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Math Help - Perpetuity

  1. #1
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    Perpetuity

    Hello.

    I've tried numerous online calculators without being able to find the answer given for this equation:

    3 500 000 = 71706.70(1-(1+i)^-60/i)

    Is it possible to determine algebraically?
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  2. #2
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    Re: Perpetuity

    It is not a perpetuity, it looks like an ordinary annuity with 60 payments in amount of 71706.70 having a present value of 3 500 000

    As far as I know, you can only find the rate using iterative methods
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  3. #3
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    Re: Perpetuity

    Using the tadRATE function https://github.com/FinancialEngineer/tadJS/

    tadJS.tadRATE( 60, -71706.70, 3500000, 0 )

    i = 0.007033437953072039
    i = 0.7033%
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  4. #4
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    Re: Perpetuity

    Quote Originally Posted by AbrahamA View Post
    Using the tadRATE function https://github.com/FinancialEngineer/tadJS/

    tadJS.tadRATE( 60, -71706.70, 3500000, 0 )

    i = 0.007033437953072039
    i = 0.7033%
    0.7033% is the monthly rate

    The annual rate is

    i x 12 = 0.084401255436864468
    i x 12 = 8.44%

    The annual effective yield is

    (1+i)^12 - 1 = ( 1 + 0.007033437953072039 )^12 - 1
    (1+i)^12 - 1 = ( 1.007033437953072039 )^12 - 1
    (1+i)^12 - 1 = 0.087743997569067497701061360044075
    (1+i)^12 - 1 = 8.77%
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  5. #5
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    Re: Perpetuity

    Thank you for your help.

    It is indeed, as you say, an ordinary annuity.

    I continued my research and it appears to be very tedious to find "i" algebraically.
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  6. #6
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    Re: Perpetuity

    Quote Originally Posted by Scienceboy View Post
    Thank you for your help.

    It is indeed, as you say, an ordinary annuity.

    I continued my research and it appears to be very tedious to find "i" algebraically.
    Did you say you found an algebraic solution!
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  7. #7
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    Re: Perpetuity

    Quote Originally Posted by Scienceboy View Post
    Hello.

    I've tried numerous online calculators without being able to find the answer given for this equation:

    3 500 000 = 71706.70(1-(1+i)^-60/i)

    Is it possible to determine algebraically?
    Your formula is bit messed up, here is the edited version of the formula for present value of ordinary annuity

    3 500 000 = 71706.70 [ 1-(1+i)^-60] / i

    You may want to see the complete formula for present value of ordinary annuity that has gradients as well

    Present value of constant, growing, shrinking, increasing & decreasing annuity

    formula for present value of annuity due

    3 524 617 = 71706.70 (1+i) [ 1-(1+i)^-60] / i

    formula for present value of ordinary perpetuity

    10 195 114 = 71706.70 / i

    formula for present value of perpetuity due

    10 266 820 = 71706.70 (1+i) / i

    formula for present value of deferred ordinary annuity by 12 months

    3 217 669 = 71706.70 (1+i)^-12 [ 1-(1+i)^-60] / i

    formula for present value of deferred annuity due by 12 months

    3 240 300 = 71706.70 (1+i)^-11 [ 1-(1+i)^-60] / i

    formula for present value of deferred ordinary perpetuity by 12 months

    9 372 715 = 71706.70 (1+i)^-12 / i

    formula for present value of deferred perpetuity due by 12 months

    9 505 022 = 71706.70 (1+i)^-11 / i
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  8. #8
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    Re: Perpetuity

    To complete the argument

    the formula for present value of a constant, growing and shrinking annuity is

    Code:
    PV = R (1+i*type) (1+i)^-d [ 1 - (1+g)^n . (1+i)^-n ] / [ i - g ]
    where

    R = annuity payment
    i = interest rate
    g = growth or shrink rate
    type = 0 for ordinary annuity, 1 for annuity due
    d = time period by which to defer the annuity
    n = number of annuity payments

    for constant annuity payments g=0 thus

    Code:
    PV = R (1+i*type) (1+i)^-d [ 1 - (1+0)^n . (1+i)^-n ] / [ i - 0 ]
    
