# Inflation calculation

• ### Partner Equity account or loan account

Hi, bit of an odd question to ask but I'm trying to see if an inflation calculation is correct and can't work out if it is or not.

The aim is to calculate a budget envelope for the 5 year period from 2023-2027. Just one number is needed at present for this 5 year period. The calculation to derive this is to take total actual expenditure for 2015-2020 and then to uplift this by 2% inflation. These are example numbers but will illustrate the method. Actual spend for 2015-2020 was £500m. Then this £500m has been uplifted as follows: 500 x 1.02^7 = 574.34.

The aim is to give an equivalent budget envelope for the 5 year period in question that has been uplifted by inflation. The 2021-22 period was deemed not relevant here but the inflation for this period will still be required. 2% inflation has been assumed across this whole 2015-2027 period which obviously is a big assumption.

My question is whether the calculation above is correct? It appears to be but not 100% sure. It's obviously a strange way of doing it by not doing it annually but that's whats been done. Feel free to flag any flaws in the method. Thanks in advance.

### Replies (38)

By D V Fields
25th Nov 2021 21:19

Much depends on your precise requirement.
Based on your calculation you appear to be taking the total £500m from six whole years (2015 to 2020 inclusive) and then applying seven compounded 2% annual increases for inflation (1.02 to power of 7) to that figure. I would just confirm that you do want a compounded increase rather than a flat 2% increase; as well as the number of years you do want to be using for consistency. Hope that helps.
Dave

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By David Ex
25th Nov 2021 23:16

I’m afraid I don’t understand the question.

And I’m intrigued to know what a “budget envelope” is.

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By lionofludesch
26th Nov 2021 06:13

David Ex wrote:

And I’m intrigued to know what a “budget envelope” is.

Cheaper version of the Chancellor's "red box"?

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By NotAnAccountant2
26th Nov 2021 06:48

The basic idea looks fine but 2023-2027 is four years while 2015-2020 is five. (or 5 and 6).

Depends on whether you want to take a geometric or arithmetic mean how you resolve this conflict.

(but 2% is small enough that the Taylor series of (1+x)^N is dominated by the first term - 500*1.14 = 570 and the 'corrections' from compounding will be swamped by natural variability)

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By paulwakefield1
26th Nov 2021 09:34

There are so many assumptions (e.g. the expenditure is at the end of each period, the annual expenditure profile over the 5 years is the same as in history, etc) that the calculation is as good as any as variations will just be lost in the wash.

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By ireallyshouldknowthisbut
26th Nov 2021 10:32

its a budget, therefore its 100% guesswork.

Therefore you can do it how you like.

but I would compound it if I was putting it in. Personally I would just do my budget without inflation unless there are very specific circumstances, such as costs on an escalator, but income not which I was trying to model and demonstrate. if you are putting them ALL up by inflation you might as well just do it in "real terms", ie ignore the issue.

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By D V Fields
27th Nov 2021 11:15

[quote=ireallyshouldknowthisbut]

its a budget, therefore its 100% guesswork.

Not with my definition of a budget…
“A plan quantified in monetary terms, prepared and approved prior to a defined period of time, usually showing planned income to be generated and/or expenditure to be incurred during that period and the capital to be employed to attain a given objective.”

Dave

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By pdrysdale
27th Nov 2021 10:23

Thanks all for the comments. Much appreciated.

I should say I mis-described it slightly as has been pointed out. The original 5 year period is 2015-19. 2020-21 was a 2 year period and then we are looking to set a budget envelope for the 2022-26 5 year period based on the original 5 year period from 2015-19 but uplifted for inflation which we're assuming is 2% (assumption I know!).

I believe it's right to compound it in the inflation calculation but could someone just describe why please? I know it makes sense but I can't seem to describe it in plain english.

