# PQ article on Index Numbers

If I told you that I ate 1,295 jam doughnuts one week, 1,358 the following week and then 1,412 in the third week you could see quite clearly that the quantity I was eating each week was increasing (along with my belly) but it is less obvious how large this growth is in percentage terms. Index numbers are a way of representing figures so that the size of changes over time are much easier to see..

# PQ article on Index Numbers

If I told you that I ate 1,295 jam doughnuts one week, 1,358 the following week and then 1,412 in the third week you could see quite clearly that the quantity I was eating each week was increasing (along with my belly) but it is less obvious how large this growth is in percentage terms. Index numbers are a way of representing figures so that the size of changes over time are much easier to see..

With index numbers we choose a point in time as a ‘base’ point which is where we start our calculations from and give this base point a starting index value of 100. We then show subsequent results as index values which can be easily compared back to this 100. So if the index value rises to 110 we can see that the increase has been 10%.

Calculating index numbers from results

If we want to work out index numbers we use the following formula:

(New result / Result in base period) x 100

This can be used to calculate index numbers for changes in volumes, prices, costs or anything else that we might be interested in. Let’s consider the quantity of jam doughnuts that I ate.

•  If we set the first week as the ‘base’ week we would give the quantity of 1,295 jam doughnuts an index value of 100. This is our starting point and will make it easy to identify the change in the next couple of weeks as percentages.
• In week 2 the quantity was 1,358 which has increased from the first week, meaning that the index value would rise above 100. Using the formula the index value will now be 1,358/1,295 x 100 = 104.9. This clearly shows that since the base period of the first week there has been an increase in quantity of 4.9%.
• In week 3 the quantity was higher again so the index will have climbed even higher. The index would now be 1,412/1,295 x 100 = 109.0 showing that there has been a 9% increase since the base period of the first week.

Using index numbers to forecast results

One of the principal uses of index numbers is to improve the quality of budgets and forecasts. If we want to forecast a result and we know the index for the future period we can use:

Forecast result = result in the base period x (New index value / 100)

If I knew that in week 4 the quantity index for my jam doughnut habit was going to climb further to 112.4, I could work out that this meant that I was going to eat 1,295 x 112.4/100 = 1,456 jam doughnuts in week 4.

In the explanations above we have been using a quantity index. Another common index that we might look at is a price index, such as RPI (the Retail Price Index which measures average changes in prices for products that are sold in the UK). This shows us the effect of inflation over time, as this can be expected to increase prices.

As prices are expected to rise over time it can mean that comparing revenues or costs at different points in time is a bit meaningless. The price of a jam doughnut today will be far higher than the price of one ten years ago due to the effect of inflation. We therefore need to make adjustments to some of the figures to make them comparable.

Let’s say we have the following information:

 Month Price index Labour cost July 145.6 £10,000 August 151.2 £10,240

You can see that the labour cost has risen from July to August but this is not a big surprise as the price index has increased, meaning that staff will have demanded higher wages. You can’t really compare the labour costs in July and August as they stand; we need to adjust one or other figure to account for the impact of the changing price index.

We have two ways to do this:

1. Inflate the July figure to ‘August prices’ to compare to the August figure
2. Deflate the August figure to ‘July prices’ to compare to the July figure

Inflating earlier cash flows

To inflate cash flows we need to increase them to account for likely rises in prices caused by inflation. We do this using the following formula:

To inflate:   cash flow x (Index in later period / Index in earlier period)

Applying this to the information in the table we could inflate the July labour cost to ‘August prices’ to add the effect of the increase in the price index. This gives £10,000 x 151.2/145.6 = £10,385.

Deflating later cash flows

To deflate cash flows we need to decrease them using a slightly different formula:

To deflate:   cash flow x (Index in earlier period / Index in later period)

Using the same figures as we did for ‘inflating’ cash flows we can deflate the August labour cost to ‘July prices’ to remove the effect of the increase in the price index. This gives £10,240 x 145.6/151.2 = £9,861.

Whether we compare the figures in ‘August prices’ (July £10,385 and August £10,240) or in ‘July prices’ (July £10,000 and August £9,861) you can see that once we account for the change in the price index the labour cost has actually fallen!

Here’s one for you to try.

Calculate each of the material costs in the table below in Year 3 prices:

 Year Price index Material cost 1 124.6 £57,500 2 131.5 £63,100 3 139.2 £65,700 4 145.7 £67,300 5 151.8 £72,900

Once you’ve had a go, watch me work my solution below: