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To solve this problem, we need to calculate the amount $ X $ that must be repaid annually on a loan of $20,000 over 10 years, where the interest rate is 11% for the first five years and 12% for the subsequent five years.

We can use the formula for the present value of an annuity to determine $ X $, since the payments are made at the end of each period.

The formula for the present value $ PV $ of an annuity (where payments are made at the end of each period) is:
\[
PV = X \cdot \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
where $ r $ is the interest rate per period, $ n $ is the number of periods, and $ X $ is the regular payment.

We need to break this calculation into two parts since the interest rate changes after the first five years:

1. Calculate the present value of the annuity for the first 5 years at 11% interest.
2. Calculate the present value of the annuity for the last 5 years at 12% interest, discounted back to present value at the beginning of the 6th year (i.e., its value at the end of the 5th year needs to be discounted back 5 more years at the 11% rate).

We sum these two present values to equal the initial loan amount of $20,000.

Let's compute $ X $.

A loan of 20,000 is repaid by a payment of X at the end of each year for 10 years. The loan has an annual effective interest rate of 11% for the first five years and 12% thereafter. Calculate X.

Learn how to calculate the annual payment X for a loan of $20,000 repaid over 10 years with changing interest rates (11% for 5 years, 12% thereafter). This solution uses the present value of annuities and discounting techniques.

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