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  • Old Exam Question

    Could somebody please help me answer this question from a previous exam.

    An annuity immediate pays an initial benefit of one per year, increasing by 10.25% every four years. The annuity is payable for 40 years.
    Using an annual effective interest rate of 5%, determine an expression for the present value of this annuity.


    According to the answer key the answer is
    (1+V^2)a20 where a20 is an annuity immediate payable for 20 yrs. I can not even come close to getting this. Any ideas will be greatly appreciated.

  • #2
    Originally posted by amf
    Could somebody please help me answer this question from a previous exam.

    An annuity immediate pays an initial benefit of one per year, increasing by 10.25% every four years. The annuity is payable for 40 years.
    Using an annual effective interest rate of 5%, determine an expression for the present value of this annuity.


    According to the answer key the answer is
    (1+V^2)a20 where a20 is an annuity immediate payable for 20 yrs. I can not even come close to getting this. Any ideas will be greatly appreciated.
    Are you sure it's 10.25% increase every four years? It looks more like every 2 years is more appropriate.

    Comment


    • #3
      Originally posted by amf
      Could somebody please help me answer this question from a previous exam.

      An annuity immediate pays an initial benefit of one per year, increasing by 10.25% every four years. The annuity is payable for 40 years.
      Using an annual effective interest rate of 5%, determine an expression for the present value of this annuity.


      According to the answer key the answer is
      (1+V^2)a20 where a20 is an annuity immediate payable for 20 yrs. I can not even come close to getting this. Any ideas will be greatly appreciated.
      Sorry - I was wrong. Every 4 years is appropriate. Here comes the solution:

      Consider the payments. The first one is v = 1/1.05. The next 3 that follow are v^2, v^3, v^4.

      For the fifth payment, it's 1*1.1025 = 1*(1.05^2). Discount it 5 periods to get (1.05^2)*v^5 = v^3. The next 3 that follow the fifth payment are v^4, v^5, v^6.

      This continues for the next 32 years. To illustrate, the 9th - 12th payments are v^5, v^6, v^7 and v^8.

      Thus, we get that the PV of the new annuity is:

      PV = (v + v^2 + v^3 + v^4) + (v^3 + v^4 + v^5 + v^6) + ....

      To simplify this expression, take the first two terms in each set of parentheses and add them together, and take the last two terms in each set of parentheses and add them. You'll end up with:

      PV = [(v + v^2) + (v^3 + v^4) + .... + (v^19 + v^20)] + [(v^3 + v^4) + (v^5 + v^6) + ... + (v^21 + v^22)]
      = a_bar20 + (v^2)*a_bar20 = (1 + v^2) * a_bar20.

      Whenever you come to a problem in annuities where there's no simple way to use formulas to evaluate the annuity, use the present values of each payment, then put them back together to make it look like an annuity.

      Comment


      • #4
        Thank you very much for your help. Part of my problem was that I had the first payment increase occur at time 4 and not time five. I guess the four years should be counted from time 1 instead of time 0 since this is an annuity immediate,

        Comment


        • #5
          Originally posted by amf
          Thank you very much for your help. Part of my problem was that I had the first payment increase occur at time 4 and not time five. I guess the four years should be counted from time 1 instead of time 0 since this is an annuity immediate,
          Yeah. It should be thought of as the increase occuring every 4 payments --> first 4 payments, increase, next 4, increase, ...

          Comment

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