The calculation I've seen at my workplace appears incorrect. They have taken an average annual amount for the 2015-19 period which let's assume is 100. They've then applied inflation to this for each of the next 7 years at 2% but this has been applied per annum and not compounded. This gives a figure of 552.25m for the 5 year period from 2022-27 (based essentially on rolling forward 100 adding 2% inflation each year). The compound approach gives a figure of 574.34m based on 500x1.02^7. This latter compound approach would seem to be correct but I can't fully describe why. The closest I can come is by saying that to roll it forward by 7 years it needs compound interest applied rather than uplifting it annual each year given the inflation would be compounded annually. But I'm sure there's a better way of describing it.

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By lionofludesch
27th Nov 2021 10:35

pdrysdale wrote:

Thanks all for the comments. Much appreciated.

I should say I mis-described it slightly as has been pointed out. The original 5 year period is 2015-19. 2020-21 was a 2 year period and then we are looking to set a budget envelope for the 2022-26 5 year period based on the original 5 year period from 2015-19 but uplifted for inflation which we're assuming is 2% (assumption I know!).

I believe it's right to compound it in the inflation calculation but could someone just describe why please? I know it makes sense but I can't seem to describe it in plain english.

Because the 2% is 2% of last year's figure, not 2% of the starting figure.

So if something cost £100 in year zero, it'll cost £102 in year 1 and £104.04, not £104, in year 2.

But, as was said early in the thread, the 4p discrepancy is dwarfed by other factors.

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By Paul Crowley
27th Nov 2021 22:17

Your work place and two Pauls + a Lion all agree
If you disagree then show your annual figures. There will only be five, so dead easy
No idea how you came to your conclusion, but you appear to be wanting to make one calculation when there are many
Compounding is annual and has been done in the calculations

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By paulwakefield1
27th Nov 2021 11:09

I agree with Lion that the level of assumption dwarfs the differences. But arithmetically I am curious as to how 552.25m is reached - it would seem to be more complex than applying simple interest.

The closest I can come is if 100m is compounded for 2 years and then each of the compounded next 5 years is accumulated. This is arguably correct at the start of the period but ignores the fact that, presumably, the expenditure in the forecast period will be spread over a 5 year period.

Edit: It could be argued as an approach if you were trying to get to a starting point for the forecast period but not actually applying inflation within the forecast.

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By Paul Crowley
27th Nov 2021 16:47

I may be wrong, but the problem as I see it is the inconsistency of assumptions
It I was doing it myself I would revert to simple annual figures
Start point is the latest actual annual unless the annual figures vary significantly
Otherwise decide the norm
In this example £100 as start point, Ye 2020
Then just apply
2020 100
2021 102
2022 104
2023 106.1
2024 108.2
2025 110.4
2026 112.6
2027 114.9
5 year total 552.2

Lions Answer, ignoring the decimals less than one tenth

Laid out that way you also show your workings and fairly easy to follow for the non numbers people
That calculation compounds annually at the end of the year
If you want to compound at beginning then add anther yeat and deduct year 1
Total 563.7

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By paulwakefield1
27th Nov 2021 18:06

Yes - that matches the calculation I did and appears to be what the OP's workplace have done. Whereas the OP's calculation is the 2027 figure x5 (i.e. taking 2015 as the base for Year 1, 2016 as the base for year 2, etc). Both are arithmetically valid approaches depending on the base assumptions. Whether they are too mechanisitic for a budget especially at those levels of expenditure is another question entirely.

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By pdrysdale
27th Nov 2021 19:23

Thanks again for comments. I think Paul Crowley you laid out how the 552.25 was reached. I should say that although these amounts are quite small they are actually quite large in grand scheme of how this applies at my workplace as the overall amount is quite large. Plus the budget requested itself is way over the target/envelope and so important that we get the envelope right

I should also say at this stage we're just interested in determining the 2022-26 budget envelope amount as a whole and not worried about the annual split within quite yet. The exam question is how to roll forward the 2015-19 budget figure of £500m with inflation of 2% applied to get to a 2022-26 budget envelope total based on 2015-19 budget plus inflation.

