Example 2.6 A business owner takes out a loan of 450'000 on January 1 at a fixed-interest rate of 9% per year. It is payable over 15 years level payments due on each December 31 for the first 14 years and a final payment of 50'000 due on December 31 of the 15th year. Calculate the annual payment amount.
Final answer is 56032.
How do the book come up with answer?
Hint: You just have to set equal the future value (end of year 15) of the loan and the future value (end of year 15) of the payments.
$$\underbrace{1.09^{15}\cdot 450,000}_{\text{future value of the loan}}=r\cdot \underbrace{\underbrace{\frac{1-1.09^{14}}{1- 1.09}}_{ \text{future value of the first 14 pmts}\\ \text{at the end of year 14 }}\cdot1.09}_{\text{future value of the first 14 pmts}\\ \text{at the end of year 15 }}+\underbrace{50,000}_{\text{future value of the last payment}}$$
The key point that the reference point in time on both sides of the equation is 15 years in the future.