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Math Help - Present/Future Value of Annuity Question

  1. #1
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    Present/Future Value of Annuity Question

    Jane Doe is 25 years old and starting next year wants to
    save and invest in order to ensure an annual income of $100kin each of the 20 years after she retires, till she is 85. She willretire at 65.
    She expects to earn 10% interest till retirement and 5%interest in the years after retirement.
    How much must she save each year from now till retirementstarting next year?


    This part is easy; PVA with pmt=100k, t=20, r=5% --> ~$1,246,200

    The second part: As it turned out, Jane spent the next 5 years on the beach inBarcelona. She is 30 now and has $4,000 in the bank. Howmuch does she need to save each year, starting next year?All other data unchanged.

    What should be my thought process behind this? I know the future value when she's 65 should be $1,246,200, interest should be 10%, and there's 35 more years.
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  2. #2
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    Re: Present/Future Value of Annuity Question

    Hey youngb11.

    What formulas do you have for the annuity?

    What you can do is either re-arrange to solve for n or use a numerical scheme if you can't re-arrange n by itself on only one side of the equation.
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    Re: Present/Future Value of Annuity Question

    Quote Originally Posted by youngb11 View Post
    Jane Doe is 25 years old and starting next year wants to
    save and invest in order to ensure an annual income of $100kin each of the 20 years after she retires, till she is 85. She willretire at 65.
    She expects to earn 10% interest till retirement and 5%interest in the years after retirement.
    How much must she save each year from now till retirementstarting next year?


    This part is easy; PVA with pmt=100k, t=20, r=5% --> ~$1,246,200

    The second part: As it turned out, Jane spent the next 5 years on the beach inBarcelona. She is 30 now and has $4,000 in the bank. Howmuch does she need to save each year, starting next year?All other data unchanged.

    What should be my thought process behind this? I know the future value when she's 65 should be $1,246,200, interest should be 10%, and there's 35 more years.
    I would give it a try, though I have had limited exposure to business math...I will do both the problems as I see that you have partly done the first problem and maybe it'll be more useful to everybody else...

    1. Jane wants $1000 per year after retirement at 5% interest, so she would want to know the present value of annuity(pva) at the point of retirement,
    pva=p*\frac{1-\frac{1}{(1+r)^n}}{r} #p=>pension & r=>interest rate & n=> no. of years
    putting the values,
    pva=100000*\frac{1-\frac{1}{(1+0.5)^{20}}}{0.5}=1246221

    Now that jane has to have $ 1246221 at retirement, it becomes a future value problem now:
    fv=c*\frac{(1+r)^n - 1}{r} # c=cash payments to bank
    since jane wants to have 1246221 as future value at retirement,
    1246221=c*\frac{(1+0.1)^{40} - 1}{0.1}
    therefore, c=2815.72
    so jane would need to save at least \$ 2815.72 and invest it, to fulfill her plans...

    2. I guess the $4000 accumulated by jane would get compounded annually and contribute in making the $ 1246221, which she requires at the point of retirement.
    A=p(1+r)^n=4000(1+0.1)^{35}=112409.74
    now we will again calculate c (cash payments to bank) like we did before but take into account the $ 1246221
    fv=1246221=112409.74 + c\frac{(1+0.1)^{35} -1}{0.1}
    =1133811.26=c*\frac{(1+0.1)^{35} -1}{0.1}
    solving for c we get c=\$ 4183.49

    note that in the second case jane's savings or cash payment to bank has increased from $2815.72 to $4183.49. This is because, Jane would have made more by saving what she had for 5 years than she had earned at barcelona.If she had saved $ 2815.72 per year for 5 years, her account would have gone to 2815.72*\frac{(1+0.1)^5 -1}{0.1} or
     \$ 17190.25 .

    So jane made a terrible decision and I have tried to make approximations upto 2 decimal places
    please correct me if i am wrong.
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