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4 May, 02:09

A loan of $10,000 is being repaid with payments of $1,000 at the end of each year for 20 years. If each payment is immediately reinvested at 5% effective, find the effective annual rate of interest earned by the lender over the 20-year period.

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  1. 4 May, 04:47
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    The effective annual interest rate is 24.62%

    Step-by-step explanation:

    for effective annual interest rate we will be using the following formula

    to calculate the present value of the annuity for the loan repayments first which is:

    Pv = C[ (1 - (1+i) ^-n) / i]

    where Pv is the present value of the annuity for $1000 loan payments.

    C is the annual payments of $1000 being done every year.

    i is the interest rate if the payments are reinvested immediately.

    then n is period the payments are made in which is 20 years.

    we use the present value formula because we want the present value that will be reinvested in yearly payments of $1000 so we will substitute the above mentioned values to the formula above:

    Pv = 1000[ ((1 - (1+5%) ^20) / 5%] then we compute

    Pv = $12462.21 which is the present value of the $1000 investment per year for 20 years.

    now to get the effective interest rate, we will calculate the interest rate between the $10000 loan and the present value investment $12462.21 because the initial value of the lumpsum investment is $10000.

    which will be $12462.21/$10000 - 1=ieff

    1.246221-1 = ieff

    0.246221 x 100 = ieff

    24.62% = ieff
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