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# Chain Index Numbers

According to the fixed base methods, the base remains the same and unchangeable throughout the series. But, as the time passes some items may be added in the series while some may be deleted. It, therefore, becomes tough to compare the result of the current conditions with that of the past period. Thus, in such a situation changing the base period is more appropriate. Chain Index Numbers method is one such method and we shall discuss it now in detail.

### Chain Index Numbers

Under this method, firstly we express the figures for each year as a percentage of the preceding year. These are known as Link Relatives. We then need to chain them together by successive multiplication to form a chain index.

Thus, unlike fixed base methods, in this method, the base year changes every year. Hence, for the year 2001, it will be 2000, for 2002 it will be 2001, and so on. Let us now study this method step by step.

### Steps in the construction of Chain Index Numbers

1. Calculate the link relatives by expressing the figures as the percentage of the preceding year. Thus,

Link Relatives of current year = $$\frac{price of current year}{price of previous year}$$ × 100

1. Calculate the chain index by applying the following formula:

Chain Index = $$\frac{Current year relative × Previous year link relative}{100}$$

### Advantages of Chain Index Numbers Method

1. This method allows the addition or introduction of the new items in the series and also the deletion of obsolete items.
2. In an organization, management usually compares the current period with the period immediately preceding it rather than any other period in the past. In this method, the base year changes every year and thus it becomes more useful to the management.

### Disadvantages of Chain Index Numbers Method

1. Under this method, if the data for any one of the year is not available then we cannot compute the chain index number for the subsequent period. This is so because we need to calculate the link relatives, which are not possible to be calculated in this case.
2. In case an error occurs in the calculation of any of the link relatives, then that error gets compounded and all the subsequent link relatives will also become incorrect. Thus, the entire series will give a misrepresented picture.

Source: freepik.com

#### Splicing

Splicing is a technique where we link the two or more index number series which contain the same items and a common overlapping year but with different base year to form a continuous series. It may be forward splicing or backward splicing. We can further understand this with the help of the table given below:

 Splicing The index number of old series The index number of new series Forward $$\frac{100}{overlapping index number of old series}$$ × Given index number of old series No change Backward No change $$\frac{Index number of old series}{100}$$ × Given index number of new series

## Solved Example on Chain Index Numbers

Q. From the following data calculate the index numbers using the Chain Index Numbers method.

 Year 2011 2012 2013 2014 2015 2016 2017 2018 Prices 120 124 130 144 150 160 164 170

Construction of Chain Index Numbers

 Year Price Link Relatives Chain indices 2011 120 100 100 2012 124 $$\frac{124}{120}$$ x 100 = 103.33 $$\frac{103.33 × 100}{100}$$ = 103.33 2013 130 $$\frac{130}{124}$$ x 100 = 104.83 $$\frac{104.83 × 103.33}{100}$$ = 108.32 2014 144 $$\frac{144}{130}$$ x 100 = 110.76 $$\frac{110.76 ×108.32}{100}$$ = 119.98 2015 150 $$\frac{150}{144}$$ x 100 = 104.16 $$\frac{104.16 × 119.98}{100}$$ = 124.97 2016 160 $$\frac{160}{150}$$ x 100 = 106.66 $$\frac{106.66 × 124.97}{100}$$ = 133.29 2017 164 $$\frac{164}{160}$$ x 100 = 102.5 $$\frac{102.5 × 133.29}{100}$$ = 136.62 2018 170 $$\frac{170}{164}$$ x 100 = 103.65 $$\frac{103.65 × 136.62}{100}$$ = 141.61

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