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February 13th, 2017, 09:24 PM   #11
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Your example works like this:
Year01 : 1454.63 yearend: 17455.56
Year02 : 1483.72 yearend: 35260.20:1454.63(1.02)= 1483.72
....
Year09 : 1704.33 yearend:170272.50
Year10 : 1738.42 yearend:191133.54 *****
Year11 : 1773.18 5months:199999.44

Looking at the 10 years period, we have the equivalent
of an annuity of 17455.56 annually earning 2% annually;
using the future value formula:
17455.56 * (1.02^10 - 1) / .02 = 191133.54 *****

Notice that "annual" can be used, since there is
no interest involved, only a percentage increase.

So if we had 10 years instead of 10 years 5 months,
there would be no problems (d = 1st annual amount):
d * (1.02^10 - 1) / .02 = 191133.54 : solve for d

But I can't see how to handle "part of year" as the
5 months in your example.
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February 14th, 2017, 06:07 AM   #12
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I'll give you more details about the actual application.

I work in the insurance industry. If someone is injured in a car accident, they are entitled to receive 1,000,000.00 in coverage from their insurance company for medical expenses. This is paid out on a monthly basis, based on the injured party's needs. For example, they could receive 5,000.00* every month until the 1,000,000.00 limit of coverage is exhausted.

*It is actually 5,000.00 for the first twelve months only, then 5,100.00 (5,000.00 x 1.02) for the next twelve months, as this benefit provides for the indexation of payments to protect against inflation.

We are sometimes asked to determine the monthly allowance if the injured party wants payment for their "lifetime". For example, the injured party has 200,000.00 remaining in coverage and their life expectancy has been calculated at 10 years and 5 months. The insurance company wants to know what initial monthly amount the injured party needs to receive so that the 200,000.00 is exhausted over 10 years and 5 months, with payments increasing each year by 2%.

I know how to calculate the total payout if I know the initial monthly amount and payment term. For example, if the initial monthly amount is 1,454.63 (which I know is the answer to my question through trial and error) and the payment term is 10 years and 5 months, I would do the following:

Year 1: 1,454.63 x 12‎ = 17,455.56
Year 2: (1,454.63 x 1.02) x 12 = 17,804.67
Year 3: (1,483.72 x 1.02) x 12 = 18,160.73
Year 4: (1,513.39 x 1.02) x 12 = ‎18,523.89
Year 5: (1,543.66 x 1.02) x 12 = 18,894.40
Year 6: (1,574.53 x 1.02) x 12 = 19,272.25
Year 7: (1,606.02 x 1.02) x 12 = 19,657.68
Year 8: (1,638.14 x 1.02) x 12 = ‎20,050.83
Year 9: (1,670.90 x 1.02) x 12 = 20,451.82
Year 10: (1,704.32 x 1.02) x 12 = 20,860.88
Year 11: (1,738.41 x 1.02) x 5 = 8,865.89
Total Payout: ‎199,998.60

The total payout should be 199,999.55, but I'm assuming this is due to inconsistent rounding.

I previously said 100.20 but that was incorrect, so may have caused confusion.

Hopefully I have explained this well enough to understand. I am hoping there is a way to solve for the initial monthly amount without time-consuming trial and error. Perhaps I need to use a program like Excel with several formulae?

Last edited by mshaughn; February 14th, 2017 at 06:11 AM.
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February 14th, 2017, 06:13 AM   #13
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Thanks, Denis. I didn't see your post before replying. That is what I'm trying to do. Is there a way to combine formulae to solve for the additional five months?
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February 14th, 2017, 06:37 AM   #14
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I am 99% confident that there is both an algorithm and an excel process for the algorithm. It may be a little convoluted, and I am busy this morning. I shall probably be able to give you a definite answer sometime this afternoon (unless denis beats me to it).
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February 14th, 2017, 10:40 AM   #15
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OK

I have written an excel spread sheet that computes the initial monthly payment without any trial and error.

It gives the exact same answer as you got.

If you send me a private message with your email address, I'll send you the code by email.

And skipjack was on the right path!!! Basically the spreadsheet assumes that the initial payment is 1 dollar, figures out what that would add up to, and divides that total into the total to be paid. Amazing that the actuaries at an insurance company did not do the same thing.
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February 14th, 2017, 11:29 AM   #16
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i = percentage increase (.02)
n = number of complete years (10)
e = number of payments in final year (5)
v = total disbursement (200,000)
a = 1st year monthly payment (?)

a = v*i / [12*(r^n - 1) + i*e*r^n] where r = 1 + i

a = 1454.634064...

That'll handle any similar case...

Note: the payments during last e months are simply:
a * (1 + i)^(n - 1) = 1738.4223695....
Thanks from JeffM1 and mshaughn

Last edited by Denis; February 14th, 2017 at 11:40 AM. Reason: Added note
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February 14th, 2017, 02:40 PM   #17
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Thank you everyone for your help! You've made my life a lot easier at work.
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February 14th, 2017, 03:43 PM   #18
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Quote:
Originally Posted by mshaughn View Post
Thank you everyone for your help! You've made my life a lot easier at work.
Thanks for not asking HOW I got that equation:
took me 3 sheets of paper!
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February 14th, 2017, 04:02 PM   #19
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I notice that mshaughn calculated 1,454.63 × 1.02 (which is 1483.7226), then multiplied that by 12 to get 17804.6712, which was rounded to 17804.67.
As the actual monthly payment was presumably 1483.72, that should have been multiplied by 12 to give 17804.64.

Should the monthly payments for year 3 be calculated as 1483.72 × 1.02 (rounded) or should they be calculated as 1.454.63 × 1.02² (rounded)?

Using the second method and starting with payments of 1,454.63, the recalculated total payment would be 199999.43.

Using the second method and starting with payments of 1,454.64, the recalculated total payment would be 200000.80.

Of course, each year's 12th payment could be adjusted to compensate for the accumulated rounding errors of the entire year...
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February 14th, 2017, 05:02 PM   #20
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Hard to tell how the insurance company handles these.

I presume they "set up" the thing by creating the equivalent
of a frozen savings account at 0% interest and round up the
payment to closest dollar.

If they send out a "statement" so recipient knows where he/she
stands, it presumably would look like:
Code:
0                   200,000.00
1  -1,455.00   198,545.00
2  -1,455.00   197,090.00
...
11 -1,455.00   183,995.00
12 -1,455.00   182,540.00
Also, there is probably some agreement where the recipient
can ask for settlement in full, at a discounted amount,
pegged to some rate like the Bank of Canada rate up here.
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