    PV = R (1+i*type) (1+i)^-d [ 1 - (1+i)^-n ] /i
    the formula for present value of a constant, growing and shrinking perpetuity is derived by taking the limit as N tends to ∞inity

    Code:
    PV = R (1+i*type) (1+i)^-d [ 1 - (1+g)^∞ . 1 / (1+i)^∞ ] / [ i - g ]
    PV = R (1+i*type) (1+i)^-d [ 1 - ∞ . 1 / ∞ ] / [ i - g ]
    PV = R (1+i*type) (1+i)^-d [ 1 - ∞ . 0 ] / [ i - g ]
    PV = R (1+i*type) (1+i)^-d [ 1 - 0 ] / [ i - g ]
    
    PV = R / [ i - g ] . [ (1+i*type) (1+i)^-d ]
    the formula for present value of a constant, increasing and decreasing annuity is

    Code:
    PV = R (1+i*type) (1+i)^-d [ 1 - (1+i)^-n ]/i + Q/i [ {(1+i*type) (1+i)^-d [ 1 - (1+i)^-n ]/i} - { n . (1+i)^-n } ]
    where

    R = annuity payment
    i = interest rate
    Q = money amount by which annuity payment increases or decreases per period
    type = 0 for ordinary annuity, 1 for annuity due
    d = time period by which to defer the annuity
    n = number of annuity payments

    for constant annuity payment Q = 0 thus

    Code:
    PV = R (1+i*type) (1+i)^-d [ 1 - (1+i)^-n ]/i + 0 / i [ {(1+i*type) (1+i)^-d [ 1 - (1+i)^-n ]/i} - { n . (1+i)^-n } ]
    
    PV = R (1+i*type) (1+i)^-d [ 1 - (1+i)^-n ]/i
    the formula for present value of a constant, increasing and decreasing perpetuity is derived by taking the limit as N tends to ∞inity

    Code:
    PV = R (1+i*type) (1+i)^-d [ 1 - 1/(1+i)^∞ ]/i + Q/i [ {(1+i*type) (1+i)^-d [ 1 - 1/(1+i)^∞ ]/i} - { ∞ . 1 / (1+i)^∞ } ]
    PV = R (1+i*type) (1+i)^-d [ 1 - 1/∞ ]/i + Q/i [ {(1+i*type) (1+i)^-d [ 1 - 1/∞ ]/i} - { ∞ . 1 / ∞ } ]
    PV = R (1+i*type) (1+i)^-d [ 1 - 0 ]/i + Q/i [ {(1+i*type) (1+i)^-d [ 1 - 0 ]/i} - { ∞ . 0 } ]
    PV = R (1+i*type) (1+i)^-d [ 1 ]/i + Q/i [ {(1+i*type) (1+i)^-d [ 1 ]/i} - { 0 } ]
    PV = R (1+i*type) (1+i)^-d / i + Q/i [ {(1+i*type) (1+i)^-d /i } ]
    
    PV = R/i [(1+i*type) (1+i)^-d]  + Q/(i^2) [(1+i*type) (1+i)^-d]
    Last edited by topsquark; May 18th 2014 at 06:50 AM.
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  9. #9
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    Re: Perpetuity

    Quote Originally Posted by AbrahamA View Post
    Using the tadRATE function https://github.com/FinancialEngineer/tadJS/

    tadJS.tadRATE( 60, -71706.70, 3500000, 0 )

    i = 0.007033437953072039
    i = 0.7033%
    Just finished me 7th beer and I get the impression that either my calculator is drunk or I am.
    Anyways, get more like i = .00703333825736278

    According to the Google calculator,
    71,706.70*[1-(1.007033437953072039)^(-60)]/.007033437953072039
    =3,499,990.15804

    ------------------------------------------------------------------------------------------------------
    Just took a shot of tequila to let it sink in for while and yet I get the same results. I must really really be enchanted. Cheers.
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  10. #10
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    Re: Perpetuity

    I'll bet dollars to donuts that AbrahamA == Scienceboy
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  11. #11
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    Re: Perpetuity

    Just finished me 7th beer and I get the impression that either my calculator is drunk or I am.
    Anyways, get more like i = .00703333825736278

    According to the Google calculator,
    71,706.70*[1-(1.007033437953072039)^(-60)]/.007033437953072039
    =3,499,990.15804
    @Jonah

    You and your calculator are both sober, nothing is wrong

    It's partly my fault, I write so much code in a day and then forget to check for errors and omissions in the code.

