Business Mathematics Brief 12th Edition by Stanley Salzman ,Gary Clendenen, Charles Miller

Exercise 81

Find the present value of this annuities. Round to the nearest cent. (See Examples.)
Finding the Present Value of an Annuity
At the end of each quarter for 5 years, the Daily News deposits $4325 in an account paying 6% compounded quarterly. The goal is to accumulate funds for a new printing press. ( a ) Use the concepts of Section to find the future value of the annuity. ( b ) Then find the lump sum (present value) that must be deposited today to accumulate the same future value.

SOLUTION
(a)
(or 1.5%) per quarter; 5 years × 4 = 20 quarters. Use the amount of an annuity table in Section to find 23.12367.
Future value = $4325 × 23.12367 = $ 100,009.87 (rounded)
(b) It is not necessary to use this future value to find the present value of the annuity. Instead, use the present value of an annuity table with 1.5, per period and 20 periods to find 17.16864.
Present value = $4325 × 17.16864 = $74,254.37
Thus, a deposit of $4325 at the end of every quarter for 5 years has a present value today of $74,254.37. If we assume 6% compounded quarterly and ignore income taxes, each of the following has exactly the same value:
1. 20 end-of-quarter deposits of $4325
2. A future value at the end of 5 years of $100,009.87
3. A present value on hand today of $74,254.37
Finding the Present Value
Tom and Brandy Barrett recently divorced. The judge gave custody of their 4-year-old son to Brandy and ruled that Tom must pay $1500 in child support to Brandy at the end of each quarter until the son turns 16. Find the lump sum that Tom must put into an account earning 6% compounded quarterly to cover the periodic payments. Find the interest earned.
SOLUTION
Payments must be made for 16 - 4 = 12 years, or for 12 × 4 = 48 quarters. The interest rate per quarter is
per quarter. Look across the top of the present value of an annuity table for 1.5% and down the side for 48 payments to find 34.04255.
Present value of annuity = $1500 × 34.04255 = $ 51,063.83
A deposit of $51,063.83 today will make 48 end-of-quarter payments of $1500 each. Interest earned during the 12 years is the sum of all payments less the original lump sum.
Interest = (48 × $1500) - $51,063.83 = $20,936.17
Quick TIP
Although the $1500 withdrawals to Brandy are at the end of each quarter, the original lump sum must be deposited at the beginning of the first year.
Finding the Present Value
An American company hires a project manager to work in Saudi Arabia. The contract states that if the manager works there for 5 years, he will receive an extra benefit of $15,000 at the end of each semiannual period for the 8 years that follow. Find the lump sum that can be deposited today to satisfy the contract, assuming 6% compounded semiannually.

SOLUTION
The project manager works from years 1 to 5. He then receives two $15,000 annuity payments each year during years 6 through 13. Solve this problem in two steps.
1. Find the present value at the beginning of year 6 of the annuity with $15,000 payments.
Use
per compounding period and 8 × 2 = 16 compounding periods to find 12.56110 in the present value of an annuity table.
Present value of annuity = $15,000 × 12.56110 = $188,416.50
This is the present value of the annuity needed at the beginning of year 6 to fund payments in years 6 through 13. But it is also the future value needed for the investment made today that will fund the eventual payments.
2. Find the lump sum needed today to accumulate the $188,416.50 by the end of year 5.
Use the table showing present value of a dollar in Section (page) with
per compounding period and 5 × 2 = 10 compounding periods to find.74409.
Present value needed today = $188,416.50 ×.74409 = $ 140,198.83
A lump sum of $140,198.83 today will grow to $188,416.50 in 5 years. The $188,416.50 at the end of year 5 is enough to make 16 semiannual payments of $15,000 each during years 6 through 13.