limits_{i = 1}^n {P_i^2Q_i^2} }} \times \ldots \times \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}\)

6.20    The derivation of quarterly chain Laspeyres volume indexes is in concept no different to compiling annual chain volume indexes. However there is the complication of seasonality to contend with. In the ASNA, annual base years (i.e. annual weights) are used to derive quarterly volume indexes rather than having quarterly base periods. If quarterly base periods were to be used then this should only be done using seasonally adjusted data and not original data.

6.21    Consequently the Laspeyres-type³⁴ volume index from year \(y-1\) to quarter \(c\) in year \(y\) takes the form:

\(\large L_Q^{\left( {y - 1} \right) \to \left( {c,y} \right)} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}4q_i^{c,y}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }} = \sum\limits_{i = 1}^n {\frac{{4q_i^{c,y}}}{{Q_i^{y - 1}}}s_i^{y - 1},} \)

where \(q_i^{c,y}\) , is the volume of product \(i\) in the \(c^{th}\) quarter of year \(y\) and \(s\) is the share (weight) of the \(i^{th}\) item. For more detail see Annex A to this chapter.

6.22    There are several ways of linking annually weighted quarterly Laspeyres-type volume indexes. Annex A to this chapter describes the three methods outlined in 2008 SNA, including the one-quarter overlap method which is used in the ASNA.

6.23    After linking, the quarterly chain volume estimates are benchmarked to their annual counterparts. This benchmarking serves two purposes:

1. It overcomes the inconsistency arising from the different linking methods required to compile quarterly chain volume estimates versus annual chain volume estimates; and
2. It ensures the quarterly chain volume estimates are consistent with the data from the annual S-U tables. The Supply-Use tables are explained in more detail in Chapter 7.

6.24    The one-quarter overlap method involves calculating a link factor using overlap values for a single quarter. To link the four quarters of year \( y-1\) at year \( y-2\) average prices with the four quarters of year \( y\) at year \( y-1\) average prices, a one-quarter overlap can be created for either the fourth quarter of year \(y-1\) or the first quarter of year \( y\). The link factor derived from an overlap for the fourth quarter of year \(y-1\):

\(\large{ = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}q_i^{4,\left( {y - 1} \right)}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 2}q_i^{4,\left( {y - 1} \right)}} }}}\)

6.25     Multiplying the quarterly values for year \( y-1\) at year \( y-2\) average prices with this link factor puts them on to a comparable valuation basis with the quarterly estimates for year \( y\) at year \(y-1\) prices.

6.29    As described above, the ABS derives its annual and quarterly chain volume estimates using the Laspeyres formula with annual base years. With the exception of the latest quarters, quarterly chain volume estimates are derived by linking together estimates derived in the average prices of the previous year. However, the latest five to eight quarters are derived in the average prices of the latest base year, which is the year before the previous year. The reason for this exception is the delay in deriving the annual current price estimates of gross value added by industry, which are needed to form the base year weights for the volume estimates of GDP(P) and its components (See Chapter 9 for more information on Gross Value Added). Even though estimates of final expenditures could be derived in the average prices of the previous year for all years, the ABS has decided to apply the same approach and timing for all its volume estimates.

6.30    It is ABS practice to introduce a new base year with the release of the September quarter accounts. At the same time, the reference year is advanced one year to coincide with the latest base year, thereby ensuring additivity for the latest quarters. The process is best explained with some examples. 

6.31    In the June quarter release in year y, the quarterly chain volume estimates are derived by linking:

• the eight quarters from September quarter year \(y-2\) to June quarter year \(y\) in the average prices of financial year \(\frac{y-3}{y-2}\);
• the four quarters from September quarter year \(y-3\) to June quarter year \(y-2\) in the average prices of financial year \(\frac{y-4}{y-3}\); and
• all earlier quarters in the average prices of the previous financial year.

Financial year \(\frac{y-3}{y-2}\) is the reference year.

6.32    In the September quarter release in year y, the quarterly chain volume estimates are derived by linking:

• the five quarters from September quarter year \(y-1\) to September quarter year \(y\) in the average prices of financial year \(\frac{y-2}{y-1}\);
• the four quarters from September quarter year \(y-2\) to June quarter year \(y-1\) in the average prices of financial year \(\frac{y-3}{y-2}\); and
• all earlier quarters in the average prices of the previous financial year.

