Brooke won $100 000 in a lottery. The price will be paid in yearly installments of $10 000 each year for 10 years. What is the present value of her winnings, if the current interest rates are 6.4% per annum, compounded annually?
P(1 + 0.064)^10 = 100000
P = 53,775.40
We are asked for the present value of an annuity
present value = $10,000(1 - 1.064)^-10 / .064
= $72,225.92
ouch! forgot about the payments!
good catch, as usual.
To calculate the present value of Brooke's winnings, we need to discount the future cash flows (the yearly installments) to their present value. The formula to calculate the present value of an annuity is as follows:
Present Value = CF × (1 - (1 + r)^(-n)) / r
Where:
CF = Cash Flow per period
r = Interest rate per period
n = Number of periods
In this case, Brooke will receive $10,000 per year for 10 years. The interest rate is 6.4% per annum (or 0.064 as a decimal). Using this information, we can calculate the present value as follows:
Present Value = $10,000 × (1 - (1 + 0.064)^(-10)) / 0.064
Now we can solve the equation step by step:
1. First, calculate the value inside the parentheses: (1 + 0.064)^(-10) = approximately 0.525939.
2. Subtract this value from 1: 1 - 0.525939 = approximately 0.474061.
3. Divide this result by 0.064 to get the present value: 0.474061 / 0.064 = approximately 7.407847.
Therefore, the present value of Brooke's winnings is approximately $74,078.47.