    It seems like the tadRATE function in tadJS library had some omissions in calculations

    Now that I fixed it, I get an IRR of

    i = 0.007033338257374002

    Checking the results

    Code:
    71,706.70*[1-(1.007033338257374002)^-60]/0.007033338257374002
    71,706.70*[1-0.65670315464511959772966724995381]/0.007033338257374002
    71,706.70*0.34329684535488040227033275004619/0.007033338257374002
    24616.683900808802541478069407737/0.007033338257374002
    3,499,999.99
    Here is correct code

    Code:
    var tadJS = {
    tadAEY: function(r, c)
    {
    	if (r==0.0)
    		return 0.0;
    	if (c==0.0)
    		return Math.exp(r) - 1;
    	else
    		return Math.pow(1.0+r*c, 1/c) - 1;
    },
    
    tadFVIF: function(r, n, c)
    {
    	c = (typeof c !== "undefined") ? c : 1;
    	if (r==0.0)
    		return 1.0;
    	if (n==0.0)
    		return 1.0;
    	return Math.pow(1.0+this.tadAEY(r,c),n);
    },
    
    tadPVIF: function(r, n, c)
    {
    	c = (typeof c !== "undefined") ? c : 1;
    	if (r==0.0)
    		return 1.0;
    	if (n==0.0)
    		return 1.0;
    	return Math.pow(1.0+this.tadAEY(r,c),-n);
    },
    
    tadPVIFbar: function(r, n, c)
    {
    	c = (typeof c !== "undefined") ? c : 1;
    	if (r==0.0)
    		return -n;
    	if (n==0.0)
    		return 0.0;
    	return -n * this.tadPVIF(r,n+c,c);
    },
    
    tadPVIFA: function(r, n, c)
    {
    	c = (typeof c !== "undefined") ? c : 1;
    	if (r==0.0)
    		return 1.0;
    	if (n==0.0)
    		return 0.0;
    	return (1.0-this.tadPVIF(r,n,c))/this.tadAEY(r,c);
    },
    
    tadPVIFAbar: function(r, n, c)
    {
    	c = (typeof c !== "undefined") ? c : 1;
    	if (r==0.0)
    		return 1.0;
    	if (n==0.0)
    		return 0.0;
    	return ( this.tadAEY(r,c) * -this.tadPVIFbar(r,n,c)  - (1.0-this.tadPVIF(r,n,c)*this.tadFVIF(r,1-c,c) ) ) / (this.tadAEY(r,c)*this.tadAEY(r,c));
    },
    
    tadRATE: function( nper, pmt, pv, fv, atype, c, guess) {
        guess = (typeof guess === "undefined") ? 0.10 : guess;
        c = (typeof c !== "undefined") ? c : 1;
        atype = (typeof atype !== "undefined") ? atype : 0;
        pmt = (typeof pmt !== "undefined") ? pmt : 0;
        pv = (typeof pv !== "undefined") ? pv : 0;
        fv = (typeof fv !== "undefined") ? fv : 0;
    
        var i;
        var x = 0.0;
        var x0 = guess;
        var f;
        var fbar;
    
        for (i=0; i<100; i++)
        {
           f = pv + pmt * this.tadPVIFA(x0, nper,c) + fv * this.tadPVIF(x0, nper,c);
           fbar =  pmt * this.tadPVIFAbar(x0, nper,c) + fv * this.tadPVIFbar(x0, nper,c);
    
           if (fbar == 0.0)
              return null;
           else
              x = x0 - f/fbar;
    
           if ( Math.abs(x-x0) < 0.000000001 )
              return x;
    
           x0 = x;
        };
    
        return null;
    }
    };
    Last edited by AbrahamA; May 18th 2014 at 05:22 PM. Reason: added the tadFVIF function to code listing
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  12. #12
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    Re: Perpetuity