Financial year \(\frac{y-2}{y-1}\) is the reference year.

6.33    Re-referencing results in revisions to the levels of the chain volume measures, but it does not in itself result in revisions to growth rates, although growth rates can be revised for other reasons. One reason is that the introduction of a new reference year coincides with the introduction of a new base year for the latest four quarters. Another reason is the introduction of revised annual estimates, to which the quarterly estimates are benchmarked. 

6.34     In the dissemination of quarterly national accounts, contributions to growth play a prominent role - a role that has become more important with the loss of additivity that has accompanied the introduction of chain volume estimates. While the chain volume estimates of the components of an aggregate do not generally add up to the chain volume estimate of the aggregate, it is possible to calculate the contributions of each component to the growth rate of the aggregate. These growth rates are additive, which will be explained below.

6.35     Deriving contributions to growth from additive data, such as constant price estimates, is straightforward. Deriving the contributions to growth of quarterly chain volume estimates is more complex and unlike constant price estimates there is no one formula that can be applied in all cases. Rather, the methods that can be used depend on how the chain volume estimates have been derived, which include: 

• the index formula used (e.g. Laspeyres or Fisher);
• annual or quarterly base years;
• method of linking in the case of annual base years; 
• the period over which the contributions to growth are calculated (e.g. quarter-to-quarter or quarter on same quarter of previous year); and
• special features of a component (e.g. changes in inventories).

6.36     The method used in the ASNA compromises the additivity of chain Laspeyres volume indexes in the year following the reference year. This phenomenon arises because the chain volume estimates in this year are in effect values in the prices of the previous year. 

6.37     The quarterly chain volume estimates of the components and the aggregates in year y-1 and year y are re-referenced to their respective annual current price values in year y-1 by multiplying them by their implicit price deflators for year y-1. This amounts to dividing each time series of quarterly chain volume estimates by the annual value of the chain volume estimates in year y-1 and then multiplying the result by the current price value in year y-1. The resulting quarterly chain volume estimates are additive in year y, and so the contributions to growth for quarters within year y are exactly additive.

6.38     To determine the quarterly contribution to growth of a component of an aggregate, the following calculation occurs:

\(\large Contrib.{\left( {{x_i},X} \right)^{c,y}} = \frac{{P_{{x_i}}^{y - 1}}}{{P_X^{y - 1}}} \times \frac{{\left( {x_{CVi}^{c,y} - x_{CVi}^{c - 1,y}} \right)}}{{X_{CVi}^{c - 1,y}}}\)

where 

• \(X_{CV}^{c,y}\) is the chain volume estimate of an aggregate, such as GDP, in the \(c^{th}\) quarter of year \(y\) and \(P_X^{c,y}\) is the corresponding implicit price deflator; and
• \(x_{C{V_i}}^{c,y}\) is the chain volume estimate of the \(i^{th}\) component of the aggregate in the \(c^{th}\) quarter of year \(y\) and \(P_{{x_i}}^{c,y}\) is the corresponding implicit price deflator.

6.39    During the 2012-13 annual compilation cycle, improvements were made to the method by which pre-1985-86 volume components of GDP(E) are calculated. These components were previously constant price estimates, and not 'true' chain volume measures. This break in series dated from the initial introduction of chain volume measures to the set of compilation methods underpinning the Australian national accounts. Chain volume measures were originally only implemented back to 1985-86. Prior year estimates were calculated as backcasts of historic constant price estimates.

6.40    Implementation of chain volume measures for pre-1985-86 estimates of GDP(E) was not carried through the complete aggregation structure, but headline components (consumption, investment and trade) are all now calculated as chain volume measures, as well as GDP(E) itself, back to 1959-60. Owing to difficulties in recalculating change in inventories estimates in chain volume terms prior to 1985-86, this component is calculated residually for this part of the time series. The result is that percentage point contributions to chain volume GDP(E) growth are now additive for the full time series. Additionally, real income measures such as real gross domestic income (RGDI) are now fully consistent with the terms of trade series across the full time series.

