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Math Help - nominal/effective interest rates

  1. #1
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    nominal/effective interest rates

    SO here's my problem that i hope you guys/gals can help me with.

    A bank offers customers 6.6% compounded quarterly on all savings accounts.


    1. If management wanted to change to an interest rate paid monthly, what nominal rate should the bank set to maintain the same effective rate?

    2. Suppose management decides to offer an interest rate with semi-annual compounding instead of monthly for these savings accounts. What would be the equivalent nominal interest rate compounded semi-annually?




    thanks in advance.
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  2. #2
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    Define:
    i^{(p)} the nominal rate convertible pthly
    i the effective annual rate

    I assume that "interest at 5% compounded monthly" is american english for "a nominal interest rate of 5% convertible monthly"


    The key thing to remember when converting effective rates to nominal rates is:

    \left(1+\frac{i^{(p)}}{p} \right)^p = 1+i

    or, equivalently

    \frac{i^{(p)}}{p}  = \left( (1+i)^\frac{1}{p} \right) -1

    or, in english
    if the nomimal rate (convertible pthly) is x, then the effective pthly rate is x/p

    Applying this to part 1:
    The effective quarterly rate is supposed to be 6.6/4 = 1.65%
    This is an annual effective rate of 1.065^4 = 1.067651543


    you want to find a monthly nominal rate that has the same annual effective rate

    \left(1+\frac{i^{(12)}}{12} \right)^{12} = 1.067651543
    solve for i^{(12)}.

    can you finish from there? (part 2 requires no new concepts)
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  3. #3
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    I hate to say it but I am so lost......
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  4. #4
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    The steps to reach a solution are:

    1. Identify the desired interest rate.
    The question says that this is 6.6%, convertible quarterly. i^{(4)} = 6.6\%

    2. Convert the desired interest rate into an effective quarterly rate
    Using the formula i gave you. the effective quarterly rate is 6.6%/4=1.65%


    3. Convert the desired effective quarterly rate into an effective annual rate
    There are 4 quarters in a year, so the effective annual rate is (1.065^4)-1 = 0.06765143 = 6.7675143%
    NB: i typoed this in my first post

    Summary
    We have worked out that the effective annual interest rate on the account should be 6.77%


    4.
    Find the nominal interest rate (compounded monthly) that gives an effective annual rate of 6.77%

    Using the formula i gave you
    \left(1+\frac{i^{(12)}}{12} \right)^{12} = 1.067651543

    \left(1+\frac{i^{(12)}}{12} \right) = 1.067651543^{\frac{1}{12}}

    \frac{i^{(12)}}{12}= \left( 1.067651543^{\frac{1}{12}} - 1 \right)

    i^{(12)}= \left( 1.067651543^{\frac{1}{12}} - 1 \right) \times 12

    i^{(12)} = 6.564 \%


    How far down can you get before you get lost?
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  5. #5
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    the last step shouldnt it be
    i^12=6.77%

    How do you solve for "i"
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  6. #6
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    to start with its important to realise that

    i^{(12)} \neq i^{12}

    i is the effective annual rate. We worked out that this should be 6.77%
    i^{(12)} is the nominal monthly rate. This is what the question asks for

    So, we dont want to solve for "i", we want to solve for i^{(12)}


    Step 4 was
    \left(1+\frac{i^{(12)}}{12} \right)^{12} = 1+i

    \left(1+\frac{i^{(12)}}{12} \right)^{12} = 1.067651543

    ...

    i^{(12)} = 6.564 \%



    PS: There were also some minor typos in my second post that i edited out about 20 minutes ago .
    PPS: This answer uses the standard international actuarial notation. You may have been taught different notation. if you post a link to siome relevant course materials, i can re-write it in the notation that your professor uses,
    Last edited by SpringFan25; June 30th 2010 at 12:55 PM.
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  7. #7
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    How do i get formulas on here cause I want to show how i did it but the end result is the same
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  8. #8
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    use [tex] tags

    http://www.mathhelpforum.com/math-he...-tutorial.html


    The important ones you are likely to want are:

    fractions: \frac{top}{bottom}

    Powers x^{stuff}

    (double click the images to see the code used to generate them)
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  9. #9
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    Perhaps this will help along with SpringFan's...

    PROBLEM: we want to solve this for r: 6.6% cpd quarterly = r% cpd monthly

    What it means is interest will be calculated AND DEPOSITED 12 times
    during a year, instead of 4 times only.

    So we have: (1 + r/12)^12 = (1 + .066/4)^4

    RULE: if a^p = b, then a = b^(1/p)
    So we have: 1 + r/12 = [(1 + .066/4)^4]^(1/12)

    RULE: (a^p)^q = a^(p*q)
    So we have: 1 + r/12 = (1 + .066/4)^(1/3)

    Move the 1: r/12 = (1 + .066/4)^(1/3) - 1

    Isolate the r: r = 12[(1 + .066/4)^(1/3) - 1]

    Use calculator: r = .06564.... or 6.56% rounded
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  10. #10
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    thanks to both of you for you help.

    I have another post here...

    http://www.mathhelpforum.com/math-he...sory-note.html

    can you guys help?

    thanks again.
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