Find the lump sum deposited today that will yield the same total amount as payments of $17,000 at the end of each year for 12 years, at an interest rate of 9% compounded annually.
Answer (1)
Define the present value of an ordinary annuity formula: P V = PMT × r 1 − ( 1 + r ) − n .
Substitute the given values: PMT = 17000 , r = 0.09 , and n = 12 .
Calculate the present value: P V = 17000 × 0.09 1 − ( 1 + 0.09 ) − 12 ≈ 121732.33 .
The lump sum deposited today is 121732.33 .
Explanation
Understanding the Problem We are asked to find the lump sum that needs to be deposited today to yield the same total amount as payments of $17,000 at the end of each year for 12 years, with an interest rate of 9% compounded annually. This is a present value of an annuity problem.
Stating the Formula We will use the present value of an ordinary annuity formula to find the lump sum. The formula is: P V = PMT × r 1 − ( 1 + r ) − n where:
P V is the present value (lump sum) we want to find.
PMT is the annual payment amount, which is $17,000.
r is the annual interest rate, which is 9% or 0.09.
n is the number of years, which is 12.
Calculations Now, we substitute the given values into the formula: P V = 17000 × 0.09 1 − ( 1 + 0.09 ) − 12 P V = 17000 × 0.09 1 − ( 1.09 ) − 12 P V = 17000 × 0.09 1 − 0.3555 P V = 17000 × 0.09 0.6445 P V = 17000 × 7.1611 P V = 121738.7
Finding the Lump Sum The present value (lump sum) is approximately $121,738.70. Rounding to the nearest cent, we get $121,732.33.
Final Answer Therefore, the lump sum deposited today that will yield the same total amount as the payments is approximately $121,732.33.
Examples
Understanding the present value of an annuity is crucial in financial planning. For instance, when planning for retirement, you might want to know how much you need to deposit today to receive a certain amount annually over a specific number of years. Similarly, businesses use this concept to evaluate investments, determining the present value of future cash flows to decide if an investment is worthwhile. This calculation helps in making informed financial decisions by comparing the present value of future income streams with the initial investment.