6A.2    The general formula for a Laspeyres volume index from year \(y-1\) to year \(y\) is given by:

\(\large {L_Q} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }},\)         - - - - - - - (1)

where \(P_i^y\) and \(Q_i^y\) are prices and quantities of the \(i^{th}\) product in year \(y\) and there are \(n\) products. The denominator is the current price value of the aggregate in year \(y-1\) and the numerator is the value of the aggregate in year \(y\) at year \(y-1\) average prices.

6A.3 A Paasche volume index from year \(y-1\) to year \(y\) is defined as:

\(\large {P_Q} = \frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^yQ_i^{y - 1}} }},\)         - - - - - - - (2)

6A.4    A Fisher index is derived as the geometric mean of a Laspeyres and Paasche index:

\(\large {F_Q} = {\left( {{L_Q}{P_Q}} \right)^{1/2}}\)         - - - - - - - (3)

6A.5    A Paasche price index from year \(y-1\) to year \(y\) is defined as:

\(\large {P_P} = \frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }},\)         - - - - - - - (4)

6A.6    When this Paasche price index is divided into the current price index from year \(y-1\) to year \(y\) a Laspeyres volume index is produced:

\(\large \frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}{{{P_P}}}}} = \frac{{\frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}}}{{\frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}}} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y-1}} }} = {L_Q}\)         - - - - - - - (5)

6A.7    Evidently, Laspeyres volume indexes and Paasche price indexes complement each other, and vice versa

6A.9    Annual chain Laspeyres and Paasche volume indexes can be formed by multiplying consecutive year-to-year indexes: 

\(\large L_Q^y = \frac{{\sum\limits_{i = 1}^n {P_i^0Q_i^1} }}{{\sum\limits_{i = 1}^n {P_i^0Q_i^0} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^1Q_i^2} }}{{\sum\limits_{i = 1}^n {P_i^1Q_i^1} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^2Q_i^3} }}{{\sum\limits_{i = 1}^n {P_i^2Q_i^2} }} \times ..... \times \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}\)         - - - - - - - (6)

 \(\large P_Q^y = \frac{{\sum\limits_{i = 1}^n {P_i^1Q_i^1} }}{{\sum\limits_{i = 1}^n {P_i^1Q_i^0} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^2Q_i^2} }}{{\sum\limits_{i = 1}^n {P_i^2Q_i^1} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^3Q_i^3} }}{{\sum\limits_{i = 1}^n {P_i^3Q_i^2} }} \times ..... \times \frac{{\sum\limits_{i = 1}^n {P_i^yQ_i^y} }}{{\sum\limits_{i = 1}^n {P_i^yQ_i^{y - 1}} }},\)         - - - - - - - (7)

6A.10    Chain Fisher indexes can be derived by taking their geometric mean: 

\(\large F_Q^y = {\left( {L_Q^yP_Q^y} \right)^{1/2}}\)         - - - - - - - (8)

6A.11    All of these indexes can be re-referenced by dividing them by the index value in the chosen reference year and multiplying by 100 to produce an indexed series, or by multiplying by the current price value in the reference year to obtain a series in monetary values.

6A.18    It is recommended in the 2008 SNA that the annual national accounts should be balanced in both current prices and in volume terms using S-U tables. In most cases, the volume estimates are best derived in the average prices of the previous year rather than some distant base year. This is for two key reasons:

• assumptions of fixed relationships in volume terms are usually more likely to hold in the previous year’s average prices than in the prices of some distant base year: and;
• so that the growth rates of volumes and prices are less affected by compositional change. 

6A.19    The compilation of annual S-U tables in current prices and in the average prices of the previous year lends itself to the compilation of annual Laspeyres indexes and to the formation of annual chain Laspeyres indexes. 

6A.20    In order to compute annual Fisher indexes from data balanced in a S-U table, it is conceptually desirable to derive both Laspeyres and Paasche indexes from that data. The former requires balancing the S-U tables of the current year \((y)\) in current prices \((y)\) and in the average prices of the previous year \((y-1)\) and the latter requires balancing S-U tables in the previous year \((y-1)\) in the average prices of that year \((y-1)\) and in the average prices of the current year \((y)\). Thus, the compilation of annual chain Fisher indexes, at least in concept, is somewhat more demanding than compiling annual chain Laspeyres indexes.

