Index Numbers (Relevan o AAT Examinaion Paper 4: Business Economics and Financial Mahemaics) Y.O. Lam, SCOPE, Ciy Universiy of Hong Kong In business world, we need o reduce a large number of facs or informaion o daa so ha we can idenify changes over ime. An index number provides a sandardized way for comparing he levels of prices, wages, volume of oupu and oher variables a given daes o heir levels a a previous dae. An index number can also be used o compare one hing o anoher. In conclusion, index numbers make i easier o inerpre daa. This aricle shows you how o presen and inerpre daa in index number forma. Example 1 Suppose he reail sales for RGE shop in 29 are $256 billion and he sales in 2 are $257 billion. The increase of $1 billion is significan. Bu if we conver he sales daa o an index which is based on sales in 29, as shown below, he increase is no ha significan as i is less han 4%. Reail sales in 2 /Reail sales in 29 $257 billion/$256 billion x Example 2.39 The GDPs of China and Hong Kong are US$4,814 billion and US$2,88 billion respecively. Wha is he GDP of China compared wih ha of Hong Kong? The index of GDP for China is 2.35 compared wih ha of Hong Kong, and is calculaed as (GDP of China / GDP of Hong Kong) (4,814 billion / 2,88 billion) 2.35 In oher words, he GDP of China is 2.35 imes ha of Hong Kong. Prices indices and quaniy indices An index can be a price index or a quaniy index. A price index measures he change in price of a group of iems over ime. A quaniy index measures he change in quaniy of a group of iems over ime. For example, he food price index is 168 in 2 compared wih 15 in 29. The index increased by 18 poins beween 29 and 2; his is a rise of (18/15) 12 percen. These figures are ofen defined as price relaives since hey indicae how he price changes over ime. From he above discussion, he index number is expressed in erm of percenages bu he percen sign is usually omied. The base period should be saed. In he above example for food prices, we used 29 as he base period. - 1 -
Weighed index In he business, we may be required o deal wih a baske of commodiies. Weighed means of relaive indices are always used. To obain a weighed mean of relaive indices, we need o calculae index relaives for each componen and hen o obain a weighed average of he relaives using he corresponding weighs. Example 3 The wholesale price index in PP shop is made up of he prices of hree iems. The price index and weighing in 29 and 2 for each iem are as follows. Table 1 Price and weigh informaion for PP shop Iem Price, 29 Price, 2 Weigh Bread $2. $4. 6 Tea $. $12. 2 Cookies $2. $2.5 4 Calculae an index for prices of PP shop for year 2. Soluion Price index: Bread 4/2 2. Tea 12/ 1.2 Cookies 2.5/2. 1.25 PP shop s price index (2. 6 1.2 2 1.25 4)/ (6 2 4) Choice of weigh 194/12 1.62 There are wo mehods in compuing a weighed index: he Laspeyres mehod and he Passche mehod. Laspeyres index Laspeyres indices use base period values as weighs. The original prices and quaniies of he iems are used in calculaing he index in erms of eiher price or quaniy consumed. - 2 -
Passche index Passche indices use curren year values as weigh. The curren prices and quaniies of he iems are used in calculaing he index in eiher erms of price or quaniy consumed. Example 4 DOC Company produces and sells four ypes of elecric appliances. The prices and quaniies in 29 and 2 are shown below: Compuaion of Laspeyres and Passche index for DOC Company, using 29 as he base period: Table 2 Price and quaniy informaion for DOC Company 29 2 Type Price ($) Quaniy ( ) Price ($) Quaniy ( ) Radio 2 12 15 Toaser 2 4 25 25 Clock 3 5 Hair dryer 225 25 Calculae he Laspeyres price index and Passche quaniy index o measure he overall changes beween 29 and 2. Soluion: Laspeyres price index p q p q 12 2 25 4 2 2 4 3 3 25 225 116.41-3 -
