solve A deferred annuity is set up that will pay $2000 at the end of each month
for 5 years at 6% annual interest rate, compounded monthly. The annuity is
deferred for 2 years rst. What lump sum should you put in the account now?
10


To solve for the lump sum needed to fund the deferred annuity, we'll use the concept of the **present value of an annuity**.

### Key Information:
- **Monthly payment (PMT):** $2000
- **Number of payments (n):** 5 years × 12 months = 60 payments
- **Annual interest rate (r):** 6% (compounded monthly, so $ \frac{6\%}{12} = 0.5\% $ monthly interest rate)
- **Interest rate per period (i):** 0.005 (monthly interest rate)
- **Deferred period:** 2 years (24 months)
- **Formula to use:** The present value of an annuity due formula and the concept of deferral.

### Solution Steps:

1. **Present value of the annuity:**
   The formula for the present value of an annuity is:

   \[
   PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}
   \]

   where:
   - $ PMT = 2000 $ is the monthly payment,
   - $ i = 0.005 $ is the monthly interest rate,
   - $ n = 60 $ is the total number of payments (5 years × 12 months).

2. **Deferral period:**
   Since the annuity is deferred for 2 years, we need to calculate the present value of the annuity at the start of the 5-year period, then discount that value back 2 years to the present.

3. **Discount the present value back by the deferral period:**
   Once the present value of the annuity at the start of the 5-year period is calculated, it must be discounted for the 2-year deferral.

Let me calculate it for you.The lump sum that you need to invest now to fund the deferred annuity is approximately **$91,780.35**.

Would you like a more detailed breakdown or have any questions?

Here are 5 related questions to deepen your understanding:

1. How does the interest rate affect the present value of an annuity?
2. What would happen if the deferral period was extended to 3 years?
3. How is the formula for the present value of an annuity derived?
4. What would be the lump sum if the interest rate were 8% instead of 6%?
5. How can the monthly payment amount be adjusted to match a different lump sum?

**Tip:** When working with deferred annuities, always account for the deferral period by discounting the present value accordingly.

A deferred annuity is set up that will pay $2000 at the end of each month for 5 years at 6% annual interest rate, compounded monthly. The annuity is deferred for 2 years first. What lump sum should you put in the account now?

Learn how to calculate the lump sum required to fund a deferred annuity with $2000 monthly payments over 5 years, deferred for 2 years, at a 6% annual interest rate compounded monthly. This guide covers the present value of an annuity and the concept of discounting future payments.

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