The Problem:
You win the grand prize on a game show. You have the following choices:
Option 1: $1-million dollars paid as a $25 000 annuity every year over 40 years.
Option 2: The present value of option 1 if the current interest rate is 4%, compounded annually.
You accept Option 2, but invest your prize money in an annuity that will still pay you $25,000 every year. Your bank offers you 5% interest, compounded annually. Over 40 years how much more money would you have earned than if you accepted Option 1?
accept Option 2, but invest your prize money in an annuity that will still pay you $25,000 ...
To find out how much more money you would have earned by choosing Option 2 and investing it in an annuity with a 5% interest rate, compounded annually, you can follow these steps:
1. Calculate the present value of Option 1 using the given interest rate of 4% compounded annually.
To calculate the present value, you can use the formula:
PV = CF / (1 + r)^n
Where:
PV = Present Value
CF = Cash Flow per year (in this case, $25,000)
r = Interest rate per period (4% or 0.04 in decimal form)
n = Number of periods (40 years)
Using the formula, you get:
PV = $25,000 / (1 + 0.04)^40
2. Once you have calculated the present value of Option 1, you can invest that amount in an annuity with a 5% interest rate, compounded annually, for 40 years.
To calculate the future value of this annuity, you can use the formula:
FV = PV * (1 + r)^n
Where:
FV = Future Value
PV = Present Value (calculated in step 1)
r = Interest rate per period (5% or 0.05 in decimal form)
n = Number of periods (40 years)
Using the formula, you get:
FV = PV * (1 + 0.05)^40
3. Finally, subtract the future value of the annuity from Option 2 from the total amount you would receive with Option 1 over 40 years.
Total amount with Option 1 = $25,000 * 40 = $1,000,000
Difference in earnings = Total amount with Option 1 - FV from Option 2
Remember to calculate the present value of Option 1 before proceeding with the calculations in steps 2 and 3.
I believe it works like this:
Present Value of the annuity. So,
PV = Pmt x (1 - 1 / (1 + i)^n) / i
PV =25000*(1-1/(1+4%)^40)/4%= 494,819.35 <= that's your answer!
Hoped it helped!
I agree with jewellry's answer for the first part
For the last part, I will use 4% as the current interest rate.
This part is not very realistic, since it highly unlikely to be able to get the 4% throughout the 40 years, just like it not likely to get the 5% form the bank for that time.
anyway ...
value of the 494819.35 after 40 years
= 494819.35(1.05)^40
= 3,483,522.62
value of the annual 25,000 calculated at 4%
= 25000( 1.04^40 - 1)/.04
= 2,375,637.89
difference = $1,107,884.73