Firs we use he quaniy in 29 as our base in calculaing Laspeyres price index for he oal amoun spen on he four iems in DOC Company. We conclude ha he price for DOC Company has increased by 16.41 percen. Passche quaniy index p q p q 12 12 15 2 25 25 25 4 5 3 25 25 9.69 The Passche quaniy index uses curren quaniy as weighs insead of base year price as weighs (as he Laspeyres index does). The resul indicaes ha here has been a decrease of 9.31 percen in he quaniy of elecrical appliances beween 29 and 2. When you are deciding which ype of index o use, he following poins should be considered. Table 3: Differences beween he Laspeyres and Passche indices Laspeyres index The index requires weighs for he base year only. The index can be compued when eiher he curren prices or quaniies are known The index value can be direcly compared over several years. The index does no reflec curren changes over ime because he weighs are ou-of-dae. Passche index The index requires weighs by ime o ime and i is cosly The index canno be compued unil he curren prices and quaniies are known. The index value can only be compared beween he curren year and he base year. The index reflecs curren informaion. I should be noed ha he Laspreyes index is more commonly used in oday's business. The weighs should be revised every few years. Time series deflaion Anoher ype of price index used for measuring he real value of a commodiy is he consumer price index, or CPI. This index is widely used in boh he privae and public secors. In he public secor, governmens use he CPI o reflec changes in he cos of living for residens. In he privae secor, employers ofen use he CPI in calculaing heir employees salary rises. The - 4 -
CPI is considered he mos relevan inflaion measure from he poin of view of consumers. For example, say he cos of a commodiy was $ in 29 and $12 in 2; his represens an increase of 2 percen. If he reail price index is 25 percen beween 29 and 2, we can argue ha he real cos of he commodiy has increased. Consider he daa shown in Table 4. Paul works for Ciy Ld. Over he las hree years, he has earned a monhly salary increase of $5 each year. Though his manager claims ha $5 is equivalen o he increase in he cos of living over he pas hree years, Paul has found ha his salary rise has no kep pace wih inflaion. The real value of Paul s salary is only $18, in year 3 compared wih $2, in year 1. Table 4 : The real salary Year Monhly salary Consumer Prices Index (CPI) Calculaion Real salary 1 $2, 15 $2, 2 $2,5 16 2,6 / 16 x 15 $19,219 3 $21, 175 21, / 175 x 15 $18, When comparing he coss and profis of company or assessing invesmen decisions, we need o consider he real money values. The bes way o analyze he siuaion is o calculae he real money values before making our decisions. Anoher example, given below, illusraes how o use ineres raes o calculae inflaion in order o assess invesmen or fuure profis. Example 5 The PPW Company is considering a hree-year projec wih he following expeced profis and inflaion index. The iniial invesmen is $3, and he required rae of reurn is 2 percen. Deermine wheher he projec should be acceped. Table 5: Inflaion index and expeced profi Year Inflaion index Expeced profi ($ ) (3) 1 95 2 2 3 111 125-5 -
Soluion The inflaion index provides an index (column 2) and his can be used o deflae he expeced profis (column 3) and conver hem o heir real values. For example, he real profi in year 1 is $9, only insead of $99,. Year (1) Inflaion index (2) Expeced ($ ) (3) profi Calculaion (4) Real profis ($ ) (5) (3) (3) 1 1 99 (99/1) x 9 2 115 138 (138/115) x 12 3 12 144 (144/12) x 12 The ne presen value is he difference beween he presen value of he cash ouflows and he presen value of fuure cash inflows. The oal profi is given in column 4 and compared wih he iniial invesmen. The ne presen value of he hree-year projec, as shown in he able, is negaive, so PPW Company should no inves in his projec. Ne presen value Year Real profis ($ ) Discoun facor a 2% Presen value (1) (2) (3) (4) (3) 1. (3) 1 9.833 74.97 2 12.694 83.28 3 12.579 69.48 NPV (76.27) - 6 -