Earl Watkins is ready to retire and has saved up $250,000 for that purpose. He places all of this money into an account which will pay him annual payments for 20 years. How large will these annual payments be if the account earns 17% compounded annually?
To calculate the annual payments that Earl Watkins will receive, we can use the present value of an annuity formula.
The present value of an annuity formula is:
PV = PMT * [(1 - (1 + r)^(-n)) / r],
Where:
PV = present value (initial amount invested)
PMT = annual payment
r = interest rate per period
n = number of periods
Given:
PV = $250,000
r = 17% (0.17) compounded annually
n = 20 years
Let's plug in the given values and calculate PMT:
250,000 = PMT * [(1 - (1 + 0.17)^(-20)) / 0.17]
Simplifying the equation:
[(1 - (1.17)^(-20)) / 0.17] = 250,000 / PMT
Using a financial calculator or spreadsheet, we find:
PMT = $9,742.44 (rounded to the nearest cent)
Therefore, Earl Watkins' annual payments will be approximately $9,742.44.
To determine the annual payments that Earl Watkins will receive over 20 years, we need to use the concept of present value of an annuity.
The present value of an annuity formula is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present value (amount in the account, which is $250,000 in this case)
PMT = Annuity payment (what we want to find)
r = Interest rate per compounding period (17% per year)
n = Number of compounding periods (20 years)
Now, let's plug in the given values into the formula:
$250,000 = PMT * (1 - (1 + 0.17)^(-20)) / 0.17
Let's solve the equation step by step:
First, simplify the exponent inside the brackets:
$250,000 = PMT * (1 - 1.17^(-20)) / 0.17
Next, calculate the exponent:
$250,000 = PMT * (1 - 0.1092497) / 0.17
Now, subtract the value inside the brackets:
$250,000 = PMT * 0.8907503 / 0.17
To isolate PMT, divide both sides by 0.8907503 / 0.17:
PMT = $250,000 / (0.8907503 / 0.17)
PMT ≈ $33,663.25
Therefore, the annual payments will be approximately $33,663.25.