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.

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

so firstly, we need to isolate the y variables to be able to solve these inequalities. firstly, add 0.5x on both sides of the first inequality and subtract 1.5x on both sides of the second inequality:

now since the slope is positive for the first inequality, this means that the line going upwards belongs to the first inequality, and the line going downwards belongs to the second inequality.

next, since y ≥ in the first inequality, this means that the shaded region will be above the first inequality's line, thus shading regions a and b.

next, since y ≤ in the second inequality, this means that the shaded region is going to be below this line, thus shading regions a and c.

since both lines shade region a, this means that region a is the solution.

step-by-step explanation:

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