Question 1172988: What is the present value of a 2-year deferred annuity at 4% interest compounded quarterly with payments of $1000 made every quarter for 3 years?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this calculation step-by-step:
**1. Understand Deferred Annuity**
* A deferred annuity is an annuity whose payments begin at some future date.
* In this case, the payments begin 2 years from now (deferred period).
* Payments are $1,000 made quarterly for 3 years.
* Interest rate is 4% per year, compounded quarterly.
**2. Calculate the Interest Rate per Period (i)**
* Annual interest rate: 4% or 0.04
* Compounding frequency: Quarterly (4 times a year)
* Interest rate per quarter: 0.04 / 4 = 0.01
**3. Calculate the Number of Payment Periods (n)**
* Payment duration: 3 years
* Payments per year: 4 (quarterly)
* Total payment periods: 3 * 4 = 12
**4. Calculate the Present Value of the Ordinary Annuity**
* We'll use the present value of an ordinary annuity formula:
* PV = PMT * [(1 - (1 + i)^-n) / i]
* Where:
* PV = Present value of the annuity
* PMT = Payment amount ($1,000)
* i = Interest rate per period (0.01)
* n = Number of periods (12)
* PV = 1000 * [(1 - (1 + 0.01)^-12) / 0.01]
* PV = 1000 * [(1 - (1.01)^-12) / 0.01]
* PV = 1000 * [(1 - 0.887449) / 0.01]
* PV = 1000 * [0.112551 / 0.01]
* PV = 1000 * 11.2551
* PV = $11,255.10
This $11,255.10 is the present value of the annuity at the *start* of the payment period, which is 2 years from now.
**5. Calculate the Present Value of the Deferred Annuity**
* Now, we need to discount this present value back to today (2 years earlier).
* Deferred period: 2 years
* Number of deferred periods: 2 * 4 = 8 quarters
* We'll use the present value formula for a lump sum:
* PV_deferred = PV / (1 + i)^deferred_periods
* Where:
* PV_deferred = Present value of the deferred annuity
* PV = Present value of the annuity at the start of payments ($11,255.10)
* i = Interest rate per period (0.01)
* deferred_periods = Number of deferred periods (8)
* PV_deferred = 11255.10 / (1 + 0.01)^8
* PV_deferred = 11255.10 / (1.01)^8
* PV_deferred = 11255.10 / 1.082857
* PV_deferred = $10,393.85 (approximately)
**Therefore, the present value of the 2-year deferred annuity is approximately $10,393.85.**
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