Question

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.

Answer 1

The effective annual interest rate is 24.62%

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