Final answer:
The present value of Peter's quarterly payments can be calculated using the formula for an ordinary annuity for end-of-period payments, and an adjusted formula for an annuity due for beginning-of-period payments, taking into account the 7 percent return rate and the 20-year period.
Step-by-step explanation:
To calculate the present value of annuity payments received at the end of each period, we use the formula for the present value of an ordinary annuity. In Peter's case, where the payments are $5,000 every quarter and the required return is 7 percent, the formula is as follows:
- PV = PMT × [(1 - (1 + r)^{-n}) / r]
- Here, PMT = payment per period ($5,000),
- r = periodic interest rate (7% annual rate divided by 4),
- n = total number of payments (20 years × 4 quarters per year).
For case (a), where payments are at the end of each quarter, we use the aforementioned formula directly.
On the other hand, when payments are at the beginning of each period, as in case (b), this is known as an annuity due. For an annuity due, we adjust the ordinary annuity formula by multiplying it by (1 + r), which accounts for the immediate first payment. So, the formula with Peter's figures would be:
- PV = PMT × [(1 - (1 + r)^{-n}) / r] × (1 + r)
Adding up all the present values for different time periods will give the final present value of the annuity. The goal is to determine what the future payments are worth in today's dollars considering the time value of money. Remember that in a real-world scenario, expected returns and interest rates can vary and aren't always a precise calculation.