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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://brianstewart.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  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: APR with fee payment Quote:
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.moneyzine.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:
 
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:
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  
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 Quote:
 
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://brianstewart.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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