Help With Math! Calculating effect of being paid weekly vs. every 6 weeks

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Messner8000
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Joined: Sun May 21, 2017 8:03 am

Help With Math! Calculating effect of being paid weekly vs. every 6 weeks

Post by Messner8000 »

I’m having a hard time wrapping my mind around something and thought others more math-inclined could help! I’ve chosen some nice round numbers, but if anyone knows how to do the math I can apply it to the real numbers.

My wife recently started her own business and submits claims through a processing company to get paid. Suppose she generates 20 claims per week (which we’ll call in week 1 claim 1A, 1B, 1C, in week 2 claim 2A, 2B, 2C, etc.), and for each claim she submits, she gets $100. She has been told that each claim (1A, 1B, etc.) costs 25 cents to submit, but she can submit 6 claims of the same type in one submission – i.e., she can submit six weeks worth of A claims for 25 cents total, but she can’t combine same week claims (such as 1A-1F).

This leaves a continuum of options we’re considering. At one extreme would be to submit all 20 claims each week. At 25 cents per claim, that is 5 dollars per week in claim fees (.25 x 20), or $260/year in fees. The benefit of this approach is she gets paid every week. If we assume $100/claim, that is $2000/week.

The other extreme is to submit claims every 6 weeks. It would still be 25 cents per claim, but she could put 6 weeks worth of claims (e.g., 1A . . . 6A) in a single submission. This would I think decrease her fees to about $43 (1/6th of $260). So, she would save about $217 per year in fees, but she would only get paid every 6 weeks. So she would get $12,000 every six weeks.

Okay…so here is my math question: how would I calculate the (opportunity?) cost of only getting paid every six weeks instead of every week (or an in between interval, like biweekly)? I’m not worried about liquidity, so it’s just a matter of trying to figure out how to crunch the numbers if the payments were put in money market account or in the market, but I don’t know how to do the math doing any rate of return. Any ideas? Thanks!!
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#Cruncher
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Re: Help With Math! Calculating effect of being paid weekly vs. every 6 weeks

Post by #Cruncher »

In the example, the fee is fixed at $5 whether claims are submitted weekly or only every six weeks. This means that the net proceeds would be $1,995 if submitted weekly or $11,995 is submitted every six weeks. Given this we can use the Excel RATE function to determine the interest rate that would make the two equivalent. I.e., $1,995 at X% per week would grow to $11,995 after six weeks. It turns out this is 4.4% on an annualized basis. So, if you would earn less than 4.4%, it would be better to only submit claims every six weeks.

Code: Select all

Row                Col A      Col B  Formula in column B
  1                Weeks          6
  2          Claims/week         20
  3        Revenue/claim        100
  4            Fixed fee          5
  5  Weekly: net payment      1,995  =B2*B3-B4
  6  At end: net payment     11,995  =B1*B2*B3-B4
  7        Weekly return    0.0834%  =RATE(B1,-B5,0,B6,0)
  8    Annualized return    4.4330%  =(1+B7)^52-1
Edited 10:55 AM to answer the question in the next post:
Messner8000 wrote: Sun Jan 23, 2022 9:01 amDoes the calculation account for the fact that if you were paid weekly, on week one you'd have $1,995 earning interest, on week 2 you'd have $3,990 earning interest, on week 3 you'd have $5,985 earning interest, etc.?
Actually the amount earning interest increases slightly more than this because the interest itself is compounded as shown here:

Code: Select all

Week  Interest  Collected    Balance
----  --------  ---------   --------
   0                            0.00
   1      0.00   1,995.00   1,995.00
   2      1.66   1,995.00   3,991.66 [*]
   3      3.33   1,995.00   5,990.00
   4      5.00   1,995.00   7,989.99
   5      6.67   1,995.00   9,991.66
   6      8.34   1,995.00  11,995.00
         -----  ---------
 Sum     25.00  11,970.00
* Example interest calculation for week 2: 1.66 = 1995 * 0.0834%
Last edited by #Cruncher on Sun Jan 23, 2022 9:56 am, edited 1 time in total.
Topic Author
Messner8000
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Joined: Sun May 21, 2017 8:03 am

Re: Help With Math! Calculating effect of being paid weekly vs. every 6 weeks

Post by Messner8000 »

Thanks so much for the response! The percentage seems much higher than I would have thought. Does the calculation account for the fact that if you were paid weekly, on week one you'd have $1,995 earning interest, on week 2 you'd have $3,990 earning interest, on week 3 you'd have $5,985 earning interest, etc.?

Thanks!!
Topic Author
Messner8000
Posts: 179
Joined: Sun May 21, 2017 8:03 am

Re: Help With Math! Calculating effect of being paid weekly vs. every 6 weeks

Post by Messner8000 »

Thanks, #Cruncher - I really appreciate it! I will admit this is still mind boggling to me - I would have thought that having all the extra money sitting in the account week in and week out over the year, it wouldn't have to earn much at all in interest to end up ahead of the fees associated with weekly payments. I guess this just goes to show why I'm a 3-fund portfolio guy---keep it simple :)
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