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Thread: question about annuities

  1. #1
    Actuary.com - Level I Poster
    Join Date
    Jul 2005

    question about annuities

    Hello -- I have a question about the valuation (present value) of an annuity that makes payments more frequently than interest is compounded. The formulas presented in Broverman and Kellison will calculate this, but from my understanding of them it is necessary for interest to be paid/collected on fractional time periods (fractional of the compounding period). Is this true?

    For example, if $100 is paid at the end of every month over the next year, and if we assume a interest of 4% compounded quarterly, the present value calculation on January 1, 2006 would have the first payment discounted by 100*(1.01)^(-1/3). But, doesn't this assume a fractional interest payment?

    If fractional interest wasn't allowed, it seems to me that the present value would just be 300[(1.01)^(-1) + (1.01)^(-2) + (1.01)^(-3) + (1.01)^(-4)]. But, the two answers are not the same. What am I missing here?
    Last edited by wowen; September 25th 2005 at 09:40 AM.

  2. #2
    Actuary.com - Level I Poster
    Join Date
    Mar 2005
    Here's what I would do. Again I am assuming a NOMINAL rate of 4% compounded quarterly. That would mean 1% per quarter and you are making $300 in deposits/quarter. So that would be PV=300a(angle 4) at 1%. If the 4% is the effective rate then use i=((1.04)^(1/4))-1 and plug that into calculator. The key is to draw your timeline and mark when the interest is compounded. Then redraw your numberline with quarterly deposits of 300 and 4 compound periods (or discount periods).

    This is the method I use anytime payments are more frequent than interest is compounded. I believe the ASM author (Which I used to pass exam FM) called this the "fusion" method. Again, my memory fails but I think that's what he called it.

    Hope this helps.
    Last edited by Mathguy36; September 24th 2005 at 03:31 PM.

  3. #3
    Author, Instructor and Seminar Provider
    Join Date
    Mar 2005
    It is generally understood, unless indicated otherwise,
    that if the payment period does not coincide with the
    interest period then interest can be regarded as compounding
    every payment period at an interest rate per payment period
    that is equivalent to the quoted rate. Equivalence here
    means compound equivalence. So in the example that
    you mention, there is a quoted rate of 1% per month
    (nominal annual 4% convertible quarterly). Since the
    payments are monthly, we can use an equivalent monthly
    interest rate (or present value factor) to value the annuity.
    The 1-month pv factor equivalent to a 3-month pv factor
    of (1.01)^(-1) is, as you have mentioned, (1.01)^(-1/3) .
    There is an implicit "fractional interest payment" per month,
    even though the situation doesn't explicitly say that
    fractional interest will be charged per month.

    This is the way amortized loan transactions work.
    Each time a payment is made, there is a calculation of
    interest due at that point, and a new outstanding balance
    is found. If payments are monthly, it doesn't matter that
    interest was quoted as quarterly, or annual effective, or something else.
    The equivalent monthly interest rate is used to determine
    the amount of interest due since the last monthly payment was made.

    It is an odd feature of Canadian law that for mortgage
    loans, lenders must quote either an annual effective rate
    of interest or a nominal annual rate of interest compounded
    semi-annually, although virtually all mortgage loans in
    Canada have monthly payments, and the equvialent
    monthly interest rate is used for calculating interest
    amounts, principal repaid and outstanding balances
    from one month to the next.

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