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January 21st, 2013, 03:59 PM   #1
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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
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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
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January 22nd, 2013, 02:15 AM   #3
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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
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January 22nd, 2013, 07:33 AM   #4
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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.
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January 22nd, 2013, 09:34 AM   #5
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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.
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January 22nd, 2013, 05:14 PM   #6
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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...
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January 22nd, 2013, 08:52 PM   #7
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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.
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January 22nd, 2013, 09:44 PM   #8
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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.
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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).
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January 23rd, 2013, 07:23 AM   #10
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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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