math

Jungle Jim owes three debts:

$500 due in one year plus interest at 6% compounded semi-annually,
$2000 due in two years,
$1000 due in three years plus interest at 5% compounded monthly.

He wishes to discharge these debts by paying $500 now and two equal but unknown payments in one and two years respectively. Find the size of the equal payments if money is, at present, worth 12% compounded quarterly. Use a focal date of two years.

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  1. You state different interest rates for the 1st and 3rd debt, but state no rate for the 2nd debt of $2000 due in 2 years.
    Other than , choose a time as a reference or focal date
    I picked the present time, so

    at present, the value of the debt = value of payments

    500(1.03)^-2 + 2000(1+i)^-? + 1000(1.0041666..)^-36 = 500 + x(1.03)^-4 + x(1.03)^-8

    once you establish what the second interest rate is, evaluate each part, the x terms can be added.

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  2. The focal date is 2 years, that is your reference.

    Old debt (owing) = New debt (paying)

    Old Debt

    First part

    500*(1 + 0.06/2)^(1*2) - The future value of the first debt

    * (1 + 0.12/4)^(1*4) - To move the debt to the two year focal date. Move focal date + 1 year. Note we use the rate 12% compound quarterly to move it to the focal date as that is what the money is worth now.

    Second part

    Just leave it a $2000. This debit is already at the focal date. That is why no interest rate is given.

    Third part

    FV = 1000*(1+0.05/12)^(3*12) - The future value of the third debt

    * (1 + 0.12/4)^(-1*4) - To move the debt to the two year focal date. Move focal date - 1 year.

    To put the old debt all together:

    Old Debt = 500(1+0.06/2)^(1*2)*(1+0.12/4)^(1*4) + 2000 + 1000(1+0.05/12)^(3*12)* (1 + 0.12/4)^(-1*4)

    Old Debt = 500(1.03)^2*(1.03)^4 + 2000 + 1000(1.004166667)^36*(1.03)^-4

    Old Debt = $3,628.979182

    New Debt (Paying) x = payment amount

    500(1*0.12/4)^(2*4) - $500 is paid now, add +2 to get it to the focal date

    x(1+0.12/4)^(1*4) - A payment is made in 1 year, add +1 to get it to the focal date

    x - Just add x. Payment is at focal date

    So, we get

    New debt = 500(1*0.12/4)^(2*4) + x(1+0.12/4)^(1*4) + x

    New Debt = 500(1.03)^8 + x(1.03)^4 + x

    Old Debt = New Debt

    $3,628.979182 = 500(1.03)^8 + x(1.03)^4 + x

    Solving for x gets you x = $1,409.35

    So, the size of each payment is $1,409.35

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