Math - Compound

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If a bank offers interest at a nominal rate of 6%, how much greater is the effective rate if interest is compounded continuously than if the compounding is quarterly?

I don't get this question at all... All I'm given is the rate and how am I suppose to compare compounded continuously and quarterly if I'm not given the initial value, time and so on...

  • Math - Compound -

    well, lets compare both to compounding once a year for n years.

    Once a year
    final/original = 1.06^n

    Four times a year
    6/4 = 1.5% per quarter
    final/original = 1.015^4n

    Continuously
    final/original = e^.06 n

    Well, lets see how four times a year compares to once a year
    compare 1 = 1.015^4n / 1.06^n
    ln compare 1 = 4 n ln 1.015 - n ln 1.06
    = n(.05955445-.0582689) = .0012855419 n
    so
    compare 1 = e^.0012855419 n
    or .12855 % better than once a year

    Now do the same for continuous
    compare 2 = e^06 n / 1.06^n
    ln compare 2 = .06 n - n ln 1.06
    = .001731 n
    compare 2 = e^.001731 n
    or .1731 % better than once a year

  • Math - Compound -

    Quarterly componding of interest after one year at 6% annual rate gives you an annual yield of
    (1 + 0.06/4)^4 - 1 = 6.136%

    Continuous compounding requires you to consider limits. The answer is

    Limit (as n approaches infinity) of
    1 + 0.06/n)^n - 1
    Calculus shows that this equals
    e^(0.06) -1 = 6.184%

    If you don't understand limits and e, consider the "daily interest" case, with n = 365. In that case the annual yield is 6.183%

  • Easier -

    Just do them each for one year
    yearly 1.06
    quarterly
    1.015^4 = 1.06136 or 6.136 % yearly
    continuously
    e^.06 = 1.06184 or 6.184 % yearly

  • Math - Compound -

    thank you both for your help... I like Damon's method as it seems straightforward and I can follow it...

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