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October 28th, 2015, 08:23 PM   #1
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2 rates

Hoping to "liven up" this section

A loan of $3000 over 24 months is obtained.

The rate will be 9% annual compounded monthly for the 1st year,
then 12% annual compounded monthly for the 2nd year.

What is the required monthly payment over the 24 months?

In case you cheat by using numeric method:
a = 3000, u = .09/12, v = .12/12, p = monthly payment
What is p in terms of a,u,v ?

(I can hear Sir Jonah smacking his lips)
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November 7th, 2015, 08:27 PM   #2
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Quote:
Originally Posted by Denis View Post
Hoping to "liven up" this section

A loan of $3000 over 24 months is obtained.

The rate will be 9% annual compounded monthly for the 1st year,
then 12% annual compounded monthly for the 2nd year.

What is the required monthly payment over the 24 months?

In case you cheat by using numeric method:
a = 3000, u = .09/12, v = .12/12, p = monthly payment
What is p in terms of a,u,v ?

(I can hear Sir Jonah smacking his lips)
Solution by Ogg

Oggs do speak, but only to those who know how to listen

Payment = ($140.03)
Total Interest Paid = $308.79
Total Principal Paid = $(3,052.05)
Balance at end of 24th month = $(52.05)



Solution by Odd (King James Version)

Whom having not seen, ye love

Payment = ($138.09)
Total Interest Paid = $314.19
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)



Solution by Atan

Better to reign in Hell than serve in Heaven

Payment = ($138.23)
Total Interest Paid = $317.40
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)



Solution by Atan (John Milton Version)

Vanity, definitely my favorite sin

Payment = ($???)
Total Interest Paid = ???
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)
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November 8th, 2015, 05:08 AM   #3
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Quote:
Originally Posted by AbrahamA View Post
Solution by Odd (King James Version)

Payment = ($138.09)
Total Interest Paid = $314.19
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)
Jimmy Da King's correct (cause it's same as mine!)
Code:
MONTH  PAYMENT  INTEREST   BALANCE
  00                                      3000.00 (9%)
  01    -138.09      22.50        2884.41 (9%)
....
  12    -138.09      12.60        1554.23 (9%)
  13    -138.09      15.54        1431.68 (12%)
....
  23    -138.09        2.72        136.72 (12%)
  24    -138.09        1.37              .00 (12%)
a = 3000
u = .0075
v = .01
p = monthly payment = ?

p = auvmn / [vn(m - 1) + u(n - 1)]
where:
m = (1 + u)^12
n = (1 + v)^12

Step 1: get FV of a:
a(1+u)^12(1+v)^12 [1]

Step 2: get FV of 1st year payments p:
p{[(1+u)^12 - 1] / u * (1+v)^12} [2]
Note:
to end 1st year, then results to end 2nd year

Step 3: get FV of 2nd year payments p:
p[(1+v)^12 - 1] / v [3]

Step 4: solve for p:
[2] + [3] = [1]

p = 138.0914495336702774404
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November 8th, 2015, 07:48 PM   #4
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Given a loan amount of \$3,000 for a period of 24 months given the rate of return in first 12 months to be 9% annual cpd monthly and the rate for last 12 months to be 12% annual cpd monthly.

There is no unique solution in determination of monthly payment.

There are an unlimited number of monthly payments determined given the data in problem specification.

All such payments lead to recovery of principal amount of \$3,000 with the remaining balance at $0.00 at the end of 24th month.

From the perspective of the borrower, the problem is one that of minimizing the monthly payment amount thus resulting in arbitrage where the market value of the loan is lower than the theoretical price (present value) of the promised cash flows.

The task is to find the lowest possible monthly payment amount resulting in maximum savings for the borrower. This turns out to be an exercise in hedging and arbitrage.

Here are a sample of monthly payments in order of lowest to highest (the last one being close to the actual monthly payment).

