Math Problem Statement
What is the present value of an annuity that pays $74 per year for 5 years and an additional $1,000 with the final payment? Use a nominal rate of 9.65%.
Solution
To calculate the present value (PV) of the annuity, we can break it into two parts:
- The annuity part: Payments of $74 per year for 5 years.
- The lump sum part: An additional payment of $1,000 at the end of 5 years.
Formula for Present Value of an Annuity
The formula for the present value of an annuity is:
Where:
- (annual payment),
- (annual nominal interest rate),
- (number of years).
Formula for Present Value of a Lump Sum Payment
The present value of a lump sum paid in the future is:
Where:
- (lump sum payment),
- (interest rate),
- (number of years).
Step-by-Step Calculation
- Present value of the annuity:
- Present value of the lump sum:
Let's calculate these values.1. Present value of the annuity: $283.04 2. Present value of the lump sum: $630.89
Total present value of the annuity:
So, the present value of the annuity is $913.94.
Would you like further details on any of these steps or calculations?
Here are some related questions:
- How does the discount rate affect the present value of an annuity?
- What is the difference between a nominal and effective interest rate?
- How would the present value change if payments were made quarterly instead of annually?
- How would inflation impact the value of future annuity payments?
- What is the future value of this annuity?
Tip: The present value of an annuity decreases as the interest rate increases.
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Math Problem Analysis
Mathematical Concepts
Present Value
Annuities
Lump Sum Payments
Time Value of Money
Formulas
Present Value of Annuity Formula: PV = P * [(1 - (1 + r)^-n) / r]
Present Value of Lump Sum: PV = F / (1 + r)^n
Theorems
Time Value of Money Principle
Suitable Grade Level
Grades 11-12 or Undergraduate Finance
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