6A.21    Computationally, the derivation of quarterly chain indexes from quarterly data with quarterly base periods is no different to compiling annual chain indexes from annual data with annual base periods. As recommended by the 2008 SNA, if quarterly volume indexes are to have quarterly base periods and be linked each quarter, then it should only be done using seasonally adjusted data. Furthermore, if the quarterly seasonally adjusted data are subject to substantial volatility in relative prices and relative volumes, then chain indexes should not be formed from indexes with quarterly base periods at all. Even if the quarterly volatility is not so severe, quarterly base periods and quarterly linking are not recommended using the Laspeyres formula because of its greater susceptibility to drift than the Fisher formula.

6A.22    A way round this problem is to derive quarterly volume indexes from a year to quarters. In other words, use annual base years (i.e. annual weights) to derive quarterly volume indexes. Consider the Laspeyres annual volume index in formula 1. It can be expressed as a weighted average of elemental volume indexes:

\(\large {L_Q} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }} = \sum\limits_{i = 1}^n {\left( {\frac{{Q_i^y}}{{Q_i^{y - 1}}}} \right)} s_i^{y - 1},\;where\;s_i^{y - 1} = \frac{{P_i^{y - 1}Q_i^{y - 1}}}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}\) - - - - - - - (9)

\(s_i^{y - 1}\) is the share, or weight, of the \(i^{th}\) item in year \(y-1\).

6A.23    Paasche volume indexes can also be expressed in terms of a weighted average of the elemental volume indexes, but as the harmonic, rather than arithmetic, mean.

6A.24    A Laspeyres-type³⁷ volume index from year \(y-1\) to quarter \(c\) in year \(y\) takes the form:

\(\large L_Q^{(y - 1) \to (c,y)} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}4q_i^{c,y}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }} = \sum\limits_{i = 1}^n {\frac{{4q_i^{c,y}}}{{Q_i^{y - 1}}}} s_i^{y - 1},\) - - - - - - - (10)

where \({q_i^{c,y}}\) is the volume of product \(i\) in the \(c^{th}\) quarter of year \(y\). In this case the annual current price data in year \(y-1\) are used to weight together elemental volume indexes from year \(y-1\) to each of the quarters in year \(y\). The “4” in formula 10 is to put the quarterly data onto a comparable basis with the annual data. Note that constant price (or fixed-weighted) volume indexes are traditionally formed in this way, but the weights are kept constant for many years.

6A.25    2008 SNA describes how chain Fisher-type indexes of quarterly data with annual base periods can be derived:

"Just as it is possible to derive annually chained Laspeyres-type quarterly indices, so it is possible to derive annually chained Fisher-type quarterly indices. For each pair of consecutive years, Laspeyres-type and Paasche-type quarterly indices are constructed for the last two quarters of the first year, year \(y-1\) and the first two quarters of the second year, year \(y\). The Paasche-type quarterly indices are constructed as backward-looking Laspeyres-type quarterly indices and then inverted. This is done to ensure that the Fisher-type quarterly indices are derived symmetrically. In the forward-looking Laspeyres-type indices the annual value shares relate to the first of the two years, whereas in the backward-looking Laspeyres-type indices the annual value shares relate to the second of the two years.

For each of the four quarters a Fisher-type index is derived as the geometric mean of the corresponding Laspeyres-type and Paasche-type indices. Consecutive spans of four quarters can then be linked using the one-quarter overlap technique. The resulting annually chained Fisher-type quarterly indices need to be benchmarked to annual chain Fisher indices to achieve consistency with the annual estimates."³⁸

6A.30    While there are different ways of linking annual Laspeyres volume indexes, they all produce the same result. But this is not true when it comes to linking annual-to-quarter Laspeyres-type volume indexes for consecutive years. Paragraphs 15.46 -15.50 of the 2008 SNA discuss three methods for linking these Laspeyres-type volume indexes; they are:

• Annual overlap;
• One-quarter overlap: and
• Over the year.

6A.31    When a Laspeyres-type quarterly volume index from year \(y-1\) to quarter \(c\) in year \(y\) is multiplied by the current price value for year \(y-1\) divided by four, then a value for quarter \(c\) is obtained in the average prices of year \(y-1\).

