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Math Help - sime annuities / determining the interest rate

  1. #1
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    sime annuities / determining the interest rate

    Hey forum, got 2 questions today worked through the first one and my answers off by a few hundred so wanna figure out what im doing wrong there and the second one i got as far as i could but then i got to doing linear interpolation and got stuck.

    First question - A man aged 40 deposits $5000 at the beginning of each year for 25 years into an RRSP paying interest at j1=7%. Starting on his 65th birthday he makes 15 annual withdrawals from the fund at the beginning of each year. During this period ( from his 65th birthday and on) the fund pays interest at j1= 6%. Determine the ammount of each withdrawal starting at age 65. Answer is supposed to be $32868.66 but im getting R= $32561.4787.

    S=5000[ (1.07^25)-1 / .07 ] = 316245.1886
    316245.1886 = R [ 1-(1.06^-15)\.06 ] =R=32561.4787
    I used the formulas S=Rsni and A=Rani


    Second Question -A television set sells for $700.Sales Tax of 7% is added to that. The TV may be purchased for $100 down and monthly payments of $60 for one year. What is the interest rate j12? what is the annual effective rate?
    Answers are 19.61%, 21.47%

    749 = 100+60ani = 649/60=ani = 10.81666667 = k
    i=[ 1-((10.81666667/12)^2) / 10.81666667 = .017334
    Dont know where to go from there if i am even doing it right, but i used the formulas A=rani and i=(1-(k/n)^2)/k


    any help is appreciated
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  2. #2
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    Question 1:
    > S=5000[ (1.07^25)-1 / .07 ] = 316245.1886
    That would be ok if 1st deposit was a year later, not "at the beginning".
    You need to use the immediate annuity formula instead: that'll give you 338,382.35

    Question 2:
    > 749 = 100+60ani = 649/60=ani = 10.81666667 = k
    > i=[ 1-((10.81666667/12)^2) / 10.81666667 = .017334
    I don't follow that!

    Using the payment formula:
    60 = 649i / [1 - 1/(1+i)^12
    The rate i must be calculated numerically, using "iteration" (look it up),
    or other exotic means...
    The 19.61 (annual cpd monthly) that you show is correct.
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  3. #3
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    Quote Originally Posted by Wilmer View Post
    Using the payment formula:
    60 = 649i / [1 - 1/(1+i)^12
    The rate i must be calculated numerically, using "iteration" (look it up),
    or other exotic means...
    The 19.61 (annual cpd monthly) that you show is correct.
    I used this online IRR calculator to find the monthly interest rate 1.63%
    (annual rate 1.63% x 12 = 19.56%)

    EAR = (1+1.63%)^12 - 1
    EAR = (1.0163)^12 - 1
    EAR = 1.21412 - 1
    EAR = 0.21412
    EAR = 21.41%

    I used this data -649 60 60 60 60 60 60 60 60 60 60 60 60

    Then I used the online irr calculation tool that uses linear interpolation to approximate the interest rate and I got the following monthly IRR Calculation

    Net Cash Flows

    CF0 = -649
    CF1 = 60
    CF2 = 60
    CF3 = 60
    CF4 = 60
    CF5 = 60
    CF6 = 60
    CF7 = 60
    CF8 = 60
    CF9 = 60
    CF10 = 60
    CF11 = 60
    CF12 = 60
    Discounted Net Cash Flows at 0%

    DCF1 = 60/(1+0%)1 = 60/1 = 60
    DCF2 = 60/(1+0%)2 = 60/1 = 60
    DCF3 = 60/(1+0%)3 = 60/1 = 60
    DCF4 = 60/(1+0%)4 = 60/1 = 60
    DCF5 = 60/(1+0%)5 = 60/1 = 60
    DCF6 = 60/(1+0%)6 = 60/1 = 60
    DCF7 = 60/(1+0%)7 = 60/1 = 60
    DCF8 = 60/(1+0%)8 = 60/1 = 60
    DCF9 = 60/(1+0%)9 = 60/1 = 60
    DCF10 = 60/(1+0%)10 = 60/1 = 60
    DCF11 = 60/(1+0%)11 = 60/1 = 60
    DCF12 = 60/(1+0%)12 = 60/1 = 60
    NPV Calculation at 0%

    NPV = 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 -649
    NPV = 720 -649
    NPV at 0% = 71
    Discounted Net Cash Flows at 2%

    DCF1 = 60/(1+2%)1 = 60/1.02 = 58.82
    DCF2 = 60/(1+2%)2 = 60/1.0404 = 57.67
    DCF3 = 60/(1+2%)3 = 60/1.06121 = 56.54
    DCF4 = 60/(1+2%)4 = 60/1.08243 = 55.43
    DCF5 = 60/(1+2%)5 = 60/1.10408 = 54.34
    DCF6 = 60/(1+2%)6 = 60/1.12616 = 53.28
    DCF7 = 60/(1+2%)7 = 60/1.14869 = 52.23
    DCF8 = 60/(1+2%)8 = 60/1.17166 = 51.21
    DCF9 = 60/(1+2%)9 = 60/1.19509 = 50.21
    DCF10 = 60/(1+2%)10 = 60/1.21899 = 49.22
    DCF11 = 60/(1+2%)11 = 60/1.24337 = 48.26
    DCF12 = 60/(1+2%)12 = 60/1.26824 = 47.31
    NPV Calculation at 2%

    NPV = 58.82 + 57.67 + 56.54 + 55.43 + 54.34 + 53.28 + 52.23 + 51.21 + 50.21 + 49.22 + 48.26 + 47.31 -649
    NPV = 634.52 -649
    NPV at 2% = -14.48
    IRR with Linear Interpolation

    iL = 0%
    iU = 2%
    npvL = 71
    npvU = -14.48
    irr = iL + [(iU-iL)(npvL)] / [npvL-npvU]
    irr = 0 + [(0.02-0)(71)] / [71--14.48]
    irr = 0 + [(0.02)(71)] / [85.48]
    irr = 0 + 1.42 / 85.48
    irr = 0 + 0.0166
    irr = 0.0166
    irr = 1.66%
    Last edited by dexteronline; November 4th 2010 at 11:04 PM.
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  4. #4
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    Quote Originally Posted by dexteronline View Post
    I used this online IRR calculator to find the monthly interest rate 1.63%
    (annual rate 1.63% x 12 = 19.56%)
    Monthly rate is really 1.63450976....%, or 19.61411721...% annual cpd monthly.

    Annual effective = [(1.0163450976)^12 - 1]*100 = 21.47708445...%
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  5. #5
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    Quote Originally Posted by Wilmer View Post
    Monthly rate is really 1.63450976....%, or 19.61411721...% annual cpd monthly.

    Annual effective = [(1.0163450976)^12 - 1]*100 = 21.47708445...%
    Its the rounding off error that is showing the difference in my calculation, my tools are set to produce accuracy up to two decimal positions

    He could have used the MS Excel Rate function to make it easy

    =RATE(12,60,-649,0,0)
    which gets an answer of 1.6345097747% ( accuracy of up to ten decimal places )
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  6. #6
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    That's OK: I'd rather pay 1.63% than 1.6345% !
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