I am inclined to think that the calculation is 500x1.02^7=574.34. This is just using the 2015-19 5 year period as a whole and is essentially assuming that to roll this forward in a symmetrical way each year needs 7 years worth of compound inflation adding. It would be good to get your view though.

There is not a 2019 budget figure (i.e. Year 5) with inflation built in that could be rolled forward on a per annum basis. This is because the budget was flat-phased (including flat-phasing the total 5 year inflation allowance) - obviously not ideal but it is what it is.

The calculation done by my workplace is to take the average annual amount (i.e. 500/5) which is 100 and then they have applied inflation to this 100 figure at 2% a year from 2020 onwards. This gives a 5 year total of 552.25. This approach would have merit, I believe, if it took the 2019/Year 5 budget figure and this figure had the full previous inflation built into it (meaning it would be above 100 assuming all else equal) but given the average figure was used I don't think this methodology is quite right as by virtue of using the 'average' for that period isn't using the year 5 amount which would have previous inflation built in. And so the base it's working off is too low. I'm not articulately this very well though so thoughts welcome.

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By Hugo Fair
27th Nov 2021 21:44

Sounds perfectly clearly articulated to me ... but if you think an example would help whoever you're trying to convince then use a wider range of figures for the 2015-19 period.
For instance, say the figures each year were 60, 80, 100, 120, 140.
So if you base on the average then that is 100 ... whereas if you base on final year of period (usually a more logical proposition) then that is 140.
A very different starting point for your forecasts of inflation-driven amounts!

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By Paul Crowley
27th Nov 2021 22:04

Assuming 500 is the total and costs are rising then the final figure should be the start point
But we are only given 500 as a 5 year figure
All calculations remain intact as dead easy to multiply 100 by 1.4 if start point is 140
What is clearly wrong is, in my opinion, taking all years as the end result
Very likely to lose credibility if people presented to can operate a spreadsheet or a calculator

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By Paul Crowley
27th Nov 2021 21:55

Your verson just ends up as the figure for for the last year
Simple test, take your figure and divide by 5

Several agree the Lion calculation, and I have shown the annual route
Your version shows a five year costing that is identical for each year
Any figures man would challenge your calculation

If you want a higher figure just change the start value or the inflation rate

For the avoidance of doubt I consider your calculation wrong
You are assuming the entire period is at the year five (or seven) accumulated compound rate which is very clearly not the case in year one

SHOW YOUR ANNUAL EXPECTED COSTS if you want people to trust your figures

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By pdrysdale
28th Nov 2021 11:08

Hi Paul, thanks for comments. Really appreciate you looking at this. The annual route is not an option here I'm afraid. I don't doubt the annual route would be more accurate but the question here is how to correctly inflate the previous 5 year figure (in aggregate) from 2015-2019 to give an equivalent 5 year value for 2022-26 using 2% inflation assumption.

My reasoning is that to roll this forward each year would need 7 years worth of compound inflation. 2015 rolled forward to 2022 needs 7 years compound inflation. Same goes for 2016. And so on. Each year of this original 5 year budget ultimately needs 7 years worth of compound inflation applied in order to roll it forward with inflation in my mind in a symmetrical/uniform way. And so my reasoning is that the total as a whole (500) should be uplifted by 1.02^7. This then gives the 574.34 figure. This essentially applies 7 years compound inflation on the whole budget.

I'm only focused on uplifting this in the 5 year aggregate figure at present as that is the ask. I don't doubt doing it annually would be more accurate but we don't have the previous budget numbers (2015-19) with inflation built in unfortunately.

What do you think to the calculation above which gives 574.34 on 5 year total basis? could you see a justification for using this or do you think this calculation is wrong?
I agree doing it annually would be better and more accurate but afraid that's not really an option at present although perhaps one I could suggest.

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By Hugo Fair
28th Nov 2021 12:52

I'm not sure whether you keep asking slightly different questions, or whether it's my understanding of them that shifts slightly with each change of wording - but:
* If you really can't elicit the true figure for 2019 (why not?), then the relevance of whatever you do is more than questionable.

