Comparing the difference between 2 compound interest rates

[zachtheoldham]

Hey everyone!

I've got a bit of a conundrum here. I need an equation to calculate the difference between 2 different compound interest rates, that will add the difference between each previous time period to the total, let me explain:

A Youtube channel has 1000 subscribers, every month that passes, the channel grows by 10%. For every 100 subscribers, the channel will generate a new client every month. Each client is worth $1000/month.

So month 1 the channel grows 100 subscribers, bringing the total to 1100. That means that the channel will generate 11 new clients, which is an extra $11,000/month.

In month 2, the channel grows 110 subscribers, bringing the total to 1210. That means that the channel will generate 12.1 new clients, which is an extra $12,100/month.

I would like to compare the difference in additional monthly revenue if the channel grew by 15% per month.

Currently this is the formula that I have:

Opportunity Cost = (s(1+NGR)t*SCR*RPC)-(s(1+CGR)t*SCR*RPC)

s=current subscribers
t=months
NGR=New Growth Rate
CGR=Current Growth Rate
SCR=Subscriber to Client Rate
RPC=Revenue Per Client

So if we were to calculate the opportunity cost for 6 months from now we would have:

s=1000
t=6
NGR=15%
CGR=10%
SCR=1%
RPC=$1000

(1000(1+0.15)6*0.01*$1000)-(1000(1+0.1)6*0.01*$1000)

(1000(1.15)6*0.01*$1000)-(1000(1.1)6*0.01*$1000)

(2313*0.01*$1000)-(1772*0.01*$1000)

($23,130)-($17,720)

=$5,410/month

So on month 6 the difference in additional revenue generated is $5,410/month (23.13 new clients vs 17.72 new clients).

My problem is, I need to sum the difference in monthly revenue for each month prior. I.e Month 1 difference + Month 2 difference + Month 3 difference + Month 4 difference + Month 5 difference + Month 6 difference = Total monthly difference.

Currently, my equation only tells me the difference in a particular month, but does not sum the difference for each month prior. Can anyone help with an equation to fix this?

Thanks!!

 

[BigBeachBanana]

I haven't gone through your calculations but, what you're trying to do can be easily implemented in Excel. Simply have a monthly difference column and do some sort of cumulative sum.

 

[Cubist]

My "spideee" senses tell me this isn't part of a math course or homework ?️ . Therefore I think this would be the, "answer at the back of the book":-

Spoiler: possible answer

This might be correct. Do your own verification before using it (at your own risk)...

[math] s\cdot \mathit{SCR}\cdot \mathit{RPC}\cdot \left(\frac{1-\left(1+\mathit{CGR}\right)^{\mathit{tn}+1}}{\mathit{CGR}}-\frac{1-\left(1+\mathit{NGR}\right)^{\mathit{tn}+1}}{\mathit{NGR}}\right) [/math]
where "tn" is the last month

As text:-
s*SCR*RPC* ((1 - (1 + CGR)^(tn + 1))/CGR - (1 - (1 + NGR)^(tn + 1))/NGR)

 

[zachtheoldham]

My "spideee" senses tell me this isn't part of a math course or homework ?️ . Therefore I think this would be the, "answer at the back of the book":-

Spoiler: possible answer

This might be correct. Do your own verification before using it (at your own risk)...

[math] s\cdot \mathit{SCR}\cdot \mathit{RPC}\cdot \left(\frac{1-\left(1+\mathit{CGR}\right)^{\mathit{tn}+1}}{\mathit{CGR}}-\frac{1-\left(1+\mathit{NGR}\right)^{\mathit{tn}+1}}{\mathit{NGR}}\right) [/math]
where "tn" is the last month

As text:-
s*SCR*RPC* ((1 - (1 + CGR)^(tn + 1))/CGR - (1 - (1 + NGR)^(tn + 1))/NGR)

Hey Cubist,

Thanks for the response!

I ran the numbers but this doesn't seem to be correct? Fyi this is the correct representation of my previous formula: ((s(1+NGR)^t)*SCR*RPC)-((s(1+CGR)^t)*SCR*RPC), it was missing a bracket and a ^. Your formula doesn't seem to stack the results of each month prior, I got a number that didn't seem correct.

Any ideas? Is there anything I can do to provide you with more information? Thanks so much for the help you're a lifesaver

 

[zachtheoldham]

I haven't gone through your calculations but, what you're trying to do can be easily implemented in Excel. Simply have a monthly difference column and do some sort of cumulative sum.

Yes it definitely can! I'd just like an equation that represents this!

 

[Cubist]

Hey Cubist,

Thanks for the response!

I ran the numbers but this doesn't seem to be correct? Fyi this is the correct representation of my previous formula: ((s(1+NGR)^t)*SCR*RPC)-((s(1+CGR)^t)*SCR*RPC), it was missing a bracket and a ^. Your formula doesn't seem to stack the results of each month prior, I got a number that didn't seem correct.

Any ideas? Is there anything I can do to provide you with more information? Thanks so much for the help you're a lifesaver

What number did you get? If I use tn=6 (and your previous settings from post#1) then I get...

s*SCR*RPC* ( (1 - (1 + CGR)^(tn + 1))/CGR - (1 - (1 + NGR)^(tn + 1))/NGR )
= s*SCR*RPC* ( (-9.487171) - (-11.066799203125) )
= s*SCR*RPC* ( 1.579628203125 )
= 15796.28203125

--

After looking at all the negative numbers above, then the expression might be better rearranged as...
s*SCR*RPC* (((1 + NGR)^(tn + 1) - 1)/NGR - ((1 + CGR)^(tn + 1) - 1)/CGR)
= s*SCR*RPC* ( 11.066799203125 - 9.487171 )
= 15796.28203125

 

[BigBeachBanana]

Yes it definitely can! I'd just like an equation that represents this!

Looks like @Cubist figured out a more compact formula for you already. My suggestion would simply putting your formula under summation, where [imath]t=[/imath] current month, and [imath]n=[/imath] ending month.
[math]\sum_{t=1}^{n}((s_t(1+NGR_t)^t)\times SCR_t\times RPC_t)-((s_t(1+CGR_t)^t)\times SCR_t\times RPC_t)[/math]

 

[zachtheoldham]

What number did you get? If I use tn=6 (and your previous settings from post#1) then I get...

s*SCR*RPC* ( (1 - (1 + CGR)^(tn + 1))/CGR - (1 - (1 + NGR)^(tn + 1))/NGR )
= s*SCR*RPC* ( (-9.487171) - (-11.066799203125) )
= s*SCR*RPC* ( 1.579628203125 )
= 15796.28203125

--

After looking at all the negative numbers above, then the expression might be better rearranged as...
s*SCR*RPC* (((1 + NGR)^(tn + 1) - 1)/NGR - ((1 + CGR)^(tn + 1) - 1)/CGR)
= s*SCR*RPC* ( 11.066799203125 - 9.487171 )
= 15796.28203125

You're a legend thank you, I had misinterpreted what you meant when you said "previous month". This is perfect thanks so much