October 28th, 2015, 08:23 PM  #1 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  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) 
November 7th, 2015, 08:27 PM  #2  
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  Quote:
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)  
November 8th, 2015, 05:08 AM  #3  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Quote:
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%) 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  
November 8th, 2015, 07:48 PM  #4 
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5 
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) 
November 8th, 2015, 08:04 PM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
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? 
November 9th, 2015, 12:14 AM  #6  
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  Quote:
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:
i(1324) = 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:
i(1324) = 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)  
November 9th, 2015, 04:25 AM  #7 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
...too tired to argue... You got too much time on your hands, which makes you a crooked lender! 
November 9th, 2015, 05:10 AM  #8 
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  
November 9th, 2015, 08:20 AM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  
November 9th, 2015, 10:48 AM  #10 
Senior Member Joined: May 2008 Posts: 301 Thanks: 81  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 desirein this case, our desire to do highquality 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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