Индекс

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How is an index calculated?[править | править код]

The index is calculated by summing the sum of price ratios (current price of items divided by their prices on the base date), multiplying by 100, and dividing with the "index divisor".

<math>\begin{align}
 \text{index} & = \sum_{i=1}^{n} \left ( \frac{p_i}{q_i} \right ) \times \frac{100}{{div}_{\text{index}}} \\
   & = \left ( \frac{p_1}{q_1} + \frac{p_2}{q_2} + \dots + \frac{p_n}{q_n} \right ) \times \frac{100}{{div}_{\text{index}}}
 \end{align}</math>


<math>\begin{align}
 \text{where:} \qquad \qquad               p & = \text{current price} \\
                                           q & = \text{base price} \\
                                           n & = \text{number of items} \\
           \sum \left ( \frac{p}{q} \right ) & = \text{sum of price ratios} \\
                        {div}_{\text{index}} & = \text{index divisor}
 \end{align}</math>


For example, let us assume the index is composed of four fictional items (A=30, B=40, C=70, D=60). At the point of the creation of the index, the index would equal to 100, and the index divisor, would be equal to the number of items included in the index.

<math>\begin{align}
 \text{index} & = \sum_{i=1}^{n} \left ( \frac{p_i}{q_i} \right ) \times \frac{100}{{div}_{\text{index}}} 
   \qquad \text{for }{A, B, C}\text{ and }{D} \\
   & = \left ( \frac{p_1}{q_1} + \frac{p_2}{q_2} + \dots + \frac{p_n}{q_n} \right ) \times \frac{100}{n} \\
   & = \left ( \frac{30}{30} + \frac{40}{40} + \frac{70}{70} + \frac{60}{60} \right ) \times \frac{100}{4} \\
   & = 100
 \end{align}</math>


After several months, the prices of the four items would have changed (A=22, B=31, C=85, D=64). When the index is re-calculated, the index is down to 94.73 by 5.27 points. As long as the items themselves are not changed (via addition and/or removal), the index divisor remains equal to the number of items.

<math>\begin{align}
 \qquad \qquad \qquad \qquad \text{index} & = \sum_{i=1}^{n} \left ( \frac{p_i}{q_i} \right ) \times \frac{100}{{div}_{\text{index}}}
   \qquad \text{for }{A, B, C}\text{ and }{D} \\
   & = \left ( \frac{22}{30} + \frac{31}{40} + \frac{85}{70} + \frac{64}{60} \right ) \times \frac{100}{4} \\
   & = 94.73
 \end{align}</math>

What happens when index items need to be changed?[править | править код]

When old items are removed and/or new items are added to the index, the index is adjusted accordingly. The index divisor needs to be re-calculated and is no longer equal to the total number of items. While the value of the index remains the same at the moment of the removal and/or addition, the index divisor is different.

<math>\begin{align}
 \text{since:} \qquad \qquad \qquad \qquad
 
 \text{index}_{\text{adj}} & = \text{index}_{\text{old}} \\
 \sum \left ( \frac{p}{q} \right )_{\text{adj}} \times \frac{\cancel{100}}{{div}_{\text{adj}}} & = \sum \left ( \frac{p}{q} \right )_{\text{old}} \times \frac{\cancel{100}}{{div}_{\text{old}}} \\
 \frac{\sum \left ( \frac{p}{q} \right )_{\text{adj}}}{{div}_{\text{adj}}} & = \frac{\sum \left ( \frac{p}{q} \right )_{\text{old}}}{{div}_{\text{old}}} \\
 \end{align}</math>


<math>\text{thus:}</math>
<math>{div}_{\text{adj}} = {div}_{\text{old}} \times \frac{\sum \left ( \frac{p}{q} \right )_{\text{adj}}}{\sum \left ( \frac{p}{q} \right )_{\text{old}}}</math>


<math>\text{where:}</math>


<math>\begin{align}
   {div}_{\text{old}} & = \text{index divisor prior to adjustment} \\
   {div}_{\text{adj}} & = \text{index divisor after adjustment} \\
   \sum \left ( \frac{p}{q} \right )_{\text{old}} & = \text{sum of price ratios prior to adjustment} \\
   \sum \left ( \frac{p}{q} \right )_{\text{adj}} & = \text{sum of price ratios after adjustment} \\
     & = \text{sum of price ratios prior to adjustment} \\
     & \quad - \text{sum of removed ratios} + \text{sum of added ratios} \\
     & = \sum \left ( \frac{p}{q} \right )_{\text{old}} - \sum \left ( \frac{p}{q} \right )_{\text{removed}} + \text{number of added items} \\
 \end{align}</math>


From the earlier example of the fictional index with four items (see above), Item B will be removed, and new items E and F are added (E=120, F=354).

<math>{div}_{\text{old}} = 4</math>
<math>\sum \left ( \frac{p}{q} \right )_{\text{old}} = \left ( \frac{22}{30} + \frac{31}{40} + \frac{85}{70} + \frac{64}{60} \right ) = 3.78928571 \qquad \left ( \text{for }{A, B, C}\text{ and }{D} \right )</math>
<math>\sum \left ( \frac{p}{q} \right )_{\text{adj}} = 3.78928571 - \left ( \frac{31}{40} \right ) + 2 = 5.01428571 \qquad \left ( \text{for }{A, C, D, E}\text{ and }{F} \right )</math>
<math>{div}_{\text{adj}} = 4 \times \frac{5.01428571}{3.78928571} = 5.2931</math>


To check if our calculations are correct, we need to check the "adjusted" index using the "adjusted" index divisor.

<math>\text{index}_{\text{old}} = 94.73</math>
<math>\text{index}_{\text{adj}} = \left ( \frac{22}{30} + \frac{85}{70} + \frac{64}{60} + \frac{120}{120} + \frac{354}{354} \right ) \times \frac{100}{5.2931} = 94.73</math>