Please how do i calculate this problem.
Your girlfriend just won the Power Ball lottery. She has the choice of $10,000,000 today or a 30-year annuity of $500,000, with the first payment coming today. What rate of return is built into the annuity?
first payment today: $10,000,000 - $500,000 = $9,500,000
-$9,500,000= PV
$500,000= PMT
29=N
CPT R= 3.078% or 3.08%
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To calculate the rate of return built into the annuity, you can use the present value of an annuity formula. The formula for the present value of an annuity is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value of the annuity
PMT = Payment (in this case, $500,000 per year)
r = Interest rate per period
n = Number of periods (in this case, 30 years)
In this problem, we need to solve for r. Rearranging the formula, we get:
(1 - (1 + r)^(-n)) / r = PV / PMT
Substituting the given values into the formula, we have:
(1 - (1 + r)^(-30)) / r = $10,000,000 / $500,000
Now, we can solve for the interest rate (r) step-by-step:
1. Multiply both sides of the equation by r:
(1 - (1 + r)^(-30)) = r * ($10,000,000 / $500,000)
2. Simplify the right side:
(1 - (1 + r)^(-30)) = 20 * r
3. Expand the left side of the equation:
1 - (1 + r)^(-30) = 20r
4. Move the terms to one side of the equation:
(1 + r)^(-30) + 20r - 1 = 0
Now, the equation is set up to solve for r numerically. You can use numerical methods, such as graphing or utilizing a spreadsheet software, to approximate the value of r.
To calculate the rate of return built into the annuity, we need to determine the present value (PV) of the annuity payments and compare it to the $10,000,000 lump sum amount.
First, let's calculate the present value of the annuity. We'll use the formula for the present value of an ordinary annuity:
PV = Payment × (1 - (1 + i)^(-n)) / i
Where:
PV is the present value of the annuity,
Payment is the periodic payment in the annuity ($500,000 in this case),
i is the interest rate per compounding period, and
n is the number of compounding periods (30 years in this case).
Since we want to find the interest rate, we need to rearrange the formula:
i = (Payment / PV) ^ (1/n) - 1
Now, substitute the given values:
Payment = $500,000
PV = $10,000,000 (since this is the lump sum amount)
n = 30
Let's calculate the rate of return:
i = ($500,000 / $10,000,000) ^ (1/30) - 1
Simplifying the equation:
i = 0.05 - 1
i ≈ 0.02 or 2%
Therefore, the rate of return built into the annuity is approximately 2%.