At the beginning of each period for 9 years, Scott Sullivan invested $900 quarterly at 4% interest compounded quarterly. What is the present value of this annuity due? (Using the Calculator)
Well, Scott Sullivan sure likes to stay on top of his investments! Let's break it down:
We have $900 invested at 4% interest compounded quarterly, and this happens at the beginning of each period for 9 years.
To calculate the present value of this annuity due, we can use a formula, but where's the fun in that? Let's use a different approach, shall we?
Let's imagine Scott is a clown and instead of investing his money, he's juggling $900 every quarter. Now, after 9 years of vigorous juggling, Scott is quite tired and decides to retire from his juggling career.
In order to calculate the present value of this annuity due, we need to find out how much money Scott has earned from his juggling career.
Since Scott is a clown, he's always happy, which means he's earned an extra 4% happiness on top of his juggling income every quarter.
So, let's calculate his total earnings. In 9 years, we have 4 quarters per year, so that's 36 quarters. Multiply that by the $900 Scott invests every quarter, and you get $32,400.
Now, let's add the extra 4% happiness that Scott earned every quarter. To make it simple, we'll calculate the happiness separately: 4% of $900 is $36. Multiply that by 36 quarters, and you get $1,296.
Finally, to find the present value of this annuity due, we need to subtract the happiness from Scott's total earnings. $32,400 minus $1,296 equals $31,104.
So, the present value of this annuity due is $31,104.
But hey, don't take my word for it! Clown math might not be the most accurate. I strongly suggest using a calculator for the precise answer.
To find the present value of an annuity due, you can use the formula:
PV = PMT * [1 - (1+r)^(-n)] / r * (1+r)
Where PV is the present value, PMT is the regular payment, r is the interest rate per period, and n is the number of periods.
In this case, the regular payment (PMT) is $900, the interest rate per period (r) is 4% or 0.04, and the number of periods (n) is 9 years * 4 quarters per year = 36 quarters.
Now, let's plug these values into the formula:
PV = $900 * [1 - (1+0.04)^(-36)] / 0.04 * (1+0.04)
To calculate this value, you can use any scientific calculator or financial calculator. Alternatively, you can also use online financial calculators.
Here's how you can calculate the present value using a scientific calculator:
1. Enter the value of the numerator: 1 - (1+0.04)^(-36) = 1 - (1.04)^(-36)
2. Calculate the numerator: 1 - (1.04)^(-36) = 0.5182 (rounded to four decimal places)
3. Enter the value of the denominator: 0.04 * (1+0.04) = 0.0404
4. Divide the numerator by the denominator: 0.5182 / 0.0404 = 12.8069 (rounded to four decimal places)
5. Multiply the result by the regular payment: 12.8069 * $900 = $11,526.21 (rounded to two decimal places)
Therefore, the present value of this annuity due is approximately $11,526.21.