Market price of the monthly payment

Payment = ($138.09)
Total Interest Paid = $314.19
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)


Arbitrage driven prices of monthly payment

Payment = ($137.95)
Total Interest Paid = $310.84
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($137.98)
Total Interest Paid = $311.55
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($138.01)
Total Interest Paid = $312.22
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($138.08)
Total Interest Paid = $313.90
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)
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November 8th, 2015, 08:04 PM   #5
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Agree....but all nice sounding theoretical stuff...

2 scenarios:

loan is granted at 12%, but lender is a good Catholic,
and decides to charge 9% for 1st year...

the rates are 11.9999999% 1st year, 12.0000001% 2nd year:
do all your scenarios still hold?
If yes, then rates are 12% 1st year and 12% 2nd year:
all your scenarios still hold?
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November 9th, 2015, 12:14 AM   #6
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Quote:
Originally Posted by Denis View Post
2 scenarios:

loan is granted at 12%, but lender is a good Catholic,
and decides to charge 9% for 1st year...
The theory of arbitrage on loan repayments by yours truly Abraham A. holds true for the said rate schedules for 1st and 2nd year.

The actual monthly payments for both scenarios is the same as if the rate for the whole period of 24 months was set to 12% annual compounded monthly.

The arbitrage driven prices follow the same trend as alluded to in previous post, so my mentor Sir Denis born in a Catholic family has to say about the word of Odd as revealed to His messenger Abraham

Quote:
Originally Posted by Denis View Post
the rates are 11.9999999% 1st year, 12.0000001% 2nd year:
do all your scenarios still hold?
i(1-12) = 11.9999999%
i(13-24) = 12.0000001%

Market price of the monthly payment

Payment = ($141.22)
Total Interest Paid = $389.29
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)


Arbitrage driven prices of monthly payment

Payment = ($141.06)
Total Interest Paid = $385.39
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($141.09)
Total Interest Paid = $386.28
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($141.13)
Total Interest Paid = $387.09
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($141.21)
Total Interest Paid = $389.12
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Quote:
Originally Posted by Denis View Post
If yes, then rates are 12% 1st year and 12% 2nd year:
all your scenarios still hold?
i(1-12) = 12%
i(13-24) = 12%

Market price of the monthly payment

Payment = ($141.22)
Total Interest Paid = $389.29
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)


Arbitrage driven prices of monthly payment

Payment = ($141.06)
Total Interest Paid = $385.39
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($141.09)
Total Interest Paid = $386.28
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($141.13)
Total Interest Paid = $387.09
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)

Payment = ($141.21)
Total Interest Paid = $389.12
Total Principal Paid = $(3,000.00)
Balance at end of 24th month = $(0.00)
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November 9th, 2015, 04:25 AM   #7
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...too tired to argue...
You got too much time on your hands,
which makes you a crooked lender!
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November 9th, 2015, 05:10 AM   #8
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Quote:
Originally Posted by Denis View Post
...too tired to argue...
You got too much time on your hands,
which makes you a crooked lender!
These were excerpts from the Book of Odd, if that's what makes us crooked then wait till I show you excerpts from the Book of Atan, I suppose that will make us wicked.
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November 9th, 2015, 08:20 AM   #9
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Quote:
Originally Posted by AbrahamA View Post
wicked
You a cricket player?
If you bowl a cricket ball and hit the wicket, you WICKED it
Most boring game on the planet...
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November 9th, 2015, 10:48 AM   #10
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Beer soaked ramblings follow.

There now children, you're forgetting the first two noble truths of the Buddha,
One: Existence is suffering.
Two: The cause of suffering is desire--in this case, our desire to do high-quality finance math.

Let's take it down a notch and get down to the reality of the common commercial practice that payments must be rounded up to the nearest cent. Such as it is, the more interesting aspect of this problem lies in determining theoretically the concluding payment of this loan. Sometimes it matches the actual spreadsheet thingamajig, sometimes it doesn't. Such things are of course to be expected due to rounding conventions. It is beautiful, isn't it?

For a monthly payment of 138.10, I get a final payment of 137.88
What say you Sir D? Sir Dexter?
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