You appear to be saying that you only have the total figure for the 5 years 2015-19 - and want an inflation-driven equivalent total figure for the 5 years 2022-26 (but without any regard to any underlying trend/changes within either figure)?

Whatever the methodology I can't see this being useful or reliable.

To retain any semblance of financial sense, you need first to establish a figure for the 2019 year and then to apply inflation to get figures for each of 2020 & 2021 ... and to proceed onwards through the next 5 years (summing those if you wish to do so for some reason).

If instead this was a Maths O-level (or whatever they're called nowadays) exam question, then your logic (or what I understand of it) would be fine ... where the two periods (of 5 years) have equal duration, you can apply a geometric increase from one figure to get the other (as the gap between year 1 of each 'group' is the same as the gap between year 2 of each 'group' and so on).
BUT, not only does this fail to produce anything useful that I can see, it fails to take account of the 2 year inter-regnum between your two 5-year figures ... and you can't simply extend your factorial to 7 because then you're not comparing the totals of two periods of equal duration.

I like Maths, but need to have either a practical objective or a pure fun theoretical issue to tackle ... and this ticks neither box.

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By Paul Crowley
28th Nov 2021 14:21

No responder yet gets any point you are making
A bit like an exam. If question incompete ask the invigilator if there is extra information, if not declare your assumptions

The 5 year figures are available but you chose not to ask

Time now to finally ask the questions you should have asked before starting the exam question

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By paulwakefield1
28th Nov 2021 14:51

I think (Think, mind you - I could be way off mark) the difference between the two interpretations is, for the OP:

Each of the 5 base years has expenditure of 100 which will be replicated in each of the 5 forecast years. So each base year will have experienced 7 years of inflation (5 years plus 2 "fallow" years) by the time the equivalent expenditure appears in the forecast. 2015 in 2022, 2016 in 2023, etc.

The workplace argument is that, at the start of the forecast period, Year 1 of the base period will have had 7 years inflation, Year 2 will have had 6 years, etc. Total 552.26.

All very fine arithmetical arguments but, as a method of budgeting substantial expenditure, rather worrying. Myriad assumptions - even expenditure, constant inflation percentage, the fallow years, etc. leave alone whether the inflation rate is actually what the business is experiencing/has expereinced nor whether historical expenditure bears any relation to what the business is doing now or expected to do in the future.

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By Hugo Fair
28th Nov 2021 17:31

I can't believe I'm replying again but, from a purely mathematical perspective, there is a basic flaw in what you seem to have proposed ...
* The total for 2015-2019 was £500m
* Your proposed calc (500 x 1.02^7 = 574.34) generates a value for a 7-year period - in this case 2020-26
* What you wanted was a total for 2022-26 - a 5-year period.

Myriad hidden assumptions have been made in getting this far, but they appear to include a straight-line rate of increase.
In which case the figure for 2022-26 should be:
* ((500 x 1.02^7) - 500) = 74.34 increase for the 7 years, or
* (74.34 * (5 / 7)) = 53.1 increase for only 5 years
Giving a total of (500 + 53.1) = 553.1

As I've said there are too many hidden assumptions for this to be guaranteed as a mathematically correct answer - and I can't see how it can be considered remotely useful or reliable within a set of financial forecasts - but it's probably better than the proposal shown in OP.

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By pdrysdale
28th Nov 2021 18:56

Thanks for reply. I'm not sure your calc is right. The first 5 year period (2015-19) total was £500m. So this is a 5 year total. I'd then like to work out the equivalent 5 year total for 2022-26 by uplifting this £500m with inflation at 2%. My reasoning is every year in this initial 5 year period needs 7 years worth of inflation to be applied to roll it forward to become a 2022-26 equivalent with inflation. This would include applying inflation for 2020 & 2021. Therefore I believe the calculation to work out 2022-26 budget equivalent with inflation is 500x1.02^7=574.34.

