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# Math and Science Solutions for Businesses/Math formula needed to solve business problem

Question
I'm trying to figure out a formula to calculate potential revenue over four years. It is a subscription product that costs \$1,000/month. I am assuming that for the first year we will get three additional clients per month (starting with the first month) and that, of these, one client will buy 12 months, one will buy 9 months and one will buy 6 months. For the second year, we will get six additional clients per month (two 6-month contracts; two 9-month contracts; two 12-month contracts). For the third year, we will get 12 additional clients per month (four 6-month contracts; four 9-month contracts; four 12-month contracts). How can I calculate the projected revenue per year? Also, how can I factor in the impact of potential variations in client retention (e.g. what are the revenue differences if we have a 50% retention rate vs 60% or 70%)?

Thank you so much for your help!!

The following are some useful formulas based on summing the number of various subscribers over the years. First, I'll define some constants and variables. I'll be using lots of subscripts and will write them like

Ai = "A sub i" = where A = some variable and i is an integer.

R = revenue per subscription = \$1000
Ai = number of new clients per month in year i, where Ai = i, i = 1,2,3,4  <-- this says that Ai just equals the integer i in this notation
Bj = length of subscription, j = 1,2,3 and B1 = 6, B2 = 9 , B3 = 12
Mk = month of year, Mk = k, k = 1,2,...,12  <-- this says that Mk just equals the integer k

Thus, the revenue for the 2nd year for a 9 month subscription in month 5 would be

Rijk = R･A2･B2･M5 = R･2･9･5 = R･90

We have to sum these terms over the periods of interest. I'll use a shorthand symbol for sum, which is the capital letter S in Greek called sigma ∑. It is used with the indices (integers i, j, k) as

∑(i=1,4)Ai = sum of Ai for i = 1, through 4 = A1+A2+A3+A4.

So, the total revenue obtained from the sum of the number of subscribers over all 12 months (Mk), for each of the 3 lengths of subscription (Bj) over the 4 years (Ai) looks like

Rtot = R･∑(i=1,4)∑(j=1,3)∑(k=1,12){ Ai･Bj･Mk }.

Note that R = constant and so can be taken outside the sums. You could write all this out and do the addition but you would have 4･3･12 = 144 terms to add. However, using the fact that the variables are independent and the sums and multiplications are linear means we can factor the sums to get

Rtot = R･[  ∑(i=1,4){ Ai }  ]･[  ∑(j=1,3){ Bj }  ]･[  ∑(k=1,12){ Mk }  ]

where I've put the sums inside brackets , [ ], to show that they are separate from each other and are multiplied together. At this point, the sums are small enough to add the terms by hand, but we can use another mathematical trick to get the sums more easily. A formula for the sum of a sequence is

∑(i=1,n){ Ai } = n(A1 + An)/2.

Using this formula for the 3 terms above gives totals of 27, 10 and 78, so that

Total number of clients in 4 years for all 3 3 lengths of subscriptions is 27･10･78 = 21,060. Multiply this by R to get the total revenue.

Using the above formulas you should be able to calculate the revenue over whatever time periods (years) you want.

Regarding client retention, you'll need to adjust your sums by "subtracting out" subscriptions that lapse. Let r = retention rate, which is a fraction between 0 and 1 (0 < r < 1). So, for instance, if only a fraction r keep their 12 month subscription, then for year 2, the number of client would be

r･∑(k=1,12){ Mk }.

For 9 and 6 month periods, you need to multiply by r after summing up only 9 and 6 months, respectively. Its a matter of bookkeeping . Of course, r is a parameter and can be varied as often as necessary.

Let me know if this works for you.
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment I really appreciate the super speedy reply. And thank you for taking the time to put together such a thoughtful and detailed response! The answer given was quite a bit over my head so I ended up not using it. But maybe I should have posted in a different section for less math-minded people. :-) Anyway, this was my first time using allexperts and I love the idea of it. Keep up the great work!!!

Math and Science Solutions for Businesses

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#### Randy Patton

##### Expertise

Questions regarding application of mathematical techniques and knowledge of physics and engineering principles to product and services design, optimization, prediction, feasibility and implementation. Examples include sales and product performance projections based on math/physics models in addition to standard regression; practical and cost effective sensor design and component configuration; optimal resource allocation using common tools (eg., MS Office); advanced data analysis techniques and implementation; simulation and "what if" analysis; and innovative applications of remote sensing.

##### Experience

26 years as professional physical scientist and project manager for elite research company providing academic quality basic and applied research for government and defense industry clients (currently retired). Projects I have been involved in include: - Notional sensor performance predictions for detecting underwater phenomena - Designing and testing guidance algorithms for multi-component system - Statistical analysis of ship tracking data and development of anomaly detector - Deployed vibration sensors in Arctic ice floes; analysis of data - Developed and tested ocean optical instrument to measure particles - Field testing of protoype sonar system - Analysis of synthetic aperture radar system data for ocean surface measurements - Redesigned dust shelters for greeters at Burning Man Festival Project management with responsibility for allocation and monitoriing of staff and equipment resources.

Publications
“A Numerical Model for Low-Frequency Equatorial Dynamics” (with Mark A. Cane), J. of Phys. Oceanogr., 14, No. 12, pp. 18531863, December 1984.

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MIT, MS Physical Oceanography, 1981 UC Berkeley, BS Applied Math, 1976

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