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Math Help - Annuity and serial loan with a twist

  1. #1
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    Question Annuity and serial loan with a twist

    Hi there!
    I'm preparing myself for my examination and have a case I'm pretty sure will be given at it - but we don't have a specific formula for this case.

    Let's assume that you take a loan, 500.000, which you are to pay yearly during the next 20 years with an interest of 5%. In short:

    K0 = 500.000 (five-hundred-thousand)
    n = 20
    r = 0,05

    To figure out what you pay can be done by:

    How much you pay every period (annuity):
    K = ( K0 * r ) / ( 1 - ( 1 + r )^n

    Total paid in interest:
    n * K - K0

    How much you pay every period by serial loan:
    (K0 * r ( n + 1 )) / 2


    Now the twist:
    Assume that you, i.e.:
    inherited a lot of money, or
    sold your property and got a higher return than what it cost you, or
    won in lottery,

    and then, five years into this you paid of your entire loan, how much did you pay in
    1) interest
    2) instalments
    3) total

    I can calculate this by calculating how much one pays every period, calculate rent for the given period, calculate instalment for the given period, by that figure out how much of the loan is left to pay for the next period, and so it continues. But doing this over and over again takes a lot of time and I was wondering if there is any formula for figuring out this an easier way.

    And if anyone know of a solution to both annuity and serial loan, I'd be very happy for it!

    If you think my english isn't all correct then you're probably right, greetings from Norway!

    Hope to get an answer soon
    -ShellSh
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  2. #2
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    Sorry, but I can't follow your formulas or what you're doing;
    not saying you're doing anything wrong!

    Formula for loan payment:
    P = A(i) / [1 - 1/(1 + i)^n]
    So for your problem:
    P = 500000(.05) / (1 - 1/1.05^20) = 40121.2936...

    Total interest is simply: Pn - A = 20(40121.2936...) - 500000 = 302425.872...

    Is that what you get?

    Next is where are we 5 years later; right?

    Steps:
    1: X = FV of 500000 (n = 5)
    2: Y = FV of 5 payments of (or annuity of) 40121.2936
    3: X - Y = owing in 5 years

    1: 500000(1.05)^5 = 638140.7812...
    2: 40121.2936(1.05^5 - 1) / .05 = 221695.4737...
    3: 638140.7812 - 221695.4737 = 416445.3075...
    OK?

    Interest: 416445.3075 - [500000 - 5(40121.2936)] = 117051.7755...

    Hope that was helpful and not too confusing
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  3. #3
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    Perfect! I can see by my calculations in Excel that you are right. I need to spend some time to look at this to get the overview of what you actually have done, but it is all correct! I'm impressed! Still, I wonder how to do that with a serial loan =)

    Edit: Though i have a problem understanding you when you write: P = 500000(.05) / (1 - 1/1.05^20) = 40121.2936...

    The 1-1 part... Is it written: P = 500000(.05) / ( 0 /1.05^20) = 40121.2936... ?

    Edit2: Figured it out =)

    Edit3: Btw, what does FV mean?
    Last edited by ShellSh; December 12th 2010 at 01:30 PM.
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  4. #4
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    FV = Future Value
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  5. #5
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    Still, I'm wondering if anyone could help me out with calculating the serial loan similar to what Wilmer did. Anyone, pretty please?
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  6. #6
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    What is wrong with what he did?
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  7. #7
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    I've given you the full answer if an annuity loan.

    Just noticed that you also need this if a "serial loan". What the heck is that?
    The only info you supplied is:

    > How much you pay every period by serial loan:
    > (Ko * r ( n + 1 )) / 2

    500000 * .05 * 21 / 2 = 262500 : makes NO SENSE...

    Please supply a FULL description of your "serial loan".

    I also noticed that you posted this:

    > How much you pay every period (annuity):
    > K = ( Ko * r ) / ( 1 - ( 1 + r )^n

    That is not correct; should be: K = ( Ko * r ) / ( 1 - ( 1 + r )^(-n))

    NOTE:
    I googled "serial loans"; very little information exists.
    Seems to be a "trick" by lenders: end result is a bit less in total interest;
    BUT the initial payments are higher: so that's not really saving on interest;
    apply a slightly higher payment on an annuity basis and you get same results!
    Anyway, I shouldn't comment on the "morality" of lending institutions!!
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  8. #8
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    It might be that it have another name than what my dictionary comes up with, but I'll try to explain it (you're pretty spot on in your description):

    A serial loan is a loan with changing payments for each period.
    The loan instalment is constant throughout all periods, while the interest costs are variable, highest in the beginning, and lower towards the end. This means that the payments vary. It lowers because the costs of interest decrease.

    Here is what an annuity loan will look like:



    And here is what a serial loan would look like:



    Interest is blue.
    Loan instalment is red.

    (pictures are taken from matematikk.net :: Emner :: Personlig konomi)

    The formula for total rates paid with serial loan is:
    Loan*i(n+1)/2

    Yes, the answer you gave for the annuity loan was exactly what I was asking for. About the annuity formula: you are right, my bad. And your comment on the "morality" of lending insitutions is interesting, never thought much of that =)
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  9. #9
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    Well...

    Ya...I see...a "constant principal" loan; yours will look like:
    Code:
    YEAR     PAYMENT     INTEREST     BALANCE
     0                              500,000.00
     1    -50,000.00    25,000.00   475,000.00
     2    -48,750.00    23,750.00   450,000.00
     3    -47,500.00    22,500.00   425,000.00
     4    -46,250.00    21,250.00   400,000.00
     5    -45,000.00    20,000.00   375,000.00
    ...
    19    -27,500.00     2,500.00    25,000.00
    20    -26,250.00     1,250.00          .00
    The INTEREST column will add to 262,500;
    Per your formula: 500000(.05)(21) / 2 = 262,500 ; so formula CORRECT!

    The "principal" portion of the payment is simply A / n : 500000 / 20 = 25,000;
    what is owing after x payments: A - x(A / n) ;
    after 5 years (x = 5) : 500000 - 5(500000 / 20) = 375,000

    A = 500000, P = 25000, i = .05, n = 20, x = 5
    To calculate the interest paid after x payments:
    [2A - P(x-1)] * i * x / 2
    = 900000 * .05 * 5 / 2
    = 112,500 (agrees with my illustration above; add the interest shown)

    SOoooooo:
    annuity basis 1st payment: ~40,121
    serial basis 1st payment: 50,000
    That's why total interest is lower with serial...or "appears" lower...
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