I accept this calculation itself contains a lot of assumptions but ultimately the aim is to work out an equivalent 5 year total with inflation applied, however flawed this approach might be otherwise.

NB. You said that "* Your proposed calc (500 x 1.02^7 = 574.34) generates a value for a 7-year period - in this case 2020-26" - this is not correct. It does not generate a value for a 7 year period but rather a 5 year period I believe since the starting point (500) is a 5 year number. It just applies 7 year worth of compound inflation on a 5 year base.

P.S. What does OP stand for?

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By Hugo Fair
28th Nov 2021 19:09

OP stands for Original Post (or sometimes Poster) ... possibly silly but standard on this site.
As for the calc, I've already explained why I think you're wrong and nothing in your most recent post (basically just repeating your perspective) changes my mind.
You are of course free to keep ignoring every response - as you appear determined to do - but it's Sunday night and I'm out.

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By lionofludesch
28th Nov 2021 19:11

pdrysdale wrote:

Thanks for reply. I'm not sure your calc is right. The first 5 year period (2015-19) total was £500m. So this is a 5 year total. I'd then like to work out the equivalent 5 year total for 2022-26 by uplifting this £500m with inflation at 2%. My reasoning is every year in this initial 5 year period needs 7 years worth of inflation to be applied to roll it forward to become a 2022-26 equivalent with inflation. This would include applying inflation for 2020 & 2021. Therefore I believe the calculation to work out 2022-26 budget equivalent with inflation is 500x1.02^7=574.34.

Surely this factor covers years 1-7, rather than the 3-7 you seek.

But I still think you are pursuing spurious accuracy.

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By pdrysdale
28th Nov 2021 19:39

Yes this uplifts for inflation for the next 7 years as the calculation is also uplifting for the 2 year gap before the 5 year period and so inflation is needed to be added for 2020 to 2026 so a 7 year period and so the inflation needs to be compounded for 7 years given the desire to fully uplift it for inflation.

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By Tax Dragon
28th Nov 2021 20:51

I have a question. (Well, two questions - why did I read this thread?! is one.)

OP, you said (11.08 28th November) "2015 rolled forward to 2022 needs 7 years compound inflation". Agreed, but most of that inflation is in the past. Are you assuming 2%, or was it 2%? If not, why not use the right figure(s)?

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By pdrysdale
28th Nov 2021 21:28

Yes fair point, would ideally use the correct inflation index figure I agree. And this is unlikely to be 2%. For calculation illustrative purposes I'm using 2% but the actual number would differ in reality I agree.

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By pdrysdale
28th Nov 2021 21:12

I've done a table below to demonstrate my approach here. Comments welcome. These are illustrative numbers I should say.

Year 1 original (2015)= 10. Year 1 equivalent (2022) with inflation = 11.5
Year 2 original (2016)= 170. Year 2 equivalent (2023) with inflation = 195.3
Year 3 original (2017)= 120. Year 3 equivalent (2024) with inflation = 137.8
Year 4 original (2018)= 50. Year 4 equivalent (2025) with inflation = 57.4
Year 5 original (2019)= 150. Year 5 equivalent (2026) with inflation = 172.3
TOTAL = 500 574.34

In each line above the formula applied has been original amount x 1.02^7. In essence the calc above is rolling these amounts forward by 7 years of compound inflation.

Whichever set of numbers is applied above is always going to give the same outcome of 574.34 as it's always rolling forward by 7 years of compound inflation so any combination will get to that same number.

This to me illustrates that the methodology I'm suggesting also works on an annual basis, albeit I accept it is not taking the year 5 amount as a starting point and rolling that forward - it is looking at it as a whole. The critical thing is we need to maintain the 5 year budget total and then add inflation to that rather than roll it forward from a year 5 base as that year 5 base will not be representative of the total budget period. If we rolled forward from the 150 figure noted above that would not be reflective of the 5 year budget figure.