    Quote Originally Posted by romsek View Post
    I'll bet dollars to donuts that AbrahamA == Scienceboy
    You can check the IP of AbrahamA and ScienceBoy to confirm the locations

    I have never heard of the chap until yesterday
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  13. #13
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    Re: Perpetuity

    The formulas I presented only allow for the trivial of present value calculations whereas the actual PV calculation are quite involved

    As a teaser, how about finding present value of series of payments where each annuity payment is itself an annuity. And some of these payments go on forever as in a perpetuity. The odd option is to allow annuity payments even after a perpetual payment

    But then there may be a schedule of discount rates rather than a single discount rate.

    And the payments may inflate or deflate per period as compared to constant annuity payments.

    For the global investor, the income in foreign denominated currency has to be converted to local currency to find the value of such an investment thus usage of exchange rates would be needed.

    No matter if they are socialist Americans or capitalist Russians or hybrid Chinese, they all must pay the extortion to the State to stay in business thus the need for applying tax rates.

    Interest may be compounded on different frequencies such as annually, semi-annually, quarterly, monthly, fortnightly, weekly, daily, infinite or continuous and even biennially or triennially.

    The length of payment periods may be other than a year such as half-yearly, quarterly, monthly, fortnightly, weekly, daily, and even biennially or triennially.

    The cash flows may be concentrated at any given time of the year other than the full year, thus the need to apply various discounting conventions such as 1st Qtr, mid-year, 3rd Qtr, full-year, biennial, triennial and so on.

    The borrowers may not be able to keep the promise of full-payment thus a need to apply possible hair-cuts on income.

    Saints are rarely found in finance, thus the possibility of rigging the discount rate to con money from the business partner

    Now what you have is 13 attributes of an investment and N such attributes making it a 13 X N matrix of data from which to find the present value

    The table shown below displays the details of an investment that has a present value of 6,883.43

    Finding the present value is possible using tadNPV function in tadXL v3.0 add-in for Excel 2007, 2010 and 2013

    Download it here so you can find the present value yourself

    http://finance.thinkanddone.com/32_b....0_en_demo.zip


    Rates 8% 7% 6% 5% 4% 3% 2% 1%
    Growth 1% 2% 3% 4% 5% 6% 7% 8%
    Exchange_Rates 1 1 1 1 1 1 1 1
    Tax_Rates 30% 31% 32% 33% 34% 35% 36% 37%
    Cash_Flows (100) 200 300 (50) 400 500 600 700
    Adjust_for_inflation 0 0 0 0 0 0 0 0
    Frequencies 10 365 24 365 INF 260 INF 5,200
    Types 1 1 1 1 1 1 1 1
    Compoundings =1/4 =1/365 =1/12 =1/365 1 =1/26 1 =1/52
    Periods =1/4 =1/365 =1/12 =1/365 1 =1/26 1 =1/52
    Concentrations 1 0.5 2 10 1 0.5 1 0.75
    Hair_Cuts 0% 20% 20% 20% 0% 20% 20% 20%
    Rate_Rigged_By 0% 15% 15% 15% 0% 15% 15% 15%
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  14. #14
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    Re: Perpetuity

    Quote Originally Posted by Scienceboy View Post
    Thank you for your help.

    It is indeed, as you say, an ordinary annuity.

    I continued my research and it appears to be very tedious to find "i" algebraically.
    Hello Scienceboy

    It would be waste of time if I did went ahead and attempted to find "i"

    Even if I did find "i", I wouldn't be the first one to do so, right!

    Scienceboy, you had me waste all my life for no reason at all

    I see no point in continue to live my life as it is at the moment and had been like this for the last 24 years

    I don't even no how to define my own identity. Sent to a land who have continuously ensured that I fail rather than succeed

    Sorry Scienceboy, you should have selected someone else for the job.

    I think I am about to take an eternal break from life.
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  15. #15
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    Re: Perpetuity

    what on Earth are you talking about?
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