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Math Help - Help with annuities!

  1. #1
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    Help with annuities!

    I'm having trouble understanding this question.
    any help is much appreciated

    Today is your 30th birthday. You intend to retire at age 60 and you want to be able to receive a 20 year, $100,000 annuity with the first payment to be received on your sixty-first birthday. You would like to save enough money over the next 15 years to achieve your objective; that is, you want to accumulate the necessary funds by your forty-fifth birthday.

    (a) If you expect your investments to earn 12 percent per year over the next 15 years and 10 percent per year thereafter, how much must you accumulate by the time you reach age 45?

    (b) What equal, annual amount must you save at the end of each of the next 15 years to achieve your objective, assuming that you currently have $10,000 available to meet your goal? Assume the conditions stated in Part a
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  2. #2
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    I'll assume youve been taught standard actuarial notation (ie v=\frac{1}{1+i} among other things):


    The scenario you are given is:
    Step 0: You have $10k
    Step 1: you save $P per year for 15 years (interest 12%) in arrear (15 payments total). At the end of this you are 45 years old.
    Step 2: At age 45, you stop paying into your savings plan. The existing value grows at 10%
    Step 3: At age 60, you use the accumulated value of your plan to buy a $100k annuity, payable in arrear for 20 years.

    The easiest way to work through these problems is to work backwards

    part (a):
    Think of the question as:
    "How much money must you have on your 45th birthday, in order to pay for $100k per year for 20 years, with the first payment delayed for 16 years".
    First note: The interest rate at all times from age 45 is 10%, so use i=0.1 when working this out


    Start by working out the lump sum required at age 60 to pay for the annuity. This is just a $100k annuity for 20 years, payable in arrear:

    Value~~Z = \frac{1-v^{20}}{i} \times 100000


    Now, work out what you need at age 45 in order to have a lump sum of S at time 60:
    Value of Z in 15 years = : v^{15}Z

    This is the answer to the question.
    Last edited by SpringFan25; May 30th 2010 at 02:43 AM.
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  3. #3
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    part b

    The way to think about this is:
    "How much money do i ahve to save each month for it to accumulate to the value i worked out in part a, at age 45".

    The question tells you that the relevant interest rate is 12%, and we will be making annual savings in arrear.

    The accumulated value of 1 paid in arrear for 15 years is:

    S_{15} = \frac{(1+i)^{15} -1}{i}

    So, define:
    K = the value of the benefit at age 45 (this is the answer of part a)
    P = the amount you save each year

    You must solve
     10000(1+i)^{15} + P \times S_{15} = K
    Last edited by SpringFan25; May 30th 2010 at 02:41 AM.
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  4. #4
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    Quote Originally Posted by SpringFan25 View Post
    The scenario you are given is:
    Step 1: you save $P per year for 15 years (interest 12%). At the end of this you are 45 years old.
    Step 2: At age 45, you stop paying into your savings plan. The existing value grows at 10%
    Step 3: At age 60, you use the accumulated value of your plan to buy a $100k annuity, payable in arrear for 20 years.
    Step 1: you deposit $10,000 (interest 12%) at age 30.
    Step 2: you save $P per year for 15 years (interest 12%) from age 31 to age 45.
    Step 3: At age 45, you stop paying into your savings plan. The existing value grows at 10%
    Step 4: At age 60, you use the accumulated value of your plan to buy a $100k annuity, payable from age 61 to age 80, at 10%
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  5. #5
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    True that, i've edited the original post.
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  6. #6
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    So, working backward:

    1: calculate value at age 60 required for annuity.
    A = 100000(1 - 1/1.10^20) / .10 = 851,356.37

    2: calculate value at age 45 required to reach A.
    B = A / 1.10^15 = 203,807.95

    3: calculate accumulation to age 45 of initial 10,000.
    C = 10000(1.12)^15 = 54,735.66

    4: calculate required payment from age 31 to 45.
    P = .12(B - C) / (1.12^15 - 1) = 3,998.73
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  7. #7
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    thank you both very much.
    You were very helpful
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  8. #8
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    Quote Originally Posted by ihatemath09 View Post
    thank you both very much.
    You were very helpful
    Change your name to ilovemath, else we don't help no more
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