What it means
When a company reports its sales for January, then February, then March, those are successive months. The same idea applies to investment returns over successive years or to a loan balance as it changes from one payment date to the next.
The key point is that each period starts where the last one ended. A balance that grows by 5% in the first year begins the second year at the higher figure, so the second year's growth is earned on a larger base.
Because of this, you cannot simply add up percentage changes across successive periods. A gain of 10% followed by a loss of 10% does not return you to the starting point, because the loss is applied to the larger amount, and the correct method is to multiply the growth factors together.
Analysts also use successive periods to spot trends and unusual changes. Comparing each quarter with the one before shows momentum, while comparing it with the same quarter a year earlier removes seasonal effects.
Averages need care across successive periods. The simple average of 10%, -10% and 20% is 6.7%, but the growth that actually compounded is smaller, about 5.9% a year, because gains and losses build on each other.
The geometric average is the right measure whenever results compound. For managers, the discipline is to use consistent period lengths and consistent accounting methods.
A change of method partway through a series makes successive periods hard to compare, and the notes to the accounts should explain any such change.
In practice
Real-world examples.
Example
A retailer reports quarterly sales of $1,200,000, $1,300,000 and $1,500,000 in three successive quarters. The finance manager calculates the growth between each pair to see whether momentum is building. Growth of 8.3% and then 15.4% shows that sales are accelerating.
Example
A fund reports annual returns of 8%, -4% and 12% over three successive years. The adviser multiplies the growth factors, 1.08 x 0.96 x 1.12 = 1.161, to find a true cumulative return of about 16.1% instead of adding the percentages to reach 16%. The gap looks small here but widens over longer runs.
Example
A subscription business tracks customers lost in each of twelve successive months. The data reveals that cancellations rise every January, so the company plans a retention campaign in December. It then compares the next January's losses with the earlier ones to see whether the campaign helped.
Formula
Calculation
Cumulative return = (1 + r1) x (1 + r2) x ... x (1 + rn) - 1
An investment of $50,000 earns 10% in year one, loses 10% in year two and earns 20% in year three. The growth factors multiply to 1.10 x 0.90 x 1.20 = 1.188, so the cumulative return is 1.188 - 1 = 18.8%. The ending value is $50,000 x 1.188 = $59,400, whereas simply adding 10% - 10% + 20% would suggest 20% and $60,000. The difference of $600 comes from the loss in year two being applied to a larger balance than the gain in year one.Case study
Seen in the real world.
Falconer Textiles is an illustrative, fictional manufacturer whose sales manager reported that revenue had grown 15% in one year, then fallen 15% the next, and said that the company was back where it started. The finance director disagreed and asked for the numbers.
Revenue began at $2,000,000, rose to $2,300,000 after the 15% increase, and then fell by 15% of $2,300,000, which is $345,000, to $1,955,000. The company was therefore $45,000 below where it began, not level.
In this illustrative story, the board changed its reporting so that every presentation showed the actual figures for each successive period beside the percentages. The change prevented repeated confusion and made the loss of ground in the second year easier to see. Managers also began asking for cumulative figures across three or more periods before accepting claims about recovery.
Watch out
Common mistakes.
- Adding percentage changes across successive periods instead of multiplying the growth factors.
- Comparing periods of different lengths, such as a four-week month with a five-week month, without adjusting.
- Changing the accounting method partway through and presenting the results as if they were comparable.
Questions
People also ask.
Why does a 10% gain followed by a 10% loss leave you worse off?
The loss is calculated on a larger balance, so it removes more money than the gain added.
What is the difference between successive and consecutive periods?
In finance the words are used in much the same way, meaning periods that follow each other in order with no gaps.
How do I find the average growth over several periods?
Use the geometric average, which takes the nth root of the cumulative growth factor, and not the simple average of the percentages, because the simple average overstates the result when returns vary.
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