# The difference between an arithmetic mean and a geometric mean. And why it matters.

Understanding the difference between an arithmetic mean and a geometric mean might sound like something that can be left to statisticians. But now, because of a change proposed in the Budget, it has become crucially important to the great majority of current and prospective pensioners. The difference means that they are all going to be worse off in their retirement.

The change is the Government’s proposal to adopt the CPI rather than the RPI for the indexation of benefits, tax credits and public service pensions from April 2011. So the change directly affects every current or future pensioner who is entitled to a State Second Pension (previously SERPS) or to a pension from a public service pension – that means almost everyone who has ever been employed.

The Consumer Prices Index (CPI) and the Retail Prices Index (RPI) are both measures of the cost of living. One important difference between them is the basket of goods and services upon which they are based, with only the RPI including housing costs. So whenever housing costs increase faster than other prices, increases in the RPI will be higher than those in the CPI. But the reverse also applies, like at present, and the CPI can be higher than the RPI. In the long term the effect should be neutral, except to the extent that people consistently spend more (or less) on housing. In the short-term housing costs are expected to increase faster than the average, as interest rates return to more normal levels, so the RPI is likely to be higher for some years to come.

But there is also a significant technical difference in the way that the RPI and the CPI are calculated. The RPI is an arithmetic mean of price changes (the increases are added together and divided by the number of increases), while the CPI is a geometric mean (the increases are multiplied together and the nth root is taken – where n is the number of increases). While this appears abstruse, of interest only to mathematicians, the Treasury have estimated that:

“… the CPI annual rate would typically have been about 0.5 percentage points higher if the elementary aggregates had been calculated using arithmetic means as in the RPI.”[1]

In other words, pension increases will in future average about 0.5% less each year, simply because of the change in the way the index is calculated.

So which index is the right one? This is a topic of much debate among statisticians and, of course, there is no single correct answer. It all depends on what you are trying to measure. But for most people, looking at what they buy themselves at the shops and elsewhere, there is little doubt. They want to know how much more their basket of goods will cost today, compared to what it cost yesterday. And this is shown by an arithmetic mean.

As a very simple example, imagine the basket of goods is a loaf of bread and a kilo of potatoes. Initially they cost 50p each so the total cost is £1.00. But then, over the year, the price of bread increases by 4%, while that for potatoes increases by 8%. So the bread costs 52p and the potatoes cost 54p, which means the basket now costs £1.06, an increase of 6%. It can be seen that this is the weighted average of 4% and 8%.

But the geometric mean of increases of 4% and 8% with equal weights is 5.98%. In this example it makes only a minor difference in the result but, as mentioned above, the Treasury estimate that in practice it means a difference on average of 0.5% per annum, when compounded across a whole basket of goods.

Table C2 in the Budget report shows the pension increases that can be expected over the next 6 years, i.e. the increases due in April of the respective year, based on the forecast price increases in the previous year. These are summarised in the following table:

Forecast Increase (%) |
|||||||

Year of increase |
2011 |
2012 |
2013 |
2014 |
2015 |
2016 |
Overall |

CPI |
2.7 | 2.4 | 1.9 | 2.0 | 2.0 | 2.0 | 13.7 |

RPI |
3.7 | 3.2 | 3.2 | 3.3 | 3.4 | 3.5 | 22.1 |

Shortfall |
1.0 | 0.8 | 1.3 | 1.3 | 1.4 | 1.5 | 7.4 |

What this shows, using the Government’s own figures, is that over the next six years increases in SERPS/S2P and public service pensions will total 13.7%, rather than the 22.1% that was previously expected. This is, in effect, a cut of 7.4%, of which I estimate that 4.3% is due to differences in the coverage of the index and 3.1% is due to the way it is calculated. So you might have thought that the question posed in the title is only of technical interest, but it is actually going to cost a large number of people significant amounts of their pension.

**GUEST POST:**Bryn Davies is the Director of Union Pension Services Ltd, a unique independent consultancy on occupational pension schemes, primarily for trade unions. Most trade unions have used his services at some stage since UPS was established in 1989 and a number retain his services on a regular basis. Other clients have included the Equal Opportunities Commission, the National Audit Commission and in Central Europe for the PHARE programme. He has been a Fellow of the Institute of Actuaries since 1974.

**[1] The Treasury, Consumer Price Indices – Technical Manual – 2007 Edition**

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