Math Problem Statement

how much should be invested in annuity earning 8% annual interest, compounded monthly to provide a regular income of $3000 for exactly 20 years. (the first payment is to be at the end of the first 12 months)

Solution

To calculate the amount that needs to be invested in an annuity to provide a regular income of $3,000 for 20 years with an 8% annual interest rate, compounded monthly, we can use the present value formula for an annuity with monthly compounding:

PV=P×(1(1+rm)n×m)rmPV = \frac{P \times \left(1 - (1 + \frac{r}{m})^{-n \times m}\right)}{\frac{r}{m}}

Where:

  • PVPV is the present value (the amount to be invested).
  • PP is the payment per period ($3,000).
  • rr is the annual interest rate (8% or 0.08).
  • mm is the number of compounding periods per year (12 for monthly compounding).
  • nn is the number of years (20 years).

Let's plug in the values and calculate:

PV=3000×(1(1+0.0812)20×12)0.0812PV = \frac{3000 \times \left(1 - (1 + \frac{0.08}{12})^{-20 \times 12}\right)}{\frac{0.08}{12}}

Now, I'll calculate the present value (PV).The amount that should be invested in the annuity to provide a regular income of $3,000 for 20 years, with an 8% annual interest rate compounded monthly, is approximately $358,662.88.

Would you like more details on this calculation, or do you have any other questions?

Here are 5 related questions:

  1. How does the duration of the annuity affect the required investment amount?
  2. What if the interest rate were compounded quarterly instead of monthly?
  3. How much should be invested if the payment amount increases annually with inflation?
  4. What if the interest rate were lower, say 6%? How would that impact the investment amount?
  5. How does the starting point of the payments (immediate vs. deferred) impact the present value?

Tip: Monthly compounding generally results in a higher accumulated value compared to annual compounding, given the same interest rate and investment period.

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Math Problem Analysis

Mathematical Concepts

Annuities
Compound Interest

Formulas

Present Value of Annuity formula

Theorems

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Suitable Grade Level

Grades 10-12