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Math Help - effective annual rate

  1. #1
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    effective annual rate

    Company B plans to give an incentive, a discount of 5% to its customers if they pay immediately, or credit terms of 50 days to settle the full amount. At what effective annual rate are these two payment options equivalent?

    I am not very sure what the question is asking about. Is it asking the cost of forgoing the discount rate?

    My work:

    5/1, net 50

    Period= 365/ (50-1) = 7.45

    cost = 5%/ 95% = 0.053

    Therefore the effective annual cost is 7.45 x 0.053 = 0.392 or 39.2%

    Am i right? Thanks.
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  2. #2
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    Re: effective annual rate

    Find the interest rate at which 0.95 now is equivalent to 1 in 50 days,

    0.95(1+i)^{\frac{50}{365.25}} = 1

    i get about 45%
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  3. #3
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    Re: effective annual rate

    This kind of "effective rate guessing" is usually kept simple (it was in my days anyway!):
    50 days = .05
    365 days = .05 / 50 * 365 = 36.5 %

    One thing is definite:
    there is no way anything precise can be determined,
    since we're not told what rate the company borrows at;
    if it can borrow at something lesser than what the 5%/50days represents,
    you can be sure it won't offer that discount!!
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    Re: effective annual rate

    Thanks again Wilmer, how is your method different from SpringFan's method?
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  5. #5
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    Re: effective annual rate

    I disagree, as long as "equivalent" means "has same NPV" the borrowing rate is not relevent.
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  6. #6
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    Re: effective annual rate

    (95).....................................[50days]100

    100 / (1 + i)^50 = 95
    i = .001026392...

    365i = .374633... ; so ~37.5 % = "effective annual rate"
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  7. #7
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    Re: effective annual rate

    Quote Originally Posted by Wilmer View Post
    (95).....................................[50days]100

    100 / (1 + i)^50 = 95
    i = .001026392...

    365i = .374633... ; so ~37.5 % = "effective annual rate"
    what does the interest rate i=.001026392 represent? The daily interest rate for the year?
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  8. #8
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    Re: effective annual rate

    Yes.
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  9. #9
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    Re: effective annual rate

    Quote Originally Posted by Wilmer View Post
    Yes.
    Thanks, does simply multiplying the daily rate with 365 take into consideration the effects of compounding interest?
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  10. #10
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    Re: effective annual rate

    Quote Originally Posted by hooke View Post
    Thanks, does simply multiplying the daily rate with 365 take into consideration the effects of compounding interest?
    No. The resulting .374633 (or ~37.5%) is the annual rate compounded daily
    (same as a car loan quoted as 12% compounded monthly; results in 12.6825% effective).

    The actual effective annual rate is .454552, or ~45.5% : as per Spring Fan's post.
    My point earlier was that the "lender" will declare 12% annual cpd. monthly,
    not 12.6825%...looks cheaper to the borrower!
    But I should have kept my trap shut and "stick to the facts"! (as SpringFan did).

    This is what it all looks like:
    Code:
    DAY        INTEREST       BALANCE
    000                      950.0000
    001         .9751        950.9751
    002         .9760        951.9511
    ...
    049        1.0243        998.9747
    050        1.0253       1000.0000 : BINGO!
    051        1.2639       1001.2639
    ...
    364        1.4150       1380.0530
    365        1.4165       1381.4695
    365.25      .3543       1381.8238 **
    ** 1381.8238 / 950 = 1.454552, hence ~45.46% effective.
    Last edited by Wilmer; August 12th 2012 at 06:30 AM.
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