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Thread: Confused on a question regarding Perpetuity

  1. #1
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    Confused on a question regarding Perpetuity

    Hi, I'm currently stuck and don't quite know how to start off this question.

    The question is: In 8 years, a new library is being opened. Operating expenses are estimated to be $52,000, payable at the end of each month. If all funds earn 14.4% compounded monthly, how much money needs to be invested today, to be able to support the library in perpetuity?

    I drew a timeline (My horrible timeline I just converted to paint, http://i.imgur.com/2Lp3Puf.png ) however I'm not sure what I would do first. I believe this question requires multiple parts but I'm not sure what exactly to do first.

    I did N=12*1000=12000, I/Y=14.4, P/Y=12, C/Y=12, FV=0, PMT=52000, PV=X however the answer I get for PV is wrong so I think I have to do something that involves the 8 years but not sure what to do.
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  2. #2
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    Re: Confused on a question regarding Perpetuity

    The way a perpetuity works is that every time the interest is calculated, it is paid out, so the balance never changes.

    Essentially what you want here is for the interest every month (which is going to be the payout) to be \$52 000.

    As we are looking at what happens every month, a simple interest calculation is all that's needed.

    $\displaystyle \begin{align*} I &= P\,R \\ P &= \frac{I}{R} \\ P &= \frac{52\,000}{\frac{14.4}{1200}} \textrm{ (do you understand why we need to divide by 12?)} \\ &= 4\,333\,333.33 \end{align*}$

    So at the time that the library is opened, we need \$4 333 333.33 invested in the perpetuity so it will pay out \$52 000 each month.

    However, we have 8 years to earn that amount. So what we want to do is work out how much money to invest NOW, so that in 8 years there is \$4 333 333.33 invested. I'll assume that there is only the initial investment and no further payments are made. This needs a compound interest calculation.

    $\displaystyle \begin{align*} A &= P\,\left( 1 + R \right) ^n \\ 4\,333\,333.33 &= P\,\left( 1 + \frac{14.4}{1200} \right) ^{ 96 } \\ P &= \frac{4\,333\,333.33}{\left( 1 + \frac{14.4}{1200} \right) ^{96}} \\ P &= 1\,378\,773.40 \end{align*}$

    So \$1 378 773.40 needs to be invested now so that in 8 years there is enough money in order to support the library in perpetuity.
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    Re: Confused on a question regarding Perpetuity

    Quote Originally Posted by Prove It View Post
    The way a perpetuity works is that every time the interest is calculated, it is paid out, so the balance never changes.

    Essentially what you want here is for the interest every month (which is going to be the payout) to be \$52 000.

    As we are looking at what happens every month, a simple interest calculation is all that's needed.

    $\displaystyle \begin{align*} I &= P\,R \\ P &= \frac{I}{R} \\ P &= \frac{52\,000}{\frac{14.4}{1200}} \textrm{ (do you understand why we need to divide by 12?)} \\ &= 4\,333\,333.33 \end{align*}$

    So at the time that the library is opened, we need \$4 333 333.33 invested in the perpetuity so it will pay out \$52 000 each month.

    However, we have 8 years to earn that amount. So what we want to do is work out how much money to invest NOW, so that in 8 years there is \$4 333 333.33 invested. I'll assume that there is only the initial investment and no further payments are made. This needs a compound interest calculation.

    $\displaystyle \begin{align*} A &= P\,\left( 1 + R \right) ^n \\ 4\,333\,333.33 &= P\,\left( 1 + \frac{14.4}{1200} \right) ^{ 96 } \\ P &= \frac{4\,333\,333.33}{\left( 1 + \frac{14.4}{1200} \right) ^{96}} \\ P &= 1\,378\,773.40 \end{align*}$

    So \$1 378 773.40 needs to be invested now so that in 8 years there is enough money in order to support the library in perpetuity.
    Oh, alright. I managed to get to 4,333,333.33 but didn't know what to do afterwards. That makes it a lot clearer, thanks!

    Just curious so I know both ways, but is there a way to do the compound interest calculation portion via calculator? I'm confused as to what N would be.
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    Re: Confused on a question regarding Perpetuity

    Sure, PMT = 0 and N = 96
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    Re: Confused on a question regarding Perpetuity

    Quote Originally Posted by Prove It View Post
    Sure, PMT = 0 and N = 96
    Thanks! Appreciate the quick help a lot.
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