My Math Forum APR with fee payment

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 January 21st, 2013, 03:59 PM #1 Newbie   Joined: Jan 2013 Posts: 3 Thanks: 0 APR with fee payment Hi Getting tripped up trying to work out the following problem... Loan of $5000 fixed rate of interest at 6.5% loan origination fee of$100 duration of 5 years What is the APR of the loan? Trying to get the same answer as the following link :- http://brian-stewart.orpheusweb.co.uk/c ... ualapr.htm Assuming the link is indeed correct and I am wrong does anyone know what they are doing? I assume that the total credit offered is $5100 and the$100 fee is paid immediately on the inception of the loan Have tried running through IRR calc in excel and rate function but cannot get the same APR's as the link
January 21st, 2013, 09:46 PM   #2
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Re: APR with fee payment

Quote:
 Originally Posted by JMATH17 I assume that the total credit offered is $5100 and the$100 fee is paid immediately on the inception of the loan
Can't "assume"! Find out.
Try here: http://www.efunda.com/formulae/finance/apr_solver.cfm

 January 22nd, 2013, 02:15 AM #3 Newbie   Joined: Jan 2013 Posts: 3 Thanks: 0 Re: APR with fee payment Thanks Although I would like the actual math so to be clear, I want to know how the following calculator arrives at the answer of ~7.3% http://www.money-zine.com/Calculators/L ... alculator/ using the same example that I gave loan $5000 rate 6.5% duration 5 years Application fee$100 other fees $0 Thanks  January 22nd, 2013, 07:33 AM #4 Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 Re: APR with fee payment Step 1: calculate monthly payment assuming no fee 5000 * .065/12 / [1 - 1/(1 + .065/12)^60] = 97.83 Step 2: calculate interest rate if 4900 (5000 - fee) is borrowed at payment of 97.83 cannot be solved directly; numeric method required; rate will be ~7.3472... Confused? Go here: http://www.google.ca/#hl=en&tbo=d&outpu ... 24&bih=571 If you have a financial calculator (enter rate, number of payments, amount borrowed: get payment amount), then proceed: you'll get 97.83 as required payment. Then replace the 5000 by 4900, and ask to calculate rate; btw, you'll notice process takes "longer" than expected.  January 22nd, 2013, 09:34 AM #5 Global Moderator Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: APR with fee payment It can be solved using the formula for a geometric series, in a pinch. I think numerical solutions are more common. January 22nd, 2013, 05:14 PM #6 Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 Re: APR with fee payment Quote:  Originally Posted by CRGreathouse It can be solved using the formula for a geometric series, in a pinch. Are you sure, CR? Don't you need a rate? Can you illustrate...  January 22nd, 2013, 08:52 PM #7 Global Moderator Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: APR with fee payment If the total to be paid is A over n periods, with payments of p (starting at the end of the first period), then A = pR + pR^2 + pR^3 + ... + pR^n where R - 1 is the interest rate per period. Since this is a geometric series, A = pR(R^n - 1)/(R - 1) which can be solved as desired. January 22nd, 2013, 09:44 PM #8 Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 Re: APR with fee payment Quote:  Originally Posted by CRGreathouse If the total to be paid is A over n periods, with payments of p (starting at the end of the first period), then A = pR + pR^2 + pR^3 + ... + pR^n where R - 1 is the interest rate per period. Since this is a geometric series, A = pR(R^n - 1)/(R - 1) which can be solved as desired. Disagree. Should be A = p(1 - 1/R^n) / (R - 1) Example:$1000 at 12% annual cpd monthly (R - 1 = .01) over 18 months : monthly payment = 60.982...

pR(R^n - 1) / (R - 1) = 60.982(1.01)(1.01^18 - 1) / (1.01 - 1) = ~1208

p(1 - 1/R^n) / (R - 1) = 60.982(1 - 1/1.01^1 / (1.01 - 1) = ~1000

Anyway, that's simply the formula for present value of a series of equal payments,
usually written out this way: A = p[1 - 1/(1+i)^n] / i , i being the interest rate per period.
And I can't see how it can eliminate numeric method required to solve for the rate.

January 23rd, 2013, 06:12 AM   #9
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Re: APR with fee payment

Quote:
 Originally Posted by Denis Disagree. Should be A = p(1 - 1/R^n) / (R - 1)
Depends on the assumptions. That would be correct if the first payment is immediate, mine is correct if the first payment is delayed a month (as stated in my post).

 January 23rd, 2013, 07:23 AM #10 Newbie   Joined: Jan 2013 Posts: 3 Thanks: 0 Re: APR with fee payment Some interesting points The original site I was looking at confused me - not sure it's correct (?!) (http://brian-stewart.orpheusweb.co.uk/c ... ualapr.htm) Slightly harder one... How do I get the APR for a loan that has a 2 year interest only period, followed by a 3 year amortising balance (interest & principal) terms are 5 year loan notional $5000 Fixed interest rate throughout the loan (6.5%)$100 loan fee (paid immediately) 2 years interest only 3 remaining years, interest and principal My thoughts were to average the payments and use that to calculate the APR but i'm not sure if that is correct...

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