Average price increase

[Mimascruff]

Hi everyone,

I’ve been trying to work this out for the past week and I’m well and truly stuck. Hopefully someone knows more than me!

I’m working on a file with various price increases, each implemented on different dates. How do I work out the average price increase taking into account the date of implementation?

E.g.
- Price increase 1 is 3% on 1st January
- Price increase 2 is 9% on 1st July
- In reality the price increases have a delayed effect in store so I’m going to be using the month after, so Feb and Aug

Based on the % and Feb/Aug dates I’ve been doing (3%/2)+(9%/8)=2.6% but that seems really low. I feel like it should be 6%ish.

Thanks in advance

 

[BigBeachBanana]

[math]\%\ \text{Change}= \frac{9\%-3\%}{3\%}=200\%\\ \%\ \text{Change as Function of Time}=\frac{200\%}{6 \text{ months}}\times 100=\frac{33.33\%}{1\text{ month}}\\[/math]Check:
[math]3\%(1+6\times 33.33\%) \approx 9\%[/math]In general, [math]\text{Average \% change per unit of time}=\frac{New-Old}{Old}\times \frac{1}{Time}\times 100[/math]

 

[Mimascruff]

[math]\%\ \text{Change}= \frac{9\%-3\%}{3\%}=200\%\\ \%\ \text{Change as Function of Time}=\frac{200\%}{6 \text{ months}}\times 100=\frac{33.33\%}{1\text{ month}}\\[/math]Check:
[math]3\%(1+6\times 33.33\%) \approx 9\%[/math]In general, [math]\text{Average \% change per unit of time}=\frac{New-Old}{Old}\times \frac{1}{Time}\times 100[/math]

Thanks for replying. Does this logic still work to calculate the cumulative price increase?

By that I mean if I have my base price X, then in January I increase X by 3% to get my new price Y. Then in August I increase Y by 9%.

Would I still be able to use the above for the average price increase over the whole year bearing in mind the increases were implemented at different times?

 

[BigBeachBanana]

Thanks for replying. Does this logic still work to calculate the cumulative price increase?

By that I mean if I have my base price X, then in January I increase X by 3% to get my new price Y. Then in August I increase Y by 9%.

Would I still be able to use the above for the average price increase over the whole year bearing in mind the increases were implemented at different times?

Yes, in that case, you're just looking for the average of the year (12 months) and you'll only need to know the old price and current price. You can ignore the intermediary increments since you only care about the average rate of change.
For example, let's say on Jan 1, 2022, the price is $5 and on Dec 31, 2022, the price is $12.
[math]\text{Average \% change per unit of time}=\frac{\$ 12-\$5}{\$5}\times \frac{1}{12\text{ months}}\times 100=\frac{11.67\%}{\text{month}}[/math]
You can apply the same logic if you want the average per year. For example, let's say on Jan 1, 2020, the price is $5 and on Jan 1, 2025, the price is $12.
[math]\text{Average \% change per unit of time}=\frac{\$ 12-\$5}{\$5}\times \frac{1}{5\text{ years}}\times 100=\frac{28\%}{\text{year}}[/math]In words, from 2020 to 2025, the average price increase is 28% per year.

 

[The Highlander]

For example, let's say on Jan 1, 2022, the price is $5 and on Dec 31, 2022, the price is $12.
[math]\text{Average \% change per unit of time}=\frac{\$ 12-\$5}{\$5}\times \frac{1}{12\text{ months}}\times 100=\frac{11.67\%}{\text{month}}[/math]
You can apply the same logic if you want the average per year. For example, let's say on Jan 1, 2020, the price is $5 and on Jan 1, 2025, the price is $12.
[math]\text{Average \% change per unit of time}=\frac{\$ 12-\$5}{\$5}\times \frac{1}{5\text{ years}}\times 100=\frac{28\%}{\text{year}}[/math]In words, from 2020 to 2025, the average price increase is 28% per year.

Er, shouldn't there be a minus 1 in there? ?
Correct me if I'm wrong (no shortage of volunteers there! ?)
Because we are trying to find the percentage price increase, I would have expected it to be:-

For example, let's say on Jan 1, 2022, the price is $5 and on Dec 31, 2022, the price is $12.

[math]\text{Average \% change per unit of time}=\left(\frac{\$ 12-\$5}{\$5}-1 \right)\times \frac{1}{12\text{ months}}\times 100 \approx \frac{3.33\%}{\text{month}}[/math]

ie: a 40% increase over the twelve months.
​

You can apply the same logic if you want the average per year. For example, let's say on Jan 1, 2020, the price is $5 and on Jan 1, 2025, the price is $12.
[math]\text{Average \% change per unit of time}=\left(\frac{\$ 12-\$5}{\$5}-1 \right)\times \frac{1}{5\text{ years}}\times 100=\frac{8\%}{\text{year}}[/math]

ie: a 40% increase over the whole five years.​

In (other?) words, from 2020 to 2025, the average price increase is 8% per year.

 

[JeffM]

Yours is a question that depends a lot on exactly what you want to know.

From a business perspective, I suspect what is likely to be most relevant is the effective average increase relative to last year’s final price.

[math]\dfrac{1 * 1 + 1.03 * 6 + (1.03)(1.09) * 5}{12} - 1 \approx 0.066 = 6.6\%[/math]
That looks a bit odd so let me explain it.

First, i am taking a weighted arithmetic mean. There are 12 months in the year. For 1 of those months, the final price from last year was in effect for practical purposes. For 6 months (F, M, A, M, June, July), the increase anounced Jan 1 was in effect for practical purposes. For 5 months (A, S, O, N, D) the increase anounced July 1 was in effect for practical purposes. That explains the factors 1, 6, and 5 in the numerator. 1 + 6 + 5 = 12, which is the number of months in the year and is thus the denominator of the fraction.

To take into account the compounding effect of having multiple increases during the year, I use relative prices. For the first month, the price is 1 times last year’s final price. Starting in February, the effective price is 1.03 times last year’s final price. Starting in August, the effective price is 1.09 times February’s price, which is 1.03 last years price so, relative to last year’s price, we are at 1.03 * 1.09.

Notice I do not need last year’s final price because we are dealing with relative prices, and last year’s final price is just 1 times last year’s final price.

After calculating the effective relative price, we subtract 1 to get the relative change and multiply by 100 to express it as a percentage.

Notice that the answer we get is close to what your intuition told you.

Now we could get a bit more sophisticated with a lot more work, but this may be good enough.

 

[BigBeachBanana]

Er, shouldn't there be a minus 1 in there? ?
Correct me if I'm wrong (no shortage of volunteers there! ?)
Because we are trying to find the percentage price increase, I would have expected it to be:-

For example, let's say on Jan 1, 2022, the price is $5 and on Dec 31, 2022, the price is $12.

[math]\text{Average \% change per unit of time}=\left(\frac{\$ 12-\$5}{\$5}-1 \right)\times \frac{1}{12\text{ months}}\times 100 \approx \frac{3.33\%}{\text{month}}[/math]

ie: a 40% increase over the twelve months.
​

You can apply the same logic if you want the average per year. For example, let's say on Jan 1, 2020, the price is $5 and on Jan 1, 2025, the price is $12.
[math]\text{Average \% change per unit of time}=\left(\frac{\$ 12-\$5}{\$5}-1 \right)\times \frac{1}{5\text{ years}}\times 100=\frac{8\%}{\text{year}}[/math]

ie: a 40% increase over the whole five years.​

In (other?) words, from 2020 to 2025, the average price increase is 8% per year.

Yes, you're correct. We're looking for the increase only.