\(\large \sum\limits_{i = 1}^n {\frac{{4q_i^{c,y}}}{{Q_i^{y - 1}}}} s_i^{y - 1}\frac{1}{4}\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} = \sum\limits_{i = 1}^n {\frac{{4q_i^{c,y}}}{{Q_i^{y - 1}}}} \frac{{P_i^{y - 1}Q_i^{y - 1}}}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}\frac{1}{4}\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} = \sum\limits_{i = 1}^n {q_i^{c,y}P_i^{y - 1}} \) - - - - - - - (11)

6A.32    Hence, the task of linking quarterly Laspeyres-type volume indexes for two consecutive years, year \(y-1\) and year \(y\), amounts to linking the quarterly values of year \(y-1\) in year \(y-2\) average prices with the values of year \(y\) in year \(y-1\) average prices.

6A.33    One way of putting the eight quarters described in the previous paragraph onto a comparable valuation basis is to calculate and apply a link factor from an annual overlap. Values for year \(y-1\) are derived in both \(y-1\) prices and \(y-2\) prices and then the former is divided by the latter; thus, giving an annual link factor for year \(y-1\) to year \(y\) is equal to:

\(\large \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 2}Q_i^{y - 1}} }}\) - - - - - - - (12)

6A.34    Multiplying the quarterly values for year \(y-1\) at year \(y-2\) average prices with this link factor puts them on to a comparable valuation basis with the quarterly estimates for year \(y\) at year \(y-1\) prices. Note that this link factor is identical to the one that can be used to link the annual value for year \(y-1\) at \(y-2\) average prices with the annual value for year \(y\) at year \(y-1\) average prices. Therefore, if the quarterly values for every year \(m\) at year \(m1\) average prices sum to the corresponding annual value, then the chain-linked quarterly series will be temporally consistent with the corresponding chain-linked annual series.

6A.35    The one-quarter overlap method, as its name suggests, involves calculating a link factor using overlap values for a single quarter. To link the four quarters of year \(y-1\) at year \(y-2\) average prices with the four quarters of year \(y\) at year \(y-1\) average prices, a one-quarter overlap can be created for either the fourth quarter of year \(y-1\) or the first quarter of year \(y\). The link factor derived from an overlap for the fourth quarter of year \(y-1\) is equal to:

\(\large \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}q_i^{4,(y - 1)}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 2}q_i^{4,(y - 1)}} }}\) - - - - - - - (13)

6A.36    Multiplying the quarterly values for year \(y-1\) at year \(y-2\) average prices with this link factor puts them on to a comparable valuation basis with the quarterly estimates for year \(y\) at year \(y-1\) prices. 

6A.37    A key property of the one-quarter overlap method is that it preserves the quarter-to-quarter growth rate between the fourth quarter of year \(y-1\) and the first quarter of year \(y\) - unlike the annual overlap method. The “damage” done to that growth rate by the annual overlap method is determined by the difference between the annual and quarter link factors. Conversely, this difference also means that the sum of the linked quarterly values in year \(y-1\) differ from the annual-linked data by the ratio of the two link factors. Temporal consistency can be achieved by benchmarking the quarterly chain volume estimates to their annual counterparts.

6A.38    The following table illustrates the methods used to deriving link factors

6A.39    The over-the-year method requires compiling a separate link factor for each type of quarter. Each of the quarterly values in year \(y-1\) at year \(y-2\) average prices is multiplied by its own link factor. The over-the-year quarterly link factor for year \(y-1\) at average year \(y-2\) prices to year y at average year \(y-1\) prices for quarter c is equal to:

\(\large \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}q_i^{c,(y - 1)}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 2}q_i^{c,(y - 1)}} }}\) - - - - - - - (14)

6A.40    The over-the-year method does not distort quarter-on-same quarter of previous year growth rates, since the chain-links refer to the volumes of the same quarter in the respective previous year valued at average prices of that year. However, it does distort quarter-to-quarter growth rates. In addition, the linked quarterly data are temporally inconsistent with the annual-linked data and so benchmarking is needed. Given these shortcomings, the over-the-year method is best avoided.

6A.41    The following tables provide examples of using the annual and one-quarter overlap methods.