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By lionofludesch
28th Nov 2021 21:23

pdrysdale wrote:

I've done a table below to demonstrate my approach here. Comments welcome. These are illustrative numbers I should say.

Year 1 original (2015)= 10. Year 1 equivalent (2022) with inflation = 11.5
Year 2 original (2016)= 170. Year 2 equivalent (2023) with inflation = 195.3
Year 3 original (2017)= 120. Year 3 equivalent (2024) with inflation = 137.8
Year 4 original (2018)= 50. Year 4 equivalent (2025) with inflation = 57.4
Year 5 original (2019)= 150. Year 5 equivalent (2026) with inflation = 172.3
TOTAL = 500 574.34

In each line above the formula applied has been original amount x 1.02^7. In essence the calc above is rolling these amounts forward by 7 years of compound inflation.

Whichever set of numbers is applied above is always going to give the same outcome of 574.34 as it's always rolling forward by 7 years of compound inflation so any combination will get to that same number.

This to me illustrates that the methodology I'm suggesting also works on an annual basis, albeit I accept it is not taking the year 5 amount as a starting point and rolling that forward - it is looking at it as a whole. The critical thing is we need to maintain the 5 year budget total and then add inflation to that rather than roll it forward from a year 5 base as that year 5 base will not be representative of the total budget period. If we rolled forward from the 150 figure noted above that would not be reflective of the 5 year budget figure.

I expect you're right.

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By Hugo Fair
28th Nov 2021 22:12

That's so dry it's been entered in the Camel's Ar3e book of world records.

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By lionofludesch
29th Nov 2021 06:57

Hugo Fair wrote:

That's so dry it's been entered in the Camel's Ar3e book of world records.

Thanks.

I was just thinking that the OP wasn't going to give up until everyone agreed with him. I did once get twenty odd folk to agree that it was possible for 19 teams to play 28 fixtures, which everyone else in the room had told me was impossible as 19 was a prime number, but it's a tough job. I thought I'd make a start.

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By john hextall
02nd Dec 2021 11:00

575 seems to be the correct answer.

In the table below, column 1 shows the compounded effect of 2% a year over 12 years. In column 3, the first 5 years add up to 500. The last 5 years add up to 575.

1 , 1 , 96.15384615
1.02 , 2 , 98.07692308
1.0404 , 3 , 100.0384615
1.061208 , 4 , 102.0392308
1.08243216 , 5 , 104.0800154
1.104080803 , 6 , 106.1616157
1.126162419 , 7 , 108.284848
1.148685668 , 8 , 110.450545
1.171659381 , 9 , 112.6595559
1.195092569 , 10 , 114.912747
1.21899442 , 11 , 117.2110019
1.243374308 , 12 , 119.555222

2% seems very low at the moment though, considering fuel prices for example.

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By Tax Dragon
02nd Dec 2021 12:00

Very good.

I still love that only five of those years are a projection forward; seven project back/are a back-calculation based on the same 2% assumption (I'm not sure there was a single 12-month period that satisfied that assumption). It's clearly so much better than looking up the actual numbers. (Has someone... Calculatorboy?... burned all the accounts and records? Is a 'budget envelope' a storage/filing container and everyone's forgotten which dark corner the old ones are buried in?)

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By Herbacious
16th Dec 2021 12:33

Yes, it is correct, ASSUMING THAT:
- inflation is 2% in all years, i.e. from 2016 through to 2027
- that you are dealing with the FIVE year period 2016 to 2020 inclusive (with a total spend of £500); and the FIVE year period 20023 to 2027 inclusive.

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By pdrysdale
16th Dec 2021 22:16

I believe there is also an implicit assumption that the underlying notional (pre inflation) cashflows are the same in the two 5 year periods.

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By pdrysdale
16th Dec 2021 22:16

I believe there is also an implicit assumption that the underlying notional (pre inflation) cashflows are the same in the two 5 year periods.

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By lionofludesch
16th Dec 2021 23:03